Mean-field theory of vector spin models on networks with arbitrary degree distributions

Abstract

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie-Weiss, the Kuramoto, and the Sherrington-Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington-Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree $c$ scales as a power $c \propto N^{b}$ ($b < 1$) of the total number $N \gg 1$ of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties

“…We point out that this picture is not in conflict with recent results [18,50] that show the existence of localized eigenvectors in the tails of the spectral density of critical random graph models. In fact, our results for the absence of localization hold for c = O(N a ) (a < 1) [17], while in critical random graphs the mean degree scales as c = O(ln N).…”

“…In condensed matter physics, models defined on random graphs represent mean-field versions of finite-dimensional lattices which mimic the effects of finite coordination number. The spin-glass transition and Anderson localization have been intensively investigated on random graph structures over the past years [14,15], and they continue to attract a lot of interest [16][17][18][19][20].…”

Volcano transition in a system of generalized Kuramoto oscillators with random frustrated interactions

“…We also confirmed this conjecture through numerical simulations. Note that another study on higher-dimensional phase oscillators with random frustrated interactions was recently reported, where the authors considered an equilibrium in the mean-field equation for vector spin models on random networks with high connectivity, featuring an arbitrary degree distribution and links with random weights [41].…”