In this paper, we establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold @math for a Galton-Watson tree is strictly positive if and only if its offspring distribution @math has an exponential tail, i.e., @math , settling a conjecture by Huang and Durrett hd18 . On the random graph with degree distribution @math , we show that if @math has an exponential tail, then for small enough @math the contact process with the all-infected initial condition survives for @math -time w.h.p. (short survival), while for large enough @math it runs over @math -time w.h.p. (long survival). When @math is subexponential, we prove that the contact process w.h.p. displays long survival for any fixed @math .
Moreover in @cite_23 , a cutoff phenomenon" of the fraction of infected vertices was established. In @cite_4 , the same result as above is proven for @math with bounded @math . For unbounded degree distribution @math , the only known result was due to Chatterjee and Durrett @cite_12 , where they showed that if @math obeys a power law, then the contact process always displays long survival for any @math , though their survival time was slightly weaker than exponential ( @math for any @math ). Our Theorems and generalize the aforementioned results to any general @math and to an exponentially long survival.
@cite_4 @cite_12 @cite_23
2170926923 2898008966 2168137408 2170002160
We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett (2009), who showed that for arbitrarily small infection parameter @math , the survival time of the process is larger than a stretched exponential function of the number of vertices, @math . We obtain sharp bounds for the typical density of infected sites in the graph, as @math is kept fixed and @math tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law. The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival @math and (ii) when it is geometric( @math ) we have @math , where the @math are much smaller than previous estimates. We also study the critical value @math for "prolonged persistence" on graphs with @math vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known @math we give estimates on the rate of convergence. Physicists tell us that @math where @math is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct. We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate @math ( @math ) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a power-law with exponent @math , then, as the number of vertices tends to infinity and with high probability, one of the infection types will occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of @math and @math . The picture is similar with multiple starting points for the infections. In this paper we consider a fundamental problem in the area of viral marketing, called T ARGET S ET S ELECTION problem. In a a viral marketing setting, social networks are modeled by graphs with potential customers of a new product as vertices and friend relationships as edges, where each vertex @math is assigned a threshold value @math . The thresholds represent the different latent tendencies of customers (vertices) to buy the new product when their friend (neighbors) do. Consider a repetitive process on social network @math where each vertex @math is associated with two states, active and inactive, which indicate whether @math is persuaded into buying the new product. Suppose we are given a target set @math . Initially, all vertices in @math are inactive. At time step 0, we choose all vertices in @math to become active. Then, at every time step @math , all vertices that were active in time step @math remain active, and we activate any vertex @math if at least @math of its neighbors were active at time step @math . The activation process terminates when no more vertices can get activated. We are interested in the following optimization problem, called T ARGET S ET S ELECTION : Finding a target set @math of smallest possible size that activates all vertices of @math . There is an important and well-studied threshold called strict majority threshold, where for every vertex @math in @math we have @math and @math is the degree of @math in @math . In this paper, we consider the T ARGET S ET S ELECTION problem under strict majority thresholds and focus on three popular regular network structures: cycle permutation graphs, generalized Petersen graphs and torus cordalis.