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.

翻译：本文研究随机图上的弱相互作用扩散过程。我们主要关注平均场极限的性质，特别是稳态的非唯一性和分岔结构。通过将经典分岔分析扩展至包含多色相互作用势和随机图结构，我们显式地识别了分岔点，并将其与图积分算子的谱特性相关联。此外，我们建立了稳态的自洽性方程，该方程恢复了主临界阈值并揭示了沿非均匀分支的次级分岔。进一步，我们将所得的McKean-Vlasov偏微分方程刻画为在适当度量下的梯度流。同时，我们提供了强有力证据表明（负的）相互作用粒子系统的相互作用能可作为自然的序参量。特别地，在超越相变点且存在多色相互作用时，我们观察到与动力学亚稳态紧密关联的能量级联现象。