Percolation on dense graph sequences.

By Béla Bollobás, Christian Borgs, Jennifer Tour Chayes, Oliver Riordan. 2007

93 citations recorded by the source corpus.

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (G n ). Let λn be the largest eigenvalue of the adjacency matrix of G n , and let G n (p n ) be the random subgraph of G n obtained by keeping each edge independently with probability p n . We show that the appearance of a giant component in G n (p n ) has a sharp threshold at p n = 1/λ n . In fact, we prove much more: if (G n ) converges to an irreducible limit, then the density of the largest component of G n (c/n) tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lovasz, S6s and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollobas, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/526df5305249e839908bd1835e91f6db69c29f69

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Convergent Sequences of Dense Graphs II. Multiway Cuts and Statistical Physics (2012, unknown): “, [2].”
Statistical physics of inference: thresholds and algorithms (2015, background): “A remarkable result concerns any sequence of dense graphs for which it was proven that the threshold is given by the inverse of the leading eigenvalue of the adjacency matrix [25].”
Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges (2009, background): “In the previous section we discussed a random graph model where the typical Laplacian and adjacency matrices had one “special” eigenvalue with multiplicity 1 and n − 1 “trivial” eigenvalues with a special focus the emergence of a giant component [10, 37, 3, 27, 52, 15].”
Network Structure and the Aggregation of Information: Theory and Evidence from Indonesia (2016, unknown): “Further, we find that the first eigenvalue of the link matrix predicts information flow, which echoes the results in the viral transmission model studied by Bollobás et al. (2010).”