Graphons are measurable functions used to describe the asymptotic behavior of convergent graph families. Originally motivated by problems in combinatorics and graph theory, graphons have found numerous applications in the modeling and analysis of dynamical processes on networks. In this work, we use graphons to formulate the Ising model on convergent graph sequences, which include many network topologies common in applications. We derive the mean-field limit for the resulting model and obtain exact results for phase transitions in such systems. Specifically, we show that the critical temperatures of the Ising model on graphons are determined by the eigenvalues of the Hilbert-Schmidt operator associated with the graph limit. For many important network topologies, these eigenvalues can be computed explicitly. We illustrate our results with three representative random network models: Erd\H{o}s-R\'{e}nyi, small-world, and power-law. In the small-world case, we demonstrate phase transitions to both ferromagnetic and antiferromagnetic phases, as well as coexistence of local minima of the free energy. The latter gives rise to multistability, as confirmed by Monte Carlo simulations. The results of this work demonstrate that the Ising model on graphons combines the analytical tractability of exactly solvable mean-field models with the ability to accommodate a broad range of network topologies. We expect that the use of graphons in spin models will lead to new insights into the statistical physics of interacting systems on complex networks.
In this paper, we derive quenched scaling limits for linear functionals and the empirical spin field of Ising models on inhomogeneous random graphs generated by a graphon (encompassing both dense and sparse graphs), in the high-temperature regime. We first prove a joint central limit theorem (CLT) for finite collections of linear statistics of the spin configurations, where the limiting covariance is characterized by the resolvent of the associated graphon integral operator. Building on this result, we establish functional CLTs for the average magnetization and for the spin field indexed by suitable classes of regular test functions. We further prove convergence of the full empirical spin field, viewed as a random generalized function in negative Sobolev spaces. These scaling limits provide applications to both Bayesian neural networks and causal inference. Specifically, for the former, we derive infinite-width Gaussian-process limits for two-layer Bayesian neural networks with Ising-dependent output-layer signs, while for the latter, we establish the asymptotic normality of H\'{a}jek estimators for average treatment effects under network interference.
In this paper, we initiate the study of colored exponential random graph models (ERGMs), a class of exponential-family models for networks with multiple types of edge relations. Using the framework of probability graphons, we first derive a variational representation for the limiting free energy, whose maximizers determine the asymptotic structure of typical samples from the model. Then we identify several general families of colored ERGMs exhibiting replica symmetry, where the variational problem has constant maximizers and the model asymptotically concentrates on product colorings with independent edges. For general colored ERGMs, we derive Euler-Lagrange fixed-point equations for the variational maximizers, which in turn yield a general high-temperature uniqueness criterion. In the complementary zero-temperature regime, we establish a two-level selection principle: the leading energy term determines the ground states, while the lower-order energy terms, combined with entropy, act as a tie-breaker to determine the asymptotic zero-temperature structure of the model. We illustrate this principle through the induced wedge and rainbow triangle ERGMs. Both models have natural interpretations in multitype networks, and their zero-temperature limits exhibit interesting structures that connect to well-known results in extremal combinatorics. We further establish finite-temperature symmetry breaking for both these models and complement the rigorous results with numerical experiments.
B. Bhattacharya, Pierfrancesco Dionigi, Ankana Ganguly et al.· 0 citations
In this paper, we investigate the integrability of Lotka-Volterra (replicator) systems arising from interaction matrices generated from corresponding graph structures, continuing work started by Visomirski and Griffin [J. Phys. A., 58:015701, 2025] and Evripidou et al. [J. Phys. A., 55:325201, 2022] (among others). In particular, we define a new family of graphs, the spoked graphs, and show that all dynamics generated from this family are integrable. In reference to Evripidou et al. (2022), we define a new anti-cloning operator and show that its action on balanced tournament graphs (with odd vertex count) generates new graphs whose dynamics are integrable. Interestingly, we provide numerical evidence that this anti-cloning operation leads to chaotic behaviour when applied to other graph families (e.g., the directed cycles that generate the classically integrable Volterra lattice). This work completes a taxonomy of all integrable dynamics generated by directed graphs with up to six vertices started by Visomirski and Griffin (2025), and suggests several future directions of study on this topic.
Matthew Visomirski, Christopher Griffin· 0 citations
The joint asymptotic distribution of any finite collection of network moments in random graphs sampled from a graphon, which includes both the nondegenerate case as well as the degenerate case, provides the higher-order fluctuation theory for subgraph counts in the graphon model.
Anirban Chatterjee, S. Dan, B. Bhattacharya· Annals of Statistics· 0 citations
We design randomized approximation schemes for the partition function of antiferromagnetic Ising models with uniform external field on random regular bipartite graphs. Our algorithm generalizes the approach of Kocurek, Oveis Gharan and Tjowasi (arXiv, 2026) for hard-core models on the same random graph model beyond the uniqueness threshold. We show that, as long as $\lambda$ is upper bounded by a constant and $\lambda(1 - \beta) \lesssim \Delta^{-1/2}$, an efficient randomized algorithm approximates the partition function with high probability. The algorithm first truncates configurations that are large on either side of the bipartition and then samples from Gibbs distributions conditioned on fixed sizes on one or both sides. To choose an optimal truncation bound, we establish concentration properties of the Gibbs distribution on random regular bipartite graphs. Then we apply high-dimensional expansion and prove trickle-down theorems to obtain fast samplers for the conditioned distributions.
These results establish a direct and quantitative link between random matrix disorder on graphs and the performance of continuous-time quantum walk search, and suggest that disorder, rather than being merely an obstacle, can be exploited as a tunable parameter in quantum search protocols.
Sabyasachi Chakraborty, T. Čadež, Sonjoy Majumder et al.· 0 citations
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