{'fraction_non_alphanumeric': 0.04295942720763723, 'fraction_numerical': 0.005727923627684964, 'mean_word_length': 4.458333333333333, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Viral Processes by Random Walks on Random Graphs

We study viral processes on random r-regular graphs. The graph G on n vertices represents the environment and a virus is carried by k ≤ nǫ (ǫ a sufficiently small constant) particles, which make independent simple random walks on the graph. Particles can be in one of three states: susceptible (S), infected (I), recovered (R). An infected particle can infect a susceptible particle, which remains infected for a fixed infectious period ξ before recovering permanently. This is known as the SIR epidemic model. When ξ = ∞ the SI model particles never go from I to R. There are two questions dealt with in this paper. (1) When ξ < ∞, how many particles ever get infected? (2) How long does the process take to complete? We address both of these questions by reducing the dynamics of the process to an Erdos-Renyi (E-R) random graph Gk,q on the set of particles, where the edge probability q is a function of the parameters of the model. Infected particles are connected components in Gk,q, and so well known results from the literature on E-R random graphs can be directly applied using our reduction to answer question (1). In particular, we show how the parameters of the model produce two phase transitions: In the subcritical regime, O(ln k) particles are infected. In the supercritical regime, for a constant C determined by the parameters of the model, Ck get infected with probability C, and O(ln k) get infected with probability (1−C). Finally, there is a regime in which all k particles are infected. Statements are with high probability (whp), that is, with probabability tending to 1 as n → ∞. Furthermore, the graph reduction assigns weights on the edges that give information about when a particle becomes infected. This information can be used for addressing question (2). As an example, in the case of ξ = ∞, we apply a result of Janson [15] on randomly-weighted graphs to deduce that completion time converges in probability to 2θrn k ln k, where θr = r−1 r−2 . This matches the expectation determined in [7] for the same example.