Asymmetric evolving random networks

TL;DR

This paper introduces a generalized evolving random graph model with arbitrary degree distributions, revealing a phase transition in component sizes and a giant component emergence, with implications for biological networks.

Contribution

It extends existing models to handle non poissonian degree distributions and analyzes the phase transition behavior in evolving networks.

Findings

Giant component emerges when variance exceeds 1/4.

Component size distribution follows a power-law below the transition.

Transition is of infinite order, unlike static graph models.

Abstract

We generalize the poissonian evolving random graph model of Bauer and Bernard to deal with arbitrary degree distributions. The motivation comes from biological networks, which are well-known to exhibit non poissonian degree distribution. 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.