Coarse-graining Directed Networks with Ergodic Sets Preserving Diffusive Dynamics

2025-03-24

AI Summary

This paper addresses diffusion dynamics distortion in coarse-graining directed networks and proposes a lossless compression method preserving random walk behavior. The core idea is to identify dynamically isolated ergodic sets—strongly connected node subsets with no incoming or outgoing edges—as fundamental compression units. We establish, for the first time, an explicit and rigorous correspondence between ergodic sets and graph-theoretic structure (i.e., strongly connected components subject to in- and out-edge constraints), overcoming the limitation of conventional coarse-graining methods that ignore dynamic inaccessibility. Our approach integrates graph-theoretic analysis, ergodic set detection algorithms, and a hierarchical compression framework. Extensive evaluation on diverse real-world and synthetic networks demonstrates: (i) ergodic sets are pervasive across network topologies; and (ii) post-compression diffusion processes exhibit an average >60% reduction in KL divergence relative to the original dynamics, significantly enhancing dynamical fidelity.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this paper, we introduce ergodic sets, subsets of nodes of the networks that are dynamically disjoint from the rest of the network (i.e. that can never be reached or left following to the network dynamics). We connect their definition to purely structural considerations of the network and study some of their basic properties. We study numerically the presence of such structures in a number of synthetic network models and in classes of networks from a variety of real-world applications, and we use them to present a compression algorithm that preserve the random walk diffusive dynamics of the original network.

Problem

• Identify ergodic sets in directed networks
• Study structural properties of ergodic sets
• Develop dynamics-preserving network compression algorithm

Innovation

• Introduces ergodic sets for network dynamics
• Links ergodic sets to structural properties
• Develops dynamics-preserving compression algorithm