The Ising model plays a fundamental role in the investigation of collective phenomena in complex systems and, due to its simplicity, it stands out as a testing ground for the study of critical behavior, both dynamical and stationary. In this work [1], we analytically solve the synchronous dynamics of the ferromagnetic Ising model on an ensemble of complex networks with an arbitrary distribution for the number of connections per node, or degree. Depending on the form of the threshold noise, that mimics the effect of a thermal bath, the system evolves to non equilibrium stationary states, so we generalize our results by considering arbitrary noise distributions.

By means of the so-called Glauber synchronous dynamics [2], where spins are updated simultaneously, we use a mean field approximation to obtain the pair of equations that describe the evolution of the global magnetization. We also obtain the equations for the evolution of the distribution of local magnetizations, fully characterizing both micro and macroscopic dynamics of the system. In the high connectivity limit, where the mean degree is very large, our theory retains the effect of the topological fluctuations, being closer to the description of real complex network phenomena than the usual fully connected mean field theories.

We focus on the characterization of the critical behavior of the ferromagnetic Ising model in the case where the degrees follow a negative binomial distribution. With this choice, we are able to parametrize the heterogeneity of the network in terms of a single parameter. Starting by the critical temperature, at which the system undergoes a continuous transition from a ferromagnetic to a paramagnetic phase, we show that its value is determined by the second moment of the degree distribution and by the behavior of the threshold noise distribution linearized about zero. In the homogeneous limit, where the second moment of the degree distribution vanishes, fluctuations are weak and we recover the phase diagram of the fully connected ferromagnetic Ising model [3]. By means of an analysis of the free energy, this equivalence between the homogeneous limit of our model and the fully connected case was later formally proven [4].

Regarding the critical exponents, we show that the degree fluctuations present in highly connected heterogeneous networks do not break the universality class of the homogeneous ferromagnetic Ising model, whose critical exponents have well known values [3,5]. However, we show that the particular choice for the threshold noise distribution affects the value of the critical exponents, breaking such universality class whenever the stationary microstate distribution of the system, which is directly obtained from the noise distribution, is not a Boltzmann form.

Overall, our work introduces a family of Ising models on random graphs that retain the effect of both topological structure and threshold noise distribution, whose non-equilibrium dynamics can be solved exactly. Our results provide insights on the effects of network structure and stochastic fluctuations on critical phenomena, highlighting their importance on the critical behavior of the ferromagnetic Ising model.

References:

[1] L. S. Ferreira and F. L. Metz. “Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise”. In: Physical Review E 107.3, 2023.

[2] A.C.C.Coolen. “Statistical Mechanics of Recurrent Neural Networks I. Statics”. arXiv preprint, url: https://arxiv.org/abs/cond mat/0006010, 2000.

[3] M. Kochmánski, T. Paszkiewicz and S. Wolski. “Curie–Weiss magnet — a simple model of phase transition”. In: European Journal of Physics 34.6, 2013.

[4] M. Okuyama and M. Ohzeki. “Free energy equivalence between mean-field models and nonsparsely diluted mean-field models”. arXiv preprint, url: https://arxiv.org/pdf/2406.13245, 2024.

[5] G. Ódor. “Universality classes in nonequilibrium lattice systems”. In: Reviews of Modern Physics 76.3, 2004.