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Register a new userEpidemics on random graphs with tunable clustering
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In this paper, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if they share at least one group.
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We study the component structure in random intersection graphs with tunable clustering, and show that the average degree works as a threshold for a phase transition for the size of the largest component. That is, if the expected degree is less than one, the size of the largest component is a.a.s. of logarithmic order, but if the average degree is greater than one, a.a.s. a single large component of linear order emerges, and the size of the second largest component is at most of logarithmic order.
The evoSIR model is a modification of the usual SIR process on a graph $G$ in which $S-I$ connections are broken at rate $rho$ and the $S$ connects to a randomly chosen vertex. The evoSI model is the same as evoSIR but recovery is impossible. In an undergraduate project at Duke the critical value for evoSIR was computed and simulations showed that when $G$ is an ErdH os-Renyi graph with mean degree 5, the system has a discontinuous phase transition, i.e., as the infection rate $lambda$ decreases to $lambda_c$, the fraction of individuals infected during the epidemic does not converge to 0. In this paper we study evoSI dynamics on graphs generated by the configuration model. We show that there is a quantity $Delta$ determined by the first three moments of the degree distribution, so that the phase transition is discontinuous if $Delta>0$ and continuous if $Delta<0$.
We consider a large class of random geometric graphs constructed from samples $mathcal{X}_n = {X_1,X_2,ldots,X_n}$ of independent, identically distributed observations of an underlying probability measure $ u$ on a bounded domain $Dsubset mathbb{R}^d$. The popular `modularity clustering method specifies a partition $mathcal{U}_n$ of the set $mathcal{X}_n$ as the solution of an optimization problem. In this paper, under conditions on $ u$ and $D$, we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions $mathcal{U}_n$ converge in a certain sense to a continuum partition $mathcal{U}$ of the underlying domain $D$, characterized as the solution of a type of Kelvins shape optimization problem.
In this paper, we are concerned with the stochastic susceptible-infectious-susceptible (SIS) epidemic model on the complete graph with $n$ vertices. This model has two parameters, which are the infection rate and the recovery rate. By utilizing the theory of density-dependent Markov chains, we give consistent estimations of the above two parameters as $n$ grows to infinity according to the sample path of the model in a finite time interval. Furthermore, we establish the central limit theorem (CLT) and the moderate deviation principle (MDP) of our estimations. As an application of our CLT, reject regions of hypothesis testings of two parameters are given. As an application of our MDP, confidence intervals with lengths converging to $0$ while confidence levels converging to $1$ are given as $n$ grows to infinity.
Consider a collection of random variables attached to the vertices of a graph. The reconstruction problem requires to estimate one of them given `far away observations. Several theoretical results (and simple algorithms) are available when their joint probability distribution is Markov with respect to a tree. In this paper we consider the case of sequences of random graphs that converge locally to trees. In particular, we develop a sufficient condition for the tree and graph reconstruction problem to coincide. We apply such condition to colorings of random graphs. Further, we characterize the behavior of Ising models on such graphs, both with attractive and random interactions (respectively, `ferromagnetic and `spin glass).
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