Spread of information on weighted and fractal lattices

The spread of information (or infection) on networks is a well-studied topic in probability and network science. Classical models often assume simple nearest-neighbour interactions on regular Euclidean lattices, but many real and theoretical networks are far more irregular. This project focuses on rigorous mathematical modelling of information spread among mobile agents on two types of complex graphs: weighted lattices (e.g. Z^d with random conductances) and fractal graphs (such as the Sierpinski gasket and carpet). These structures go beyond the homogeneous nearest-neighbour lattice, introducing inhomogeneity and non-integer dimensional geometry that pose new challenges for analysis. <br /> <br /> In a weighted lattice, each edge is assigned a random weight or conductance (uniformly elliptic, bounded away from 0 and infinity). This models networks with variable connection strengths or traversal times. Fractal lattices like the Sierpinski gasket have a hierarchical, self-similar structure with anomalous geometry – random walks on these graphs are sub-diffusive, meaning they spread slower than on regular lattices. Understanding information spread on such graphs is both of practical significance (modelling communication in irregular networks, epidemics in heterogeneous environments) and theoretical importance (extending percolation and probability theory to new settings). Recent work has laid the groundwork for this study. Gracar and Stauffer (2019) developed a multi-scale Lipschitz percolation framework for particles moving on a weighted lattice (random conductance graph). In their model, agents perform independent random walks from a Poisson point process initial condition, and an "infection" passes when agents meet. They proved that even in a highly irregular, weighted environment, an infection can spread with positive speed(linearly growing infection frontier) with high probability. More recently, Drewitz, Gallo, and Gracar (2023) extended these ideas to fractal graphs. Focusing on the Sierpinski gasket and carpet, they generalized the Lipschitz percolation method to the fractal space-time setting and showed that if mobile agents infect each other upon contact and recover independently at rate gamma, then on these fractals an initially infected particle can lead to an everlasting infection when gamma is sufficiently small. Notably, random walks on the Sierpinski graphs have anomalous diffusion, yet the infection can still percolate through the network in the supercritical regime. <br /> <br /> Research Aim: Building on this state-of-the-art knowledge, the aim of this PhD project is to investigate new theoretical questions and refine existing frameworks for information spread on irregular graphs, with a focus on weighted lattices and fractal structures.

<h2 style="margin-top: 24px; margin-bottom: 5px;">Research Goals</h2> <ul> <li style="margin-top: 24px; margin-bottom: 5px;">Analyze the Robustness of Infection Spread in Weighted Lattices: Building on the established positive speed of infection spread, investigate the sensitivity of spread dynamics to perturbations in the weight distribution, such as heavy-tailed or correlated conductances. Determine to what extent the results persist under relaxed assumptions or under adversarial environments.</li> <li style="margin-top: 24px; margin-bottom: 5px;">Extend Fractal Graph Frameworks Beyond Sierpinski-Type Structures: Generalize the infection percolation framework to other families of fractals or nested hierarchical graphs. Study the minimal geometric or spectral conditions required to ensure survival or extinction of the infection process.</li> <li style="margin-top: 24px; margin-bottom: 5px;">Establish a Shape Theorem: Formulate and prove a shape theorem for the spread of infection or information in weighted and fractal lattices. This involves identifying deterministic limiting shapes for the infected region over time and quantifying deviations, thereby characterizing the geometry of large-scale spread.</li> </ul> <p style="margin-top: 24px; margin-bottom: 5px;">The outcomes will deepen our theoretical understanding of spreading processes in heterogeneous networks. This has implications for epidemiology (spread of disease in heterogeneous environments), information dissemination in communication networks with irregular topology, and mathematical physics (e.g. spread of a “fire” or disturbance in disordered media). By focusing on rigorous proofs and general frameworks, the project will contribute lasting tools to probability theory and network science.</p> <h2>Literature</h2> <ul> <li>Drewitz, Gallo, Gracar - Lipschitz cutset for fractal graphs and applications to the spread of infections, <em><a href="https://imstat.org/journals-and-publications/annales-de-linstitut-henri-poincare/annales-de-linstitut-henri-poincare-accepted-papers/">Annales de l’Institut Henri Poincaré, Probabilités et Statistiques</a></em>, to appear</li> <li>Gracar, Stauffer - Random walks in random conductances decoupling and spread of infection, <em><a href="https://doi.org/10.1016/j.spa.2018.09.016">Stochastic Processes and their Applications</a></em>, 129: 3547-3569 (2019)</li> <li>Gracar, Stauffer - Multi-scale Lipschitz percolation of increasing events for Poisson random walks, <em><a href="https://projecteuclid.org/euclid.aoap/1544000432">Annals of Applied Probability</a></em>, 29: 376-433 (2019)</li> </ul>

<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 section that you are applying for <em><strong>PHD Statistics FT</strong></em>, in the research information section that the research degree you wish to be considered for is <strong><em>Spread of information on weighted and fractal lattices</em></strong> 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 <em><strong>in the finance section, please state clearly the</strong></em> <strong><em>funding source that you are applying for, if you are self-funding or externally sponsored.</em></strong></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><strong>Please note that you must provide the following documents in support of your application 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>

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<p><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>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>(now closed for October 2025 entry) </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>(now closed for October 2025 entry)</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>(now closed for October 2025 entry)</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 (£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 <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:phd@engineering.leeds.ac.uk">phd@engineering.leeds.ac.uk</a>.</p>