Projects per year
Abstract
A bootstrap percolation process on a graph with infection threshold r >0 is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex that has at least r infected neighbours becomes infected and remains so forever.
Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finitevolume phase transitions in \mathbb{Z}^d that are caused by the accumulation of small local islands of infected vertices. They were also observed in the case of dense (homogeneous) random graphs by Janson, \L uczak, Turova and Vallier (2012). In this paper, we consider the class of inhomogeneous random graphs known as the ChungLu model: each vertex is equipped with a positive weight and each pair of vertices appears as an edge with probability proportional to the product of the weights. In particular, we focus on the sparse regime, where the number of edges is proportional to the number of vertices.
The main results of this paper determine those weight sequences for which a critical phenomenon occurs: there is a critical density of vertices that are infected at the beginning of the process, above which a small (sublinear) set of infected vertices creates an avalanche of infections that in turn leads to an outbreak. We show that this occurs essentially only when the tail of the weight distribution dominates a power law with exponent 3 and we determine
the critical density in this case.
Original language  English 

Pages (fromto)  9901051 
Number of pages  47 
Journal  Annals of Applied Probability 
Volume  28 
Issue number  2 
Publication status  Published  11 Apr 2018 
Keywords
 bootstrap percolation
 inhomogeneous random graphs
 critical phenomena
Fingerprint
Dive into the research topics of 'A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs'. Together they form a unique fingerprint.Projects
 1 Finished

Inhomogeneity and generalised bootstrap percolation in stochastic networks
Engineering & Physical Science Research Council
1/09/13 → 31/08/15
Project: Research Councils