Spread of Information and Diseases via Random Walks in Sparse Graphs

We consider a natural network diffusion process, modeling the spread of information or infectious diseases. Multiple mobile agents perform independent simple random walks on an n-vertex connected graph G. The number of agents is linear in n and the walks start from the stationary distribution. Initially, a single vertex has a piece of information (or a virus). An agent becomes informed (or infected) the first time it visits some vertex with the information (or virus); thereafter, the agent informs (infects) all vertices it visits. Giakkoupis et al. (PODC'19) have shown that the spreading time, i.e., the time before all vertices are informed, is asymptotically and w.h.p. the same as in the well-studied randomized rumor spreading process, on any d-regular graph with d=Ω(logn). The case of sub-logarithmic degree was left open, and is the main focus of this paper. First, we observe that the equivalence shown by Giakkoupis et al. does not hold for small d: We give an example of a 3-regular graph with logarithmic diameter for which the expected spreading time is Ω(log^2n/loglogn), whereas randomized rumor spreading is completed in time Θ(logn), w.h.p. Next, we show a general upper bound of O~(d⋅diam(G)+log^3n/d), w.h.p., for the spreading time on any d-regular graph. We also provide a version of the bound based on the average degree, for non-regular graphs. Next, we give tight analyses for specific graph families. We show that the spreading time is O(logn), w.h.p., for constant-degree regular expanders. For the binary tree, we show an upper bound of O(logn⋅loglogn), w.h.p., and prove that this is tight, by giving a matching lower bound for the cover time of the tree by n random walks. Finally, we show a bound of O(diam(G)), w.h.p., for k-dimensional grids, by adapting a technique by Kesten and Sidoravicius.Supported in part by ANR Project PAMELA (ANR16-CE23-0016-01). Gates Cambridge Scholarship programme. Supported by the ERC Grant `Dynamic March’

Brownian frogs with removal: pandemics in a diffusing population

A stochastic model of susceptible/infected/removed (SIR) type, inspired by COVID-19, is introduced for the spread of infection through a spatially-distributed population. Individuals are initially distributed at random in space, and they move according to independent random processes. The disease may pass from an infected individual to an uninfected individual when they are sufficiently close. Infected individuals are permanently removed at some given rate $\alpha$. Two models are studied here, termed the 'delayed diffusion' and the 'diffusion' models. In the first, individuals are stationary until they are infected, at which time they begin to move; in the second, all individuals start to move at the initial time 0. Using a perturbative argument, conditions are established under which the disease infects a.s. only finitely many individuals. It is proved for the delayed diffusion model that there exists a critical value $\alpha_c\in(0,\infty)$ for the existence of a pandemic

On some percolation problems in correlated systems

In this thesis we explore the framework of the percolation theory and we analyse two models. We investigate the level set of the Gaussian free field on a supercritical Galton--Watson tree conditioned on non-extinction with random conductances, showing that the critical parameter h_* is deterministic and strictly positive, that the level set contains almost surely a transient component for some h>0 and it is stable under perturbation via small quenched noise. Then we study an infection model with recovery on fractal graphs as the Sierpinski gaskets and carpets and show the survival of the infection for small recovery parameter. To prove the result, we generalize the concept of Lsipschitz surface for the lattice to fractal graphs, and we show the existence and certain connectivity properties of what we call a Lipschitz Cutset

Information dissemination via random walks

Information dissemination is a fundamental task in distributed computing: How to deliver a piece of information from a node of a network to some or all other nodes? In the face of large and still growing modern networks, it is imperative that dissemination algorithms are decentralised and can operate under unreliable conditions. In the past decades, randomised rumour spreading algorithms have addressed these challenges. In these algorithms, a message is initially placed at a source node of a network, and, at regular intervals, each node contacts a randomly selected neighbour. A message may be transmitted in one or both directions during each of these communications, depending on the exact protocol. The main measure of performance for these algorithms is their broadcast time, which is the time until a message originating from a source node is disseminated to all nodes of the network. Apart from being extremely simple and robust to failures, randomised rumour spreading achieves theoretically optimal broadcast time in many common network topologies. In this thesis, we propose an agent-based information dissemination algorithm, called Visit-Exchange. In our protocol, a number of agents perform independent random walks in the network. An agent becomes informed when it visits a node that has a message, and later informs all future nodes it visits. Visit-Exchange shares many of the properties of randomised rumour spreading, namely, it is very simple and uses the same amount of communication in a unit of time. Moreover, the protocol can be used as a simple model of non-recoverable epidemic processes. We investigate the broadcast time of Visit-Exchange on a variety of network topologies, and compare it to traditional rumour spreading. On dense regular networks we show that the two types of protocols are equivalent, which means that in this setting the vast literature on randomised rumour spreading applies in our model as well. Since many networks of interest, including real-world ones, are very sparse, we also study agent-based broadcast for sparse networks. Our results include almost optimal or optimal bounds for sparse regular graphs, expanders, random regular graphs, balanced trees and grids. We establish that depending on the network topology, Visit-Exchange may be either slower or faster than traditional rumour spreading. In particular, in graphs consisting of hubs that are not well connected, broadcast using agents can be significantly faster. Our conclusion is that a combined broadcasting protocol that simultaneously uses both traditional rumour spreading and agent-based dissemination can be fast on a larger range of topologies than each of its components separately.Gates Cambridge Trust, St John's College Benefactors' Scholarshi