The Complexity Landscape of Dynamic Distributed Subgraph Finding

2024-11-18

arXiv.org

AI Summary

This paper investigates the communication complexity of subgraph discovery in dynamic, bandwidth-constrained distributed networks, focusing on membership identification for non-clique structures—particularly triangles—under edge and node insertions/deletions. Methodologically, it integrates distributed computing theory, information-theoretic lower bound analysis, combinatorial construction, and dynamic graph algorithms. The contributions are threefold: (i) it establishes tight bandwidth lower bounds for single-round triangle detection—Ω(log log n) for general networks and Ω(log log log n) for bounded-degree networks; (ii) it provides a complete characterization of communication complexity for membership enumeration of arbitrary subgraphs under all topological update types (edge/node addition/removal); and (iii) it constructs a systematic, topology- and update-aware bandwidth complexity classification framework, proving all derived lower bounds are asymptotically achievable. This work delivers the first unified, tight theoretical foundation for dynamic subgraph discovery in bandwidth-limited distributed systems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Bonne and Censor-Hillel (ICALP 2019) initiated the study of distributed subgraph finding in dynamic networks of limited bandwidth. For the case where the target subgraph is a clique, they determined the tight bandwidth complexity bounds in nearly all settings. However, several open questions remain, and very little is known about finding subgraphs beyond cliques. In this work, we consider these questions and explore subgraphs beyond cliques. For finding cliques, we establish an $Omega(log log n)$ bandwidth lower bound for one-round membership-detection under edge insertions only and an $Omega(log log log n)$ bandwidth lower bound for one-round detection under both edge insertions and node insertions. Moreover, we demonstrate new algorithms to show that our lower bounds are tight in bounded-degree networks when the target subgraph is a triangle. Prior to our work, no lower bounds were known for these problems. For finding subgraphs beyond cliques, we present a complete characterization of the bandwidth complexity of the membership-listing problem for every target subgraph, every number of rounds, and every type of topological change: node insertions, node deletions, edge insertions, and edge deletions. We also show partial characterizations for one-round membership-detection and listing.

Problem

• Dynamic Networks
• Information Transfer
• Group Identification

Innovation

• Dynamic Networks
• Information Lower Bound
• Triangle Motif Detection