Entropy of the ensemble of acyclic graphs

Abstract

The structural phase transition in complex network models is known to yield knowledge relevant for several problems of practical application. Examples include resilience of artificial networks, dynamics of epidemic spreading, among others. The model of random graphs is probably the simplest model of complex networks, and the solution of this model falls into the universality class mean field percolation. In this work, we concentrate on a similar problem, namely, the structural phase transition in the ensemble of random acyclic graphs. It should be noted that several approaches to the problem of random graphs, such as generating functions or the Molloy–Reed criterion, rely on the fact that before the critical point, cycles should not be present in random graphs. In this way, up to the critical point our solution should produce results that are equivalent to these other methods. Our approach takes advantage of the fact that acyclic graphs allow for an exact combinatorial enumeration of the whole ensemble, what leads to an exact expression for the entropy of this system. With this definition of entropy we can determine the onset of the critical transition as well as the critical exponents associated with the transition. Our results are illustrated with Monte-Carlo results and are discussed within the context of general random graphs, as well as in comparison with another model of acyclic graphs.