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 [2], who showed that for arbitrarily small infection parameter λ, the survival time of the process is larger than a stretched exponential function of the number of vertices, n. We obtain sharp bounds for the typical density of infected sites in the graph, as λ is kept fixed and n tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.

International audienceWe study the extinction time τ of the contact process started with full occupancy on finite trees of bounded degree. We show that, if the infection rate is larger than the critical rate for the contact process on Z, then, uniformly over all trees of degree bounded by a given number, the expectation of τ grows exponentially with the number of vertices. Additionally, for any increasing sequence of trees of bounded degree, τ divided by its expectation converges in distribution to the unitary exponential distribution. These results also hold if one considers a sequence of graphs having spanning trees with uniformly bounded degree, and provide the basis for powerful coarse-graining arguments. To demonstrate this, we consider the contact process on a random graph with vertex degrees following a power law. Improving a result of Chatterjee and Durrett [CD09], we show that, for any non-zero infection rate, the extinction time for the contact process on this graph grows exponentially with the number of vertices

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Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with small average graph distance between vertices, but adding an edge comes at a cost measured according to the geometry of the ambient physical space. In most cases, we identify the order of magnitude of the average graph distance as a function of the parameters of the model. As the proofs reveal, hierarchical structures naturally emerge from our simple modeling assumptions. Moreover, a critical regime exhibits an infinite number of discontinuous phase transitions. MSC 2010: 82C22; 05C80

We consider a symmetric, finite-range contact process with two types of infection; both have the same (supercritical) infection rate and heal at rate 1, but sites infected by Infection 1 are immune to Infection 2. We take the initial configuration where sites in $(-\infty,0]$ have Infection 1 and sites in $[1,\infty)$ have Infection 2, then consider the process $\rho_t$ defined as the size of the interface area between the two infections at time $t$. We show that the distribution of $\rho_t$ is tight, thus proving a conjecture posed by Cox and Durrett in [Bernoulli 1 (1995) 343--370].Comment: Published in at http://dx.doi.org/10.3150/09-BEJ236 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm