Phase transitions for interacting particle systems on random graphs

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.