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.<em> <a href="https://arxiv.org/abs/2208.08346">The contact process on scale-free geometric random graphs</a>,</em>&nbsp;2022. (ArXiv preprint)</li> <li>E. Jacob, A. Linker, and P. M&ouml;rters.<em> <a href="https://arxiv.org/abs/2206.01073">The contact process on dynamical scale-free networks</a></em>, 2022.&nbsp;(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&#39;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&#39;s minimum English language requirements (below).</p> <p>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.</p> <p>Applications will be considered on an ongoing basis. &nbsp;Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. &nbsp;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>Please note that you must provide the following documents in support of your application by the closing date of 3 April 2024 for Leeds Opportunity Research Scholarship or&nbsp;8 April 2024 for Leeds Doctoral Scholarship:</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>

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<p><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>UK </strong>&ndash; The <a href="https://phd.leeds.ac.uk/funding/209-leeds-doctoral-scholarships-2022">Leeds Doctoral Scholarships</a>,&nbsp;<a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a>&nbsp;and <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship">School of Mathematics Scholarships</a>&nbsp;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>&ndash; 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 (now closed for 2024/25 entry). 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><strong>Important:</strong> &nbsp;Any costs associated with your arrival at the University of Leeds to start your PhD including flights, immigration health surcharge/medical insurance and Visa costs are <strong>not </strong>covered under these studentships.</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 Doctoral College Admissions by email to <a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a></p>