Phase transitions for interacting particle systems on random graphs

TL;DR

This work analyzes phase transitions for weakly interacting diffusion processes on random graphs by embedding the mean-field limit in a graphon framework. It develops a gradient-flow formulation with a modified Wasserstein metric and a free-energy functional, then derives explicit primary and secondary bifurcation thresholds through spectral analysis of the graphon operator and a self-consistency approach. The paper shows how multichromatic interaction potentials generate rich non-uniform equilibria and dynamical metastability, with rigorous results complemented by numerical simulations on ER, SW, and PL graphs. The findings illuminate how network topology and interaction structure dictate synchronization-like transitions and energy cascades in large interacting particle systems, with broad implications for collective dynamics in complex networks.

Abstract

In this paper, we study weakly interacting diffusion processes on random graphs. Our main focus is on the properties of the mean-field limit and, in particular, on the nonuniqueness and bifurcation structure of stationary states. By extending classical bifurcation analysis to include multichromatic interaction potentials and random graph structures, we explicitly identify bifurcation points and relate them to the spectral properties of the graphon integral operator. In addition, we develop a self-consistency formulation of stationary states that recovers the primary critical threshold and reveals secondary bifurcations along non-uniform branches. Furthermore, we characterize the resulting McKean-Vlasov PDE as a gradient flow with respect to a suitable metric. In addition, we provide strong evidence that (minus) the interaction energy of the interacting particle system serves as a natural order parameter. In particular, beyond the transition point and for multichromatic interactions, we observe an energy cascade that is strongly linked to dynamical metastability.

Figures (5)

• Figure 1: (a): Phase Diagrams for the Kuramoto model. (b): (top panel) Time evolution of a typical trajectory for $U(t)$ after the phase transition, for a PL graph for $\theta / \theta_c \approx 2$. (bottom panel) Empirical measure $\rho_N$ of the system at selected times $t=0,100$, represented as red vertical dashed lines in the top panel.
• Figure 2: (a): Phase Diagrams for the bi-harmonic interaction potential. (b): (top panel) Time evolution of a typical trajectory for $U(t)$ after the phase transition, for a PL graph for $\theta / \theta_c \approx 1.96$. (bottom panel) Empirical measure $\rho_N$ of the system at selected times $t=100,1500,3000$. Graphical conventions as in Figure \ref{['fig: Kuramoto']}.
• Figure 3: Phase diagram for the bi-harmonic interaction potential on Erdős-Rényi graph with a finer interaction strength grid. The vertical dashed line represent the analytical value of the secondary bifurcation point. Other graphical conventions as in Figure \ref{['fig: Kuramoto']}.
• Figure 4: (a): Phase Diagrams for the quadri-harmonic interaction potential. (b): (top panel) Time evolution of a typical trajectory for $U(t)$ after the phase transition, for a PL graph for $\theta / \theta_c \approx 1.96$. (bottom panel) Empirical measure $\rho_N$ of the system at selected times $t=100,800,3000$. Graphical conventions as in Figure \ref{['fig: Kuramoto']}.
• Figure 5: Metastability features for the quadri-harmonic potential on ER graphs. Panel (a): Energy of the system as a function of time. Panel (b): Empirical measure of the system at the end of the simulation time $T=5000$ for selected trajectories. Trajectories that have (not yet) transitioned to the final one-peak state are represented in red (blue). Here, $\theta / \theta_c \approx 1.14$

Theorems & Definitions (13)

• Proposition 2.1
• proof
• Definition 2.2
• Proposition 2.3
• proof
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• proof
• Theorem 3.4