Heterogeneous spreading processes on graphs as models of contagion

Hunter-Frankland, Ethan Thomas Pius (2026) Heterogeneous spreading processes on graphs as models of contagion. PhD thesis, University of Glasgow.

Abstract

Spreading processes on graphs model real-world contagion phenomena such as disease transmission, information flow and fire spread. To respond to a particular contagion, we can introduce interventions to these processes that model, for example, vaccination, educational campaigns or fire retardant. By asking whether we can protect a given number of vertices, safeguard a specified set or optimise over these targets, we turn these processes into decision and optimisation problems. These problems are typically computationally hard; restricted cases admit efficient algorithms and simulation is widely used when analysis is infeasible. This dissertation advances the study of spreading processes by introducing heterogeneous variants of the Firefighter Problem and by analysing the feasibility of an exact dynamical approach to compartmental models on graphs.

We first define the Cost Function Firefighter Problem, in which heterogeneity is introduced by allowing the cost of defending a vertex to vary over time and with the state of the system. We then study the Infectious Vaccination problem, in which defence propagates across the network analogously to contagion – in particular, we empirically study the impact of heterogeneity by varying the rates of contagion and defence propagation. For both models, we prove NP-hardness even on graph classes where the classical Firefighter Problem is tractable, identify tractable subclasses and obtain fixed-parameter tractability results. Specifically, we give monadic second-order formulations on bounded treewidth graphs and provide fixed-parameter tractable algorithms parameterised by structural measures including vertex cover number and neighbourhood diversity. We complement these results with simulations that compare the numbers of vertices saved by natural heuristic strategies on random and real-world networks.

Finally, we analyse a dynamical approach to compartmental epidemic models on graphs, which describes contagion by exact systems of ordinary differential equations. We give an exponential upper bound on system size in general, prove linear scaling for trees and polynomial scaling in model complexity and extend existing moment-closure results from cut vertices to arbitrary cutsets. We implement a general equation-generation framework and, via runtime experiments, show that Monte Carlo simulation is typically faster, with the dynamical approach being preferable mainly when exact, deterministic results are required.

Together, these contributions extend the theoretical understanding of spreading processes on graphs, introduce new heterogeneous models and delineate the limits of exact dynamical and algorithmic approaches. Beyond theory, they clarify when exact dynamics, parameterised algorithms or heuristic strategies coupled with simulation are best suited to predicting and mitigating contagion and belief change in heterogeneous networks.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
• Colleges/Schools: College of Science and Engineering > School of Computing Science
• Supervisor's Name: Enright, Dr. Jessica and Miller, Professor Alice
• Date of Award: 2026
• Depositing User: Theses Team
• Unique ID: glathesis:2026-86104
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 14 Jul 2026 13:05
• Last Modified: 14 Jul 2026 14:54
• Thesis DOI: 10.5525/gla.thesis.86104
• URI: https://theses.gla.ac.uk/id/eprint/86104