Abstract
Bootstrap percolation is a process that is used to describe the spread of an infection on a given graph. In the model considered here each vertex is equipped with an individual threshold. As soon as the number of infected neighbors exceeds that threshold, the vertex gets infected as well and remains so forever. We perform a thorough analysis of bootstrap percolation on a novel model of directed and inhomogeneous random graphs, where the distribution of the edges is specied by assigning two distinct weights to each vertex, describing the tendency of it to receive edges from or to send edges to other vertices. Under the mild assumption that the limiting degree distribution of the graph is integrable we determine the typical fraction of infected vertices. Our model allows us to study a variety of settings, in particular the prominent case in which the degree distribution has an unbounded variance. As a second main contribution, we quantify the notion of "systemic risk", that is, we characterize to what extent tiny initial infections can propagate to large parts of the graph through a cascade, and discover novel features that make graphs prone/resilient to initially small infections.
Highlights
In this paper we study bootstrap percolation, which is a classical mathematical model that is used to describe how a certain activity disperses on a given finite graph
Many important properties of it were studied in a broad variety of different settings, including for example the case where the underlying graph is the d-dimensional finite grid [n]d, see [22, 23, 8] and [7], the extensive study for ErdosRenyi random graphs [25] and random regular graphs [10], and the cases of tori [6] and infinite trees [17, 9]
We denote the resulting random graph by Gn(w−(n), w+(n)) and we abbreviate it with Gn(w−, w+)
Summary
In this paper we study bootstrap percolation, which is a classical mathematical model that is used to describe how a certain activity disperses on a given finite graph. The graphs in the two previous examples, as well as many others that appear in a variety of similar contexts, have three relevant characteristics They are heterogeneous, in the sense that the degree distribution (i.e., the probability that a uniformly chosen random vertex has a given number of neighbors) is far from uniform – it typically has a heavy tail. The model that we study encompasses these characteristics It contains two basic ingredients: a model for random directed graphs, where the degree distribution regarding both incoming and outgoing edges can be prescribed, and a model for bootstrap percothe electronic journal of combinatorics 26(3) (2019), #P3.12 lation, where each vertex has its own individual infection threshold.
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