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class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%AC%CE%B4%CE%BF%CE%BE%CE%BF_%CF%84%CE%B7%CF%82_%CF%86%CE%B9%CE%BB%CE%AF%CE%B1%CF%82" title="Παράδοξο της φιλίας – Greek" lang="el" hreflang="el" data-title="Παράδοξο της φιλίας" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%A7%D8%B1%D8%A7%D8%AF%D9%88%DA%A9%D8%B3_%D8%AF%D9%88%D8%B3%D8%AA%DB%8C" title="پارادوکس دوستی – Persian" lang="fa" hreflang="fa" data-title="پارادوکس دوستی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Paradoxe_de_l%27amiti%C3%A9" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Phenomenon that most people have fewer friends than their friends have, on average</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Moreno_Sociogram_2nd_Grade.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Moreno_Sociogram_2nd_Grade.png/290px-Moreno_Sociogram_2nd_Grade.png" decoding="async" width="290" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Moreno_Sociogram_2nd_Grade.png/435px-Moreno_Sociogram_2nd_Grade.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b6/Moreno_Sociogram_2nd_Grade.png 2x" data-file-width="525" data-file-height="525" /></a><figcaption>Diagram of a social network of 7-8-year-old children, mapped by asking each child to indicate two others they would like to sit next to in class. The majority of children have fewer connections than the average of those they are connected to.</figcaption></figure> <p>The <b>friendship paradox</b> is the phenomenon first observed by the sociologist <a href="/w/index.php?title=Scott_L._Feld&action=edit&redlink=1" class="new" title="Scott L. Feld (page does not exist)">Scott L. Feld</a> in 1991 that on average, an individual's friends have more friends than that individual.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It can be explained as a form of <a href="/wiki/Sampling_bias" title="Sampling bias">sampling bias</a> in which people with more friends are more likely to be in one's own friend group. In other words, one is less likely to be friends with someone who has very few friends. In contradiction to this, most people believe that they have more friends than their friends have.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The same observation can be applied more generally to <a href="/wiki/Social_network" title="Social network">social networks</a> defined by other relations than friendship: for instance, most people's sexual partners have had (on the average) a greater number of sexual partners than they have.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The friendship paradox is an example of how network structure can significantly distort an individual's local observations.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Mathematical_explanation">Mathematical explanation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friendship_paradox&action=edit&section=1" title="Edit section: Mathematical explanation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In spite of its apparently <a href="/wiki/Paradox" title="Paradox">paradoxical</a> nature, the phenomenon is real, and can be explained as a consequence of the general mathematical properties of <a href="/wiki/Social_network" title="Social network">social networks</a>. The mathematics behind this are directly related to the <a href="/wiki/Inequality_of_arithmetic_and_geometric_means" class="mw-redirect" title="Inequality of arithmetic and geometric means">arithmetic-geometric mean inequality</a> and the <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Formally, Feld assumes that a social network is represented by an <a href="/wiki/Undirected_graph" class="mw-redirect" title="Undirected graph">undirected graph</a> <span class="texhtml"><i>G</i> = (<i>V</i>, <i>E</i>)</span>, where the set <span class="texhtml mvar" style="font-style:italic;">V</span> of <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertices</a> corresponds to the people in the social network, and the set <span class="texhtml mvar" style="font-style:italic;">E</span> of <a href="/wiki/Edge_(graph_theory)" class="mw-redirect" title="Edge (graph theory)">edges</a> corresponds to the friendship relation between pairs of people. That is, he assumes that friendship is a <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric relation</a>: if <span class="texhtml mvar" style="font-style:italic;">x</span> is a friend of <span class="texhtml mvar" style="font-style:italic;">y</span>, then <span class="texhtml mvar" style="font-style:italic;">y</span> is a friend of <span class="texhtml mvar" style="font-style:italic;">x</span>. The friendship between <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> is therefore modeled by the edge <span class="texhtml">{<i>x</i>, <i>y</i>},</span> and the number of friends an individual has corresponds to a vertex's <a href="/wiki/Degree_(graph_theory)" title="Degree (graph theory)">degree</a>. The average number of friends of a person in the social network is therefore given by the average of the degrees of the <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertices</a> in the graph. That is, if vertex <span class="texhtml mvar" style="font-style:italic;">v</span> has <span class="texhtml"><i>d</i>(<i>v</i>)</span> edges touching it (representing a person who has <span class="texhtml"><i>d</i>(<i>v</i>)</span> friends), then the average number <span class="texhtml"><i>μ</i></span> of friends of a random person in the graph is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ={\frac {\sum _{v\in V}d(v)}{|V|}}={\frac {2|E|}{|V|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ={\frac {\sum _{v\in V}d(v)}{|V|}}={\frac {2|E|}{|V|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3686f540e1107e782f2a7578def5b930b0d5d860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.533ex; height:6.676ex;" alt="{\displaystyle \mu ={\frac {\sum _{v\in V}d(v)}{|V|}}={\frac {2|E|}{|V|}}.}"></span></dd></dl> <p>The average number of friends that a typical friend has can be modeled by choosing a random person (who has at least one friend), and then calculating how many friends their friends have on average. This amounts to choosing, uniformly at random, an edge of the graph (representing a pair of friends) and an endpoint of that edge (one of the friends), and again calculating the degree of the selected endpoint. The probability of a certain vertex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> to be chosen is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d(v)}{|E|}}{\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d(v)}{|E|}}{\frac {1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a680abfe1a67ee99f0803ec83bd56945f55ceeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:7.634ex; height:6.509ex;" alt="{\displaystyle {\frac {d(v)}{|E|}}{\frac {1}{2}}.}"></span></dd></dl> <p>The first factor corresponds to how likely it is that the chosen edge contains the vertex, which increases when the vertex has more friends. The halving factor simply comes from the fact that each edge has two vertices. So the expected value of the number of friends of a (randomly chosen) friend is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{v}\left({\frac {d(v)}{|E|}}{\frac {1}{2}}\right)d(v)={\frac {\sum _{v}d(v)^{2}}{2|E|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>v</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{v}\left({\frac {d(v)}{|E|}}{\frac {1}{2}}\right)d(v)={\frac {\sum _{v}d(v)^{2}}{2|E|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374e1d354042235e88d84af851bf061b75f4ee06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.35ex; height:7.009ex;" alt="{\displaystyle \sum _{v}\left({\frac {d(v)}{|E|}}{\frac {1}{2}}\right)d(v)={\frac {\sum _{v}d(v)^{2}}{2|E|}}.}"></span></dd></dl> <p>We know from the definition of variance that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sum _{v}d(v)^{2}}{|V|}}=\mu ^{2}+\sigma ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>v</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sum _{v}d(v)^{2}}{|V|}}=\mu ^{2}+\sigma ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18e7aee79492b52854e9eac4bb455d467239663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.34ex; height:6.676ex;" alt="{\displaystyle {\frac {\sum _{v}d(v)^{2}}{|V|}}=\mu ^{2}+\sigma ^{2},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span> is the variance of the degrees in the graph. This allows us to compute the desired expected value as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sum _{v}d(v)^{2}}{2|E|}}={\frac {|V|}{2|E|}}(\mu ^{2}+\sigma ^{2})={\frac {\mu ^{2}+\sigma ^{2}}{\mu }}=\mu +{\frac {\sigma ^{2}}{\mu }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>v</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>μ</mi> </mfrac> </mrow> <mo>=</mo> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>μ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sum _{v}d(v)^{2}}{2|E|}}={\frac {|V|}{2|E|}}(\mu ^{2}+\sigma ^{2})={\frac {\mu ^{2}+\sigma ^{2}}{\mu }}=\mu +{\frac {\sigma ^{2}}{\mu }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a9db0abb45cad84fd1d593b2306bce33988f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.394ex; height:6.676ex;" alt="{\displaystyle {\frac {\sum _{v}d(v)^{2}}{2|E|}}={\frac {|V|}{2|E|}}(\mu ^{2}+\sigma ^{2})={\frac {\mu ^{2}+\sigma ^{2}}{\mu }}=\mu +{\frac {\sigma ^{2}}{\mu }}.}"></span></dd></dl> <p>For a graph that has vertices of varying degrees (as is typical for social networks), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sigma }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sigma }^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8987169794d05f7664cc4cdc0ebab06d9296dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle {\sigma }^{2}}"></span> is strictly positive, which implies that the average degree of a friend is strictly greater than the average degree of a random node. </p><p>Another way of understanding how the first term came is as follows. For each friendship <span class="texhtml mvar" style="font-style:italic;">(u, v)</span>, a node <span class="texhtml mvar" style="font-style:italic;"> u</span> mentions that <span class="texhtml mvar" style="font-style:italic;">v</span> is a friend and <span class="texhtml mvar" style="font-style:italic;">v</span> has <span class="texhtml mvar" style="font-style:italic;">d(v)</span> friends. There are <span class="texhtml mvar" style="font-style:italic;">d(v)</span> such friends who mention this. Hence the square of <span class="texhtml mvar" style="font-style:italic;">d(v)</span> term. We add this for all such friendships in the network from both the <span class="texhtml mvar" style="font-style:italic;">u</span>'s and <span class="texhtml mvar" style="font-style:italic;">v</span>'s perspective, which gives the numerator. The denominator is the number of total such friendships, which is twice the total edges in the network (one from the <span class="texhtml mvar" style="font-style:italic;">u</span>'s perspective and the other from the <span class="texhtml mvar" style="font-style:italic;">v</span>'s). </p><p>After this analysis, Feld goes on to make some more qualitative assumptions about the statistical correlation between the number of friends that two friends have, based on theories of social networks such as <a href="/wiki/Assortative_mixing" title="Assortative mixing">assortative mixing</a>, and he analyzes what these assumptions imply about the number of people whose friends have more friends than they do. Based on this analysis, he concludes that in real social networks, most people are likely to have fewer friends than the average of their friends' numbers of friends. However, this conclusion is not a mathematical certainty; there exist undirected graphs (such as the graph formed by removing a single edge from a large <a href="/wiki/Complete_graph" title="Complete graph">complete graph</a>) that are unlikely to arise as social networks but in which most vertices have higher degree than the average of their neighbors' degrees. </p><p>The Friendship Paradox may be restated in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> terms as "the average degree of a randomly selected node in a network is less than the average degree of neighbors of a randomly selected node", but this leaves unspecified the exact mechanism of averaging (i.e., macro vs micro averaging). Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(V,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=(V,E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644a8d85ee410b6159ca2bdb5dcb9097e2c8f182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.331ex; height:2.843ex;" alt="{\displaystyle G=(V,E)}"></span> be an undirected graph with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V|=N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |V|=N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd0ae42424bcee5cba3e29cc5cf411a68151a5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.243ex; height:2.843ex;" alt="{\displaystyle |V|=N}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|=M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E|=M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/364761f453e89587cca4fff0a30a2c267614d2c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.61ex; height:2.843ex;" alt="{\displaystyle |E|=M}"></span>, having no isolated nodes. Let the set of neighbors of node <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> be denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {nbr} (u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {nbr} (u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0b07a42a2b18e22d1f61b65f32865477aa25d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.636ex; height:2.843ex;" alt="{\displaystyle \operatorname {nbr} (u)}"></span>. The average degree is then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ={\frac {1}{N}}\sum _{u\in V}|\operatorname {nbr} (u)|={\frac {2M}{N}}\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>∈</mo> <mi>V</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ={\frac {1}{N}}\sum _{u\in V}|\operatorname {nbr} (u)|={\frac {2M}{N}}\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d4b434461ac0dce50cd94711d4091f1a3478bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.646ex; height:6.509ex;" alt="{\displaystyle \mu ={\frac {1}{N}}\sum _{u\in V}|\operatorname {nbr} (u)|={\frac {2M}{N}}\geq 1}"></span>. Let the number of "friends of friends" of node <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> be denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {FF} (u)=\sum _{v\in \operatorname {nbr} (u)}|\operatorname {nbr} (v)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>FF</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>∈</mo> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {FF} (u)=\sum _{v\in \operatorname {nbr} (u)}|\operatorname {nbr} (v)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfad1154ec43d28e4c009e2f4eace860253f1cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.36ex; height:6.009ex;" alt="{\displaystyle \operatorname {FF} (u)=\sum _{v\in \operatorname {nbr} (u)}|\operatorname {nbr} (v)|}"></span>. Note that this can count 2-hop neighbors multiple times, but so does Feld's analysis. We have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {FF} (u)\geq |\operatorname {nbr} (u)|\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>FF</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {FF} (u)\geq |\operatorname {nbr} (u)|\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52818536dbcde93b80d14dab3340f6b9f17603ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.85ex; height:2.843ex;" alt="{\displaystyle \operatorname {FF} (u)\geq |\operatorname {nbr} (u)|\geq 1}"></span>. Feld considered the following "micro average" quantity. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MicroAvg}}={\frac {\sum _{u\in V}\operatorname {FF} (u)}{\sum _{u\in V}|\operatorname {nbr} (u)|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MicroAvg</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>∈</mo> <mi>V</mi> </mrow> </munder> <mi>FF</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>∈</mo> <mi>V</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MicroAvg}}={\frac {\sum _{u\in V}\operatorname {FF} (u)}{\sum _{u\in V}|\operatorname {nbr} (u)|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf457ecadc10a7ca28fcce5302a85e29d93ad34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.642ex; height:6.843ex;" alt="{\displaystyle {\text{MicroAvg}}={\frac {\sum _{u\in V}\operatorname {FF} (u)}{\sum _{u\in V}|\operatorname {nbr} (u)|}}}"></span></dd></dl> <p>However, there is also the (equally legitimate) "macro average" quantity, given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MacroAvg}}={\frac {1}{N}}\sum _{u\in V}{\frac {\operatorname {FF} (u)}{|\operatorname {nbr} (u)|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MacroAvg</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>∈</mo> <mi>V</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>FF</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MacroAvg}}={\frac {1}{N}}\sum _{u\in V}{\frac {\operatorname {FF} (u)}{|\operatorname {nbr} (u)|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f326f7d80e0e458cb1498ab0439042a3613b1800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.813ex; height:7.009ex;" alt="{\displaystyle {\text{MacroAvg}}={\frac {1}{N}}\sum _{u\in V}{\frac {\operatorname {FF} (u)}{|\operatorname {nbr} (u)|}}}"></span></dd></dl> <p>The computation of MacroAvg can be expressed as the following pseudocode. </p> <div style="border:1px solid #cccccc; background-color:#f8f8f8; padding:4px;"> <pre><b>Algorithm</b> MacroAvg </pre> <dl><dd><ol><li>for each node <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636dd20088dea1139b38b3c04053ccf508bbed8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle u\in V}"></span> <ol><li>initialize <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(u)\leftarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">←</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(u)\leftarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bf3e119ca23f46139aa584e533082ccca92bec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.754ex; height:2.843ex;" alt="{\displaystyle Q(u)\leftarrow 0}"></span></li></ol></li> <li>for each edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{u,v\}\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">}</mo> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{u,v\}\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2a46c8b9d1078c9f1922a12e069c781925c1f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.432ex; height:2.843ex;" alt="{\displaystyle \{u,v\}\in E}"></span> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(u)\leftarrow Q(u)+{\frac {|\operatorname {nbr} (v)|}{|\operatorname {nbr} (u)|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">←</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(u)\leftarrow Q(u)+{\frac {|\operatorname {nbr} (v)|}{|\operatorname {nbr} (u)|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5045adb48dabc621533700fa475dc6ed59ce18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.562ex; height:6.509ex;" alt="{\displaystyle Q(u)\leftarrow Q(u)+{\frac {|\operatorname {nbr} (v)|}{|\operatorname {nbr} (u)|}}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(v)\leftarrow Q(v)+{\frac {|\operatorname {nbr} (u)|}{|\operatorname {nbr} (v)|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">←</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(v)\leftarrow Q(v)+{\frac {|\operatorname {nbr} (u)|}{|\operatorname {nbr} (v)|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4391b53a92d909d79a495c38832ff68b603b23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.158ex; height:6.509ex;" alt="{\displaystyle Q(v)\leftarrow Q(v)+{\frac {|\operatorname {nbr} (u)|}{|\operatorname {nbr} (v)|}}}"></span></li></ol></li> <li>return <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{N}}\sum _{u\in V}Q(u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>∈</mo> <mi>V</mi> </mrow> </munder> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{N}}\sum _{u\in V}Q(u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b667d6c36fb1ef0e77088de21780014c485ee2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.006ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{N}}\sum _{u\in V}Q(u)}"></span></li></ol></dd></dl> <ul><li><small>"←" denotes <a href="/wiki/Assignment_(computer_science)" title="Assignment (computer science)">assignment</a>. For instance, "<i>largest</i> ← <i>item</i>" means that the value of <i>largest</i> changes to the value of <i>item</i>.</small></li> <li><small>"<b>return</b>" terminates the algorithm and outputs the following value.</small></li></ul> </div> <p>Each edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{u,v\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{u,v\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f507af3d28091510daed6d4241af30d88c1c2c92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.816ex; height:2.843ex;" alt="{\displaystyle \{u,v\}}"></span> contributes to MacroAvg the quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|\operatorname {nbr} (v)|}{|\operatorname {nbr} (u)|}}+{\frac {|\operatorname {nbr} (u)|}{|\operatorname {nbr} (v)|}}\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>nbr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|\operatorname {nbr} (v)|}{|\operatorname {nbr} (u)|}}+{\frac {|\operatorname {nbr} (u)|}{|\operatorname {nbr} (v)|}}\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dedc0dc842098f6206316fe6e3fcce253bb31b13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.406ex; height:6.509ex;" alt="{\displaystyle {\frac {|\operatorname {nbr} (v)|}{|\operatorname {nbr} (u)|}}+{\frac {|\operatorname {nbr} (u)|}{|\operatorname {nbr} (v)|}}\geq 2}"></span>, because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{a,b>0}{\frac {a}{b}}+{\frac {b}{a}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min _{a,b>0}{\frac {a}{b}}+{\frac {b}{a}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2334c2d1219e1e64a924f88b4e4c832c0a16fb3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.753ex; height:5.843ex;" alt="{\displaystyle \min _{a,b>0}{\frac {a}{b}}+{\frac {b}{a}}=2}"></span>. We thus get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MacroAvg}}={\frac {1}{N}}\sum _{u\in V}Q(u)\geq {\frac {1}{N}}\cdot M\cdot 2={\frac {2M}{N}}=\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MacroAvg</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>∈</mo> <mi>V</mi> </mrow> </munder> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <mo>⋅</mo> <mi>M</mi> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MacroAvg}}={\frac {1}{N}}\sum _{u\in V}Q(u)\geq {\frac {1}{N}}\cdot M\cdot 2={\frac {2M}{N}}=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0218a28a5ec2b0e30b5a9d686cabefa200181f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.638ex; height:6.509ex;" alt="{\displaystyle {\text{MacroAvg}}={\frac {1}{N}}\sum _{u\in V}Q(u)\geq {\frac {1}{N}}\cdot M\cdot 2={\frac {2M}{N}}=\mu }"></span>.</dd></dl> <p>Thus, we have both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MicroAvg}}\geq \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MicroAvg</mtext> </mrow> <mo>≥</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MicroAvg}}\geq \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eccc1f621a2dd9700d76495fb59575ab8561b44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.517ex; height:2.676ex;" alt="{\displaystyle {\text{MicroAvg}}\geq \mu }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MacroAvg}}\geq \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MacroAvg</mtext> </mrow> <mo>≥</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MacroAvg}}\geq \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb5e81dce39ae75fe242c2b3e716cebe398e643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.033ex; height:2.676ex;" alt="{\displaystyle {\text{MacroAvg}}\geq \mu }"></span>, but no inequality holds between them.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>In a 2023 paper, a parallel paradox, but for negative, antagonistic, or animosity ties, termed the "enmity paradox," was defined and demonstrated by Ghasemian and <a href="/wiki/Nicholas_Christakis" title="Nicholas Christakis">Christakis</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> In brief, one's enemies have more enemies than one does, too. This paper also documented diverse phenomena is "mixed worlds" of both hostile and friendly ties. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friendship_paradox&action=edit&section=2" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The analysis of the friendship paradox implies that the friends of randomly selected individuals are likely to have higher than average <a href="/wiki/Centrality" title="Centrality">centrality</a>. This observation has been used as a way to forecast and slow the course of <a href="/wiki/Epidemic" title="Epidemic">epidemics</a>, by using this random selection process to choose individuals to immunize or monitor for infection while avoiding the need for a complex computation of the centrality of all nodes in the network.<sup id="cite_ref-:2_13-0" class="reference"><a href="#cite_note-:2-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> In a similar manner, in polling and election forecasting, friendship paradox has been exploited in order to reach and query well-connected individuals who may have knowledge about how numerous other individuals are going to vote.<sup id="cite_ref-:0_16-0" class="reference"><a href="#cite_note-:0-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> However, when utilized in such contexts, the friendship paradox inevitably introduces bias by over-representing individuals with many friends, potentially skewing resulting estimates.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:3_18-0" class="reference"><a href="#cite_note-:3-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>A study in 2010 by Christakis and Fowler showed that flu outbreaks can be detected almost two weeks before traditional surveillance measures would do so by using the friendship paradox in monitoring the infection in a social network.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> They found that using the friendship paradox to analyze the health of <a href="/wiki/Centrality" title="Centrality">central</a> friends is "an ideal way to predict outbreaks, but detailed information doesn't exist for most groups, and to produce it would be time-consuming and costly."<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> This extends to the spread of ideas as well, with evidence that the friendship paradox can be used to track and predict the spread of ideas and misinformation through networks.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_13-1" class="reference"><a href="#cite_note-:2-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> This observation has been explained with the argument that individuals with more social connections may be the driving forces behind the spread of these ideas and beliefs, and as such can be used as early-warning signals.<sup id="cite_ref-:3_18-1" class="reference"><a href="#cite_note-:3-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Friendship paradox based sampling (i.e., sampling random friends) has been theoretically and empirically shown to outperform classical uniform sampling for the purpose of estimating the <a href="/wiki/Power_law" title="Power law">power-law degree distributions</a> of <a href="/wiki/Scale-free_network" title="Scale-free network">scale-free networks</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_24-0" class="reference"><a href="#cite_note-:1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The reason is that sampling the network uniformly will not collect enough samples from the characteristic <a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">heavy tail</a> part of the power-law degree distribution to properly estimate it. However, sampling random friends incorporates more nodes from the tail of the degree distribution (i.e., more high degree nodes) into the sample. Hence, friendship paradox based sampling captures the characteristic heavy tail of a power-law degree distribution more accurately and reduces the bias and variance of the estimation.<sup id="cite_ref-:1_24-1" class="reference"><a href="#cite_note-:1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>The "generalized friendship paradox" states that the friendship paradox applies to other characteristics as well. For example, one's co-authors are on average likely to be more prominent, with more publications, more citations and more collaborators,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> or one's followers on Twitter have more followers.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> The same effect has also been demonstrated for Subjective Well-Being by Bollen et al. (2017),<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> who used a large-scale Twitter network and longitudinal data on subjective well-being for each individual in the network to demonstrate that both a Friendship and a "happiness" paradox can occur in online social networks. </p><p>The friendship paradox has also been used as a means to identify structurally influential nodes within social networks, so as to magnify <a href="/wiki/Social_contagion" title="Social contagion">social contagion</a> of diverse practices relevant to human welfare and public health. This has been shown to be possible in several large-scale randomized controlled field trials conducted by <a href="/wiki/Nicholas_Christakis" title="Nicholas Christakis">Christakis</a> et al., with respect to the adoption of multivitamins<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> or maternal and child health practices<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> in Honduras, or of iron-fortified salt in India.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> This technique is valuable because, by exploiting the friendship paradox, one can identify such influential nodes without the expense and delay of actually mapping the whole network. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friendship_paradox&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_paradoxes#Mathematics" title="List of paradoxes">List of paradoxes#Mathematics</a> – List of statements that appear to contradict themselves</li> <li><a href="/wiki/Second_neighborhood_problem" title="Second neighborhood problem">Second neighborhood problem</a> – Unsolved problem about oriented graphs</li> <li><a href="/wiki/Self-evaluation_maintenance_theory" title="Self-evaluation maintenance theory">Self-evaluation maintenance theory</a> – Concept in social psychology</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friendship_paradox&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFeld1991" class="citation cs2">Feld, Scott L. (1991), "Why your friends have more friends than you do", <i><a href="/wiki/American_Journal_of_Sociology" title="American Journal of Sociology">American Journal of Sociology</a></i>, <b>96</b> (6): 1464–1477, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F229693">10.1086/229693</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2781907">2781907</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:56043992">56043992</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Sociology&rft.atitle=Why+your+friends+have+more+friends+than+you+do&rft.volume=96&rft.issue=6&rft.pages=1464-1477&rft.date=1991&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A56043992%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2781907%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1086%2F229693&rft.aulast=Feld&rft.aufirst=Scott+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZuckermanJost2001" class="citation cs2">Zuckerman, Ezra W.; Jost, John T. (2001), <a rel="nofollow" class="external text" href="http://www.psych.nyu.edu/jost/Zuckerman%20&%20Jost%20(2001)%20What%20Makes%20You%20Think%20You%27re%20So%20Popular1.pdf">"What makes you think you're so popular? Self evaluation maintenance and the subjective side of the "friendship paradox"<span class="cs1-kern-right"></span>"</a> <span class="cs1-format">(PDF)</span>, <i>Social Psychology Quarterly</i>, <b>64</b> (3): 207–223, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3090112">10.2307/3090112</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3090112">3090112</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Social+Psychology+Quarterly&rft.atitle=What+makes+you+think+you%27re+so+popular%3F+Self+evaluation+maintenance+and+the+subjective+side+of+the+%22friendship+paradox%22&rft.volume=64&rft.issue=3&rft.pages=207-223&rft.date=2001&rft_id=info%3Adoi%2F10.2307%2F3090112&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3090112%23id-name%3DJSTOR&rft.aulast=Zuckerman&rft.aufirst=Ezra+W.&rft.au=Jost%2C+John+T.&rft_id=http%3A%2F%2Fwww.psych.nyu.edu%2Fjost%2FZuckerman%2520%26%2520Jost%2520%282001%29%2520What%2520Makes%2520You%2520Think%2520You%2527re%2520So%2520Popular1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcRaney2012" class="citation cs2">McRaney, David (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9Oc_hdvqk50C&pg=PA160"><i>You are Not So Smart</i></a>, Oneworld Publications, p. 160, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-78074-104-8" title="Special:BookSources/978-1-78074-104-8"><bdi>978-1-78074-104-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=You+are+Not+So+Smart&rft.pages=160&rft.pub=Oneworld+Publications&rft.date=2012&rft.isbn=978-1-78074-104-8&rft.aulast=McRaney&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9Oc_hdvqk50C%26pg%3DPA160&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFelmleeFaris2013" class="citation cs2">Felmlee, Diane; Faris, Robert (2013), "Interaction in social networks", in DeLamater, John; Ward, Amanda (eds.), <i>Handbook of Social Psychology</i> (2nd ed.), Springer, pp. 439–464, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9400767720" title="Special:BookSources/978-9400767720"><bdi>978-9400767720</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Interaction+in+social+networks&rft.btitle=Handbook+of+Social+Psychology&rft.pages=439-464&rft.edition=2nd&rft.pub=Springer&rft.date=2013&rft.isbn=978-9400767720&rft.aulast=Felmlee&rft.aufirst=Diane&rft.au=Faris%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span>. See in particular "Friendship ties", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hXY8AAAAQBAJ&pg=PA452">p. 452</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLau2011" class="citation cs2">Lau, J. Y. F. (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KCOeRQW2s0cC&pg=PA191"><i>An Introduction to Critical Thinking and Creativity: Think More, Think Better</i></a>, John Wiley & Sons, p. 191, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-03343-2" title="Special:BookSources/978-1-118-03343-2"><bdi>978-1-118-03343-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Critical+Thinking+and+Creativity%3A+Think+More%2C+Think+Better&rft.pages=191&rft.pub=John+Wiley+%26+Sons&rft.date=2011&rft.isbn=978-1-118-03343-2&rft.aulast=Lau&rft.aufirst=J.+Y.+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKCOeRQW2s0cC%26pg%3DPA191&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKanazawa2009" class="citation cs2"><a href="/wiki/Satoshi_Kanazawa" title="Satoshi Kanazawa">Kanazawa, Satoshi</a> (2009), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091107211753/http://www.psychologytoday.com/print/34391">"The Scientific Fundamentalist: A Look at the Hard Truths About Human Nature – Why your friends have more friends than you do"</a>, <i><a href="/wiki/Psychology_Today" title="Psychology Today">Psychology Today</a></i>, archived from <a rel="nofollow" class="external text" href="http://www.psychologytoday.com/blog/the-scientific-fundamentalist/200911/why-your-friends-have-more-friends-you-do">the original</a> on 2009-11-07</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychology+Today&rft.atitle=The+Scientific+Fundamentalist%3A+A+Look+at+the+Hard+Truths+About+Human+Nature+%E2%80%93+Why+your+friends+have+more+friends+than+you+do&rft.date=2009&rft.aulast=Kanazawa&rft.aufirst=Satoshi&rft_id=http%3A%2F%2Fwww.psychologytoday.com%2Fblog%2Fthe-scientific-fundamentalist%2F200911%2Fwhy-your-friends-have-more-friends-you-do&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurkeman2010" class="citation cs2"><a href="/wiki/Oliver_Burkeman" title="Oliver Burkeman">Burkeman, Oliver</a> (30 January 2010), <a rel="nofollow" class="external text" href="https://www.theguardian.com/lifeandstyle/2010/jan/30/change-your-life-friends-popular">"This column will change your life: Ever wondered why your friends seem so much more popular than you are? There's a reason for that"</a>, <i><a href="/wiki/The_Guardian" title="The Guardian">The Guardian</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Guardian&rft.atitle=This+column+will+change+your+life%3A+Ever+wondered+why+your+friends+seem+so+much+more+popular+than+you+are%3F+There%27s+a+reason+for+that&rft.date=2010-01-30&rft.aulast=Burkeman&rft.aufirst=Oliver&rft_id=https%3A%2F%2Fwww.theguardian.com%2Flifeandstyle%2F2010%2Fjan%2F30%2Fchange-your-life-friends-popular&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLermanYanWu2016" class="citation journal cs1"><a href="/wiki/Kristina_Lerman" title="Kristina Lerman">Lerman, Kristina</a>; Yan, Xiaoran; Wu, Xin-Zeng (2016-02-17). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4757419">"The "Majority Illusion" in Social Networks"</a>. <i>PLOS ONE</i>. <b>11</b> (2): e0147617. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1506.03022">1506.03022</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016PLoSO..1147617L">2016PLoSO..1147617L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0147617">10.1371/journal.pone.0147617</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1932-6203">1932-6203</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4757419">4757419</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26886112">26886112</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+ONE&rft.atitle=The+%22Majority+Illusion%22+in+Social+Networks&rft.volume=11&rft.issue=2&rft.pages=e0147617&rft.date=2016-02-17&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4757419%23id-name%3DPMC&rft_id=info%3Abibcode%2F2016PLoSO..1147617L&rft_id=info%3Aarxiv%2F1506.03022&rft.issn=1932-6203&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0147617&rft_id=info%3Apmid%2F26886112&rft.aulast=Lerman&rft.aufirst=Kristina&rft.au=Yan%2C+Xiaoran&rft.au=Wu%2C+Xin-Zeng&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4757419&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlipourfardNettasingheAbeliukKrishnamurthy2020" class="citation journal cs1">Alipourfard, Nazanin; Nettasinghe, Buddhika; Abeliuk, Andrés; Krishnamurthy, Vikram; Lerman, Kristina (2020-02-05). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1038/s41467-020-14394-x">"Friendship paradox biases perceptions in directed networks"</a>. <i>Nature Communications</i>. <b>11</b> (1): 707. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1905.05286">1905.05286</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020NatCo..11..707A">2020NatCo..11..707A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41467-020-14394-x">10.1038/s41467-020-14394-x</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2041-1723">2041-1723</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7002371">7002371</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32024843">32024843</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Communications&rft.atitle=Friendship+paradox+biases+perceptions+in+directed+networks&rft.volume=11&rft.issue=1&rft.pages=707&rft.date=2020-02-05&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7002371%23id-name%3DPMC&rft_id=info%3Abibcode%2F2020NatCo..11..707A&rft_id=info%3Aarxiv%2F1905.05286&rft.issn=2041-1723&rft_id=info%3Adoi%2F10.1038%2Fs41467-020-14394-x&rft_id=info%3Apmid%2F32024843&rft.aulast=Alipourfard&rft.aufirst=Nazanin&rft.au=Nettasinghe%2C+Buddhika&rft.au=Abeliuk%2C+Andr%C3%A9s&rft.au=Krishnamurthy%2C+Vikram&rft.au=Lerman%2C+Kristina&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1038%2Fs41467-020-14394-x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBen_SlimanKohli2019" class="citation cs2">Ben Sliman, Malek; Kohli, Rajeev (2019), <a rel="nofollow" class="external text" href="https://ssrn.com/abstract=3395317">"The extended directed friendship paradox"</a>, <i>SSRN</i>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2139%2Fssrn.3395317">10.2139/ssrn.3395317</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:219376223">219376223</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SSRN&rft.atitle=The+extended+directed+friendship+paradox&rft.date=2019&rft_id=info%3Adoi%2F10.2139%2Fssrn.3395317&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A219376223%23id-name%3DS2CID&rft.aulast=Ben+Sliman&rft.aufirst=Malek&rft.au=Kohli%2C+Rajeev&rft_id=https%3A%2F%2Fssrn.com%2Fabstract%3D3395317&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuptaChakrabarti2021" class="citation cs2">Gupta, Yash; Chakrabarti, Soumen (2021), <a rel="nofollow" class="external text" href="https://www.cse.iitb.ac.in/~soumen/doc/2021_Friends/FriendsOfFriends.pdf"><i>Friends of friends</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Friends+of+friends&rft.date=2021&rft.aulast=Gupta&rft.aufirst=Yash&rft.au=Chakrabarti%2C+Soumen&rft_id=https%3A%2F%2Fwww.cse.iitb.ac.in%2F~soumen%2Fdoc%2F2021_Friends%2FFriendsOfFriends.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGhasemianChristakis2023" class="citation journal cs1">Ghasemian, Amir; Christakis, Nicholas A. (2023-11-16). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10654772">"The enmity paradox"</a>. <i>Scientific Reports</i>. <b>13</b> (1): 20040. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2304.10076">2304.10076</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023NatSR..1320040G">2023NatSR..1320040G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41598-023-47167-9">10.1038/s41598-023-47167-9</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2045-2322">2045-2322</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10654772">10654772</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/37973933">37973933</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=The+enmity+paradox&rft.volume=13&rft.issue=1&rft.pages=20040&rft.date=2023-11-16&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10654772%23id-name%3DPMC&rft_id=info%3Abibcode%2F2023NatSR..1320040G&rft_id=info%3Aarxiv%2F2304.10076&rft.issn=2045-2322&rft_id=info%3Adoi%2F10.1038%2Fs41598-023-47167-9&rft_id=info%3Apmid%2F37973933&rft.aulast=Ghasemian&rft.aufirst=Amir&rft.au=Christakis%2C+Nicholas+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC10654772&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-:2-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohenHavlinben-Avraham2003" class="citation cs2">Cohen, Reuven; Havlin, Shlomo; ben-Avraham, Daniel (2003), "Efficient immunization strategies for computer networks and populations", <i>Phys. Rev. Lett.</i>, <b>91</b> (24), 247901, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0207387">cond-mat/0207387</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003PhRvL..91x7901C">2003PhRvL..91x7901C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.91.247901">10.1103/PhysRevLett.91.247901</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/14683159">14683159</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:919625">919625</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+Lett.&rft.atitle=Efficient+immunization+strategies+for+computer+networks+and+populations&rft.volume=91&rft.issue=24&rft.pages=247901&rft.date=2003&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A919625%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2003PhRvL..91x7901C&rft_id=info%3Aarxiv%2Fcond-mat%2F0207387&rft_id=info%3Apmid%2F14683159&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.91.247901&rft.aulast=Cohen&rft.aufirst=Reuven&rft.au=Havlin%2C+Shlomo&rft.au=ben-Avraham%2C+Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristakisFowler2010" class="citation cs2">Christakis, N. A.; Fowler, J. H. (2010), "Social network sensors for early detection of contagious outbreaks", <i>PLOS ONE</i>, <b>5</b> (9), e12948, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1004.4792">1004.4792</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010PLoSO...512948C">2010PLoSO...512948C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0012948">10.1371/journal.pone.0012948</a></span>, <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2939797">2939797</a></span>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20856792">20856792</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+ONE&rft.atitle=Social+network+sensors+for+early+detection+of+contagious+outbreaks&rft.volume=5&rft.issue=9&rft.pages=e12948&rft.date=2010&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2939797%23id-name%3DPMC&rft_id=info%3Abibcode%2F2010PLoSO...512948C&rft_id=info%3Aarxiv%2F1004.4792&rft_id=info%3Apmid%2F20856792&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0012948&rft.aulast=Christakis&rft.aufirst=N.+A.&rft.au=Fowler%2C+J.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilson2010" class="citation cs2">Wilson, Mark (November 2010), "Using the friendship paradox to sample a social network", <i>Physics Today</i>, <b>63</b> (11): 15–16, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010PhT....63k..15W">2010PhT....63k..15W</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.3518199">10.1063/1.3518199</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Using+the+friendship+paradox+to+sample+a+social+network&rft.volume=63&rft.issue=11&rft.pages=15-16&rft.date=2010-11&rft_id=info%3Adoi%2F10.1063%2F1.3518199&rft_id=info%3Abibcode%2F2010PhT....63k..15W&rft.aulast=Wilson&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span>.</span> </li> <li id="cite_note-:0-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNettasingheKrishnamurthy2019" class="citation journal cs1">Nettasinghe, Buddhika; Krishnamurthy, Vikram (2019). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1109/tkde.2019.2940914">"<span class="cs1-kern-left"></span>"What Do Your Friends Think?": Efficient Polling Methods for Networks Using Friendship Paradox"</a>. <i>IEEE Transactions on Knowledge and Data Engineering</i>: 1. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1802.06505">1802.06505</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftkde.2019.2940914">10.1109/tkde.2019.2940914</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1041-4347">1041-4347</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3335133">3335133</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Knowledge+and+Data+Engineering&rft.atitle=%22What+Do+Your+Friends+Think%3F%22%3A+Efficient+Polling+Methods+for+Networks+Using+Friendship+Paradox&rft.pages=1&rft.date=2019&rft_id=info%3Aarxiv%2F1802.06505&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3335133%23id-name%3DS2CID&rft.issn=1041-4347&rft_id=info%3Adoi%2F10.1109%2Ftkde.2019.2940914&rft.aulast=Nettasinghe&rft.aufirst=Buddhika&rft.au=Krishnamurthy%2C+Vikram&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1109%2Ftkde.2019.2940914&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeldMcGail2020" class="citation journal cs1">Feld, Scott L.; McGail, Alec (September 2020). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/product/identifier/S2050124220000053/type/journal_article">"Egonets as systematically biased windows on society"</a>. <i>Network Science</i>. <b>8</b> (3): 399–417. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fnws.2020.5">10.1017/nws.2020.5</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2050-1242">2050-1242</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:216301650">216301650</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Network+Science&rft.atitle=Egonets+as+systematically+biased+windows+on+society&rft.volume=8&rft.issue=3&rft.pages=399-417&rft.date=2020-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A216301650%23id-name%3DS2CID&rft.issn=2050-1242&rft_id=info%3Adoi%2F10.1017%2Fnws.2020.5&rft.aulast=Feld&rft.aufirst=Scott+L.&rft.au=McGail%2C+Alec&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fproduct%2Fidentifier%2FS2050124220000053%2Ftype%2Fjournal_article&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-:3-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-:3_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:3_18-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalesicBruine_de_BruinDalegeFeld2021" class="citation journal cs1">Galesic, Mirta; Bruine de Bruin, Wändi; Dalege, Jonas; Feld, Scott L.; Kreuter, Frauke; Olsson, Henrik; Prelec, Drazen; Stein, Daniel L.; van der Does, Tamara (July 2021). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/s41586-021-03649-2">"Human social sensing is an untapped resource for computational social science"</a>. <i>Nature</i>. <b>595</b> (7866): 214–222. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021Natur.595..214G">2021Natur.595..214G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41586-021-03649-2">10.1038/s41586-021-03649-2</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1476-4687">1476-4687</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34194037">34194037</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:235697772">235697772</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Human+social+sensing+is+an+untapped+resource+for+computational+social+science&rft.volume=595&rft.issue=7866&rft.pages=214-222&rft.date=2021-07&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A235697772%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2021Natur.595..214G&rft.issn=1476-4687&rft_id=info%3Adoi%2F10.1038%2Fs41586-021-03649-2&rft_id=info%3Apmid%2F34194037&rft.aulast=Galesic&rft.aufirst=Mirta&rft.au=Bruine+de+Bruin%2C+W%C3%A4ndi&rft.au=Dalege%2C+Jonas&rft.au=Feld%2C+Scott+L.&rft.au=Kreuter%2C+Frauke&rft.au=Olsson%2C+Henrik&rft.au=Prelec%2C+Drazen&rft.au=Stein%2C+Daniel+L.&rft.au=van+der+Does%2C+Tamara&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2Fs41586-021-03649-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristakisFowler2010" class="citation journal cs1">Christakis, Nicholas A.; Fowler, James H. (September 15, 2010). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2939797">"Social Network Sensors for Early Detection of Contagious Outbreaks"</a>. <i>PLOS ONE</i>. <b>5</b> (9): e12948. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1004.4792">1004.4792</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010PLoSO...512948C">2010PLoSO...512948C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0012948">10.1371/journal.pone.0012948</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2939797">2939797</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20856792">20856792</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+ONE&rft.atitle=Social+Network+Sensors+for+Early+Detection+of+Contagious+Outbreaks&rft.volume=5&rft.issue=9&rft.pages=e12948&rft.date=2010-09-15&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2939797%23id-name%3DPMC&rft_id=info%3Abibcode%2F2010PLoSO...512948C&rft_id=info%3Aarxiv%2F1004.4792&rft_id=info%3Apmid%2F20856792&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0012948&rft.aulast=Christakis&rft.aufirst=Nicholas+A.&rft.au=Fowler%2C+James+H.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2939797&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchnirring2010" class="citation news cs1">Schnirring, Lisa (Sep 16, 2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130506173925/http://www.cidrap.umn.edu/cidrap/content/influenza/general/news/sep1610friends.html">"Study: Friend 'sentinels' provide early flu warning"</a>. <i>CIDRAP News</i>. Archived from <a rel="nofollow" class="external text" href="http://www.cidrap.umn.edu/cidrap/content/influenza/general/news/sep1610friends.html">the original</a> on May 6, 2013<span class="reference-accessdate">. Retrieved <span class="nowrap">August 14,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=CIDRAP+News&rft.atitle=Study%3A+Friend+%27sentinels%27+provide+early+flu+warning&rft.date=2010-09-16&rft.aulast=Schnirring&rft.aufirst=Lisa&rft_id=http%3A%2F%2Fwww.cidrap.umn.edu%2Fcidrap%2Fcontent%2Finfluenza%2Fgeneral%2Fnews%2Fsep1610friends.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarcia-HerranzMoroCebrianChristakis2014" class="citation journal cs1">Garcia-Herranz, Manuel; Moro, Esteban; Cebrian, Manuel; Christakis, Nicholas A.; Fowler, James H. (2014-04-09). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3981694">"Using Friends as Sensors to Detect Global-Scale Contagious Outbreaks"</a>. <i>PLOS ONE</i>. <b>9</b> (4): e92413. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014PLoSO...992413G">2014PLoSO...992413G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0092413">10.1371/journal.pone.0092413</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1932-6203">1932-6203</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3981694">3981694</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24718030">24718030</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+ONE&rft.atitle=Using+Friends+as+Sensors+to+Detect+Global-Scale+Contagious+Outbreaks&rft.volume=9&rft.issue=4&rft.pages=e92413&rft.date=2014-04-09&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3981694%23id-name%3DPMC&rft_id=info%3Abibcode%2F2014PLoSO...992413G&rft_id=info%3Apmid%2F24718030&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0092413&rft.issn=1932-6203&rft.aulast=Garcia-Herranz&rft.aufirst=Manuel&rft.au=Moro%2C+Esteban&rft.au=Cebrian%2C+Manuel&rft.au=Christakis%2C+Nicholas+A.&rft.au=Fowler%2C+James+H.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3981694&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKumarKrackhardtFeld2021" class="citation arxiv cs1">Kumar, Vineet; Krackhardt, David; Feld, Scott (2021-05-18). "Interventions with Inversity in Unknown Networks Can Help Regulate Contagion". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2105.08758">2105.08758</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.SI">cs.SI</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Interventions+with+Inversity+in+Unknown+Networks+Can+Help+Regulate+Contagion&rft.date=2021-05-18&rft_id=info%3Aarxiv%2F2105.08758&rft.aulast=Kumar&rft.aufirst=Vineet&rft.au=Krackhardt%2C+David&rft.au=Feld%2C+Scott&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEomJo2015" class="citation journal cs1">Eom, Young-Ho; Jo, Hang-Hyun (2015-05-11). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1038/srep09752">"Tail-scope: Using friends to estimate heavy tails of degree distributions in large-scale complex networks"</a>. <i>Scientific Reports</i>. <b>5</b> (1): 9752. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1411.6871">1411.6871</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015NatSR...5E9752E">2015NatSR...5E9752E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fsrep09752">10.1038/srep09752</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2045-2322">2045-2322</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4426729">4426729</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25959097">25959097</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=Tail-scope%3A+Using+friends+to+estimate+heavy+tails+of+degree+distributions+in+large-scale+complex+networks&rft.volume=5&rft.issue=1&rft.pages=9752&rft.date=2015-05-11&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4426729%23id-name%3DPMC&rft_id=info%3Abibcode%2F2015NatSR...5E9752E&rft_id=info%3Aarxiv%2F1411.6871&rft.issn=2045-2322&rft_id=info%3Adoi%2F10.1038%2Fsrep09752&rft_id=info%3Apmid%2F25959097&rft.aulast=Eom&rft.aufirst=Young-Ho&rft.au=Jo%2C+Hang-Hyun&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1038%2Fsrep09752&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-:1-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_24-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNettasingheKrishnamurthy2021" class="citation journal cs1">Nettasinghe, Buddhika; Krishnamurthy, Vikram (2021-05-19). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1145/3451166">"Maximum Likelihood Estimation of Power-law Degree Distributions via Friendship Paradox-based Sampling"</a>. <i>ACM Transactions on Knowledge Discovery from Data</i>. <b>15</b> (6): 1–28. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1908.00310">1908.00310</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F3451166">10.1145/3451166</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1556-4681">1556-4681</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:199064540">199064540</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Transactions+on+Knowledge+Discovery+from+Data&rft.atitle=Maximum+Likelihood+Estimation+of+Power-law+Degree+Distributions+via+Friendship+Paradox-based+Sampling&rft.volume=15&rft.issue=6&rft.pages=1-28&rft.date=2021-05-19&rft_id=info%3Aarxiv%2F1908.00310&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A199064540%23id-name%3DS2CID&rft.issn=1556-4681&rft_id=info%3Adoi%2F10.1145%2F3451166&rft.aulast=Nettasinghe&rft.aufirst=Buddhika&rft.au=Krishnamurthy%2C+Vikram&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1145%2F3451166&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEomJo2014" class="citation cs2">Eom, Young-Ho; Jo, Hang-Hyun (2014), "Generalized friendship paradox in complex networks: The case of scientific collaboration", <i>Scientific Reports</i>, <b>4</b>, 4603, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1401.1458">1401.1458</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014NatSR...4E4603E">2014NatSR...4E4603E</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fsrep04603">10.1038/srep04603</a>, <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3980335">3980335</a></span>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24714092">24714092</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=Generalized+friendship+paradox+in+complex+networks%3A+The+case+of+scientific+collaboration&rft.volume=4&rft.pages=4603&rft.date=2014&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3980335%23id-name%3DPMC&rft_id=info%3Abibcode%2F2014NatSR...4E4603E&rft_id=info%3Aarxiv%2F1401.1458&rft_id=info%3Apmid%2F24714092&rft_id=info%3Adoi%2F10.1038%2Fsrep04603&rft.aulast=Eom&rft.aufirst=Young-Ho&rft.au=Jo%2C+Hang-Hyun&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrund2014" class="citation cs2">Grund, Thomas U. (2014), <a rel="nofollow" class="external text" href="http://www.sociologicalscience.com/download/volume%201/april/why-your-friends-are-more-important-and-special-than-you-think.pdf">"Why Your Friends Are More Important And Special Than You Think"</a> <span class="cs1-format">(PDF)</span>, <i>Sociological Science</i>, <b>1</b>: 128–140, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.15195%2Fv1.a10">10.15195/v1.a10</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sociological+Science&rft.atitle=Why+Your+Friends+Are+More+Important+And+Special+Than+You+Think&rft.volume=1&rft.pages=128-140&rft.date=2014&rft_id=info%3Adoi%2F10.15195%2Fv1.a10&rft.aulast=Grund&rft.aufirst=Thomas+U.&rft_id=http%3A%2F%2Fwww.sociologicalscience.com%2Fdownload%2Fvolume%25201%2Fapril%2Fwhy-your-friends-are-more-important-and-special-than-you-think.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDickerson2014" class="citation web cs1">Dickerson, Kelly (16 January 2014). <a rel="nofollow" class="external text" href="http://www.slate.com/blogs/business_insider/2014/01/16/friendship_paradox_why_are_my_friends_better_off_than_me.html">"Why Your Friends Are Probably More Popular, Richer, and Happier Than You"</a>. <i><a href="/wiki/Slate_Magazine" class="mw-redirect" title="Slate Magazine">Slate Magazine</a></i>. <a href="/wiki/The_Slate_Group" title="The Slate Group">The Slate Group</a><span class="reference-accessdate">. Retrieved <span class="nowrap">17 January</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Slate+Magazine&rft.atitle=Why+Your+Friends+Are+Probably+More+Popular%2C+Richer%2C+and+Happier+Than+You&rft.date=2014-01-16&rft.aulast=Dickerson&rft.aufirst=Kelly&rft_id=http%3A%2F%2Fwww.slate.com%2Fblogs%2Fbusiness_insider%2F2014%2F01%2F16%2Ffriendship_paradox_why_are_my_friends_better_off_than_me.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodasKooti,_FarshadLerman,_Kristina2013" class="citation arxiv cs1">Hodas, Nathan; Kooti, Farshad; <a href="/wiki/Kristina_Lerman" title="Kristina Lerman">Lerman, Kristina</a> (May 2013). "Friendship Paradox Redux: Your Friends are More Interesting than You". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1304.3480">1304.3480</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.SI">cs.SI</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Friendship+Paradox+Redux%3A+Your+Friends+are+More+Interesting+than+You&rft.date=2013-05&rft_id=info%3Aarxiv%2F1304.3480&rft.aulast=Hodas&rft.aufirst=Nathan&rft.au=Kooti%2C+Farshad&rft.au=Lerman%2C+Kristina&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBollenGoncalvesVan_de_LeemputGuanchen2017" class="citation cs2">Bollen, Johan; Goncalves, Bruno; Van de Leemput, Ingrid; Guanchen, Ruan (2017), "The happiness paradox: your friends are happier than you", <i>EPJ Data Science</i>, <b>6</b>, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1602.02665">1602.02665</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1140%2Fepjds%2Fs13688-017-0100-1">10.1140/epjds/s13688-017-0100-1</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2044182">2044182</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=EPJ+Data+Science&rft.atitle=The+happiness+paradox%3A+your+friends+are+happier+than+you&rft.volume=6&rft.date=2017&rft_id=info%3Aarxiv%2F1602.02665&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2044182%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1140%2Fepjds%2Fs13688-017-0100-1&rft.aulast=Bollen&rft.aufirst=Johan&rft.au=Goncalves%2C+Bruno&rft.au=Van+de+Leemput%2C+Ingrid&rft.au=Guanchen%2C+Ruan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKimHwongStaffordHughes2015" class="citation journal cs1">Kim, David A.; Hwong, Alison R.; Stafford, Derek; Hughes, D. Alex; O'Malley, A. James; Fowler, James H.; Christakis, Nicholas A. (2015-07-11). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4638320">"Social network targeting to maximise population behaviour change: a cluster randomised controlled trial"</a>. <i>Lancet</i>. <b>386</b> (9989): 145–153. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0140-6736%2815%2960095-2">10.1016/S0140-6736(15)60095-2</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1474-547X">1474-547X</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4638320">4638320</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25952354">25952354</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Lancet&rft.atitle=Social+network+targeting+to+maximise+population+behaviour+change%3A+a+cluster+randomised+controlled+trial&rft.volume=386&rft.issue=9989&rft.pages=145-153&rft.date=2015-07-11&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4638320%23id-name%3DPMC&rft.issn=1474-547X&rft_id=info%3Apmid%2F25952354&rft_id=info%3Adoi%2F10.1016%2FS0140-6736%2815%2960095-2&rft.aulast=Kim&rft.aufirst=David+A.&rft.au=Hwong%2C+Alison+R.&rft.au=Stafford%2C+Derek&rft.au=Hughes%2C+D.+Alex&rft.au=O%27Malley%2C+A.+James&rft.au=Fowler%2C+James+H.&rft.au=Christakis%2C+Nicholas+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4638320&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAiroldiChristakis2024" class="citation journal cs1">Airoldi, Edoardo M.; Christakis, Nicholas A. (2024-05-03). <a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.adi5147">"Induction of social contagion for diverse outcomes in structured experiments in isolated villages"</a>. <i>Science</i>. <b>384</b> (6695). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.adi5147">10.1126/science.adi5147</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0036-8075">0036-8075</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/38696582">38696582</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Induction+of+social+contagion+for+diverse+outcomes+in+structured+experiments+in+isolated+villages&rft.volume=384&rft.issue=6695&rft.date=2024-05-03&rft.issn=0036-8075&rft_id=info%3Apmid%2F38696582&rft_id=info%3Adoi%2F10.1126%2Fscience.adi5147&rft.aulast=Airoldi&rft.aufirst=Edoardo+M.&rft.au=Christakis%2C+Nicholas+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1126%252Fscience.adi5147&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShakyaStaffordHughesKeegan2017" class="citation journal cs1">Shakya, Holly B.; Stafford, Derek; Hughes, D. Alex; Keegan, Thomas; Negron, Rennie; Broome, Jai; McKnight, Mark; Nicoll, Liza; Nelson, Jennifer; Iriarte, Emma; Ordonez, Maria; Airoldi, Edo; Fowler, James H.; Christakis, Nicholas A. (2017-03-01). <a rel="nofollow" class="external text" href="https://bmjopen.bmj.com/content/7/3/e012996">"Exploiting social influence to magnify population-level behaviour change in maternal and child health: study protocol for a randomised controlled trial of network targeting algorithms in rural Honduras"</a>. <i>BMJ Open</i>. <b>7</b> (3): e012996. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1136%2Fbmjopen-2016-012996">10.1136/bmjopen-2016-012996</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2044-6055">2044-6055</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5353315">5353315</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28289044">28289044</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BMJ+Open&rft.atitle=Exploiting+social+influence+to+magnify+population-level+behaviour+change+in+maternal+and+child+health%3A+study+protocol+for+a+randomised+controlled+trial+of+network+targeting+algorithms+in+rural+Honduras&rft.volume=7&rft.issue=3&rft.pages=e012996&rft.date=2017-03-01&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5353315%23id-name%3DPMC&rft.issn=2044-6055&rft_id=info%3Apmid%2F28289044&rft_id=info%3Adoi%2F10.1136%2Fbmjopen-2016-012996&rft.aulast=Shakya&rft.aufirst=Holly+B.&rft.au=Stafford%2C+Derek&rft.au=Hughes%2C+D.+Alex&rft.au=Keegan%2C+Thomas&rft.au=Negron%2C+Rennie&rft.au=Broome%2C+Jai&rft.au=McKnight%2C+Mark&rft.au=Nicoll%2C+Liza&rft.au=Nelson%2C+Jennifer&rft.au=Iriarte%2C+Emma&rft.au=Ordonez%2C+Maria&rft.au=Airoldi%2C+Edo&rft.au=Fowler%2C+James+H.&rft.au=Christakis%2C+Nicholas+A.&rft_id=https%3A%2F%2Fbmjopen.bmj.com%2Fcontent%2F7%2F3%2Fe012996&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexanderForastiereGuptaChristakis2022" class="citation journal cs1">Alexander, Marcus; Forastiere, Laura; Gupta, Swati; Christakis, Nicholas A. (2022-07-26). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9335263">"Algorithms for seeding social networks can enhance the adoption of a public health intervention in urban India"</a>. <i>Proceedings of the National Academy of Sciences</i>. <b>119</b> (30): e2120742119. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022PNAS..11920742A">2022PNAS..11920742A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.2120742119">10.1073/pnas.2120742119</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9335263">9335263</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35862454">35862454</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&rft.atitle=Algorithms+for+seeding+social+networks+can+enhance+the+adoption+of+a+public+health+intervention+in+urban+India&rft.volume=119&rft.issue=30&rft.pages=e2120742119&rft.date=2022-07-26&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC9335263%23id-name%3DPMC&rft_id=info%3Abibcode%2F2022PNAS..11920742A&rft_id=info%3Apmid%2F35862454&rft_id=info%3Adoi%2F10.1073%2Fpnas.2120742119&rft.issn=0027-8424&rft.aulast=Alexander&rft.aufirst=Marcus&rft.au=Forastiere%2C+Laura&rft.au=Gupta%2C+Swati&rft.au=Christakis%2C+Nicholas+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC9335263&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friendship_paradox&action=edit&section=5" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrogatz2012" class="citation news cs1">Strogatz, Steven (September 17, 2012). <a rel="nofollow" class="external text" href="http://opinionator.blogs.nytimes.com/2012/09/17/friends-you-can-count-on/">"Friends You Can Count On"</a>. <i>New York Times</i><span class="reference-accessdate">. Retrieved <span class="nowrap">17 January</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+York+Times&rft.atitle=Friends+You+Can+Count+On&rft.date=2012-09-17&rft.aulast=Strogatz&rft.aufirst=Steven&rft_id=http%3A%2F%2Fopinionator.blogs.nytimes.com%2F2012%2F09%2F17%2Ffriends-you-can-count-on%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriendship+paradox" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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href="/wiki/Clique" title="Clique">Clique</a> <ul><li><a href="/wiki/Adolescent_clique" title="Adolescent clique">Adolescent</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Networks</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#F4F0E;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Corporate_social_media" title="Corporate social media">Corporate social media</a></li> <li><a href="/wiki/Distributed_social_network" title="Distributed social network">Distributed social network</a> (<a href="/wiki/Comparison_of_software_and_protocols_for_distributed_social_networking" title="Comparison of software and protocols for distributed social networking"><i>list</i></a>)</li> <li><a href="/wiki/Enterprise_social_networking" title="Enterprise social networking">Enterprise social networking</a></li> <li><a href="/wiki/Enterprise_social_software" title="Enterprise social software">Enterprise social software</a></li> <li><a href="/wiki/Mobile_social_network" title="Mobile social network">Mobile social network</a></li> <li><a href="/wiki/Personal_knowledge_networking" title="Personal knowledge networking">Personal knowledge networking</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Social_networking_service" title="Social networking service">Services</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_social_networking_services" title="List of social networking services">List of social networking services</a></li> <li><a href="/wiki/List_of_virtual_communities_with_more_than_1_million_users" title="List of virtual communities with more than 1 million users">List of virtual communities with more than 1 million users</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts and<br /> theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#F4F0E;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ambient_awareness" title="Ambient awareness">Ambient awareness</a></li> <li><a href="/wiki/Assortative_mixing" title="Assortative mixing">Assortative mixing</a></li> <li><a href="/wiki/Attention_inequality" title="Attention inequality">Attention inequality</a></li> <li><a href="/wiki/Bridge_(interpersonal)" title="Bridge (interpersonal)">Interpersonal bridge</a></li> <li><a href="/wiki/Organizational_network_analysis" title="Organizational network analysis">Organizational network analysis</a></li> <li><a href="/wiki/Small-world_experiment" title="Small-world experiment">Small-world experiment</a></li> <li><a href="/wiki/Social_aspects_of_television" title="Social aspects of television">Social aspects of television</a></li> <li><a href="/wiki/Social_capital" title="Social capital">Social capital</a></li> <li><a href="/wiki/Social_data_revolution" title="Social data revolution">Social data revolution</a></li> <li><a href="/wiki/Social_exchange_theory" title="Social exchange theory">Social exchange theory</a></li> <li><a href="/wiki/Social_identity_theory" title="Social identity theory">Social identity theory</a></li> <li><a href="/wiki/Social_media_and_psychology" title="Social media and psychology">Social media and psychology</a></li> <li><a href="/wiki/Social_media_intelligence" title="Social media intelligence">Social media intelligence</a></li> <li><a href="/wiki/Social_media_mining" title="Social media mining">Social media mining</a></li> <li><a href="/wiki/Social_media_optimization" title="Social media optimization">Social media optimization</a></li> <li><a href="/wiki/Social_network_analysis" title="Social network analysis">Social network analysis</a></li> <li><a href="/wiki/Social_web" title="Social web">Social web</a></li> <li><a href="/wiki/Structural_endogamy" title="Structural endogamy">Structural endogamy</a></li> <li><a href="/wiki/Virtual_collective_consciousness" title="Virtual collective consciousness">Virtual collective consciousness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Models and<br /> processes</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Account_verification" title="Account verification">Account verification</a></li> <li><a href="/wiki/Social_network_aggregation" title="Social network aggregation">Aggregation</a></li> <li><a href="/wiki/Social_network_change_detection" class="mw-redirect" title="Social network change detection">Change detection</a></li> <li><a href="/wiki/Blockmodeling" title="Blockmodeling">Blockmodeling</a></li> <li><a href="/wiki/Collaboration_graph" title="Collaboration graph">Collaboration graph</a></li> <li><a href="/wiki/Collaborative_consumption" title="Collaborative consumption">Collaborative consumption</a></li> <li><a href="/wiki/Giant_Global_Graph" title="Giant Global Graph">Giant Global Graph</a></li> <li><a href="/wiki/Lateral_communication" title="Lateral communication">Lateral communication</a></li> <li><a href="/wiki/Reputation_system" title="Reputation system">Reputation system</a></li> <li><a href="/wiki/Social_bot" title="Social bot">Social bot</a></li> <li><a href="/wiki/Social_graph" title="Social graph">Social graph</a></li> <li><a href="/wiki/Social_media_analytics" title="Social media analytics">Social media analytics</a></li> <li><a href="/wiki/Social_network_analysis_software" title="Social network analysis software">Social network analysis software</a></li> <li><a href="/wiki/Social_networking_potential" class="mw-redirect" title="Social networking potential">Social networking potential</a></li> <li><a href="/wiki/Social_television" title="Social television">Social television</a></li> <li><a href="/wiki/Structural_cohesion" title="Structural cohesion">Structural cohesion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Economics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#F4F0E;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affinity_fraud" title="Affinity fraud">Affinity fraud</a></li> <li><a href="/wiki/Attention_economy" title="Attention economy">Attention economy</a></li> <li><a href="/wiki/Collaborative_finance" title="Collaborative finance">Collaborative finance</a></li> <li><a href="/wiki/Creator_economy" title="Creator economy">Creator economy</a></li> <li><a href="/wiki/Influencer_marketing" title="Influencer marketing">Influencer marketing</a></li> <li><a href="/wiki/Narrowcasting" title="Narrowcasting">Narrowcasting</a></li> <li><a href="/wiki/Sharing_economy" title="Sharing economy">Sharing economy</a></li> <li><a href="/wiki/Social_commerce" title="Social commerce">Social commerce</a></li> <li><a href="/wiki/Social_sorting" title="Social sorting">Social sorting</a></li> <li><a href="/wiki/Viral_marketing" title="Viral marketing">Viral marketing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algorithmic_radicalization" title="Algorithmic radicalization">Algorithmic radicalization</a></li> <li><a href="/wiki/Community_recognition" title="Community recognition">Community recognition</a></li> <li><a href="/wiki/Complex_contagion" title="Complex contagion">Complex contagion</a></li> <li><a href="/wiki/Computer_addiction" title="Computer addiction">Computer addiction</a></li> <li><a href="/wiki/Consequential_strangers" title="Consequential strangers">Consequential strangers</a></li> <li><a href="/wiki/Friend_of_a_friend" title="Friend of a friend">Friend of a friend</a></li> <li><a href="/wiki/Friending_and_following" title="Friending and following">Friending and following</a></li> <li><a class="mw-selflink selflink">Friendship paradox</a></li> <li><a href="/wiki/Influence-for-hire" title="Influence-for-hire">Influence-for-hire</a></li> <li><a href="/wiki/Internet_addiction" class="mw-redirect" title="Internet addiction">Internet addiction</a></li> <li><a href="/wiki/Information_overload" title="Information overload">Information overload</a></li> <li><a href="/wiki/Overchoice" title="Overchoice">Overchoice</a></li> <li><a href="/wiki/Six_degrees_of_separation" title="Six degrees of separation">Six degrees of separation</a></li> <li><a href="/wiki/Social_media_addiction" class="mw-redirect" title="Social media addiction">Social media addiction</a></li> <li><a href="/wiki/Social_media_and_suicide" title="Social media and suicide">Social media and suicide</a></li> <li><a href="/wiki/Social_invisibility" title="Social invisibility">Social invisibility</a></li> <li><a href="/wiki/Social_network_game" title="Social network game">Social network game</a></li> <li><a href="/wiki/Suicide_and_the_Internet" title="Suicide and the Internet">Suicide and the Internet</a></li> <li><a href="/wiki/Tribe_(internet)" title="Tribe (internet)">Tribe</a></li> <li><a href="/wiki/Viral_phenomenon" title="Viral phenomenon">Viral phenomenon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#F4F0E;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Friendship_recession" title="Friendship recession">Friendship recession</a></li> <li><a href="/wiki/Peer_pressure" title="Peer pressure">Peer pressure</a></li> <li><a href="/wiki/List_of_social_network_researchers" class="mw-redirect" title="List of social network 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