Finite-Time Node Separation in Recurrent Graph Neural Networks with Persistent Gaussian Perturbations

Persistent Gaussian perturbations have been shown to prevent asymptotic oversmoothing in recurrent Graph Neural Networks (GNNs) by ensuring a positive stationary Dirichlet energy. However, this global energy bound does not guarantee that individual node representations remain distinct at finite depths. In this paper, we provide a complementary finite-time analysis of the same persistent-noise architecture. Let d denote the representation dimension and σ the noise standard deviation. We first prove an exact second-moment decomposition for the expected squared distance between any two node representations, yielding the universal lower bound 2σ squared d at every positive time step without contraction or stationarity assumptions. More precisely, conditional pairwise distances have a noncentral chi-square characterization: the noncentrality parameter is the deterministic message-passing separation normalized by 2σ squared. This yields dynamics-aware fixed-time and finite-horizon near-collision bounds that retain information discarded by the central worst-case analysis. The earlier central Gaussian bound is recovered as the worst-case zero-separation case. We additionally prove almost-sure pairwise noncollision, derive a uniform finite-horizon guarantee, and establish permutation equivariance in distribution for the stochastic dynamics and permutation-invariant graph outputs. Our results complement the asymptotic energy analysis of prior work and provide rigorous finite-time guarantees on node-level representation separation.