Mobile Geometric Scale-Free Random Graphs

In recent years scale-free geometric random graphs have been used extensively to study real world networks, such as telecommunication or social (media) networks. These models and their applications tend to be static in nature, with the nodes of the network (and their respective power) fixed in space and the connections between them not updating. Recent work has been done in order to introduce dynamics into the system, first by making the connections update over time (in non-geometric/non-spatial versions of such models) and more recently by making the nodes of the network move in space. Such dynamics break many of the standing assumptions for the static case and introduce correlations that make standard random graph theory techniques difficult to apply. Luckily, techniques developed for interacting particle systems provide a solid foundation on which random graph theory can be of use again. This project aims to continue the work in bridging the two worlds in order to explore many of the interesting and natural questions that arise from the setup, such as the following.<br /> <br /> - The existence of connections (hereupon edges) between the nodes (hereupon vertices) is often more then just a deterministic function of the location of the vertices and their respective weights; it is instead random itself, with the probability of an edge existing increasing with proximity and the weight/influence of the two respective vertices. If vertices are mobile, the question of when and how edges update becomes relevant and warrants study.<br /> - In the static case topological properties of the graph are of interest, such as the emergence of an infinite connected component and typical distances in the graph. If one looks at snapshots in the mobile case these properties are retained. It is however not clear whether like in the case of dynamical percolation, exceptional times with different properties can exist.<br /> - The contact process has been studied on static scale-free geometric random graphs. For non-geometric models, results about the contact process are also known when edges update at various rates. First steps have been made for the mobile scale-free geometric random graphs in studying instantaneous propagation of information which indicate that many of the known techniques should be applicable here as well, but it is unclear whether the motion of the particles helps or hinders the contact process in its survival.

<h2>Literature</h2> <ul> <li>P. Gracar and A. Grauer.<a href="https://arxiv.org/abs/2404.15124"><em> Geometric scale-free random graphs on mobile vertices: broadcast and percolation times</em></a>, 2024. (ArXiv preprint)</li> <li>P. Gracar and A. Grauer. <em><a href="https://doi.org/10.1016/j.spa.2024.104360">The contact process on scale-free geometric random graphs</a></em>. Stochastic Processes and their Applications, 173:104360, 2024.</li> <li>E. Jacob, A. Linker, and P. Mörters.<em> <a href="https://arxiv.org/abs/2206.01073">The contact process on dynamical scale-free networks</a></em>, 2022. (ArXiv preprint)</li> <li>Y. Peres, A. Sinclair, P. Sousi, and A. Stauffer. <a href="https://epubs.siam.org/doi/10.1137/1.9781611973082.33"><em>Mobile geometric graphs: Detection, coverage and percolation</em></a>. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, 2011.</li> </ul>

<p>Formal applications for research degree study should be made online through the University's website. Please state clearly in the Planned Course of Study section that you are applying for <strong><em>PHD Statistics FT</em></strong> and in the research information section that the research degree you wish to be considered for is <em><strong>Mobile Geometric Scale-Free Random Graphs</strong></em> as well as <a href="https://eps.leeds.ac.uk/maths/staff/13156/dr-peter-gracar">Dr. Peter Gracar</a> as your proposed supervisor and in the finance section, please state clearly <em><strong>the funding source that you are applying for, if you are self-funding or externally sponsored (including the name of your sponsor)</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><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> <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 class="MsoNoSpacing"><strong>Please note that you must provide the following documents in support of your application by the closing date of Monday 6 January 2025 if applying for the China Scholarship Council-University of Leeds Scholarship, Monday 3 February 2025 if applying for Leeds Doctoral Scholarship or Tuesday 1 April 2025 for Leeds Opportunity Research Scholarship.</strong></p> <p><strong>If you are applying for the School of Mathematics Scholarship 2025/26, or 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</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul>

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 style="margin-bottom:12px"><strong>Self-Funded or externally sponsored students are welcome to apply.</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: Monday 3 February 2025)</strong>, <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> <strong>(closing date: Tuesday 1 April 2025)</strong> and <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> <strong>(open from October 2024)</strong> are available to UK applicants.</p> <p><strong>Non-UK</strong> – <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> <strong>(open from October 2024)</strong> are available to all International applicants.  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> <strong>(closing date: Monday 6 January 2025)</strong> is available to nationals of China. 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>You will be responsible for paying the overtime fee in full in your writing up/overtime year (£320 in Session 2024/25), 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 <em><strong>not</strong></em> 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 this project, please contact Dr Peter Gracar by email to <a href="mailto:p.gracar@leeds.ac.uk">p.gracar@leeds.ac.uk</a></p> <p>For further information about your application, please contact PGR Admissions by email to <a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a></p>