Ensemble inequivalence and phase transitions in unlabeled networks

TL;DR

This paper reveals a first-order phase transition in random unlabeled networks with a fixed average number of links, showing ensemble inequivalence and implications for modeling real-world networks.

Contribution

It uncovers a phase transition caused by nonconcavity of entropy in unlabeled networks and discusses ensemble inequivalence, contrasting with labeled network behavior.

Findings

First-order phase transition in unlabeled networks

Ensemble inequivalence below critical point

Absence of percolation transition in unlabeled networks

Abstract

We discover a first-order phase transition in the canonical ensemble of random unlabeled networks with a prescribed average number of links. The transition is caused by the nonconcavity of microcanonical entropy. Above the critical point coinciding with the graph symmetry phase transition, the canonical and microcanonical ensembles are equivalent and have a well-behaved thermodynamic limit. Below the critical point, the ensemble equivalence is broken, and the canonical ensemble is a mixture of phases: empty networks and networks with average degrees diverging logarithmically with the network size. As a consequence, networks with bounded average degrees do not survive in the thermodynamic limit, decaying into the empty phase. The celebrated percolation transition in labeled networks is thus absent in unlabeled networks. In view of these differences between labeled and unlabeled ensembles, the question of which one should be used as a null model of different real-world networks cannot be ignored.