Scale-freeness for networks as a degenerate ground state: A Hamiltonian formulation

TL;DR

This paper models the emergence of scale-free networks using a statistical mechanics approach, deriving a Hamiltonian to explain how such distributions correspond to a degenerate ground state with specific stability properties.

Contribution

It introduces a Hamiltonian formulation for network degree distributions, linking scale-freeness to a degenerate ground state in a statistical mechanical model.

Findings

Scale-free distribution corresponds to a degenerate ground state.

The ground state has small fluctuations but high entropy.

Implications for network stability and evolution are discussed.

Abstract

The origin of scale-free degree distributions in the context of networks is addressed through an analogous non-network model in which the node degree corresponds to the number of balls in a box and the rewiring of links to balls moving between the boxes. A statistical mechanical formulation is presented and the corresponding Hamiltonian is derived. The energy, the entropy, as well as the degree distribution and its fluctuations are investigated at various temperatures. The scale-free distribution is shown to correspond to the degenerate ground state, which has small fluctuations in the degree distribution and yet a large entropy. We suggest an implication of our results from the viewpoint of the stability in evolution of networks.