Beyond the Laplacian: Doubly Stochastic Matrices for Graph Neural Networks

2026-04-16

AI Summary

This work addresses the limitations of conventional graph neural networks (GNNs), which rely on adjacency or Laplacian matrices for message passing and struggle to model continuous multi-hop proximity and local centrality while being prone to oversmoothing. The authors introduce, for the first time, a doubly stochastic matrix (DSM) into GNNs by constructing it from a regularized inverse of the Laplacian matrix and propose DsmNet—a scalable architecture based on truncated Neumann series. To preserve row stochasticity and compensate for probability mass loss due to truncation, they devise a residual quality compensation mechanism. Theoretical analysis delineates the applicability boundaries of DSM on both homophilic and heterophilic graphs. The method achieves O(K|E|) computational efficiency, significantly mitigates oversmoothing, and improves performance on homophilic benchmarks, while also demonstrating the versatility of DSM as a continuous structural encoding within graph Transformers.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Graph Neural Networks (GNNs) conventionally rely on standard Laplacian or adjacency matrices for structural message passing. In this work, we substitute the traditional Laplacian with a Doubly Stochastic graph Matrix (DSM), derived from the inverse of the modified Laplacian, to naturally encode continuous multi-hop proximity and strict local centrality. To overcome the intractable $O(n^3)$ complexity of exact matrix inversion, we first utilize a truncated Neumann series to scalably approximate the DSM, which serves as the foundation for our proposed DsmNet. Furthermore, because algebraic truncation inherently causes probability mass leakage, we introduce DsmNet-compensate. This variant features a mathematically rigorous Residual Mass Compensation mechanism that analytically re-injects the truncated tail mass into self-loops, strictly restoring row-stochasticity and structural dominance. Extensive theoretical and empirical analyses demonstrate that our decoupled architectures operate efficiently in $O(K|E|)$ time and effectively mitigate over-smoothing by bounding Dirichlet energy decay, providing robust empirical validation on homophilic benchmarks. Finally, we establish the theoretical boundaries of the DSM on heterophilic topologies and demonstrate its versatility as a continuous structural encoding for Graph Transformers.

Problem

• Graph Neural Networks
• Laplacian
• Doubly Stochastic Matrix
• Over-smoothing
• Structural Message Passing

Innovation

• Doubly Stochastic Matrix
• Graph Neural Networks
• Neumann Series Approximation
• Residual Mass Compensation
• Dirichlet Energy