Statistical mechanics of scale-free networks at a critical point: Complexity without irreversibility?

TL;DR

This paper extends classical statistical mechanics to networks, revealing a critical temperature at which scale-free, hierarchical networks emerge in equilibrium, challenging the notion that complexity requires irreversibility.

Contribution

It introduces a microscopic Hamiltonian for networks based on node utility, demonstrating equilibrium formation of scale-free networks at a critical point.

Findings

Existence of a critical temperature $T_c$ inducing topological transition.

Emergence of scale-free networks with hierarchical topology at $T_c$.

Networks form in equilibrium, fulfilling detailed balance, without irreversibility.

Abstract

Based on a rigorous extension of classical statistical mechanics to networks, we study a specific microscopic network Hamiltonian. The form of this Hamiltonian is derived from the assumption that individual nodes increase/decrease their utility by linking to nodes with a higher/lower degree than their own. We interpret utility as an equivalent to energy in physical systems and discuss the temperature dependence of the emerging networks. We observe the existence of a critical temperature $T_c$ where total energy (utility) and network-architecture undergo radical changes. Along this topological transition we obtain scale-free networks with complex hierarchical topology. In contrast to models for scale-free networks introduced so far, the scale-free nature emerges within equilibrium, with a clearly defined microcanonical ensemble and the principle of detailed balance strictly fulfilled. This provides clear evidence that 'complex' networks may arise without irreversibility. The results presented here should find a wide variety of applications in socio-economic statistical systems.