Network complexity and topological phase transitions

TL;DR

This paper explores how the topology of complex fractal networks induces unique collective excitations and phase transitions, with detailed analysis of an Ising model revealing robust spin correlations due to network structure.

Contribution

It introduces a novel type of collective excitation driven by network topology and analytically demonstrates phase transitions in Ising systems on fractal networks.

Findings

Topological excitations cause phase transitions in complex networks.

Analytic computation of magnetization and correlations on fractal networks.

Robust spin correlations are maintained despite thermal fluctuations.

Abstract

A new type of collective excitations, due exclusively to the topology of a complex random network that can be characterized by a fractal dimension $D_F$, is investigated. We show analytically that these excitations generate phase transitions due to the non-periodic topology of the $D_F>1$ complex network. An Ising system, with long range interactions over such a network, is studied in detail to support the claim. The analytic treatment is possible because the evaluation of the partition function can be decomposed into closed factor loops, in spite of the architectural complexity. This way we compute the magnetization distribution, magnetization loops, and the two point correlation function; and relate them to the network topology. In summary, the removal of the infrared divergences leads to an unconventional phase transition, where spin correlations are robust against thermal fluctuations.