Topological phase transitions of random networks

TL;DR

This paper develops a statistical mechanical framework to analyze topological phase transitions in random networks, revealing rich phenomena including scale-free networks at critical points.

Contribution

It introduces a novel approach linking network topology transitions to equilibrium statistical mechanics, enabling detailed analysis of phase changes.

Findings

Identifies various topological phase transitions in network models.

Establishes a connection between network rewiring and lattice gas dynamics.

Shows emergence of scale-free networks at critical points.

Abstract

To provide a phenomenological theory for the various interesting transitions in restructuring networks we employ a statistical mechanical approach with detailed balance satisfied for the transitions between topological states. This enables us to establish an equivalence between the equilibrium rewiring problem we consider and the dynamics of a lattice gas on the edge-dual graph of a fully connected network. By assigning energies to the different network topologies and defining the appropriate order parameters, we find a rich variety of topological phase transitions, defined as singular changes in the essential feature(s) of the global connectivity as a function of a parameter playing the role of the temperature. In the ``critical point'' scale-free networks can be recovered.