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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Exploring the dynamics of self-citations and their role in shaping scientific impact</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.09811v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S1" title="In Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S2" title="In Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span>Methodology</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S2.SS1" title="In II Methodology ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.1 </span>Model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S2.SS2" title="In II Methodology ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.2 </span>Self-citations</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S2.SS3" title="In II Methodology ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.3 </span>Data</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3" title="In Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span>Results</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.SS1" title="In III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III.1 </span>Analysis of whole dataset</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.SS2" title="In III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III.2 </span>Analysis of individual authors</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.SS3" title="In III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III.3 </span>Comparisons with 3DSI model</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S4" title="In Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span>Discussion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S5" title="In Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span>Code availability</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S6" title="In Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VI </span>Data availability</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line" lang="en"> <h1 class="ltx_title ltx_title_document">Exploring the dynamics of self-citations and their role in shaping scientific impact</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Maciej J. Mrowinski<sup class="ltx_sup" id="id6.6.id1"><span class="ltx_text ltx_font_italic" id="id6.6.id1.1">1,∗</span></sup>, Aleksandra Buczek<sup class="ltx_sup" id="id7.7.id2"><span class="ltx_text ltx_font_italic" id="id7.7.id2.1">1</span></sup>, Agata Fronczak<sup class="ltx_sup" id="id8.8.id3"><span class="ltx_text ltx_font_italic" id="id8.8.id3.1">1</span></sup> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id9.9.id1"><span class="ltx_text ltx_font_italic" id="id9.9.id1.1">1</span></sup> Warsaw University of Technology, Faculty of Physics, <br class="ltx_break"/>ul. Koszykowa 75, 00-662 Warsaw, Poland <br class="ltx_break"/><sup class="ltx_sup" id="id10.10.id2"><span class="ltx_text ltx_font_italic" id="id10.10.id2.1">∗</span></sup> Corresponding author; email: maciej.mrowinski@pw.edu.pl </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id11.id1"><span class="ltx_text" id="id11.id1.1">Understanding the mechanisms driving the distribution of scientific citations is a key challenge in assessing the scientific impact of authors. We investigate the influence of the preferential attachment rule (PAR) in this process by analyzing individual citation events from the DBLP dataset, enabling us to estimate the probability of citations being assigned preferentially. Our findings reveal that, for the aggregated dataset, PAR dominates the citation distribution process, with approximately 70% of citations adhering to this mechanism. However, analysis at the individual level shows significant variability, with some authors experiencing a greater prevalence of preferential citations, particularly in the context of external citations. In contrast, self-citations exhibit notably different behaviour, with only 20% following PAR. We also demonstrate that the prominence of PAR increases with an author’s citability (average citations per paper), suggesting that more citable authors are preferentially cited, while less-cited authors experience more random citation patterns. Furthermore, we show that self-citations may influence bibliometric indexes. Our results emphasise the distinct dynamics of self-citations compared to external citations, raising questions about the mechanisms driving self-citation patterns. These findings provide new insights into citation behaviours and highlight the limitations of existing approaches in capturing the nuances of scientific impact.</span></p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">There are numerous models of citation distribution - models that aim to recreate citation vectors (vectors whose elements represent the number of citations received by each paper) based on selected bibliometric parameters describing authors. For instance, in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib1" title="">1</a>]</cite>, Georgia Ionescu and Bastien Chopard introduced a model that distinguishes between self-citations and external citations, assuming that both are distributed according to the preferential attachment rule (PAR). PAR, often referred to as the ”rich get richer” principle <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib3" title="">3</a>]</cite>, posits that articles with many citations are more likely to attract additional citations. However, studies suggest that self-citations may not follow this rule. For example, authors might be more likely to self-cite their most recent articles <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib5" title="">5</a>]</cite>. It is also worth noting that the Ionescu-Chopard model does not explicitly describe the dynamics of citation distribution. Instead, it serves as an artificial procedure designed to reproduce the citation vector at a specific point in time.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">An interesting extension of the Ionescu-Chopard model is the Three Dimensions of Scientific Impact model (3DSI), introduced in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib6" title="">6</a>]</cite>. This model does not differentiate between external and self-citations. Instead, it assumes that a fraction of all citations is distributed preferentially while the remainder is distributed randomly. Although this model reproduces citation vectors effectively, it has important limitations, which we will discuss later. The 3DSI model proposes that each scientist can be characterized by three parameters: the number of papers, the total number of citations, and the fraction of citations distributed according to PAR (hence the three dimensions of scientific impact). The first two parameters are standard bibliometric metrics that can be easily obtained, while the third parameter is derived by optimizing the model - identifying the fraction of preferential citations that provides the best fit between the model and the actual citation vector.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">As we began exploring various citation distribution models, we observed that many, including those discussed above, rely on some form of the preferential attachment rule. Intuitively, this seems reasonable. When compiling bibliographies or searching for relevant literature, authors often use scientific databases, likely favouring highly cited papers over less cited ones. Moreover, the citation network is scale-free <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib7" title="">7</a>]</cite>, a property frequently attributed to preferential attachment. Nevertheless, this raises a critical question: how significant is PAR in the citation distribution process? One might assume that the 3DSI model provides an answer, as it allows for the calculation of the fraction of citations distributed preferentially. However, does this fraction accurately reflect the role of preferential attachment in the real citation process, or is it merely a parameter resulting from the model’s assumptions? In this paper, we aim to determine the true fraction of preferential citations and address this question.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Another key question concerns self-citations, a topic that is both fascinating and contentious in the scientific community. Some view self-citations as a hallmark of productive authors <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib8" title="">8</a>]</cite>, arguing that they can genuinely enhance the visibility of one’s work <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib9" title="">9</a>]</cite>, which in turn boosts external citations with little downside <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib10" title="">10</a>]</cite>. However, while there are many legitimate reasons for self-citation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib12" title="">12</a>]</cite>, some are less benign. Self-citations can be exploited to artificially inflate bibliometric indicators <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib14" title="">14</a>]</cite>, a practice that should be considered when evaluating scientific impact <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib15" title="">15</a>]</cite>. Policymakers must also exercise caution when designing systems for career advancement and evaluation, as such policies can unintentionally influence self-citation behaviour <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib18" title="">18</a>]</cite>. Interestingly, self-citation issues extend beyond individual authors and can affect entire journals <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib20" title="">20</a>]</cite>, where they may be used to manipulate journal impact factors. Given their importance, it is crucial to understand the properties of self-citation distribution. Is it similar to external citation distribution? Does preferential attachment play a key role? These are questions we will address in this paper.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.2">To determine the fraction <math alttext="\rho" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mi id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><ci id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_ρ</annotation></semantics></math> of citations governed by the preferential attachment rule and to answer the questions posed above, we adopted a largely data-driven approach. Using the extensive DBLP database, we reconstructed citation histories and vectors for a large set of authors, tracking the evolution of citations over time. By adapting a methodology previously employed to study preferential attachment in complex networks<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib21" title="">21</a>]</cite>, we quantified the fraction <math alttext="\rho" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mi id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><ci id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">italic_ρ</annotation></semantics></math> of citations governed by PAR.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_img_portrait" height="1174" id="S1.F1.g1" src="x1.png" width="830"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>The process of calculating the value of <math alttext="\rho" class="ltx_Math" display="inline" id="S1.F1.3.m1.1"><semantics id="S1.F1.3.m1.1b"><mi id="S1.F1.3.m1.1.1" xref="S1.F1.3.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S1.F1.3.m1.1c"><ci id="S1.F1.3.m1.1.1.cmml" xref="S1.F1.3.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.3.m1.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S1.F1.3.m1.1e">italic_ρ</annotation></semantics></math> for a specific scientist involves the following steps: 1) Each citation is processed individually, in the order it was received. 2) The probability of each citation is computed based on the assumptions of the model. 3) These probabilities are combined to form the log-likelihood. 4) The log-likelihood is maximized, resulting in the value of <math alttext="\rho" class="ltx_Math" display="inline" id="S1.F1.4.m2.1"><semantics id="S1.F1.4.m2.1b"><mi id="S1.F1.4.m2.1.1" xref="S1.F1.4.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S1.F1.4.m2.1c"><ci id="S1.F1.4.m2.1.1.cmml" xref="S1.F1.4.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.4.m2.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S1.F1.4.m2.1e">italic_ρ</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">Our approach involves calculating the likelihood of each citation event based on the state of an author’s citation vector at the previous time step, incorporating both preferential attachment and random allocation mechanisms. The parameter <math alttext="\rho" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><mi id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><ci id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">italic_ρ</annotation></semantics></math>, which controls the balance between these mechanisms, is estimated by maximising the log-likelihood for individual authors (as illustrated schematically in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S1.F1" title="Figure 1 ‣ I Introduction ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">1</span></a>) and for the dataset as a whole. This analysis was conducted for all types of citations combined, as well as separately for self-citations and external citations, providing a detailed characterisation of citation dynamics.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span>Methodology</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.1 </span>Model</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1">One of the most significant and profound discoveries in the study of complex networks was the realization that many of their properties can be explained by the preferential attachment rule <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib22" title="">22</a>]</cite>. However, while it is possible to construct models that replicate certain characteristics of real systems, this does not necessarily mean that the dynamics and assumptions of these models align with those of the actual systems they aim to describe.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.1">In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib21" title="">21</a>]</cite>, the authors employed a simple yet ingenious method to investigate preferential attachment in networks. They analyzed the temporal evolution of a real network, link by link, and calculated the probability of each link forming based on the assumptions of several proposed models. This approach allowed them to aggregate the probabilities of the network’s formation under different models and to compare the likelihood of each model. We have adapted this methodology to examine the role of the preferential attachment rule in the process of citation distribution. This subsection provides a detailed description of the procedure we employed.</p> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.1">Scientist <math alttext="j" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mi id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><ci id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_j</annotation></semantics></math> is characterised by a citation vector</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math 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id="S2.E1.m1.8.8.1.1.2.2.2.2.2.2.cmml" xref="S2.E1.m1.8.8.1.1.2.2.2.2.2.2">𝑋</ci><ci id="S2.E1.m1.3.3.1.1.cmml" xref="S2.E1.m1.3.3.1.1">𝑗</ci></apply><cn id="S2.E1.m1.8.8.1.1.2.2.2.2.3.cmml" type="integer" xref="S2.E1.m1.8.8.1.1.2.2.2.2.3">2</cn></apply><ci id="S2.E1.m1.6.6.cmml" xref="S2.E1.m1.6.6">𝑡</ci></apply><ci id="S2.E1.m1.7.7.cmml" xref="S2.E1.m1.7.7">…</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.8c">X^{(j)}(t)=\left[X^{(j)}_{1}(t),X^{(j)}_{2}(t),\dots\right],</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.8d">italic_X start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( italic_t ) = [ italic_X start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , italic_X start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , … ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.5">where <math alttext="X^{(j)}_{i}(t)" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m1.2"><semantics id="S2.SS1.p3.2.m1.2a"><mrow id="S2.SS1.p3.2.m1.2.3" xref="S2.SS1.p3.2.m1.2.3.cmml"><msubsup id="S2.SS1.p3.2.m1.2.3.2" xref="S2.SS1.p3.2.m1.2.3.2.cmml"><mi id="S2.SS1.p3.2.m1.2.3.2.2.2" xref="S2.SS1.p3.2.m1.2.3.2.2.2.cmml">X</mi><mi id="S2.SS1.p3.2.m1.2.3.2.3" xref="S2.SS1.p3.2.m1.2.3.2.3.cmml">i</mi><mrow id="S2.SS1.p3.2.m1.1.1.1.3" xref="S2.SS1.p3.2.m1.2.3.2.cmml"><mo id="S2.SS1.p3.2.m1.1.1.1.3.1" stretchy="false" xref="S2.SS1.p3.2.m1.2.3.2.cmml">(</mo><mi id="S2.SS1.p3.2.m1.1.1.1.1" xref="S2.SS1.p3.2.m1.1.1.1.1.cmml">j</mi><mo id="S2.SS1.p3.2.m1.1.1.1.3.2" stretchy="false" xref="S2.SS1.p3.2.m1.2.3.2.cmml">)</mo></mrow></msubsup><mo id="S2.SS1.p3.2.m1.2.3.1" xref="S2.SS1.p3.2.m1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.2.m1.2.3.3.2" xref="S2.SS1.p3.2.m1.2.3.cmml"><mo id="S2.SS1.p3.2.m1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p3.2.m1.2.3.cmml">(</mo><mi id="S2.SS1.p3.2.m1.2.2" xref="S2.SS1.p3.2.m1.2.2.cmml">t</mi><mo id="S2.SS1.p3.2.m1.2.3.3.2.2" stretchy="false" xref="S2.SS1.p3.2.m1.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m1.2b"><apply id="S2.SS1.p3.2.m1.2.3.cmml" xref="S2.SS1.p3.2.m1.2.3"><times id="S2.SS1.p3.2.m1.2.3.1.cmml" xref="S2.SS1.p3.2.m1.2.3.1"></times><apply id="S2.SS1.p3.2.m1.2.3.2.cmml" xref="S2.SS1.p3.2.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.2.m1.2.3.2.1.cmml" xref="S2.SS1.p3.2.m1.2.3.2">subscript</csymbol><apply id="S2.SS1.p3.2.m1.2.3.2.2.cmml" xref="S2.SS1.p3.2.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.2.m1.2.3.2.2.1.cmml" xref="S2.SS1.p3.2.m1.2.3.2">superscript</csymbol><ci id="S2.SS1.p3.2.m1.2.3.2.2.2.cmml" xref="S2.SS1.p3.2.m1.2.3.2.2.2">𝑋</ci><ci id="S2.SS1.p3.2.m1.1.1.1.1.cmml" xref="S2.SS1.p3.2.m1.1.1.1.1">𝑗</ci></apply><ci id="S2.SS1.p3.2.m1.2.3.2.3.cmml" xref="S2.SS1.p3.2.m1.2.3.2.3">𝑖</ci></apply><ci id="S2.SS1.p3.2.m1.2.2.cmml" xref="S2.SS1.p3.2.m1.2.2">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m1.2c">X^{(j)}_{i}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m1.2d">italic_X start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> denotes the number of citations of the <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m2.1"><semantics id="S2.SS1.p3.3.m2.1a"><mi id="S2.SS1.p3.3.m2.1.1" xref="S2.SS1.p3.3.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m2.1b"><ci id="S2.SS1.p3.3.m2.1.1.cmml" xref="S2.SS1.p3.3.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m2.1d">italic_i</annotation></semantics></math>-th article at the <math alttext="t" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m3.1"><semantics id="S2.SS1.p3.4.m3.1a"><mi id="S2.SS1.p3.4.m3.1.1" xref="S2.SS1.p3.4.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m3.1b"><ci id="S2.SS1.p3.4.m3.1.1.cmml" xref="S2.SS1.p3.4.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m3.1d">italic_t</annotation></semantics></math>-th time step. 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Due to the granularity of the available data, one time step corresponds to one year.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.2">The author receives a number of citations during each time step. 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xref="S2.E2.m1.8.8.1.1.1.1.1"><times id="S2.E2.m1.8.8.1.1.1.1.1.1.cmml" xref="S2.E2.m1.8.8.1.1.1.1.1.1"></times><apply id="S2.E2.m1.8.8.1.1.1.1.1.2.cmml" xref="S2.E2.m1.8.8.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E2.m1.8.8.1.1.1.1.1.2.1.cmml" xref="S2.E2.m1.8.8.1.1.1.1.1.2">subscript</csymbol><apply id="S2.E2.m1.8.8.1.1.1.1.1.2.2.cmml" xref="S2.E2.m1.8.8.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E2.m1.8.8.1.1.1.1.1.2.2.1.cmml" xref="S2.E2.m1.8.8.1.1.1.1.1.2">superscript</csymbol><ci id="S2.E2.m1.8.8.1.1.1.1.1.2.2.2.cmml" xref="S2.E2.m1.8.8.1.1.1.1.1.2.2.2">𝐶</ci><ci id="S2.E2.m1.2.2.1.1.cmml" xref="S2.E2.m1.2.2.1.1">𝑗</ci></apply><cn id="S2.E2.m1.8.8.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.E2.m1.8.8.1.1.1.1.1.2.3">1</cn></apply><ci id="S2.E2.m1.5.5.cmml" xref="S2.E2.m1.5.5">𝑡</ci></apply><apply id="S2.E2.m1.8.8.1.1.2.2.2.cmml" xref="S2.E2.m1.8.8.1.1.2.2.2"><times id="S2.E2.m1.8.8.1.1.2.2.2.1.cmml" xref="S2.E2.m1.8.8.1.1.2.2.2.1"></times><apply id="S2.E2.m1.8.8.1.1.2.2.2.2.cmml" xref="S2.E2.m1.8.8.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S2.E2.m1.8.8.1.1.2.2.2.2.1.cmml" xref="S2.E2.m1.8.8.1.1.2.2.2.2">subscript</csymbol><apply id="S2.E2.m1.8.8.1.1.2.2.2.2.2.cmml" xref="S2.E2.m1.8.8.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S2.E2.m1.8.8.1.1.2.2.2.2.2.1.cmml" xref="S2.E2.m1.8.8.1.1.2.2.2.2">superscript</csymbol><ci id="S2.E2.m1.8.8.1.1.2.2.2.2.2.2.cmml" xref="S2.E2.m1.8.8.1.1.2.2.2.2.2.2">𝐶</ci><ci id="S2.E2.m1.3.3.1.1.cmml" xref="S2.E2.m1.3.3.1.1">𝑗</ci></apply><cn id="S2.E2.m1.8.8.1.1.2.2.2.2.3.cmml" type="integer" xref="S2.E2.m1.8.8.1.1.2.2.2.2.3">2</cn></apply><ci id="S2.E2.m1.6.6.cmml" xref="S2.E2.m1.6.6">𝑡</ci></apply><ci id="S2.E2.m1.7.7.cmml" xref="S2.E2.m1.7.7">…</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.8c">C^{(j)}(t)=\left[C^{(j)}_{1}(t),C^{(j)}_{2}(t),\dots\right],</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.8d">italic_C start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( italic_t ) = [ italic_C start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , italic_C start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , … ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p4.1">where <math alttext="C^{(j)}_{i}(t)" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.2"><semantics id="S2.SS1.p4.1.m1.2a"><mrow id="S2.SS1.p4.1.m1.2.3" xref="S2.SS1.p4.1.m1.2.3.cmml"><msubsup id="S2.SS1.p4.1.m1.2.3.2" xref="S2.SS1.p4.1.m1.2.3.2.cmml"><mi id="S2.SS1.p4.1.m1.2.3.2.2.2" xref="S2.SS1.p4.1.m1.2.3.2.2.2.cmml">C</mi><mi id="S2.SS1.p4.1.m1.2.3.2.3" 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id="S2.SS1.p4.1.m1.2.3.2.cmml" xref="S2.SS1.p4.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.1.m1.2.3.2.1.cmml" xref="S2.SS1.p4.1.m1.2.3.2">subscript</csymbol><apply id="S2.SS1.p4.1.m1.2.3.2.2.cmml" xref="S2.SS1.p4.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.1.m1.2.3.2.2.1.cmml" xref="S2.SS1.p4.1.m1.2.3.2">superscript</csymbol><ci id="S2.SS1.p4.1.m1.2.3.2.2.2.cmml" xref="S2.SS1.p4.1.m1.2.3.2.2.2">𝐶</ci><ci id="S2.SS1.p4.1.m1.1.1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1.1.1">𝑗</ci></apply><ci id="S2.SS1.p4.1.m1.2.3.2.3.cmml" xref="S2.SS1.p4.1.m1.2.3.2.3">𝑖</ci></apply><ci id="S2.SS1.p4.1.m1.2.2.cmml" xref="S2.SS1.p4.1.m1.2.2">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.2c">C^{(j)}_{i}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.2d">italic_C start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is the index in the citation vector of the article that received the citation.</p> </div> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.4">If we define a model <math alttext="p(C|X,\rho)" class="ltx_Math" display="inline" id="S2.SS1.p5.1.m1.3"><semantics id="S2.SS1.p5.1.m1.3a"><mrow id="S2.SS1.p5.1.m1.3.3" xref="S2.SS1.p5.1.m1.3.3.cmml"><mi id="S2.SS1.p5.1.m1.3.3.3" xref="S2.SS1.p5.1.m1.3.3.3.cmml">p</mi><mo id="S2.SS1.p5.1.m1.3.3.2" xref="S2.SS1.p5.1.m1.3.3.2.cmml">⁢</mo><mrow id="S2.SS1.p5.1.m1.3.3.1.1" xref="S2.SS1.p5.1.m1.3.3.1.1.1.cmml"><mo id="S2.SS1.p5.1.m1.3.3.1.1.2" stretchy="false" xref="S2.SS1.p5.1.m1.3.3.1.1.1.cmml">(</mo><mrow id="S2.SS1.p5.1.m1.3.3.1.1.1" xref="S2.SS1.p5.1.m1.3.3.1.1.1.cmml"><mi id="S2.SS1.p5.1.m1.3.3.1.1.1.2" xref="S2.SS1.p5.1.m1.3.3.1.1.1.2.cmml">C</mi><mo fence="false" id="S2.SS1.p5.1.m1.3.3.1.1.1.1" xref="S2.SS1.p5.1.m1.3.3.1.1.1.1.cmml">|</mo><mrow id="S2.SS1.p5.1.m1.3.3.1.1.1.3.2" xref="S2.SS1.p5.1.m1.3.3.1.1.1.3.1.cmml"><mi id="S2.SS1.p5.1.m1.1.1" xref="S2.SS1.p5.1.m1.1.1.cmml">X</mi><mo id="S2.SS1.p5.1.m1.3.3.1.1.1.3.2.1" xref="S2.SS1.p5.1.m1.3.3.1.1.1.3.1.cmml">,</mo><mi id="S2.SS1.p5.1.m1.2.2" xref="S2.SS1.p5.1.m1.2.2.cmml">ρ</mi></mrow></mrow><mo id="S2.SS1.p5.1.m1.3.3.1.1.3" stretchy="false" xref="S2.SS1.p5.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.1.m1.3b"><apply id="S2.SS1.p5.1.m1.3.3.cmml" xref="S2.SS1.p5.1.m1.3.3"><times id="S2.SS1.p5.1.m1.3.3.2.cmml" xref="S2.SS1.p5.1.m1.3.3.2"></times><ci id="S2.SS1.p5.1.m1.3.3.3.cmml" xref="S2.SS1.p5.1.m1.3.3.3">𝑝</ci><apply id="S2.SS1.p5.1.m1.3.3.1.1.1.cmml" xref="S2.SS1.p5.1.m1.3.3.1.1"><csymbol cd="latexml" id="S2.SS1.p5.1.m1.3.3.1.1.1.1.cmml" xref="S2.SS1.p5.1.m1.3.3.1.1.1.1">conditional</csymbol><ci id="S2.SS1.p5.1.m1.3.3.1.1.1.2.cmml" xref="S2.SS1.p5.1.m1.3.3.1.1.1.2">𝐶</ci><list id="S2.SS1.p5.1.m1.3.3.1.1.1.3.1.cmml" xref="S2.SS1.p5.1.m1.3.3.1.1.1.3.2"><ci id="S2.SS1.p5.1.m1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1">𝑋</ci><ci id="S2.SS1.p5.1.m1.2.2.cmml" xref="S2.SS1.p5.1.m1.2.2">𝜌</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.1.m1.3c">p(C|X,\rho)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.1.m1.3d">italic_p ( italic_C | italic_X , italic_ρ )</annotation></semantics></math> that specifies, given the citation vector <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p5.2.m2.1"><semantics id="S2.SS1.p5.2.m2.1a"><mi id="S2.SS1.p5.2.m2.1.1" xref="S2.SS1.p5.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.2.m2.1b"><ci id="S2.SS1.p5.2.m2.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.2.m2.1d">italic_X</annotation></semantics></math> and some parameter <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.p5.3.m3.1"><semantics id="S2.SS1.p5.3.m3.1a"><mi id="S2.SS1.p5.3.m3.1.1" xref="S2.SS1.p5.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.3.m3.1b"><ci id="S2.SS1.p5.3.m3.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.3.m3.1d">italic_ρ</annotation></semantics></math>, the probability of article <math alttext="C" class="ltx_Math" display="inline" id="S2.SS1.p5.4.m4.1"><semantics id="S2.SS1.p5.4.m4.1a"><mi id="S2.SS1.p5.4.m4.1.1" xref="S2.SS1.p5.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.4.m4.1b"><ci id="S2.SS1.p5.4.m4.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.4.m4.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.4.m4.1d">italic_C</annotation></semantics></math> receiving a citation, we can calculate the log-likelihood for this model in the following way:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ell^{(j)}(\rho)=\sum_{t}\sum_{C\in C^{(j)}(t)}\ln p(C|X^{(j)}(t-1),\rho)." class="ltx_Math" display="block" id="S2.E3.m1.7"><semantics id="S2.E3.m1.7a"><mrow id="S2.E3.m1.7.7.1" xref="S2.E3.m1.7.7.1.1.cmml"><mrow id="S2.E3.m1.7.7.1.1" xref="S2.E3.m1.7.7.1.1.cmml"><mrow id="S2.E3.m1.7.7.1.1.3" xref="S2.E3.m1.7.7.1.1.3.cmml"><msup id="S2.E3.m1.7.7.1.1.3.2" xref="S2.E3.m1.7.7.1.1.3.2.cmml"><mi id="S2.E3.m1.7.7.1.1.3.2.2" mathvariant="normal" xref="S2.E3.m1.7.7.1.1.3.2.2.cmml">ℓ</mi><mrow id="S2.E3.m1.1.1.1.3" xref="S2.E3.m1.7.7.1.1.3.2.cmml"><mo id="S2.E3.m1.1.1.1.3.1" stretchy="false" xref="S2.E3.m1.7.7.1.1.3.2.cmml">(</mo><mi id="S2.E3.m1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml">j</mi><mo id="S2.E3.m1.1.1.1.3.2" stretchy="false" xref="S2.E3.m1.7.7.1.1.3.2.cmml">)</mo></mrow></msup><mo id="S2.E3.m1.7.7.1.1.3.1" xref="S2.E3.m1.7.7.1.1.3.1.cmml">⁢</mo><mrow id="S2.E3.m1.7.7.1.1.3.3.2" xref="S2.E3.m1.7.7.1.1.3.cmml"><mo id="S2.E3.m1.7.7.1.1.3.3.2.1" stretchy="false" xref="S2.E3.m1.7.7.1.1.3.cmml">(</mo><mi id="S2.E3.m1.5.5" xref="S2.E3.m1.5.5.cmml">ρ</mi><mo id="S2.E3.m1.7.7.1.1.3.3.2.2" stretchy="false" xref="S2.E3.m1.7.7.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.7.7.1.1.2" rspace="0.111em" xref="S2.E3.m1.7.7.1.1.2.cmml">=</mo><mrow id="S2.E3.m1.7.7.1.1.1" xref="S2.E3.m1.7.7.1.1.1.cmml"><munder id="S2.E3.m1.7.7.1.1.1.2" xref="S2.E3.m1.7.7.1.1.1.2.cmml"><mo id="S2.E3.m1.7.7.1.1.1.2.2" movablelimits="false" rspace="0em" xref="S2.E3.m1.7.7.1.1.1.2.2.cmml">∑</mo><mi id="S2.E3.m1.7.7.1.1.1.2.3" xref="S2.E3.m1.7.7.1.1.1.2.3.cmml">t</mi></munder><mrow id="S2.E3.m1.7.7.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.cmml"><munder id="S2.E3.m1.7.7.1.1.1.1.2" xref="S2.E3.m1.7.7.1.1.1.1.2.cmml"><mo id="S2.E3.m1.7.7.1.1.1.1.2.2" movablelimits="false" xref="S2.E3.m1.7.7.1.1.1.1.2.2.cmml">∑</mo><mrow id="S2.E3.m1.3.3.2" xref="S2.E3.m1.3.3.2.cmml"><mi id="S2.E3.m1.3.3.2.4" xref="S2.E3.m1.3.3.2.4.cmml">C</mi><mo id="S2.E3.m1.3.3.2.3" xref="S2.E3.m1.3.3.2.3.cmml">∈</mo><mrow id="S2.E3.m1.3.3.2.5" xref="S2.E3.m1.3.3.2.5.cmml"><msup id="S2.E3.m1.3.3.2.5.2" xref="S2.E3.m1.3.3.2.5.2.cmml"><mi id="S2.E3.m1.3.3.2.5.2.2" xref="S2.E3.m1.3.3.2.5.2.2.cmml">C</mi><mrow id="S2.E3.m1.2.2.1.1.1.3" xref="S2.E3.m1.3.3.2.5.2.cmml"><mo id="S2.E3.m1.2.2.1.1.1.3.1" stretchy="false" xref="S2.E3.m1.3.3.2.5.2.cmml">(</mo><mi id="S2.E3.m1.2.2.1.1.1.1" xref="S2.E3.m1.2.2.1.1.1.1.cmml">j</mi><mo id="S2.E3.m1.2.2.1.1.1.3.2" stretchy="false" xref="S2.E3.m1.3.3.2.5.2.cmml">)</mo></mrow></msup><mo id="S2.E3.m1.3.3.2.5.1" xref="S2.E3.m1.3.3.2.5.1.cmml">⁢</mo><mrow id="S2.E3.m1.3.3.2.5.3.2" xref="S2.E3.m1.3.3.2.5.cmml"><mo id="S2.E3.m1.3.3.2.5.3.2.1" stretchy="false" xref="S2.E3.m1.3.3.2.5.cmml">(</mo><mi id="S2.E3.m1.3.3.2.2" xref="S2.E3.m1.3.3.2.2.cmml">t</mi><mo id="S2.E3.m1.3.3.2.5.3.2.2" stretchy="false" xref="S2.E3.m1.3.3.2.5.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.E3.m1.7.7.1.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.1.cmml"><mrow id="S2.E3.m1.7.7.1.1.1.1.1.3" xref="S2.E3.m1.7.7.1.1.1.1.1.3.cmml"><mi id="S2.E3.m1.7.7.1.1.1.1.1.3.1" xref="S2.E3.m1.7.7.1.1.1.1.1.3.1.cmml">ln</mi><mo id="S2.E3.m1.7.7.1.1.1.1.1.3a" lspace="0.167em" xref="S2.E3.m1.7.7.1.1.1.1.1.3.cmml">⁡</mo><mi id="S2.E3.m1.7.7.1.1.1.1.1.3.2" xref="S2.E3.m1.7.7.1.1.1.1.1.3.2.cmml">p</mi></mrow><mo id="S2.E3.m1.7.7.1.1.1.1.1.2" xref="S2.E3.m1.7.7.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.7.7.1.1.1.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E3.m1.7.7.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.3" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.3.cmml">C</mi><mo fence="false" id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.2" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.2.cmml">|</mo><mrow id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.2.cmml"><mrow id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.cmml"><msup id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">X</mi><mrow id="S2.E3.m1.4.4.1.3" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S2.E3.m1.4.4.1.3.1" stretchy="false" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.cmml">(</mo><mi id="S2.E3.m1.4.4.1.1" xref="S2.E3.m1.4.4.1.1.cmml">j</mi><mo id="S2.E3.m1.4.4.1.3.2" stretchy="false" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.cmml">)</mo></mrow></msup><mo id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">t</mi><mo id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.2.cmml">,</mo><mi id="S2.E3.m1.6.6" xref="S2.E3.m1.6.6.cmml">ρ</mi></mrow></mrow><mo id="S2.E3.m1.7.7.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><mo id="S2.E3.m1.7.7.1.2" lspace="0em" xref="S2.E3.m1.7.7.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.m1.7b"><apply id="S2.E3.m1.7.7.1.1.cmml" xref="S2.E3.m1.7.7.1"><eq id="S2.E3.m1.7.7.1.1.2.cmml" xref="S2.E3.m1.7.7.1.1.2"></eq><apply id="S2.E3.m1.7.7.1.1.3.cmml" xref="S2.E3.m1.7.7.1.1.3"><times id="S2.E3.m1.7.7.1.1.3.1.cmml" xref="S2.E3.m1.7.7.1.1.3.1"></times><apply id="S2.E3.m1.7.7.1.1.3.2.cmml" xref="S2.E3.m1.7.7.1.1.3.2"><csymbol cd="ambiguous" id="S2.E3.m1.7.7.1.1.3.2.1.cmml" xref="S2.E3.m1.7.7.1.1.3.2">superscript</csymbol><ci id="S2.E3.m1.7.7.1.1.3.2.2.cmml" xref="S2.E3.m1.7.7.1.1.3.2.2">ℓ</ci><ci id="S2.E3.m1.1.1.1.1.cmml" xref="S2.E3.m1.1.1.1.1">𝑗</ci></apply><ci id="S2.E3.m1.5.5.cmml" xref="S2.E3.m1.5.5">𝜌</ci></apply><apply id="S2.E3.m1.7.7.1.1.1.cmml" xref="S2.E3.m1.7.7.1.1.1"><apply id="S2.E3.m1.7.7.1.1.1.2.cmml" xref="S2.E3.m1.7.7.1.1.1.2"><csymbol cd="ambiguous" id="S2.E3.m1.7.7.1.1.1.2.1.cmml" xref="S2.E3.m1.7.7.1.1.1.2">subscript</csymbol><sum id="S2.E3.m1.7.7.1.1.1.2.2.cmml" xref="S2.E3.m1.7.7.1.1.1.2.2"></sum><ci id="S2.E3.m1.7.7.1.1.1.2.3.cmml" xref="S2.E3.m1.7.7.1.1.1.2.3">𝑡</ci></apply><apply id="S2.E3.m1.7.7.1.1.1.1.cmml" xref="S2.E3.m1.7.7.1.1.1.1"><apply id="S2.E3.m1.7.7.1.1.1.1.2.cmml" xref="S2.E3.m1.7.7.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E3.m1.7.7.1.1.1.1.2.1.cmml" xref="S2.E3.m1.7.7.1.1.1.1.2">subscript</csymbol><sum id="S2.E3.m1.7.7.1.1.1.1.2.2.cmml" xref="S2.E3.m1.7.7.1.1.1.1.2.2"></sum><apply id="S2.E3.m1.3.3.2.cmml" xref="S2.E3.m1.3.3.2"><in id="S2.E3.m1.3.3.2.3.cmml" xref="S2.E3.m1.3.3.2.3"></in><ci id="S2.E3.m1.3.3.2.4.cmml" xref="S2.E3.m1.3.3.2.4">𝐶</ci><apply id="S2.E3.m1.3.3.2.5.cmml" xref="S2.E3.m1.3.3.2.5"><times id="S2.E3.m1.3.3.2.5.1.cmml" xref="S2.E3.m1.3.3.2.5.1"></times><apply 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id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.3">𝐶</ci><list id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1"><apply id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1"><times id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2"></times><apply id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3.2">𝑋</ci><ci id="S2.E3.m1.4.4.1.1.cmml" xref="S2.E3.m1.4.4.1.1">𝑗</ci></apply><apply id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1"><minus id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1"></minus><ci id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2">𝑡</ci><cn id="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.E3.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply><ci id="S2.E3.m1.6.6.cmml" xref="S2.E3.m1.6.6">𝜌</ci></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.7c">\ell^{(j)}(\rho)=\sum_{t}\sum_{C\in C^{(j)}(t)}\ln p(C|X^{(j)}(t-1),\rho).</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.7d">roman_ℓ start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( italic_ρ ) = ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_C ∈ italic_C start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( italic_t ) end_POSTSUBSCRIPT roman_ln italic_p ( italic_C | italic_X start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( italic_t - 1 ) , italic_ρ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p5.7">The first summation in this equation is taken over the entire career of author <math alttext="j" class="ltx_Math" display="inline" id="S2.SS1.p5.5.m1.1"><semantics id="S2.SS1.p5.5.m1.1a"><mi id="S2.SS1.p5.5.m1.1.1" xref="S2.SS1.p5.5.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.5.m1.1b"><ci id="S2.SS1.p5.5.m1.1.1.cmml" xref="S2.SS1.p5.5.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.5.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.5.m1.1d">italic_j</annotation></semantics></math>, or up to the most recent point for which data is available. This log-likelihood can then be numerically maximized, yielding the value of the parameter <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.p5.6.m2.1"><semantics id="S2.SS1.p5.6.m2.1a"><mi id="S2.SS1.p5.6.m2.1.1" xref="S2.SS1.p5.6.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.6.m2.1b"><ci id="S2.SS1.p5.6.m2.1.1.cmml" xref="S2.SS1.p5.6.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.6.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.6.m2.1d">italic_ρ</annotation></semantics></math> that best describes the citation patterns of author <math alttext="j" class="ltx_Math" display="inline" id="S2.SS1.p5.7.m3.1"><semantics id="S2.SS1.p5.7.m3.1a"><mi id="S2.SS1.p5.7.m3.1.1" xref="S2.SS1.p5.7.m3.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.7.m3.1b"><ci id="S2.SS1.p5.7.m3.1.1.cmml" xref="S2.SS1.p5.7.m3.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.7.m3.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.7.m3.1d">italic_j</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p6"> <p class="ltx_p" id="S2.SS1.p6.3">It is important to note that the distribution of citations at the <math alttext="t" class="ltx_Math" display="inline" id="S2.SS1.p6.1.m1.1"><semantics id="S2.SS1.p6.1.m1.1a"><mi id="S2.SS1.p6.1.m1.1.1" xref="S2.SS1.p6.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.1.m1.1b"><ci id="S2.SS1.p6.1.m1.1.1.cmml" xref="S2.SS1.p6.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.1.m1.1d">italic_t</annotation></semantics></math>-th time step depends on the state of the citation vector from the previous time step, <math alttext="t-1" class="ltx_Math" display="inline" id="S2.SS1.p6.2.m2.1"><semantics id="S2.SS1.p6.2.m2.1a"><mrow id="S2.SS1.p6.2.m2.1.1" xref="S2.SS1.p6.2.m2.1.1.cmml"><mi id="S2.SS1.p6.2.m2.1.1.2" xref="S2.SS1.p6.2.m2.1.1.2.cmml">t</mi><mo id="S2.SS1.p6.2.m2.1.1.1" xref="S2.SS1.p6.2.m2.1.1.1.cmml">−</mo><mn id="S2.SS1.p6.2.m2.1.1.3" xref="S2.SS1.p6.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.2.m2.1b"><apply id="S2.SS1.p6.2.m2.1.1.cmml" xref="S2.SS1.p6.2.m2.1.1"><minus id="S2.SS1.p6.2.m2.1.1.1.cmml" xref="S2.SS1.p6.2.m2.1.1.1"></minus><ci id="S2.SS1.p6.2.m2.1.1.2.cmml" xref="S2.SS1.p6.2.m2.1.1.2">𝑡</ci><cn id="S2.SS1.p6.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.p6.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.2.m2.1c">t-1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.2.m2.1d">italic_t - 1</annotation></semantics></math>. However, articles published during the <math alttext="t" class="ltx_Math" display="inline" id="S2.SS1.p6.3.m3.1"><semantics id="S2.SS1.p6.3.m3.1a"><mi id="S2.SS1.p6.3.m3.1.1" xref="S2.SS1.p6.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.3.m3.1b"><ci id="S2.SS1.p6.3.m3.1.1.cmml" xref="S2.SS1.p6.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.3.m3.1d">italic_t</annotation></semantics></math>-th time step are added to this vector with their citation count initialised to zero. This approach accounts for the fact that we can only determine the year of publication, and some articles receive citations within the same year they are published. In fact, due to the peculiarities of online publishing, some articles even receive citations before their publication date, but we disregard these cases.</p> </div> <div class="ltx_para" id="S2.SS1.p7"> <p class="ltx_p" id="S2.SS1.p7.2">The log-likelihood for the entire dataset can be calculated as the sum of log-likelihoods for all authors:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ell(\rho)=\sum_{j}\ell^{(j)}(\rho)." class="ltx_Math" display="block" id="S2.E4.m1.4"><semantics id="S2.E4.m1.4a"><mrow id="S2.E4.m1.4.4.1" xref="S2.E4.m1.4.4.1.1.cmml"><mrow id="S2.E4.m1.4.4.1.1" xref="S2.E4.m1.4.4.1.1.cmml"><mrow id="S2.E4.m1.4.4.1.1.2" xref="S2.E4.m1.4.4.1.1.2.cmml"><mi id="S2.E4.m1.4.4.1.1.2.2" mathvariant="normal" xref="S2.E4.m1.4.4.1.1.2.2.cmml">ℓ</mi><mo id="S2.E4.m1.4.4.1.1.2.1" 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xref="S2.E4.m1.4.4.1.1.3.2.2.2">ℓ</ci><ci id="S2.E4.m1.1.1.1.1.cmml" xref="S2.E4.m1.1.1.1.1">𝑗</ci></apply><ci id="S2.E4.m1.3.3.cmml" xref="S2.E4.m1.3.3">𝜌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.4c">\ell(\rho)=\sum_{j}\ell^{(j)}(\rho).</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.4d">roman_ℓ ( italic_ρ ) = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_ℓ start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( italic_ρ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p7.1">This value can also be maximised, yielding <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.p7.1.m1.1"><semantics id="S2.SS1.p7.1.m1.1a"><mi 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xref="S2.E5.m1.3.3.1.1">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.5c">p(C|X(t-1),\rho)=\rho\frac{X_{C}(t-1)}{\sum_{i}X_{i}(t-1)}+(1-\rho)\frac{1}{N(% t)}.</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.5d">italic_p ( italic_C | italic_X ( italic_t - 1 ) , italic_ρ ) = italic_ρ divide start_ARG italic_X start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_t - 1 ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t - 1 ) end_ARG + ( 1 - italic_ρ ) divide start_ARG 1 end_ARG start_ARG italic_N ( italic_t ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p8.6">We emphasise one more time that while the model depends on the state of the citation vector from the previous time step, we add to it articles published in the current time step with their number of citations initialised to <math alttext="0" class="ltx_Math" display="inline" id="S2.SS1.p8.1.m1.1"><semantics id="S2.SS1.p8.1.m1.1a"><mn id="S2.SS1.p8.1.m1.1.1" xref="S2.SS1.p8.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.1.m1.1b"><cn id="S2.SS1.p8.1.m1.1.1.cmml" type="integer" xref="S2.SS1.p8.1.m1.1.1">0</cn></annotation-xml></semantics></math> - hence <math alttext="N(t)" class="ltx_Math" display="inline" id="S2.SS1.p8.2.m2.1"><semantics id="S2.SS1.p8.2.m2.1a"><mrow id="S2.SS1.p8.2.m2.1.2" xref="S2.SS1.p8.2.m2.1.2.cmml"><mi id="S2.SS1.p8.2.m2.1.2.2" xref="S2.SS1.p8.2.m2.1.2.2.cmml">N</mi><mo id="S2.SS1.p8.2.m2.1.2.1" xref="S2.SS1.p8.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p8.2.m2.1.2.3.2" xref="S2.SS1.p8.2.m2.1.2.cmml"><mo id="S2.SS1.p8.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.p8.2.m2.1.2.cmml">(</mo><mi id="S2.SS1.p8.2.m2.1.1" xref="S2.SS1.p8.2.m2.1.1.cmml">t</mi><mo id="S2.SS1.p8.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.p8.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.2.m2.1b"><apply id="S2.SS1.p8.2.m2.1.2.cmml" xref="S2.SS1.p8.2.m2.1.2"><times id="S2.SS1.p8.2.m2.1.2.1.cmml" xref="S2.SS1.p8.2.m2.1.2.1"></times><ci id="S2.SS1.p8.2.m2.1.2.2.cmml" xref="S2.SS1.p8.2.m2.1.2.2">𝑁</ci><ci id="S2.SS1.p8.2.m2.1.1.cmml" xref="S2.SS1.p8.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.2.m2.1c">N(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.2.m2.1d">italic_N ( italic_t )</annotation></semantics></math> instead of <math alttext="N(t-1)" class="ltx_Math" display="inline" id="S2.SS1.p8.3.m3.1"><semantics id="S2.SS1.p8.3.m3.1a"><mrow id="S2.SS1.p8.3.m3.1.1" xref="S2.SS1.p8.3.m3.1.1.cmml"><mi id="S2.SS1.p8.3.m3.1.1.3" xref="S2.SS1.p8.3.m3.1.1.3.cmml">N</mi><mo id="S2.SS1.p8.3.m3.1.1.2" xref="S2.SS1.p8.3.m3.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p8.3.m3.1.1.1.1" xref="S2.SS1.p8.3.m3.1.1.1.1.1.cmml"><mo id="S2.SS1.p8.3.m3.1.1.1.1.2" stretchy="false" xref="S2.SS1.p8.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p8.3.m3.1.1.1.1.1" xref="S2.SS1.p8.3.m3.1.1.1.1.1.cmml"><mi id="S2.SS1.p8.3.m3.1.1.1.1.1.2" xref="S2.SS1.p8.3.m3.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS1.p8.3.m3.1.1.1.1.1.1" xref="S2.SS1.p8.3.m3.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS1.p8.3.m3.1.1.1.1.1.3" xref="S2.SS1.p8.3.m3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS1.p8.3.m3.1.1.1.1.3" stretchy="false" xref="S2.SS1.p8.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.3.m3.1b"><apply id="S2.SS1.p8.3.m3.1.1.cmml" xref="S2.SS1.p8.3.m3.1.1"><times id="S2.SS1.p8.3.m3.1.1.2.cmml" xref="S2.SS1.p8.3.m3.1.1.2"></times><ci id="S2.SS1.p8.3.m3.1.1.3.cmml" xref="S2.SS1.p8.3.m3.1.1.3">𝑁</ci><apply id="S2.SS1.p8.3.m3.1.1.1.1.1.cmml" xref="S2.SS1.p8.3.m3.1.1.1.1"><minus id="S2.SS1.p8.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS1.p8.3.m3.1.1.1.1.1.1"></minus><ci id="S2.SS1.p8.3.m3.1.1.1.1.1.2.cmml" xref="S2.SS1.p8.3.m3.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS1.p8.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p8.3.m3.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.3.m3.1c">N(t-1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.3.m3.1d">italic_N ( italic_t - 1 )</annotation></semantics></math> in the random term. Also, we do not artificially add <math alttext="1" class="ltx_Math" display="inline" id="S2.SS1.p8.4.m4.1"><semantics id="S2.SS1.p8.4.m4.1a"><mn id="S2.SS1.p8.4.m4.1.1" xref="S2.SS1.p8.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.4.m4.1b"><cn id="S2.SS1.p8.4.m4.1.1.cmml" type="integer" xref="S2.SS1.p8.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.4.m4.1d">1</annotation></semantics></math> to the number of citations in order to kickstart the preferential attachment mechanism. What follows is that the first citation of an article must come from the random term. We believe it is a reasonable assumption, but it also means a pure preferential model with <math alttext="\rho=1" class="ltx_Math" display="inline" id="S2.SS1.p8.5.m5.1"><semantics id="S2.SS1.p8.5.m5.1a"><mrow id="S2.SS1.p8.5.m5.1.1" xref="S2.SS1.p8.5.m5.1.1.cmml"><mi id="S2.SS1.p8.5.m5.1.1.2" xref="S2.SS1.p8.5.m5.1.1.2.cmml">ρ</mi><mo id="S2.SS1.p8.5.m5.1.1.1" xref="S2.SS1.p8.5.m5.1.1.1.cmml">=</mo><mn id="S2.SS1.p8.5.m5.1.1.3" xref="S2.SS1.p8.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.5.m5.1b"><apply id="S2.SS1.p8.5.m5.1.1.cmml" xref="S2.SS1.p8.5.m5.1.1"><eq id="S2.SS1.p8.5.m5.1.1.1.cmml" xref="S2.SS1.p8.5.m5.1.1.1"></eq><ci id="S2.SS1.p8.5.m5.1.1.2.cmml" xref="S2.SS1.p8.5.m5.1.1.2">𝜌</ci><cn id="S2.SS1.p8.5.m5.1.1.3.cmml" type="integer" xref="S2.SS1.p8.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.5.m5.1c">\rho=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.5.m5.1d">italic_ρ = 1</annotation></semantics></math> is not valid - the probability of each first citation, and consequently the entire likelihood, would be <math alttext="0" class="ltx_Math" display="inline" id="S2.SS1.p8.6.m6.1"><semantics id="S2.SS1.p8.6.m6.1a"><mn id="S2.SS1.p8.6.m6.1.1" xref="S2.SS1.p8.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.6.m6.1b"><cn id="S2.SS1.p8.6.m6.1.1.cmml" type="integer" xref="S2.SS1.p8.6.m6.1.1">0</cn></annotation-xml></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p9"> <p class="ltx_p" id="S2.SS1.p9.5">Finally, we must address an important issue related to the interpretation of the parameter <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.p9.1.m1.1"><semantics id="S2.SS1.p9.1.m1.1a"><mi id="S2.SS1.p9.1.m1.1.1" xref="S2.SS1.p9.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.1.m1.1b"><ci id="S2.SS1.p9.1.m1.1.1.cmml" xref="S2.SS1.p9.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.1.m1.1d">italic_ρ</annotation></semantics></math>. While it might be tempting to view <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.p9.2.m2.1"><semantics id="S2.SS1.p9.2.m2.1a"><mi id="S2.SS1.p9.2.m2.1.1" xref="S2.SS1.p9.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.2.m2.1b"><ci id="S2.SS1.p9.2.m2.1.1.cmml" xref="S2.SS1.p9.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.2.m2.1d">italic_ρ</annotation></semantics></math> as a parameter ranging from pure randomness (for <math alttext="\rho=0" class="ltx_Math" display="inline" id="S2.SS1.p9.3.m3.1"><semantics id="S2.SS1.p9.3.m3.1a"><mrow id="S2.SS1.p9.3.m3.1.1" xref="S2.SS1.p9.3.m3.1.1.cmml"><mi id="S2.SS1.p9.3.m3.1.1.2" xref="S2.SS1.p9.3.m3.1.1.2.cmml">ρ</mi><mo id="S2.SS1.p9.3.m3.1.1.1" xref="S2.SS1.p9.3.m3.1.1.1.cmml">=</mo><mn id="S2.SS1.p9.3.m3.1.1.3" xref="S2.SS1.p9.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.3.m3.1b"><apply id="S2.SS1.p9.3.m3.1.1.cmml" xref="S2.SS1.p9.3.m3.1.1"><eq id="S2.SS1.p9.3.m3.1.1.1.cmml" xref="S2.SS1.p9.3.m3.1.1.1"></eq><ci id="S2.SS1.p9.3.m3.1.1.2.cmml" xref="S2.SS1.p9.3.m3.1.1.2">𝜌</ci><cn id="S2.SS1.p9.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p9.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.3.m3.1c">\rho=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.3.m3.1d">italic_ρ = 0</annotation></semantics></math>) to pure preferential attachment (for <math alttext="\rho=1" class="ltx_Math" display="inline" id="S2.SS1.p9.4.m4.1"><semantics id="S2.SS1.p9.4.m4.1a"><mrow id="S2.SS1.p9.4.m4.1.1" xref="S2.SS1.p9.4.m4.1.1.cmml"><mi id="S2.SS1.p9.4.m4.1.1.2" xref="S2.SS1.p9.4.m4.1.1.2.cmml">ρ</mi><mo id="S2.SS1.p9.4.m4.1.1.1" xref="S2.SS1.p9.4.m4.1.1.1.cmml">=</mo><mn id="S2.SS1.p9.4.m4.1.1.3" xref="S2.SS1.p9.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.4.m4.1b"><apply id="S2.SS1.p9.4.m4.1.1.cmml" xref="S2.SS1.p9.4.m4.1.1"><eq id="S2.SS1.p9.4.m4.1.1.1.cmml" xref="S2.SS1.p9.4.m4.1.1.1"></eq><ci id="S2.SS1.p9.4.m4.1.1.2.cmml" xref="S2.SS1.p9.4.m4.1.1.2">𝜌</ci><cn id="S2.SS1.p9.4.m4.1.1.3.cmml" type="integer" xref="S2.SS1.p9.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.4.m4.1c">\rho=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.4.m4.1d">italic_ρ = 1</annotation></semantics></math>), this interpretation oversimplifies the situation. Although a random component is undoubtedly present in the citation distribution process, many additional factors are likely at play, and a more complex model than the one in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S2.E5" title="In II.1 Model ‣ II Methodology ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">5</span></a>) could be constructed. Therefore, it is more prudent to interpret the scale of <math alttext="\rho" class="ltx_Math" display="inline" id="S2.SS1.p9.5.m5.1"><semantics id="S2.SS1.p9.5.m5.1a"><mi id="S2.SS1.p9.5.m5.1.1" xref="S2.SS1.p9.5.m5.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.5.m5.1b"><ci id="S2.SS1.p9.5.m5.1.1.cmml" xref="S2.SS1.p9.5.m5.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.5.m5.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.5.m5.1d">italic_ρ</annotation></semantics></math> as spanning the range between non-PAR and PAR, with the random term in the model functioning as a placeholder to account for citations that cannot be explained by PAR. This interpretation will be adopted throughout the paper.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.2 </span>Self-citations</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">The definition of self-citations is not straightforward, as they can be defined and counted in various ways <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib23" title="">23</a>]</cite>, which may lead to confusion <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib24" title="">24</a>]</cite>. In its simplest (direct) form, a self-citation is counted as any citation by an author to one of their own previous papers. Additionally, the definition can be extended to include citations made by coauthors of a paper, often referred to as co-author self-citations. The most challenging - and in some cases nearly impossible - type of self-citations to identify are coerced self-citations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib25" title="">25</a>]</cite>. These occur indirectly, such as when reviewers request that authors include references to specific articles during the peer review process. In the results presented in this paper, we focus exclusively on direct self-citations.</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.3 </span>Data</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">We utilized the 12th version of the DBLP Citation Network Dataset for our study <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib26" title="">26</a>]</cite>. This dataset is a comprehensive collection of metadata for approximately 4 million scientific articles, primarily in the field of computer science. The metadata includes information such as the year of publication, the list of authors, and the list of references, enabling the reconstruction of the citation network - a network where nodes represent articles and directed edges represent citations.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.1">The citation network derived from the DBLP dataset captures the citation histories of around 3 million authors, approximately 200,000 of whom have published 10 or more papers. It is worth noting that, while extensive, the DBLP dataset represents only a sample of the complete citation network. On average, it covers about 60% of all citations for each article. Nevertheless, we believe this is sufficient for qualitative analysis, especially given the reasonable assumption that the missing citations are of a similar nature across all articles.</p> </div> <figure class="ltx_figure" id="S2.F2"> <p class="ltx_p ltx_align_center" id="S2.F2.2">(a) <img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="241" id="S2.F2.1.g1" src="x2.png" width="332"/> (b) <img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="246" id="S2.F2.2.g2" src="x3.png" width="332"/></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Panel (a) presents a histogram of the fraction of self-citations (among all citations). Panel (b) shows the average fraction of self-citations for authors grouped by their total citation counts and number of published papers.</figcaption> </figure> <div class="ltx_para" id="S2.SS3.p3"> <p class="ltx_p" id="S2.SS3.p3.1">Given that we will examine the distribution of self-citations, it is worthwhile to explore some characteristics of self-citation behaviour among authors in the DBLP dataset. After all, self-citations could represent only a small fraction of total citations and might be negligible, at least for this group of authors. Panel (a) of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S2.F2" title="Figure 2 ‣ II.3 Data ‣ II Methodology ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">2</span></a> presents a histogram of the fraction of self-citations (the percentage of self-citations among all citations) for authors who have published 10 or more papers. The average fraction of self-citations is 16%, and as shown, some authors exhibit significantly higher percentages of self-citations than the average. This indicates that self-citations are far from negligible.</p> </div> <div class="ltx_para" id="S2.SS3.p4"> <p class="ltx_p" id="S2.SS3.p4.1">This observation is corroborated by panel (b) of the same figure, which depicts the average fraction of self-citations for authors with varying numbers of published papers and total citations received. The data suggest that authors with fewer total citations relative to the number of published papers tend to have a higher fraction of self-citations. This result is intuitive, as external citations are expected to grow - and, except in extreme cases, should grow - at a higher rate over time. Additionally, this panel further highlights that self-citations are not negligible, as they constitute a significant proportion of total citations for many authors, particularly those with lower overall citation counts.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span>Results</h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">III.1 </span>Analysis of whole dataset</h3> <figure class="ltx_figure" id="S3.F3"> <p class="ltx_p ltx_align_center" id="S3.F3.3">(a)<img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="191" id="S3.F3.1.g1" src="x4.png" width="249"/> (b)<img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="191" id="S3.F3.2.g2" src="x5.png" width="249"/> (c)<img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="195" id="S3.F3.3.g3" src="x6.png" width="249"/></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Log-likelihood for the entire dataset (aggregating all authors). Panel (a) presents the log-likelihood for the combined set of external and self-citations. Panel (b) displays the log-likelihood for external citations only, while panel (c) focuses on the log-likelihood for self-citations.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.2">Firstly, we analyzed the process of citation distribution across the entire dataset. Specifically, we calculated the log-likelihood for each author individually and then aggregated these values to compute the log-likelihood for the entire sample. This combined log-likelihood can be maximized to determine a value of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mi id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><ci id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_ρ</annotation></semantics></math> (the fraction of preferential citations) that characterizes the dataset as a whole. The results are shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.F3" title="Figure 3 ‣ III.1 Analysis of whole dataset ‣ III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">3</span></a>. In panel (a), the log-likelihood was calculated for the combined set of external and self-citations. As evident, there is a strong bias toward preferential attachment, with the maximum log-likelihood corresponding to <math alttext="\rho\approx 0.68" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mrow id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml">ρ</mi><mo id="S3.SS1.p1.2.m2.1.1.1" xref="S3.SS1.p1.2.m2.1.1.1.cmml">≈</mo><mn id="S3.SS1.p1.2.m2.1.1.3" xref="S3.SS1.p1.2.m2.1.1.3.cmml">0.68</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><approx id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1"></approx><ci id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2">𝜌</ci><cn id="S3.SS1.p1.2.m2.1.1.3.cmml" type="float" xref="S3.SS1.p1.2.m2.1.1.3">0.68</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">\rho\approx 0.68</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_ρ ≈ 0.68</annotation></semantics></math>. This strongly supports the hypothesis that the ”rich get richer” mechanism plays a significant role in the dynamics of citation distribution.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.3">An interesting extension of this analysis is to focus exclusively on either external or self-citations. This can be achieved by reconstructing the citation vectors as before - event by event - while restricting the log-likelihood calculations to a single type of citation. Specifically, this involves limiting the innermost summation in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S2.E3" title="In II.1 Model ‣ II Methodology ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">3</span></a>) to one type of citation, while still retaining both types of citations in the citation vector <math alttext="X" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">italic_X</annotation></semantics></math>. Panel (b) of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.F3" title="Figure 3 ‣ III.1 Analysis of whole dataset ‣ III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">3</span></a> shows the results for external citations. The maximum log-likelihood shifts further to the right compared to panel (a), corresponding to <math alttext="\rho\approx 0.73" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2" xref="S3.SS1.p2.2.m2.1.1.2.cmml">ρ</mi><mo id="S3.SS1.p2.2.m2.1.1.1" xref="S3.SS1.p2.2.m2.1.1.1.cmml">≈</mo><mn id="S3.SS1.p2.2.m2.1.1.3" xref="S3.SS1.p2.2.m2.1.1.3.cmml">0.73</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><approx id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1"></approx><ci id="S3.SS1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2">𝜌</ci><cn id="S3.SS1.p2.2.m2.1.1.3.cmml" type="float" xref="S3.SS1.p2.2.m2.1.1.3">0.73</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">\rho\approx 0.73</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_ρ ≈ 0.73</annotation></semantics></math>. This observation leads to two conclusions. First, preferential attachment plays a more pronounced role in the distribution of external citations. Second, the shift in the maximum log-likelihood between panels (a) and (b) suggests that self-citations are associated with smaller values of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><mi id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><ci id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">italic_ρ</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.1">This hypothesis is confirmed in panel (c), which displays the log-likelihood calculated exclusively for self-citations. Here, we observe an almost inverse relationship to the trends in the previous panels. The maximum log-likelihood is positioned toward the left of the plot, corresponding to <math alttext="\rho\approx 0.18" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mrow id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml"><mi id="S3.SS1.p3.1.m1.1.1.2" xref="S3.SS1.p3.1.m1.1.1.2.cmml">ρ</mi><mo id="S3.SS1.p3.1.m1.1.1.1" xref="S3.SS1.p3.1.m1.1.1.1.cmml">≈</mo><mn id="S3.SS1.p3.1.m1.1.1.3" xref="S3.SS1.p3.1.m1.1.1.3.cmml">0.18</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><apply id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1"><approx id="S3.SS1.p3.1.m1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1.1"></approx><ci id="S3.SS1.p3.1.m1.1.1.2.cmml" xref="S3.SS1.p3.1.m1.1.1.2">𝜌</ci><cn id="S3.SS1.p3.1.m1.1.1.3.cmml" type="float" xref="S3.SS1.p3.1.m1.1.1.3">0.18</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">\rho\approx 0.18</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">italic_ρ ≈ 0.18</annotation></semantics></math>. This indicates that while preferential attachment is a dominant mechanism for external citations, it is far less influential for self-citations. Instead, self-citations appear to follow a different set of distribution rules, distinct from those governing preferential attachment.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">III.2 </span>Analysis of individual authors</h3> <figure class="ltx_figure" id="S3.F4"> <p class="ltx_p" id="S3.F4.4">(a)<img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="241" id="S3.F4.1.g1" src="x7.png" width="332"/> (b)<img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="243" id="S3.F4.2.g2" src="x8.png" width="332"/> <br class="ltx_break"/>(c)<img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="241" id="S3.F4.3.g3" src="x9.png" width="332"/> (d)<img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="247" id="S3.F4.4.g4" src="x10.png" width="332"/></p> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Distributions of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F4.11.m1.1"><semantics id="S3.F4.11.m1.1b"><mi id="S3.F4.11.m1.1.1" xref="S3.F4.11.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F4.11.m1.1c"><ci id="S3.F4.11.m1.1.1.cmml" xref="S3.F4.11.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.11.m1.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F4.11.m1.1e">italic_ρ</annotation></semantics></math> for individual authors. Panel (a) presents the distribution of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F4.12.m2.1"><semantics id="S3.F4.12.m2.1b"><mi id="S3.F4.12.m2.1.1" xref="S3.F4.12.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F4.12.m2.1c"><ci id="S3.F4.12.m2.1.1.cmml" xref="S3.F4.12.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.12.m2.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F4.12.m2.1e">italic_ρ</annotation></semantics></math> for the combined set of external and self-citations. Panel (b) displays the distribution of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F4.13.m3.1"><semantics id="S3.F4.13.m3.1b"><mi id="S3.F4.13.m3.1.1" xref="S3.F4.13.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F4.13.m3.1c"><ci id="S3.F4.13.m3.1.1.cmml" xref="S3.F4.13.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.13.m3.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F4.13.m3.1e">italic_ρ</annotation></semantics></math> for external citations only, while panel (c) focuses on the distribution of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F4.14.m4.1"><semantics id="S3.F4.14.m4.1b"><mi id="S3.F4.14.m4.1.1" xref="S3.F4.14.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F4.14.m4.1c"><ci id="S3.F4.14.m4.1.1.cmml" xref="S3.F4.14.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.14.m4.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F4.14.m4.1e">italic_ρ</annotation></semantics></math> for self-citations. Panel (d) displays a 2D histogram of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F4.15.m5.1"><semantics id="S3.F4.15.m5.1b"><mi id="S3.F4.15.m5.1.1" xref="S3.F4.15.m5.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F4.15.m5.1c"><ci id="S3.F4.15.m5.1.1.cmml" xref="S3.F4.15.m5.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.15.m5.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F4.15.m5.1e">italic_ρ</annotation></semantics></math> (the probability of preferential citation) as a function of the number of citations per article, with the black solid line indicating the average <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F4.16.m6.1"><semantics id="S3.F4.16.m6.1b"><mi id="S3.F4.16.m6.1.1" xref="S3.F4.16.m6.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F4.16.m6.1c"><ci id="S3.F4.16.m6.1.1.cmml" xref="S3.F4.16.m6.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.16.m6.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F4.16.m6.1e">italic_ρ</annotation></semantics></math> for a given number of citations per article.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.3">The analysis of the log-likelihood presented in the previous section can also be performed at the level of individual scientists. This involves processing the citation events for a single scientist, constructing the log-likelihood based on these events, and maximizing it to determine the optimal value of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mi id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><ci id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">italic_ρ</annotation></semantics></math>. The results of this procedure are shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.F4" title="Figure 4 ‣ III.2 Analysis of individual authors ‣ III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">4</span></a>, which presents histograms of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><mi id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.1b"><ci id="S3.SS2.p1.2.m2.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">italic_ρ</annotation></semantics></math> for all scientists who received more than 50 citations and published at least 10 papers (with the exception of the last panel). This sample restriction was applied to exclude authors with short publication histories, as such authors could skew the results - most would likely fall into the non-PAR category based on their <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><mi id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><ci id="S3.SS2.p1.3.m3.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">italic_ρ</annotation></semantics></math> values.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.6">Panel (a) of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.F4" title="Figure 4 ‣ III.2 Analysis of individual authors ‣ III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">4</span></a> displays the distribution of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mi id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><ci id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_ρ</annotation></semantics></math> when both external and self-citations are included in the log-likelihood. The distribution is biased toward higher values of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mi id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><ci id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_ρ</annotation></semantics></math>, supporting the findings from the previous section based on the aggregate log-likelihood for the entire dataset. The mean value of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m3.1"><semantics id="S3.SS2.p2.3.m3.1a"><mi id="S3.SS2.p2.3.m3.1.1" xref="S3.SS2.p2.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m3.1b"><ci id="S3.SS2.p2.3.m3.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.3.m3.1d">italic_ρ</annotation></semantics></math> is <math alttext="0.59" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m4.1"><semantics id="S3.SS2.p2.4.m4.1a"><mn id="S3.SS2.p2.4.m4.1.1" xref="S3.SS2.p2.4.m4.1.1.cmml">0.59</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m4.1b"><cn id="S3.SS2.p2.4.m4.1.1.cmml" type="float" xref="S3.SS2.p2.4.m4.1.1">0.59</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m4.1c">0.59</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m4.1d">0.59</annotation></semantics></math>. However, a natural question arises: why is there a notable discrepancy between the average <math alttext="\rho=0.59" class="ltx_Math" display="inline" id="S3.SS2.p2.5.m5.1"><semantics id="S3.SS2.p2.5.m5.1a"><mrow id="S3.SS2.p2.5.m5.1.1" xref="S3.SS2.p2.5.m5.1.1.cmml"><mi id="S3.SS2.p2.5.m5.1.1.2" xref="S3.SS2.p2.5.m5.1.1.2.cmml">ρ</mi><mo id="S3.SS2.p2.5.m5.1.1.1" xref="S3.SS2.p2.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS2.p2.5.m5.1.1.3" xref="S3.SS2.p2.5.m5.1.1.3.cmml">0.59</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.5.m5.1b"><apply id="S3.SS2.p2.5.m5.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1"><eq id="S3.SS2.p2.5.m5.1.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1.1"></eq><ci id="S3.SS2.p2.5.m5.1.1.2.cmml" xref="S3.SS2.p2.5.m5.1.1.2">𝜌</ci><cn id="S3.SS2.p2.5.m5.1.1.3.cmml" type="float" xref="S3.SS2.p2.5.m5.1.1.3">0.59</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.5.m5.1c">\rho=0.59</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.5.m5.1d">italic_ρ = 0.59</annotation></semantics></math> calculated for individual scientists and the maximum <math alttext="\rho=0.68" class="ltx_Math" display="inline" id="S3.SS2.p2.6.m6.1"><semantics id="S3.SS2.p2.6.m6.1a"><mrow id="S3.SS2.p2.6.m6.1.1" xref="S3.SS2.p2.6.m6.1.1.cmml"><mi id="S3.SS2.p2.6.m6.1.1.2" xref="S3.SS2.p2.6.m6.1.1.2.cmml">ρ</mi><mo id="S3.SS2.p2.6.m6.1.1.1" xref="S3.SS2.p2.6.m6.1.1.1.cmml">=</mo><mn id="S3.SS2.p2.6.m6.1.1.3" xref="S3.SS2.p2.6.m6.1.1.3.cmml">0.68</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.6.m6.1b"><apply id="S3.SS2.p2.6.m6.1.1.cmml" xref="S3.SS2.p2.6.m6.1.1"><eq id="S3.SS2.p2.6.m6.1.1.1.cmml" xref="S3.SS2.p2.6.m6.1.1.1"></eq><ci id="S3.SS2.p2.6.m6.1.1.2.cmml" xref="S3.SS2.p2.6.m6.1.1.2">𝜌</ci><cn id="S3.SS2.p2.6.m6.1.1.3.cmml" type="float" xref="S3.SS2.p2.6.m6.1.1.3">0.68</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.6.m6.1c">\rho=0.68</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.6.m6.1d">italic_ρ = 0.68</annotation></semantics></math> obtained from optimizing the log-likelihood of the entire dataset? The answer lies in the aggregation process. In the previous subsection, the log-likelihood of each citation event contributed to the total value being optimized - often multiple times, as each citation event was processed for every author of the cited article. Consequently, authors with many citations had a larger influence on the aggregate log-likelihood than those with fewer citations. In contrast, the individual-author approach in this subsection treats all authors equally, regardless of their citation counts, leading to the observed discrepancy.</p> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.6">The distribution of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mi id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><ci id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_ρ</annotation></semantics></math> for external citations, shown in panel (b), aligns with the results from the previous subsection. It is skewed toward higher values, with an average <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mi id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><ci id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">italic_ρ</annotation></semantics></math> of <math alttext="0.66" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><mn id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml">0.66</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m3.1b"><cn id="S3.SS2.p3.3.m3.1.1.cmml" type="float" xref="S3.SS2.p3.3.m3.1.1">0.66</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">0.66</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">0.66</annotation></semantics></math>, indicating that preferential attachment plays a significant role in the distribution of external citations. Conversely, the distribution of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p3.4.m4.1"><semantics id="S3.SS2.p3.4.m4.1a"><mi id="S3.SS2.p3.4.m4.1.1" xref="S3.SS2.p3.4.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.4.m4.1b"><ci id="S3.SS2.p3.4.m4.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.4.m4.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.4.m4.1d">italic_ρ</annotation></semantics></math> for self-citations, depicted in panel (c), is concentrated around lower values, with an average of <math alttext="0.19" class="ltx_Math" display="inline" id="S3.SS2.p3.5.m5.1"><semantics id="S3.SS2.p3.5.m5.1a"><mn id="S3.SS2.p3.5.m5.1.1" xref="S3.SS2.p3.5.m5.1.1.cmml">0.19</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.5.m5.1b"><cn id="S3.SS2.p3.5.m5.1.1.cmml" type="float" xref="S3.SS2.p3.5.m5.1.1">0.19</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.5.m5.1c">0.19</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.5.m5.1d">0.19</annotation></semantics></math>. Notably, a substantial fraction of scientists (approximately 40%) are characterized by <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p3.6.m6.1"><semantics id="S3.SS2.p3.6.m6.1a"><mi id="S3.SS2.p3.6.m6.1.1" xref="S3.SS2.p3.6.m6.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.6.m6.1b"><ci id="S3.SS2.p3.6.m6.1.1.cmml" xref="S3.SS2.p3.6.m6.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.6.m6.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.6.m6.1d">italic_ρ</annotation></semantics></math> values close to 0 (as shown in the first bin of the histogram). These findings further reinforce the conclusion that self-citations are fundamentally different from external citations, as they do not follow the ”rich get richer” dynamic.</p> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.4">Lastly, it is reasonable to assume that different groups of scientists - for instance, those with varying levels of prominence - are characterized by distinct <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.1"><semantics id="S3.SS2.p4.1.m1.1a"><mi id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.1b"><ci id="S3.SS2.p4.1.m1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.1d">italic_ρ</annotation></semantics></math> values. However, quantifying prominence introduces challenges. While many bibliometric indicators exist for this purpose <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib27" title="">27</a>]</cite>, the most well-known being the Hirsch index (or <math alttext="h" class="ltx_Math" display="inline" id="S3.SS2.p4.2.m2.1"><semantics id="S3.SS2.p4.2.m2.1a"><mi id="S3.SS2.p4.2.m2.1.1" xref="S3.SS2.p4.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.2.m2.1b"><ci id="S3.SS2.p4.2.m2.1.1.cmml" xref="S3.SS2.p4.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m2.1d">italic_h</annotation></semantics></math>-index) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib28" title="">28</a>]</cite>, such indicators are often controversial. Nonetheless, they remain important factors in decisions regarding promotions and funding allocation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib31" title="">31</a>]</cite>. That said, calculating bibliometric indices on a limited subset of the citation network may be problematic. For example, this issue is mitigated in studies like <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib6" title="">6</a>]</cite>, where the number of citations is derived directly from article metadata rather than reconstructed from the network. To avoid potential misinterpretations stemming from inconsistent citation counts, we opted for a straightforward measure of prominence (or, more accurately in this context, popularity or ”citability”): the average number of citations per paper. Panel (d) of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.F4" title="Figure 4 ‣ III.2 Analysis of individual authors ‣ III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">4</span></a> illustrates how <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p4.3.m3.1"><semantics id="S3.SS2.p4.3.m3.1a"><mi id="S3.SS2.p4.3.m3.1.1" xref="S3.SS2.p4.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.3.m3.1b"><ci id="S3.SS2.p4.3.m3.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.3.m3.1d">italic_ρ</annotation></semantics></math> changes, on average, with this measure. As shown, <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p4.4.m4.1"><semantics id="S3.SS2.p4.4.m4.1a"><mi id="S3.SS2.p4.4.m4.1.1" xref="S3.SS2.p4.4.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.4.m4.1b"><ci id="S3.SS2.p4.4.m4.1.1.cmml" xref="S3.SS2.p4.4.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.4.m4.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.4.m4.1d">italic_ρ</annotation></semantics></math> increases with the average number of citations per paper. This suggests that the more citable a scientist is, the more preferential the citation of their papers becomes. This intuitive result will be revisited in the Discussion section.</p> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">III.3 </span>Comparisons with 3DSI model</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.7">In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib6" title="">6</a>]</cite>, the authors introduced a citation distribution model known as the Three Dimensions of Scientific Impact (3DSI). This model is, on one hand, a simplification of the Ionescu-Chopard model and, on the other, a generalization of it. 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start_ARG italic_C end_ARG start_ARG italic_N end_ARG</annotation></semantics></math> citations are assigned uniformly across the published papers, and the remaining <math alttext="n_{p}=\rho\tfrac{C}{N}" class="ltx_Math" display="inline" id="S3.SS3.p1.7.m7.1"><semantics id="S3.SS3.p1.7.m7.1a"><mrow id="S3.SS3.p1.7.m7.1.1" xref="S3.SS3.p1.7.m7.1.1.cmml"><msub id="S3.SS3.p1.7.m7.1.1.2" xref="S3.SS3.p1.7.m7.1.1.2.cmml"><mi id="S3.SS3.p1.7.m7.1.1.2.2" xref="S3.SS3.p1.7.m7.1.1.2.2.cmml">n</mi><mi id="S3.SS3.p1.7.m7.1.1.2.3" xref="S3.SS3.p1.7.m7.1.1.2.3.cmml">p</mi></msub><mo id="S3.SS3.p1.7.m7.1.1.1" xref="S3.SS3.p1.7.m7.1.1.1.cmml">=</mo><mrow id="S3.SS3.p1.7.m7.1.1.3" xref="S3.SS3.p1.7.m7.1.1.3.cmml"><mi id="S3.SS3.p1.7.m7.1.1.3.2" xref="S3.SS3.p1.7.m7.1.1.3.2.cmml">ρ</mi><mo id="S3.SS3.p1.7.m7.1.1.3.1" xref="S3.SS3.p1.7.m7.1.1.3.1.cmml">⁢</mo><mfrac id="S3.SS3.p1.7.m7.1.1.3.3" xref="S3.SS3.p1.7.m7.1.1.3.3.cmml"><mi id="S3.SS3.p1.7.m7.1.1.3.3.2" xref="S3.SS3.p1.7.m7.1.1.3.3.2.cmml">C</mi><mi id="S3.SS3.p1.7.m7.1.1.3.3.3" xref="S3.SS3.p1.7.m7.1.1.3.3.3.cmml">N</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.7.m7.1b"><apply id="S3.SS3.p1.7.m7.1.1.cmml" xref="S3.SS3.p1.7.m7.1.1"><eq id="S3.SS3.p1.7.m7.1.1.1.cmml" xref="S3.SS3.p1.7.m7.1.1.1"></eq><apply id="S3.SS3.p1.7.m7.1.1.2.cmml" xref="S3.SS3.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p1.7.m7.1.1.2.1.cmml" xref="S3.SS3.p1.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS3.p1.7.m7.1.1.2.2.cmml" xref="S3.SS3.p1.7.m7.1.1.2.2">𝑛</ci><ci id="S3.SS3.p1.7.m7.1.1.2.3.cmml" xref="S3.SS3.p1.7.m7.1.1.2.3">𝑝</ci></apply><apply id="S3.SS3.p1.7.m7.1.1.3.cmml" xref="S3.SS3.p1.7.m7.1.1.3"><times id="S3.SS3.p1.7.m7.1.1.3.1.cmml" xref="S3.SS3.p1.7.m7.1.1.3.1"></times><ci id="S3.SS3.p1.7.m7.1.1.3.2.cmml" xref="S3.SS3.p1.7.m7.1.1.3.2">𝜌</ci><apply id="S3.SS3.p1.7.m7.1.1.3.3.cmml" xref="S3.SS3.p1.7.m7.1.1.3.3"><divide id="S3.SS3.p1.7.m7.1.1.3.3.1.cmml" xref="S3.SS3.p1.7.m7.1.1.3.3"></divide><ci id="S3.SS3.p1.7.m7.1.1.3.3.2.cmml" xref="S3.SS3.p1.7.m7.1.1.3.3.2">𝐶</ci><ci id="S3.SS3.p1.7.m7.1.1.3.3.3.cmml" xref="S3.SS3.p1.7.m7.1.1.3.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.7.m7.1c">n_{p}=\rho\tfrac{C}{N}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.7.m7.1d">italic_n start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = italic_ρ divide start_ARG italic_C end_ARG start_ARG italic_N end_ARG</annotation></semantics></math> are distributed according to the preferential attachment rule (PAR).</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.3">To optimize this model, the authors compare the sorted citation vector of a scientist (or even an entire journal <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib32" title="">32</a>]</cite>) with the model’s output to identify the value of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><mi id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><ci id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">italic_ρ</annotation></semantics></math> that produces the best fit. While the model delivers accurate results, it is important to emphasise that it operates on sorted citation vectors. Specifically, it estimates the number of citations <math alttext="\hat{X}_{k}" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.1"><semantics id="S3.SS3.p2.2.m2.1a"><msub id="S3.SS3.p2.2.m2.1.1" xref="S3.SS3.p2.2.m2.1.1.cmml"><mover accent="true" id="S3.SS3.p2.2.m2.1.1.2" xref="S3.SS3.p2.2.m2.1.1.2.cmml"><mi id="S3.SS3.p2.2.m2.1.1.2.2" xref="S3.SS3.p2.2.m2.1.1.2.2.cmml">X</mi><mo id="S3.SS3.p2.2.m2.1.1.2.1" xref="S3.SS3.p2.2.m2.1.1.2.1.cmml">^</mo></mover><mi id="S3.SS3.p2.2.m2.1.1.3" xref="S3.SS3.p2.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.m2.1b"><apply id="S3.SS3.p2.2.m2.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.2.m2.1.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1">subscript</csymbol><apply id="S3.SS3.p2.2.m2.1.1.2.cmml" xref="S3.SS3.p2.2.m2.1.1.2"><ci id="S3.SS3.p2.2.m2.1.1.2.1.cmml" xref="S3.SS3.p2.2.m2.1.1.2.1">^</ci><ci id="S3.SS3.p2.2.m2.1.1.2.2.cmml" xref="S3.SS3.p2.2.m2.1.1.2.2">𝑋</ci></apply><ci 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end_ARG start_ARG italic_ρ end_ARG divide start_ARG italic_C end_ARG start_ARG italic_N end_ARG ( ∏ start_POSTSUBSCRIPT italic_i = italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT divide start_ARG italic_i end_ARG start_ARG italic_i - italic_ρ end_ARG - 1 ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p2.4">but it does not account for the timing of when each paper was published. This is a common limitation of models lacking explicit time dependency, which often results in the oldest papers accumulating the most citations.</p> </div> <figure class="ltx_figure" id="S3.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="648" id="S3.F5.g1" src="x11.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Comparison of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F5.7.m1.1"><semantics id="S3.F5.7.m1.1b"><mi id="S3.F5.7.m1.1.1" xref="S3.F5.7.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F5.7.m1.1c"><ci id="S3.F5.7.m1.1.1.cmml" xref="S3.F5.7.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.7.m1.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F5.7.m1.1e">italic_ρ</annotation></semantics></math> values calculated using the 3DSI model and those obtained through our approach. The colours represent a histogram of occurrences for each (<math alttext="\rho" class="ltx_Math" display="inline" id="S3.F5.8.m2.1"><semantics id="S3.F5.8.m2.1b"><mi id="S3.F5.8.m2.1.1" xref="S3.F5.8.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F5.8.m2.1c"><ci id="S3.F5.8.m2.1.1.cmml" xref="S3.F5.8.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.8.m2.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F5.8.m2.1e">italic_ρ</annotation></semantics></math>, 3DSI <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F5.9.m3.1"><semantics id="S3.F5.9.m3.1b"><mi id="S3.F5.9.m3.1.1" xref="S3.F5.9.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F5.9.m3.1c"><ci id="S3.F5.9.m3.1.1.cmml" xref="S3.F5.9.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.9.m3.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F5.9.m3.1e">italic_ρ</annotation></semantics></math>) pair, while the black solid line indicates the average 3DSI <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F5.10.m4.1"><semantics id="S3.F5.10.m4.1b"><mi id="S3.F5.10.m4.1.1" xref="S3.F5.10.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F5.10.m4.1c"><ci id="S3.F5.10.m4.1.1.cmml" xref="S3.F5.10.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.10.m4.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F5.10.m4.1e">italic_ρ</annotation></semantics></math> for a given <math alttext="\rho" class="ltx_Math" display="inline" id="S3.F5.11.m5.1"><semantics id="S3.F5.11.m5.1b"><mi id="S3.F5.11.m5.1.1" xref="S3.F5.11.m5.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.F5.11.m5.1c"><ci id="S3.F5.11.m5.1.1.cmml" xref="S3.F5.11.m5.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.11.m5.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.F5.11.m5.1e">italic_ρ</annotation></semantics></math> value. The dashed line represents <math alttext="Y=X" class="ltx_Math" display="inline" id="S3.F5.12.m6.1"><semantics id="S3.F5.12.m6.1b"><mrow id="S3.F5.12.m6.1.1" xref="S3.F5.12.m6.1.1.cmml"><mi id="S3.F5.12.m6.1.1.2" xref="S3.F5.12.m6.1.1.2.cmml">Y</mi><mo id="S3.F5.12.m6.1.1.1" xref="S3.F5.12.m6.1.1.1.cmml">=</mo><mi id="S3.F5.12.m6.1.1.3" xref="S3.F5.12.m6.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.12.m6.1c"><apply id="S3.F5.12.m6.1.1.cmml" xref="S3.F5.12.m6.1.1"><eq id="S3.F5.12.m6.1.1.1.cmml" xref="S3.F5.12.m6.1.1.1"></eq><ci id="S3.F5.12.m6.1.1.2.cmml" xref="S3.F5.12.m6.1.1.2">𝑌</ci><ci id="S3.F5.12.m6.1.1.3.cmml" xref="S3.F5.12.m6.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.12.m6.1d">Y=X</annotation><annotation encoding="application/x-llamapun" id="S3.F5.12.m6.1e">italic_Y = italic_X</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.4">Given that, at first glance, the parameter <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><mi id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><ci id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">italic_ρ</annotation></semantics></math> in our model (as defined in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S2.E5" title="In II.1 Model ‣ II Methodology ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">5</span></a>)) appears equivalent to the same parameter in the 3DSI model, it is worthwhile to compare the two. To this end, we calculated <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS3.p3.2.m2.1"><semantics id="S3.SS3.p3.2.m2.1a"><mi id="S3.SS3.p3.2.m2.1.1" xref="S3.SS3.p3.2.m2.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.2.m2.1b"><ci id="S3.SS3.p3.2.m2.1.1.cmml" xref="S3.SS3.p3.2.m2.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.2.m2.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.2.m2.1d">italic_ρ</annotation></semantics></math> using the approach outlined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib6" title="">6</a>]</cite>. However, for consistency, we used citation counts directly from the citation network rather than those provided in the metadata. It is worth noting that since the publication of the original paper, alternative methods for estimating <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS3.p3.3.m3.1"><semantics id="S3.SS3.p3.3.m3.1a"><mi id="S3.SS3.p3.3.m3.1.1" xref="S3.SS3.p3.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.3.m3.1b"><ci id="S3.SS3.p3.3.m3.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.3.m3.1d">italic_ρ</annotation></semantics></math> have been developed <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib35" title="">35</a>]</cite>, and the model itself has been generalized to accommodate negative values of <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS3.p3.4.m4.1"><semantics id="S3.SS3.p3.4.m4.1a"><mi id="S3.SS3.p3.4.m4.1.1" xref="S3.SS3.p3.4.m4.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.4.m4.1b"><ci id="S3.SS3.p3.4.m4.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.4.m4.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.4.m4.1d">italic_ρ</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.1">The comparison is shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.F5" title="Figure 5 ‣ III.3 Comparisons with 3DSI model ‣ III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">5</span></a>. While there is a correlation between the <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS3.p4.1.m1.1"><semantics id="S3.SS3.p4.1.m1.1a"><mi id="S3.SS3.p4.1.m1.1.1" xref="S3.SS3.p4.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.1.m1.1b"><ci id="S3.SS3.p4.1.m1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.1.m1.1d">italic_ρ</annotation></semantics></math> values from the 3DSI model and those obtained through our approach, there is also noticeable variance. Additionally, the 3DSI model appears to underestimate the influence of preferential attachment. We will revisit and discuss these findings further in the Discussion section.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span>Discussion</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this manuscript, we investigated whether the preferential attachment rule (commonly referred to as the ”rich get richer” or Matthew effect) is the primary driving force behind the distribution of citations to scientific articles. By examining individual citations one by one, we calculated the probability that a citation is assigned according to PAR. Our analysis was conducted on a large combined set of authors as well as for individual scientists. This was made possible through the DBLP dataset, a standard source of data for such studies.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">Our findings indicate that, at least for the aggregated dataset of multiple scientists, the preferential attachment rule is indeed the dominant factor in the citation distribution process, with the probability of preferential citations close to 70%. This reinforces the foundational assumption behind models such as the Ionescu-Chopard model, which often incorporate some form of PAR. However, when focusing on individual scientists rather than the combined dataset, a more nuanced picture emerges. Specifically, we observe a spectrum where some scientists attract significantly more preferential citations than others, particularly with respect to external citations. Nonetheless, the average probability of a preferential external citation remains close to 70%.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.1">When the analysis is restricted to self-citations, a noticeably different behaviour becomes apparent. For the aggregated dataset, the probability of a preferential self-citation is approximately 20%. The difference between external and self-citations becomes even more pronounced at the individual level. A substantial group of authors exhibits a probability of preferential self-citation close to 0, and only rarely does this probability exceed 50%. These findings strongly suggest that self-citations represent a distinct category of citations, governed by rules that differ from those of external citations.</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.1">Interestingly, the prominence of the preferential attachment rule appears to increase with an author’s citability, defined here as the average number of citations per paper. It seems that more citable authors are cited in a more preferential manner, while the citation patterns for less-cited authors tend to involve greater randomness. This result is intuitive: highly citable authors likely produce influential work, some of which frequently appears in bibliographies. Scientists who encounter these papers repeatedly are naturally inclined to cite them in their own manuscripts, thereby amplifying the Matthew effect. Additionally, articles with higher citation counts are more likely to be recommended by search engines. Conversely, less-cited authors producing important but specialised work are more likely to experience citation patterns that appear less preferential and more uniform, as their work is cited primarily within specific contexts. Finally, the expectation to include a sufficient number of references in bibliographies may lead to the occasional inclusion of less-relevant citations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib36" title="">36</a>]</cite>, which could partially explain the randomness in the citation patterns of less-citable authors <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib37" title="">37</a>]</cite>. Let those without sin cast the first stone…</p> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.1">However, self-citations once again diverge from this pattern. When we repeated the analysis shown in panel (d) of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S3.F4" title="Figure 4 ‣ III.2 Analysis of individual authors ‣ III Results ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">4</span></a> for self-citations, no meaningful correlation similar to that observed for the full citation set or external citations could be detected. This suggests that, irrespective of an author’s citability, the preferential attachment rule does not explain the distribution of self-citations.</p> </div> <div class="ltx_para" id="S4.p6"> <p class="ltx_p" id="S4.p6.1">We also compared our results with those from the 3DSI model, a citation distribution model that assumes citations are allocated based on a mixture of PAR and uniform distribution. While 3DSI is highly effective at replicating citation vectors, its probability of preferential citation should not be interpreted as describing the actual citation characteristics of an author. This is because 3DSI relies on sorted citation vectors, disregarding time information and an author’s citation history - it simply distributes the same number of citations at each time step. In contrast, our approach is entirely data-driven, processing citation events in chronological order. As a result, the probabilities we estimate are based on actual citation histories. Our findings show that 3DSI underestimates the influence of PAR in the citation process and exhibits significant variance. Therefore, the probability of preferential citation estimated by 3DSI should be treated as a model-specific parameter rather than an intrinsic bibliometric property of authors.</p> </div> <figure class="ltx_figure" id="S4.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="646" id="S4.F6.g1" src="x12.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Results of citation distribution simulations. The <math alttext="Y" class="ltx_Math" display="inline" id="S4.F6.7.m1.1"><semantics id="S4.F6.7.m1.1b"><mi id="S4.F6.7.m1.1.1" xref="S4.F6.7.m1.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S4.F6.7.m1.1c"><ci id="S4.F6.7.m1.1.1.cmml" xref="S4.F6.7.m1.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.7.m1.1d">Y</annotation><annotation encoding="application/x-llamapun" id="S4.F6.7.m1.1e">italic_Y</annotation></semantics></math>-axis represents the <math alttext="h" class="ltx_Math" display="inline" id="S4.F6.8.m2.1"><semantics id="S4.F6.8.m2.1b"><mi id="S4.F6.8.m2.1.1" xref="S4.F6.8.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.F6.8.m2.1c"><ci id="S4.F6.8.m2.1.1.cmml" xref="S4.F6.8.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.8.m2.1d">h</annotation><annotation encoding="application/x-llamapun" id="S4.F6.8.m2.1e">italic_h</annotation></semantics></math>-index calculated after distributing both external and self-citations, while the <math alttext="X" class="ltx_Math" display="inline" id="S4.F6.9.m3.1"><semantics id="S4.F6.9.m3.1b"><mi id="S4.F6.9.m3.1.1" xref="S4.F6.9.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.F6.9.m3.1c"><ci id="S4.F6.9.m3.1.1.cmml" xref="S4.F6.9.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.9.m3.1d">X</annotation><annotation encoding="application/x-llamapun" id="S4.F6.9.m3.1e">italic_X</annotation></semantics></math>-axis represents the <math alttext="h" class="ltx_Math" display="inline" id="S4.F6.10.m4.1"><semantics id="S4.F6.10.m4.1b"><mi id="S4.F6.10.m4.1.1" xref="S4.F6.10.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.F6.10.m4.1c"><ci id="S4.F6.10.m4.1.1.cmml" xref="S4.F6.10.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.10.m4.1d">h</annotation><annotation encoding="application/x-llamapun" id="S4.F6.10.m4.1e">italic_h</annotation></semantics></math>-index calculated using only external citations. Colours indicate a histogram of occurrences for each <math alttext="h" class="ltx_Math" display="inline" id="S4.F6.11.m5.1"><semantics id="S4.F6.11.m5.1b"><mi id="S4.F6.11.m5.1.1" xref="S4.F6.11.m5.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.F6.11.m5.1c"><ci id="S4.F6.11.m5.1.1.cmml" xref="S4.F6.11.m5.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.11.m5.1d">h</annotation><annotation encoding="application/x-llamapun" id="S4.F6.11.m5.1e">italic_h</annotation></semantics></math>-index pair, with the black solid line showing the average. The dashed line corresponds to <math alttext="Y\!=\!X" class="ltx_Math" display="inline" id="S4.F6.12.m6.1"><semantics id="S4.F6.12.m6.1b"><mrow id="S4.F6.12.m6.1.1" xref="S4.F6.12.m6.1.1.cmml"><mi id="S4.F6.12.m6.1.1.2" xref="S4.F6.12.m6.1.1.2.cmml">Y</mi><mo id="S4.F6.12.m6.1.1.1" lspace="0.108em" rspace="0.108em" xref="S4.F6.12.m6.1.1.1.cmml">=</mo><mi id="S4.F6.12.m6.1.1.3" xref="S4.F6.12.m6.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.12.m6.1c"><apply id="S4.F6.12.m6.1.1.cmml" xref="S4.F6.12.m6.1.1"><eq id="S4.F6.12.m6.1.1.1.cmml" xref="S4.F6.12.m6.1.1.1"></eq><ci id="S4.F6.12.m6.1.1.2.cmml" xref="S4.F6.12.m6.1.1.2">𝑌</ci><ci id="S4.F6.12.m6.1.1.3.cmml" xref="S4.F6.12.m6.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.12.m6.1d">Y\!=\!X</annotation><annotation encoding="application/x-llamapun" id="S4.F6.12.m6.1e">italic_Y = italic_X</annotation></semantics></math>. For aesthetic purposes, bins with a count of 0 are displayed in the same colour as those with a count of 1.</figcaption> </figure> <div class="ltx_para" id="S4.p7"> <p class="ltx_p" id="S4.p7.1">Finally, we would like to address an important issue related to self-citations. Our results clearly demonstrate that self-citations represent a distinct category of citations, governed by rules different from those of external citations. However, how significant are they in the broader context? Scientists can use self-citations to enhance certain bibliometric measures - potentially, though not necessarily, in a problematic manner. One could argue that self-citations could simply be excluded from citation vectors before calculating these measures. To assess the potential impact of self-citations, we conducted additional simulations using an alternative dataset: the full citation histories of approximately 15,000 scientists who published a paper in Nature in 2019, obtained from Scopus.</p> </div> <div class="ltx_para" id="S4.p8"> <p class="ltx_p" id="S4.p8.1">For each scientist, we reconstructed (or rather, in this case, simulated) their citation vector by processing incoming citations sequentially, year by year, in the same manner used to estimate <math alttext="\rho" class="ltx_Math" display="inline" id="S4.p8.1.m1.1"><semantics id="S4.p8.1.m1.1a"><mi id="S4.p8.1.m1.1.1" xref="S4.p8.1.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S4.p8.1.m1.1b"><ci id="S4.p8.1.m1.1.1.cmml" xref="S4.p8.1.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p8.1.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S4.p8.1.m1.1d">italic_ρ</annotation></semantics></math>. However, instead of assigning citations to actual cited papers based on real data, we distributed citations according to predefined rules. We assumed that external citations were always distributed according to the preferential attachment rule - an extreme assumption but justified by the results presented in this paper. In contrast, self-citations were distributed uniformly across the scientist’s previously published papers.</p> </div> <div class="ltx_para" id="S4.p9"> <p class="ltx_p" id="S4.p9.5">We performed two types of simulations. In the first, we omitted self-citations entirely and distributed only external citations. In the second, we distributed both external and self-citations but removed the self-citations from the final simulated citation vector. At the conclusion of each simulation, we calculated the Hirsch index (<math alttext="h" class="ltx_Math" display="inline" id="S4.p9.1.m1.1"><semantics id="S4.p9.1.m1.1a"><mi id="S4.p9.1.m1.1.1" xref="S4.p9.1.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.p9.1.m1.1b"><ci id="S4.p9.1.m1.1.1.cmml" xref="S4.p9.1.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p9.1.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.p9.1.m1.1d">italic_h</annotation></semantics></math>-index) for each scientist, effectively using only external citations (as self-citations were either not distributed or subtracted from the citation vector). Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#S4.F6" title="Figure 6 ‣ IV Discussion ‣ Exploring the dynamics of self-citations and their role in shaping scientific impact"><span class="ltx_text ltx_ref_tag">6</span></a> presents the results, where the <math alttext="Y" class="ltx_Math" display="inline" id="S4.p9.2.m2.1"><semantics id="S4.p9.2.m2.1a"><mi id="S4.p9.2.m2.1.1" xref="S4.p9.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S4.p9.2.m2.1b"><ci id="S4.p9.2.m2.1.1.cmml" xref="S4.p9.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p9.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S4.p9.2.m2.1d">italic_Y</annotation></semantics></math>-axis corresponds to the <math alttext="h" class="ltx_Math" display="inline" id="S4.p9.3.m3.1"><semantics id="S4.p9.3.m3.1a"><mi id="S4.p9.3.m3.1.1" xref="S4.p9.3.m3.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.p9.3.m3.1b"><ci id="S4.p9.3.m3.1.1.cmml" xref="S4.p9.3.m3.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p9.3.m3.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.p9.3.m3.1d">italic_h</annotation></semantics></math>-index calculated after distributing both external and self-citations, and the <math alttext="X" class="ltx_Math" display="inline" id="S4.p9.4.m4.1"><semantics id="S4.p9.4.m4.1a"><mi id="S4.p9.4.m4.1.1" xref="S4.p9.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.p9.4.m4.1b"><ci id="S4.p9.4.m4.1.1.cmml" xref="S4.p9.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p9.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.p9.4.m4.1d">italic_X</annotation></semantics></math>-axis corresponds to the <math alttext="h" class="ltx_Math" display="inline" id="S4.p9.5.m5.1"><semantics id="S4.p9.5.m5.1a"><mi id="S4.p9.5.m5.1.1" xref="S4.p9.5.m5.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.p9.5.m5.1b"><ci id="S4.p9.5.m5.1.1.cmml" xref="S4.p9.5.m5.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p9.5.m5.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.p9.5.m5.1d">italic_h</annotation></semantics></math>-index calculated using only external citations.</p> </div> <div class="ltx_para" id="S4.p10"> <p class="ltx_p" id="S4.p10.1">Two important conclusions can be drawn from this figure. First, the Hirsch index is, on average, higher when self-citations are included, in some extreme cases by nearly 50%. This suggests that self-citations can significantly enhance the visibility and perceived scientific impact of a researcher’s work. While this process can occur naturally - as in our simulations, where a benign distribution scheme was assumed - it is easy to imagine scenarios where self-citations are strategically manipulated to target specific bibliometric measures.</p> </div> <div class="ltx_para" id="S4.p11"> <p class="ltx_p" id="S4.p11.1">The second conclusion, which provides valuable insight for regulators, is that self-citations cannot simply be removed from citation vectors. While they can be excluded arithmetically, doing so merely removes their numerical contribution and does not account for the potential influence self-citations have on the distribution of external citations. The existence and distribution of self-citations may shape the patterns of external citations in ways that are both indirect and significant, and this influence cannot be ignored when assessing bibliometric indices.</p> </div> <div class="ltx_para" id="S4.p12"> <p class="ltx_p" id="S4.p12.1">Looking ahead, many questions about self-citations remain unanswered. While we have shown that their distribution is not governed by preferential attachment, the exact mechanisms by which they are assigned to articles remain unclear. Scientists are known to favour their own work when compiling bibliographies for their papers (the so-called self-citation bias; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib11" title="">11</a>]</cite>), and self-citations can account for a significant fraction of total citations. As noted earlier, this makes them a potential tool for artificially inflating bibliometric indices such as the <math alttext="h" class="ltx_Math" display="inline" id="S4.p12.1.m1.1"><semantics id="S4.p12.1.m1.1a"><mi id="S4.p12.1.m1.1.1" xref="S4.p12.1.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.p12.1.m1.1b"><ci id="S4.p12.1.m1.1.1.cmml" xref="S4.p12.1.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p12.1.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.p12.1.m1.1d">italic_h</annotation></semantics></math>-index <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib38" title="">38</a>]</cite>. Developing models in which authors aim to maximise specific bibliometric indicators could be a promising avenue for understanding the distribution of self-citations and identifying authors who engage in such practices. Moreover, research indicates that most self-citations occur shortly after publication and that their influence wanes more rapidly than external citations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib39" title="">39</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib40" title="">40</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib41" title="">41</a>]</cite>. Incorporating a time-dependent component, such as manuscript ageing, into our model could enhance its ability to capture these temporal dynamics. Additionally, exploring whether the asymmetry of interactions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib42" title="">42</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib43" title="">43</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib44" title="">44</a>]</cite> - both within scientific collaboration networks and citation networks - can be linked to and explain the citation behaviours of authors would be an intriguing direction for future research.</p> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span>Code availability</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">The source code is available on the GitHub repository: https://github.com/brokenpumpernickel/citation-distributions.</p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VI </span>Data availability</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">The 12th version of the DBLP Citation Network Dataset <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.09811v1#bib.bib26" title="">26</a>]</cite> is publicly and freely available at https://www.aminer.cn/citation. 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