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

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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 \(2σ^2 d\) 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 \(2σ^2\). 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.