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>To apply for this project you will need to make a formal application for research degree study through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University website</a>. You will need to create a login ID with a username and PIN. </p> <p>•    For <strong>Application type</strong> please select <strong>Research Degrees – Research Postgraduate</strong>. <br /> •    The admission year for this project is <strong>2027/28</strong> Academic Year. <br /> •    You will need to select your <strong>Planned Course of Study</strong> from a drop-down menu. For this project, scroll down and select <strong>PHP Statistics Full Time.</strong> <br /> •    The project start date for this project is <strong>1 October 2027</strong>, please use this as your <strong>Proposed Start Date of Research</strong>. <br /> •    Please state clearly in the<strong> Research Information Section</strong> that the research degree you wish to be considered for is <strong>Hirsch's Citation Index and Limit Shape of Random Partitions</strong> as well as <a href="https://eps.leeds.ac.uk/maths/staff/4008/professor-leonid-bogachev">Professor Leonid Bogachev</a> as your proposed supervisor.</p> <p><strong>Please state clearly in the Finance section</strong>, <strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong>.</p> <p>More information on how to apply is available on our website <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">here</a>. You will be required to provide a personal statement which outlines your interest in the project you are applying for, why you have chosen it and how your skills map onto the requirements of the project.</p> <p>We will assess applications continuously as we receive them. We welcome and strongly encourage any potential applicants to contact the supervisor(s) for an informal discussion, prior to applying, and recommend submitting your application early.</p> <p><strong>If you are applying 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>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <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>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<p>This is a non funded PhD project. Applicants are expected to be self funded or to secure external funding.</p> <p>You can find out more about our Funding Opportunities on our <a href="https://phd.leeds.ac.uk/">Postgraduate Research Opportunities</a> website.</p> <p><strong>Important: </strong>Please note that all costs associated with your arrival at Leeds (<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>For questions related to the application process and eligibility, please contact PGR Admissions by email to <a href="mailto:epspgradmissions@leeds.ac.uk">epspgradmissions@leeds.ac.uk</a></p> <p>For further information about this project, please contact Professor Leonid Bogachev by email to <a href="mailto:L.V.Bogachev@leeds.ac.uk">L.V.Bogachev@leeds.ac.uk</a></p>