The end time of SIS epidemics driven by random walks on edge-transitive graphs

Network epidemics is a ubiquitous model that can represent different phenomena and finds applications in various domains. Among its various characteristics, a fundamental question concerns the time when an epidemic stops propagating. We investigate this characteristic on a SIS epidemic induced by agents that move according to independent continuous time random walks on a finite graph: Agents can either be infected (I) or susceptible (S), and infection occurs when two agents with different epidemic states meet in a node. After a random recovery time, an infected agent returns to state S and can be infected again. The End of Epidemic (EoE) denotes the first time where all agents are in state S, since after this moment no further infections can occur and the epidemic stops. For the case of two agents on edge-transitive graphs, we characterize EoE as a function of the network structure by relating the Laplace transform of EoE to the Laplace transform of the meeting time of two random walks. Interestingly, this analysis shows a separation between the effect of network structure and epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically in the size of the graph) under different parameter scalings, identifying regimes where EoE converges in distribution to a proper random variable or to infinity. We also highlight the impact of different graph structures on EoE, characterizing it under complete graphs, complete bipartite graphs, and rings

First-order transition in Potts models with "invisible' states: Rigorous proofs

In some recent papers by Tamura, Tanaka and Kawashima [arXiv:1102.5475, arXiv:1012.4254], a class of Potts models with "invisible" states was introduced, for which the authors argued by numerical arguments and by a mean-field analysis that a first-order transition occurs. Here we show that the existence of this first-order transition can be proven rigorously, by relatively minor adaptations of existing proofs for ordinary Potts models. In our argument we present a random-cluster representation for the model, which might be of independent interest

Metastates in Finite-type Mean-field Models:Visibility, Invisibility, and Random Restoration of Symmetry

We consider a general class of disordered mean-field models where both the spin variables and disorder variables take finitely many values. To investigate the size-dependence in the phase-transition regime we construct the metastate describing the probabilities to find a large system close to a particular convex combination of the pure infinite-volume states. We show that, under a non-degeneracy assumption, only pure states are seen, with non-random probability weights for which we derive explicit expressions in terms of interactions and distributions of the disorder variables. We provide a geometric construction distinguishing invisible states (having zero weights) from visible ones. As a further consequence we show that, in the case where precisely two pure states are available, these must necessarily occur with the same weight, even if the model has no obvious symmetry relating the two.Comment: 30 pages, 2 figure

Self-switching random walks on Erd\"os-R\'enyi random graphs feel the phase transition

We study random walks on Erd\"os-R\'enyi random graphs in which, every time the random walk returns to the starting point, first an edge probability is independently sampled according to a priori measure $\mu$, and then an Erd\"os-R\'enyi random graph is sampled according to that edge probability. When the edge probability $p$ does not depend on the size of the graph $n$ (dense case), we show that the proportion of time the random walk spends on different values of $p$ -- {\it occupation measure} -- converges to the a priori measure $\mu$ as $n$ goes to infinity. More interestingly, when $p=\lambda/n$ (sparse case), we show that the occupation measure converges to a limiting measure with a density that is a function of the survival probability of a Poisson branching process. This limiting measure is supported on the supercritial values for the Erd\"os-R\'enyi random graphs, showing that self-witching random walks can detect the phase transition

Transient and slim versus recurrent and fat: Random walks and the trees they grow

International audienceNetwork growth models that embody principles such as preferential attachment and local attachment rules have received much attention over the last decade. Among various approaches, random walks have been leveraged to capture such principles. In this paper we consider the No Restart Random Walk (NRRW) model where a walker builds its graph (tree) while moving around. In particular, the walker takes s steps (a parameter) on the current graph. A new node with degree one is added to the graph and connected to the node currently occupied by the walker. The walker then resumes, taking another s steps, and the process repeats. We analyze this process from the perspective of the walker and the network, showing a fundamental dichotomy between transience and recurrence for the walker as well as power law and exponential degree distribution for the network. More precisely, we prove the following results: s = 1: the random walk is transient and the degree of every node is bounded from above by a geometric distribution. s even: the random walk is recurrent and the degree of non-leaf nodes is bounded from below by a power law distribution with exponent decreasing in s. We also provide a lower bound for the fraction of leaves in the graph, and for s = 2 our bound implies that the fraction of leaves goes to one as the graph size goes to infinity. NRRW exhibits an interesting mutual dependency between graph building and random walking that is fundamentally influenced by the parity of s. Understanding this kind of coupled dynamics is an important step towards modeling more realistic network growth processes