If we consider the contact process with infection rate λ on a random graph on n vertices with power law degree distributions, mean field calculations suggest that the critical value λc of the infection rate is positive if the power α > 3. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399-1404. Here, we show that the critical value λc is zero for any value of α > 3, and the contact process starting from all vertices infected, with a probability tending to 1 as n → ∞, maintains a positive density of infected sites for time at least exp(n 1−δ ) for any δ > 0. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability ρ(λ). It is expected that ρ(λ) ∼ Cλ β as λ → 0. Here we show that α − 1 ≤ β ≤ 2α − 3, and so β > 2 for α > 3. Thus even though the graph is locally tree-like, β does not take the mean field critical value β = 1.

We consider a model of long-range first-passage percolation on the ddimensional square lattice Z d in which any two distinct vertices x; y 2 Z d are connected by an edge having exponentially distributed passage time with mean kx yk˛C o.1/ , where˛> 0 is a fixed parameter and k k is the`1-norm on Z d . We analyze the asymptotic growth rate of the set B t , which consists of all x 2 Z d such that the first-passage time between the origin 0 and x is at most t as t ! 1. We show that depending on the values of˛there are four growth regimes: (i) instantaneous growth for˛< d , (ii) stretched exponential growth for˛2 .d; 2d /, (iii) superlinear growth for˛2 .2d; 2d C 1/, and finally (iv) linear growth for˛> 2d C 1 like the nearest-neighbor first-passage percolation model corresponding to˛D 1. LONG-RANGE FIRST-PASSAGE PERCOLATION207 used certain versions of LRFPP for modeling biological invasion of species [17,24,42,51]. Along with many other factors they use dispersal kernels r. / with heavy tails as part of their models for dispersal mechanisms of biological objects (such as seeds, pollen, fungi, etc.). However, most of their conclusions are based on simulations in two-dimensional grid and nonrigorous heuristics. In [17], followed by [42], the authors recognized two phases of spatiotemporal behavior, which they call long-distance dispersal and short-distance dispersal, based on whether the second moment of the dispersal kernel is infinite or finite. They argue that under finite second-moment conditions (short-distance dispersal regime) the growth behavior of the region reachable within time t is same as that in nearest-neighbor (or finiterange) FPP. On the other hand, the authors in [24] recognized one additional phase, which they call medium-distance dispersal, but they didn't specify where the transitions between different phases occur. As we will prove here, the situation is much more delicate, and there are at least four distinct phases (with three critical points in between) depending on the heavy tail index of the dispersal kernel.Aldous [2] considered communication of continuously arriving information through a finite agent network in a certain game-theoretic setup. In one of the cases, where the network topology is a two-dimensional discrete torus and the communication cost between any two agents is a nondecreasing function of the euclidean distance between them, the main technical tool to understand the time evolution of the fraction of informed agents is the analysis of the LRFPP model, which we propose here, on a large two-dimensional discrete torus. Aldous proposed a simplified version of this LRFPP model, which he named short-long FPP, in which agent network topology is a discrete torus, each pair of nearest-neighbor agents communicate at rate 1, and all other pairs of agents communicate at a rate that depends only on the size of the torus regardless of the distance between the agents. The continuous analogue of the short-long FPP model has been analyzed rigorously on a (two-dimensional) real torus [18] and...

We introduce a new kind of percolation on finite graphs called jigsaw percolation. This model attempts to capture networks of people who innovate by merging ideas and who solve problems by piecing together solutions. Each person in a social network has a unique piece of a jigsaw puzzle. Acquainted people with compatible puzzle pieces merge their puzzle pieces. More generally, groups of people with merged puzzle pieces merge if the groups know one another and have a pair of compatible puzzle pieces. The social network solves the puzzle if it eventually merges all the puzzle pieces. For an Erdős-Rényi social network with n vertices and edge probability pn, we define the critical value pc(n) for a connected puzzle graph to be the pn for which the chance of solving the puzzle equals 1/2. We prove that for the n-cycle (ring) puzzle, pc(n) = Θ(1/ log n), and for an arbitrary connected puzzle graph with bounded maximum degree, pc(n) = O(1/ log n) and ω(1/n b ) for any b > 0. Surprisingly, with probability tending to 1 as the network size increases to infinity, social networks with a power-law degree distribution cannot solve any bounded-degree puzzle. This model suggests a mechanism for recent empirical claims that innovation increases with social density, and it might begin to show what social networks stifle creativity and what networks collectively innovate.

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