具有持久高斯扰动的递归图神经网络中的有限时间节点分离

中文摘要 AI 辅助

持久高斯扰动已被证明可以通过确保正的平稳狄利克雷能量来防止递归图神经网络（GNN）中的渐近过平滑。然而，这种全局能量界限并不能保证在有限深度下各个节点表示保持不同。在本文中，我们对相同的持久噪声架构进行了补充性的有限时间分析。设 \\(d\\) 表示表示维度，\\(\sigma\\) 表示噪声标准差。我们首先证明了任意两个节点表示之间的期望平方距离的精确二阶矩分解，从而在没有任何收缩或平稳性假设的情况下，在每个正时间步上得出通用下界 \\(2\sigma^2 d\\)。更精确地说，条件成对距离具有非中心卡方特征：非中心参数是由 \\(2\sigma^2\\) 归一化的确定性消息传递分离度。这产生了动态感知的固定时间和有限时间范围近碰撞界限，保留了中心最坏情况分析所丢弃的信息。早期的中心高斯界限被恢复为最坏情况下的零分离情况。我们还证明了几乎必然的成对非碰撞，推导出统一的有限时间范围保证，并建立了随机动力学在分布上的置换等变性和图输出的置换不变性。我们的结果补充了先前工作的渐近能量分析，并为节点级表示分离提供了严格的有限时间保证。

英文摘要

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.

发表机构

• Amirkabir University of Technology (Tehran Polytechnic)（阿米尔卡比尔理工大学（德黑兰理工大学））

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