Evolution of a modified binomial random graph by agglomeration

Mihyun Kang, Angelica Pachón, Pablo M. Rodriguez

Published 2013-10-11, updated 2014-10-21Version 3

In the classical Erd\"os-R\'enyi random graph G(n, p) there are n vertices and each of the possible edges is independently present with probability p. One of the most known results for this model is the threshold for connectedness, a phenomenon which is tightly related to the nonexistence of isolated vertices. Numerous random graphs inspired in real networks are inhomogeneous in the sense that not all vertices have the same characteristic, which may influence the connection probability between pairs of vertices. The random graph G(n, p) is homogeneous in this respect. Thus, with the aim to study real networks, new random graph models have been introduced and analyzed recently. The purpose of this paper is to contribute to this task by proposing a new inhomogeneous random graph model which is obtained in a constructive way from the classical Erd\"os-R\'enyi model. We consider an initial configuration of subsets of vertices and we call each subset a super-vertex, then the random graph model is defined by letting that two super-vertices be connected if there is at least one edge between them in G(n,p). Our main result concerns to the threshold for connectedness. We also analyze the phase transition for the emergence of the giant component and the degree distribution. We point out that even though our model begin from G(n, p), it assumes the existence of community structure and under certain conditions it exhibits a power law degree distribution, both important properties for real applications.