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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 λ c 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… (More)

- Shirshendu Chatterjee, Rick Durrett
- Random Struct. Algorithms
- 2011

We consider a model for gene regulatory networks that is a modification of Kauff-mann's (1969) random Boolean networks. There are three parameters: n = the number of nodes, r = the number of inputs to each node, and p = the expected fraction of 1's in the Boolean functions at each site. Following a standard practice in the physics literature , we use a… (More)

Aldous (2007) defined a gossip process in which space is a discrete N × N torus, and the state of the process at time t is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate N −α to a site chosen at random from the torus. We will be interested in the case in which α < 3,… (More)

We consider the discrete time threshold-θ contact process on a random r-regular graph. We show that if θ ≥ 2, r ≥ θ + 2, 1 is small and p ≥ p 1 (1), then starting from all vertices occupied the fraction of occupied vertices is ≥ 1 − 2 1 up to time exp(γ 1 (r)n) with high probability. We also show that for p 2 < 1 there is an 2 (p 2) > 0 so that if p ≤ p 2… (More)

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… (More)

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… (More)

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