Phase Transitions in Finite Random Networks

Abstract

We analyze ensembles of random networks with fixed numbers of edges, triangles, and nodes. In the limit as the number of nodes goes to infinity, this model is known to exhibit sharp phase transitions as the density of edges and triangles is varied. In this paper we study finite size effects in ensembles where the number of nodes is between 30 and 66. Despite the small number of nodes, the phases and phase transitions are clearly visible. Concentrating on 54 nodes and one particular phase transition, we use spectral data to examine the structure of a typical graph in each phase, and see that is it very similar to the graphon that describes the system as n diverges. We further investigate how graphs change their structure from that of one phase to the other under a natural edge flip dynamics in which the number of triangles is either forced downwards or allowed to drift upwards. In each direction, spectral data shows that the change occurs in three stages: a very fast stage of about 100 edge flips, in which the triangle count reaches the targeted value, a slower second stage of several thousand edge flips in which the graph adopts the qualitative features of the new phase, and a very slow third stage, requiring hundreds of thousands of edge flips, in which the fine details of the structure equilibrate.