🤖 AI Summary
This work addresses the limitation of traditional graph Laplacians in capturing memory effects in complex network diffusion, particularly their inability to model the influence of backtracking paths. The authors propose a novel class of walk-based Laplacian operators that flexibly incorporate memory by either excluding or reweighting backtracking walks, yielding three variants: fully traversing, non-backtracking, and continuously interpolated. This framework uniquely integrates non-backtracking and backtracking-weighted mechanisms into the Laplacian formalism, enabling controllable modeling of path memory while preserving certain classical spectral properties. Leveraging Krylov subspace methods and GPU acceleration, the approach scales efficiently to large-scale networks. Experiments on multiple real-world datasets demonstrate its superior expressiveness and computational efficiency, significantly enhancing the modeling of diffusion dynamics in complex networks.
📝 Abstract
We develop a novel framework for modeling diffusion on complex networks by constructing Laplacian-like operators based on walks around a graph. Our approach introduces a parametric family of walk-based Laplacians that naturally incorporate memory effects by excluding or downweighting backtracking trajectories, where walkers immediately revisit nodes. The framework includes: (i) walk-based Laplacians that count all traversals in the network; (ii) nonbacktracking variants that eliminate immediate reversals; and (iii) backtrack-downweighted variants that provide a continuous interpolation between these two regimes. We establish that these operators extend the definition of the standard Laplacian and also preserve some of its properties. We present efficient algorithms using Krylov subspace methods for computing them, ensuring applicability of our proposed framework to large networks. Extensive numerical experiments on real-world networks validate the modeling flexibility of our approach and demonstrate the computational efficiency of the proposed algorithms, including GPU acceleration.