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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Citation_metrics" title="Category:Citation metrics">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle" style="font-size:200%;font-weight:normal;"><a href="/wiki/Citation_impact" title="Citation impact">Citation metrics</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Locations_of_papers_in_a_map_of_science_and_locations_of_the_key_papers_for_Nobel_prizes.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Locations_of_papers_in_a_map_of_science_and_locations_of_the_key_papers_for_Nobel_prizes.tif/lossy-page1-110px-Locations_of_papers_in_a_map_of_science_and_locations_of_the_key_papers_for_Nobel_prizes.tif.jpg" decoding="async" width="110" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Locations_of_papers_in_a_map_of_science_and_locations_of_the_key_papers_for_Nobel_prizes.tif/lossy-page1-165px-Locations_of_papers_in_a_map_of_science_and_locations_of_the_key_papers_for_Nobel_prizes.tif.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Locations_of_papers_in_a_map_of_science_and_locations_of_the_key_papers_for_Nobel_prizes.tif/lossy-page1-220px-Locations_of_papers_in_a_map_of_science_and_locations_of_the_key_papers_for_Nobel_prizes.tif.jpg 2x" data-file-width="2220" data-file-height="1356" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Altmetrics" title="Altmetrics">Altmetrics</a></li> <li><a href="/wiki/Article-level_metrics" title="Article-level metrics">Article-level</a></li> <li><a href="/wiki/Author-level_metrics" title="Author-level metrics">Author-level</a> <ul><li><a href="/wiki/Eigenfactor" title="Eigenfactor">Eigenfactor</a></li> <li><a href="/wiki/G-index" title="G-index">G-index</a></li> <li><a href="/wiki/H-index" title="H-index">H-index</a></li></ul></li> <li><a href="/wiki/Bibliographic_coupling" title="Bibliographic coupling">Bibliographic coupling</a></li> <li><a href="/wiki/Citation" title="Citation">Citation</a> <ul><li><a href="/wiki/Citation_analysis" title="Citation analysis">Analysis</a></li> <li><a class="mw-selflink selflink">Dynamics</a></li> <li><a href="/wiki/Citation_index" title="Citation index">Index</a></li> <li><a href="/wiki/Citation_graph" title="Citation graph">Graph</a></li></ul></li> <li><a href="/wiki/Co-citation" title="Co-citation">Co-citation</a> <ul><li><a href="/wiki/Co-citation_Proximity_Analysis" title="Co-citation Proximity Analysis">Proximity Analysis</a></li></ul></li> <li><a href="/wiki/Coercive_citation" title="Coercive citation">Coercive citation</a></li> <li><a href="/wiki/Citation_cartel" title="Citation cartel">Citation cartel</a></li> <li><a href="/wiki/Initiative_for_Open_Citations" title="Initiative for Open Citations">I4OC</a></li> <li><a href="/wiki/Journal_ranking" title="Journal ranking">Journal-level</a> <ul><li><a href="/wiki/CiteScore" title="CiteScore">CiteScore</a></li> <li><a href="/wiki/Impact_factor" title="Impact factor">Impact factor</a></li> <li><a href="/wiki/SCImago_Journal_Rank" title="SCImago Journal Rank">SCImago</a></li></ul></li> <li><a href="/wiki/Kardashian_Index" title="Kardashian Index">Kardashian Index</a></li></ul> </div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Citation_metrics" title="Template:Citation metrics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Citation_metrics" title="Template talk:Citation metrics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Citation_metrics" title="Special:EditPage/Template:Citation metrics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Yearly_citation_for_research_paper.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Yearly_citation_for_research_paper.png/220px-Yearly_citation_for_research_paper.png" decoding="async" width="220" height="236" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Yearly_citation_for_research_paper.png/330px-Yearly_citation_for_research_paper.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Yearly_citation_for_research_paper.png/440px-Yearly_citation_for_research_paper.png 2x" data-file-width="864" data-file-height="926" /></a><figcaption>Yearly citation c(t) for a research paper from the e Physical Review corpus.</figcaption></figure> <p><b>Citation dynamics</b> describes the number of <a href="/wiki/References" class="mw-redirect" title="References">references</a> received by the <a href="/wiki/Article_(publishing)" title="Article (publishing)">article</a> or other scientific work over <a href="/wiki/Time" title="Time">time</a>. The <a href="/wiki/Citation" title="Citation">citation</a> dynamics is usually described by the bang, that take place two–three years after the work has been published, and the burst size spans several <a href="/wiki/Order_of_magnitude" title="Order of magnitude">orders of magnitude</a>. The presence of bursts is not consistent with other models based on <a href="/wiki/Preferential_attachment" title="Preferential attachment">preferential attachment</a>. Those models are able to account for the skewed citation distribution but their reference accumulation is gradual.<sup id="cite_ref-eom_1-0" class="reference"><a href="#cite_note-eom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The dynamics of scientific production has changed significantly over the past years. Due to <a href="/wiki/Technological_change" title="Technological change">technological progress</a>, the number of published papers has been increasing exponentially until now. This, along with a much shorter time needed for the article to be published, has affected the <a href="/wiki/Citation" title="Citation">citation</a> dynamics of the modern papers. Furthermore, if the <a href="/wiki/Reference_list" class="mw-redirect" title="Reference list">reference list</a> of the study includes papers published in different years, older papers tend to have more citations. This may not necessarily because they are better but just because they had more time to accumulate those references. </p> <div class="mw-heading mw-heading2"><h2 id="Model">Model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Citation_dynamics&action=edit&section=1" title="Edit section: Model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It has been found that citation distributions are best described by a shifted <a href="/wiki/Power_law" title="Power law">power-law</a>.<sup id="cite_ref-eom_1-1" class="reference"><a href="#cite_note-eom-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Probability" title="Probability">probability</a> that paper <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> is cited at time <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> after publication 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 \Pi _{i}(t)=\eta _{i}c_{i}^{t}P_{i}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{i}(t)=\eta _{i}c_{i}^{t}P_{i}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b264003a7a7183fc7f8ee4420a18b70904864d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.019ex; height:3.176ex;" alt="{\displaystyle \Pi _{i}(t)=\eta _{i}c_{i}^{t}P_{i}(t)}"></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 \Pi _{i}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{i}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef2bf4b99043bd6f6911e7f32ffe162518c2606" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.192ex; height:2.843ex;" alt="{\displaystyle \Pi _{i}(t)}"></span> serves as the outcome variable for each particular paper <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> at time <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. Fitness, <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 \eta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b336ed51c970728280e38f5a131ac52f69833c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.955ex; height:2.176ex;" alt="{\displaystyle \eta _{i}}"></span>, captures the inherent differences between papers, accounting for the perceived <a href="/wiki/Novelty" title="Novelty">novelty</a> and importance of a discovery. <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 c_{i}^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9afdcd29327456ab39deb41766e2b5d5b7816f25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.833ex; height:3.176ex;" alt="{\displaystyle c_{i}^{t}}"></span> represents the cumulative number of citations acquired by a paper <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> at time <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></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 P_{i}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e637de47071664a29005349746e88acf940f2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.941ex; height:2.843ex;" alt="{\displaystyle P_{i}(t)}"></span> is a log-normal survival probability.<sup id="cite_ref-Barabasi_2-0" class="reference"><a href="#cite_note-Barabasi-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The probability is equal </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 P_{i}(t)={\frac {1}{{\sqrt {2\pi }}\sigma _{i}t}}exp\left\lbrack -{\frac {(lnt-\mu _{i})^{2}}{2\sigma _{i}^{2}}}\right\rbrack }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>e</mi> <mi>x</mi> <mi>p</mi> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>l</mi> <mi>n</mi> <mi>t</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}(t)={\frac {1}{{\sqrt {2\pi }}\sigma _{i}t}}exp\left\lbrack -{\frac {(lnt-\mu _{i})^{2}}{2\sigma _{i}^{2}}}\right\rbrack }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdea11fea3b77a9016f4824ed75c6e1574a67ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.43ex; height:7.509ex;" alt="{\displaystyle P_{i}(t)={\frac {1}{{\sqrt {2\pi }}\sigma _{i}t}}exp\left\lbrack -{\frac {(lnt-\mu _{i})^{2}}{2\sigma _{i}^{2}}}\right\rbrack }"></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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> is time; <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" 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 \sigma }"></span> is <a href="/wiki/Longevity" title="Longevity">longevity</a>, capturing the decay rate; 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> indicates immediacy, governing the time for a paper to reach its citation peak. The ultimate impact <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 c_{i}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74ddbd7cc02c2ae333d91f1e02db2963252e6e09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.882ex; height:2.843ex;" alt="{\displaystyle c_{i}^{\infty }}"></span> represents the total number or citations that the paper receives during its lifetime. </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 c_{i}^{\infty }=m(e^{\lambda _{i}}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>=</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}^{\infty }=m(e^{\lambda _{i}}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe8ef575e402bce35af6d8f6671757d5d783a05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.732ex; height:3.343ex;" alt="{\displaystyle c_{i}^{\infty }=m(e^{\lambda _{i}}-1)}"></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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is a global parameter that has the same value for all publications. <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 \lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}"></span> represents the relative fitness of the paper. From the above formula, we can see that the total number of references that the paper can receive during its lifetime depends only on its relative fitness which is very hard to quantify. </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=Citation_dynamics&action=edit&section=2" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Citation_index" title="Citation index">Citation index</a></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=Citation_dynamics&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-eom-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-eom_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-eom_1-1"><sup><i><b>b</b></i></sup></a></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="CITEREFEomFortunato2011" class="citation journal cs1">Eom, Young-Ho; Fortunato, Santo (2011). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3178574">"Characterizing and Modeling Citation Dynamics"</a>. <i>PLOS ONE</i>. <b>6</b> (9): e24926. <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/1110.2153">1110.2153</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/2011PLoSO...624926E">2011PLoSO...624926E</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.0024926">10.1371/journal.pone.0024926</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/PMC3178574">3178574</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/21966387">21966387</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=Characterizing+and+Modeling+Citation+Dynamics&rft.volume=6&rft.issue=9&rft.pages=e24926&rft.date=2011&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3178574%23id-name%3DPMC&rft_id=info%3Abibcode%2F2011PLoSO...624926E&rft_id=info%3Aarxiv%2F1110.2153&rft_id=info%3Apmid%2F21966387&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0024926&rft.aulast=Eom&rft.aufirst=Young-Ho&rft.au=Fortunato%2C+Santo&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3178574&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACitation+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Barabasi-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Barabasi_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangSongBarabási2013" class="citation journal cs1">Wang, Dashun; Song, Chaoming; Barabási, Albert-László (4 Oct 2013). <a rel="nofollow" class="external text" href="https://www.science.org/doi/pdf/10.1126/science.1237825">"Quantifying Long-Term Scientific Impact"</a>. <i>Science</i>. <b>342</b> (6154): 127–132. <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/1306.3293">1306.3293</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/2013Sci...342..127W">2013Sci...342..127W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1237825">10.1126/science.1237825</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/24092745">24092745</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:260558492">260558492</a><span class="reference-accessdate">. Retrieved <span class="nowrap">18 May</span> 2017</span>.</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=Quantifying+Long-Term+Scientific+Impact&rft.volume=342&rft.issue=6154&rft.pages=127-132&rft.date=2013-10-04&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A260558492%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2013Sci...342..127W&rft_id=info%3Aarxiv%2F1306.3293&rft_id=info%3Apmid%2F24092745&rft_id=info%3Adoi%2F10.1126%2Fscience.1237825&rft.aulast=Wang&rft.aufirst=Dashun&rft.au=Song%2C+Chaoming&rft.au=Barab%C3%A1si%2C+Albert-L%C3%A1szl%C3%B3&rft_id=https%3A%2F%2Fwww.science.org%2Fdoi%2Fpdf%2F10.1126%2Fscience.1237825&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACitation+dynamics" class="Z3988"></span></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Citation_dynamics&oldid=1210091183">https://en.wikipedia.org/w/index.php?title=Citation_dynamics&oldid=1210091183</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Academic_publishing" title="Category:Academic publishing">Academic publishing</a></li><li><a href="/wiki/Category:Citation_metrics" title="Category:Citation metrics">Citation metrics</a></li><li><a href="/wiki/Category:Reference" title="Category:Reference">Reference</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 24 February 2024, at 23:36<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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