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A bootstrap percolation process on a graph with infection threshold $r\ge 1$is a dissemination process that evolves in time steps. The process begins witha subset of infected vertices and in each subsequent step every uninfectedvertex that has at least $r$ infected neighbours becomes infected and remainsso forever.  Critical phenomena in bootstrap percolation processes were originallyobserved by Aizenman and Lebowitz in the late 1980s as finite-volume phasetransitions in $\mathbb{Z}^d$ that are caused by the accumulation of smalllocal islands of infected vertices. They were also observed in the case ofdense (homogeneous) random graphs by Janson, \L uczak, Turova and Valier(2012). In this paper, we consider the class of inhomogeneous random graphsknown as the Chung-Lu model: each vertex is equipped with a positive weight andeach pair of vertices appears as an edge with probability proportional to theproduct of the weights. In particular, we focus on the sparse regime, where thenumber of edges is proportional to the number of vertices.  The main results of this paper determine those weight sequences for which acritical phenomenon occurs: there is a critical density of vertices that areinfected at the beginning of the process, above which a small (sublinear) setof infected vertices creates an avalanche of infections that in turn leads toan outbreak. We show that this occurs essentially only when the tail of theweight distribution dominates a power law with exponent 3 and we determine thecritical density in this case.

A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs