Eur. Phys. J. B 35, 377-389 (2003)
https://doi.org/10.1140/epjb/e2003-00290-4

Asymmetric evolving random networks

Service de Physique Théorique de Saclay, Laboratoire de la Direction des Sciences de la Matière du Commisariat à l'Énergie Atomique, URA2306 du CNRS, CEA Saclay, 91191 Gif-sur-Yvette, France

Corresponding authors: ^a coulomb@spht.saclay.cea.fr - ^b bauer@spht.saclay.cea.fr

Received: 20 December 2003
Published online: 15 October 2003

Abstract

We generalize the Poissonian evolving random graph model of M. Bauer and D. Bernard (2003), to deal with arbitrary degree distributions. The motivation comes from biological networks, which are well-known to exhibit non Poissonian degree distributions. A node is added at each time step and is connected to the rest of the graph by oriented edges emerging from older nodes. This leads to a statistical asymmetry between incoming and outgoing edges. The law for the number of new edges at each time step is fixed but arbitrary. Thermodynamical behavior is expected when this law has a large time limit. Although (by construction) the incoming degree distributions depend on this law, this is not the case for most qualitative features concerning the size distribution of connected components, as long as the law has a finite variance. As the variance grows above 1/4, the average being <1/2, a giant component emerges, which connects a finite fraction of the vertices. Below this threshold, the distribution of component sizes decreases algebraically with a continuously varying exponent. The transition is of infinite order, in sharp contrast with the case of static graphs. The local-in-time profiles for the components of finite size allow to give a refined description of the system.