Phase transitions on Markovian bipartite graphs - an application of the zero-range process

TL;DR

This paper investigates phase transitions and the emergence of giant components in a Markovian bipartite graph model where one set's edges evolve via a zero-range process, combining rigorous analysis and simulations.

Contribution

It provides a rigorous analysis of phase diagrams and critical behavior in a bipartite graph model with zero-range dynamics, including the Evans interaction case.

Findings

Identification of phase diagram types below the zero-range condensation point

Determination of critical exponents for giant component growth

Support for Landau theory applicability through simulations

Abstract

We analyze the existence and the size of the giant component in the stationary state of a Markovian model for bipartite multigraphs, in which the movement of the edge ends on one set of vertices of the bipartite graph is a zero-range process, the degrees being static on the other set. The analysis is based on approximations by independent variables and on the results of Molloy and Reed for graphs with prescribed degree sequences. The possible types of phase diagrams are identified by studying the behavior below the zero-range condensation point. As a specific example, we consider the so-called Evans interaction. In particular, we examine the values of a critical exponent, describing the growth of the giant component as the value of the dilution parameter controlling the connectivity is increased above the critical threshold. Rigorous analysis spans a large portion of the parameter space of the model exactly at the point of zero-range condensation. These results, supplemented with conjectures supported by Monte Carlo simulations, suggest that the phenomenological Landau theory for percolation on graphs is not broken by the fluctuations.