Construction of equilibrium networks with an energy function

TL;DR

This paper introduces a novel energy-based model for constructing equilibrium networks that exhibit phase transitions from random to highly connected structures, analyzed through mean-field theory and simulations.

Contribution

It presents a new topological energy function for equilibrium network construction and analyzes the resulting phase transitions using both theoretical and numerical methods.

Findings

Network undergoes a topological phase transition at low temperatures.

First-order phase transitions occur, with transition temperature decreasing logarithmically with system size.

Discrepancies between mean-field predictions and simulations are discussed.

Abstract

We construct equilibrium networks by introducing an energy function depending on the degree of each node as well as the product of neighboring degrees. With this topological energy function, networks constitute a canonical ensemble, which follows the Boltzmann distribution for given temperature. It is observed that the system undergoes a topological phase transition from a random network to a star or a fully-connected network as the temperature is lowered. Both mean-field analysis and numerical simulations reveal strong first-order phase transitions at temperatures which decrease logarithmically with the system size. Quantitative discrepancies of the simulation results from the mean-field prediction are discussed in view of the strong first-order nature.