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

The structure of a complex network influences any dynamical processes occurring within it. In particular, strong heterogeneities in network structure often underlie nontrivial dynamical properties of networked systems. Crucial to the study of such systems are minimal spin models describing networks of coupled state variables, either scalars or continuous vectors. Such models help clarify the influence of heterogeneous structures on the cooperative behaviour of many interacting degrees of freedom.
In studying cooperative behaviour, mean-field theories work on the simplifying assumption of the statistical equivalence of all spins, each spin experiencing an effective random field drawn from the same distribution. This reduces the original problem of many interacting elements to one a single spin coupled to an effective field. It’s generally expected that such mean-field theories provide a universal description of spin models on high-connectivity networks, for which the mean degree is infinitely large. Intuitively, it would appear that spin models on networks gradually evolve toward fully-connected behaviour as the mean degree increases, since the detailed structure of a network becomes increasingly irrelevant in the high-connectivity limit. Whether this intuition is mathematically well-founded, however, has remained as an open question.
In a new pre-print, LML External Fellow Fernando Metz and Thomas Peron now provide a comprehensive solution to this problem. Surprisingly, they demonstrate that the high-connectivity limit of spin models on networks is not universal, but instead depends on the full degree distribution. Hence, traditional, fully-connected mean-field theories cannot generally predict the macroscopic behaviour of spin models on high-connectivity networks. As Metz and Peron show, this non-universal character is intimately related to the breakdown of the central limit theorem as applied to the distribution of effective local fields. This breakdown has been studied for more than seventy years in probability theory, but its importance for network spin models has remained unclear before now.
The central result of the new work is a set of equations describing the equilibrium behaviour of spin models on high-connectivity networks with an arbitrary degree distribution. The spins are continuous vectors with finite dimension, while the random pairwise interactions between spins follow an arbitrary distribution. The analysis reveals that, even in the high-connectivity limit, the behaviour of spin models on heterogeneous graphs is non-universal and depends strongly on the degree distribution. The authors also illustrate the effects of degree fluctuations on the mean-field behaviour of spin models by focusing on two examples – the Kuramoto model of synchronization and the Ising model of spin-glasses – obtaining the complete phase diagrams for both models.
The paper is available at

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