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.

arXiv comment20 pages, 7 figures

References in corpus (55)

• Statistical physics of social dynamics
• Random graphs with arbitrary degree distributions and their applications
• Synchronization in complex networks
• Critical phenomena in complex networks
• The Kuramoto model in complex networks
• Emergence of Cooperation and Organization in an Evolutionary Game
• The Bethe lattice spin glass revisited
• Random graphs with clustering
• Ising Model on Networks with an Arbitrary Distribution of Connections
• The onset of synchronization in large networks of coupled oscillators
• Synchronization of Kuramoto Oscillators in Scale-Free Networks
• Ferromagnetic ordering in graphs with arbitrary degree distribution
• Frequency synchronization in random oscillator network
• Crises and collective socio-economic phenomena: simple models and challenges
• Conservation laws for the voter model in complex networks
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Optimisation problems and replica symmetry breaking in finite connectivity spin-glasses
• Edge-Based Compartmental Modeling for Infectious Disease Spread Part III: Disease and Population Structure
• Continuous versus Discontinuous Transitions in the $D$-Dimensional Generalized Kuramoto Model: Odd $D$ is Different
• Dynamics of random recurrent networks with correlated low-rank structure
• Emergence of polarized ideological opinions in multidimensional topic spaces
• First- and second-order phase transitions in scale-free networks
• Curie-Weiss magnet: a simple model of phase transition
• The weighted random graph model
• Finite Connectivity Attractor Neural Networks
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Belief propagation for networks with loops
• Approximate solution to the stochastic Kuramoto model
• The Cavity Approach to Parallel Dynamics of Ising Spins on a Graph
• Ising model on two connected Barabasi-Albert networks
• Onset of Synchronization in Complex Networks of Noisy Oscillators
• Finitely connected vector spin systems with random matrix interactions
• Data-driven contact structures: from homogeneous mixing to multilayer networks
• Analytic solution of attractor neural networks on scale-free graphs
• Spectra of sparse regular graphs with loops
• Statistical Mechanics of Recurrent Neural Networks II. Dynamics
• Parallel dynamics of disordered Ising spin systems on finitely connected directed random graphs with arbitrary degree distributions
• Cavity approach for real variables on diluted graphs and application to synchronization in small-world lattices
• Linear stability analysis for large dynamical systems on directed random graphs
• Near optimal configurations in mean field disordered systems
• Statistical physics of nonlinear wave interaction
• The phase diagram of Lévy spin glasses
• Fluctuation spectra of large random dynamical systems reveal hidden structure in ecological networks
• Self-starting nonlinear mode-locking in random lasers
• On the onset of synchronization of Kuramoto oscillators in scale-free networks
• The spectral density of dense random networks and the breakdown of the Wigner semicircle law
• The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism
• Nonlinear $XY$ and $p$-clock models on sparse random graphs: mode-locking transition of localized waves
• Condensation and extremes for a fluctuating number of independent random variables
• Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass
• Comparison of Gabay-Toulouse and de Almeida-Thouless instabilities for the spin glass XY model in a field on sparse random graphs
• The full replica symmetry breaking in the Ising spin glass on random regular graph
• Ising model with invisible states on scale-free networks
• Games in rigged economies
• Feed-forward chains of recurrent attractor neural networks with finite dilution near saturation