Cliques in geometric inhomogeneous random graphs

Cliques in dense inhomogeneous random graphs

Geometric Inhomogeneous Random Graphs

For the theoretical study of real-world networks, we propose a model of scale-free randomgraphs with underlying geometry that we call geometric inhomogeneous random graphs (GIRGs).GIRGs generalize hyperbolic random graphs, which are a popular model to test algorithms forsocial and technological networks. Our generalization overcomes some limitations of hyperbolicrandom graphs, w...

Bootstrap Percolation on Geometric Inhomogeneous Random Graphs

Geometric inhomogeneous random graphs (GIRGs) are a model for scale-free networks with underlying geometry. We study bootstrap percolation on these graphs, which is a process modelling the spread of an infection of vertices starting within a (small) local region. We show that the process exhibits a phase transition in terms of the initial infection rate in this region. We determine the speed of...

Cliques in random graphs

1. Introduction. Let 0 < p < 1 be fixed and denote by G a random graph with point set N, the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables ei5 , 1 < i < j, defined by _ 1 if (i, j) is an edge of G, et '-0 if (i, j) is not an edge of G, are independent r .v.'s with P(e i, = 1) = p, P(eij = 0) = 1-p. De...

Sampling Geometric Inhomogeneous Random Graphs in Linear Time

Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for real-world networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models...