Hirsch's Citation Index and Limit Shape of Random Partitions

SUMMARY. The h-index, introduced by Hirsch (2005) as a measure of a researcher's publication output, is defined as the largest integer h such that the person has h papers with at least h citations each. Yong (2014) proposed a statistical approach to modelling the h-index by interpreting citations as parts of an integer partition and leveraging the so-called limit shape of a suitable class of random partitions. However, the limit shape may strongly depend on the distribution of partitions, which would also affect the asymptotics of the h-index. The aim of the project is to explore this approach, for instance by trying out the Markov Chain Monte Carlo (MCMC) techniques, using the uniform distribution as an uninformative prior. These and similar ideas have a potential to be extended beyond the citation topic, and may offer an interesting blend of theoretical and more applied issues, with a possible gateway to further applications of discrete probability and statistics in social sciences involving count data.

<p>The <em>h</em>-index of citations, introduced by Hirsch [3] as a measure of the quality of a researcher's publication output, is defined as the largest integer <em>h</em> such that the person has <em>h</em> papers with at least <em>h</em> citations each. The h-index has become quite popular (see Google Scholar or Web of Science). Yong [6] proposed a statistical approach to modelling the h-index by interpreting citations as parts of an integer partition [1] and leveraging the so-called <em>limit shape</em> of a suitable class of random partitions [5]. Namely, identifying an integer partition with its <em>Young diagram</em> (with blocks representing parts), it is clear that the h-index is the size of the largest <em>h</em> x <em>h</em> square that fits in. If partitions of a given integer <em>N</em> are treated as random, with uniform distribution (i.e., all such partitions are assumed to be equally likely), then their Young diagrams have the limit shape (under the suitable scaling), first identified by Vershik [5]. In particular, it follows that the typical value of the h-index for someone with a large number <em>N</em> of citations should be close to 0.54 <em>N</em>. However, the assumption of uniform distribution on partitions is of course rather arbitrary, and needs to be tested statistically. This issue is important since the limit shape may strongly depend on the distribution of partitions [2], which would also affect the asymptotics of the h-index.</p> <p>Thus, the aim of this project is to explore such an extension of Yong's approach. To this end, one might try and apply Markov Chain Monte Carlo (MCMC) techniques [4], whereby the uniform distribution may serve as an uninformative prior. These and similar ideas have a potential to be extended beyond the citation topic, and may offer an interesting blend of theoretical and more applied issues, with a possible gateway to further applications of discrete probability and statistics in social sciences involving count data.</p> <p>Successful candidates should have a good degree in mathematics and/or statistics. Programming skills to carry out MCMC simulations would be useful but not essential, as the appropriate training will be provided.</p> <p>You will be based within a strong research group in <a href="https://eps.leeds.ac.uk/maths-statistics/doc/probability-financial-mathematics">Probability and Financial Mathematics</a>.</p> <p><strong>References</strong></p> <ol> <li>Andrews, G.E. and Eriksson, K<em>. Integer Partitions. </em>Cambridge Univ. Press, Cambridge, 2004.</li> <li>Bogachev, L.V. Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts<em>. Random Struct. Algorithms, </em><strong>47</strong> (2015), 227–266. (<a href="https://doi.org/10.1002/rsa.20540">doi:10.1002/rsa.20540</a>)</li> <li>Hirsch, J.E. An index to quantify an individual's scientific research output.<em> Proc. Natl. Acad. Sci. USA, </em><strong>102</strong> (2005), 16569–16572. (<a href="https://doi.org/10.1073/pnas.0507655102">doi:10.1073/pnas.0507655102</a>)</li> <li><em>Markov Chain Monte Carlo in Practice </em>(W.R. Gilks, S. Richardson and D.J. Spiegelhalter, eds.).<em> </em>Chapman & Hall/CRC, London, 1996.</li> <li>Vershik, A.M. Asymptotic combinatorics and algebraic analysis. In:<em> Proc. Intern. Congress Math. 1994, vol. 2. </em>Birkhäuser, Basel, 1995, pp. 1384–1394. (<a href="https://doi.org/10.1007/978-3-0348-9078-6_133"><span id="doi-url">doi:10.1007/978-3-0348-9078-6_133</span></a>)</li> <li>Yong, A. Critique of Hirsch's citation index: a combinatorial Fermi problem.<em> Notices Amer. Math. Soc.,</em> <strong>61</strong> (2014), 1040–1050. (<a href="https://doi.org//10.1090/noti1164">doi:/10.1090/noti1164</a>)</li> </ol>

<p>Formal applications for research degree study should be made online through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University's website</a>. Please state clearly in the Planned Course of Study that you are applying for <em><strong>PHD Statistics FT</strong></em> and in the research information section that the research degree you wish to be considered for is <em><strong>Hirsch's Citation Index and Limit Shape of Random Partitions</strong></em> as well as <a href="https://eps.leeds.ac.uk/maths/staff/4008/dr-leonid-bogachev">Dr Leonid Bogachev</a> as your proposed supervisor and in the finance section, please state clearly <em><strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong></em>.</p> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p>Applications will be considered on an ongoing basis.  Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application.  We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p><strong>If you are applying for University or School Scholarships for 2026/27 entry, with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date including the grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p>

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<p><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>Scholarship opportunities open from October 2025</strong></p> <p><strong>UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/138-leeds-doctoral-scholarship-2025-faculty-of-engineering-and-physical-sciences#:~:text=Key%20facts&text=One%20Leeds%20Doctoral%20Scholarship%20is,rata%20for%20part%2Dtime%20study.">Leeds Doctoral Scholarship</a> <strong>(closing date: February 2026)</strong> and <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> <strong>(closing date: April 2026)</strong> are available to UK applicants.  <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/48-china-scholarship-council-university-of-leeds-scholarships-2021">China Scholarship Council - University of Leeds Scholarship</a> is available to nationals of China <strong>(closing date: January 2026)</strong>. The <a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a> is available to support US citizens. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p>Please note that if you are successful in securing an academic offer for PhD study, this does not mean that you have been successful in securing an offer of funding.</p> <p>If you are applying for the Leeds Doctoral Scholarship, Leeds Opportunity Research Scholarship, China Scholarship Council-University of Leeds Scholarship or Leeds Marshall Scholarship, you will need to complete a separate application, specific to these scholarships, to be considered for funding.</p> <p>You will be responsible for paying the overtime fee in full in your writing up/overtime year (£340 in Session 2025/26), but the scholarship maintenance allowance will continue to be paid for up to 6 months in the final year of award.</p> <p><strong>Important: </strong>Please note that that the award does not cover the costs associated with moving to the UK.  All such costs (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source. </p> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<p>For further information about your application, please contact PGR Admissions by email to <a href="mailto:phd@engineering.leeds.ac.uk">phd@engineering.leeds.ac.uk</a></p> <p>For further information about this project, please contact Dr Leonid Bogachev by email to <a href="mailto:L.V.Bogachev@leeds.ac.uk">L.V.Bogachev@leeds.ac.uk</a></p>