Abstract The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random graphs with an arbitrary degree distribution in the high-connectivity limit, from which we perform a thorough analysis of the impact of degree fluctuations on the spectral density, the inverse participation ratio, and the distribution of the local density of states (LDOSs). For random graphs with a negative binomial degree distribution, we show that all eigenvectors are extended and that the spectral density exhibits a logarithmic or a power-law divergence when the variance of the degree distribution is large enough. We elucidate this singular behaviour by showing that the distribution of the LDOSs at the centre of the spectrum displays a power-law tail controlled by the variance of the degree distribution. In the regime of weak degree fluctuations the spectral density has a finite support, which promotes the stability of large complex systems on random graphs. 

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

The Ising model on networks plays a fundamental role as a testing ground for understanding cooperative phenomena in complex systems. Here we solve the synchronous dynamics of the Ising model on random graphs with an arbitrary degree distribution in the high-connectivity limit. Depending on the distribution of the threshold noise that governs the microscopic dynamics, the model evolves to nonequilibrium stationary states. We obtain an exact dynamical equation for the distribution of local magnetizations, from which we find the critical line that separates the paramagnetic from the ferromagnetic phase. For random graphs with a negative binomial degree distribution, we demonstrate that the stationary critical behavior as well as the long-time critical dynamics of the first two moments of the local magnetizations depend on the distribution of the threshold noise. In particular, for an algebraic threshold noise, these critical properties are determined by the power-law tails of the distribution of thresholds. We further show that the relaxation time of the average magnetization inside each phase exhibits the standard mean-field critical scaling. The values of all critical exponents considered here are independent of the variance of the negative binomial degree distribution. Our work highlights the importance of certain details of the microscopic dynamics for the critical behavior of nonequilibrium spin systems. 

Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise

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. 

K White

Papers

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

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