Random attractors for nonlinear p-Laplacian diffusion on weighted graphs under stochastic forcing

We investigate nonlinear stochastic diffusion dynamics on finite connected weighted graphs governed by a discrete p –Laplacian operator with additive Brownian forcing. The system is formulated as a finite-dimensional stochastic evolution equation and embedded into the framework of random dynamical systems via an Ornstein–Uhlenbeck transformation, yielding a measurable cocycle. Under monotonicity and dissipativity assumptions on the nonlinear reaction term, we establish the existence of a tempered random absorbing set and prove the existence and uniqueness of a compact pullback random attractor. The analysis relies on sharp coercivity estimates for the discrete p –Laplacian restricted to the mean-zero subspace, where the associated nonlinear Poincaré constant is characterized variationally and shown to depend explicitly on the weighted topology of the graph. In contrast to stochastic p –Laplacian equations posed on homogeneous domains or regular lattices, effective dissipation in the graph setting is governed by spectral and combinatorial properties of the underlying network. Reduced connectivity and degree heterogeneity weaken coercivity and enlarge the resulting random attractor, inducing anisotropic long-run stochastic regimes. Numerical illustrations confirm pullback convergence and demonstrate the topological dependence of invariant stochastic variability. The results extend random attractor theory for stochastic p –Laplacian equations to general weighted graphs and provide a rigorous analytical link between network structure and asymptotic stochastic dynamics.