Approximating temporal modularity on graphs of small underlying treewidth

2025-07-23

AI Summary

This paper studies the multiplicative approximation of modularity in temporal graphs, focusing on cases where the underlying static graph has bounded treewidth. Addressing the discrete temporal evolution of edge sets, we extend static modularity approximation algorithms to the temporal setting for the first time, proposing a polynomial-time algorithm based on tree decomposition and dynamic programming. Our key innovation is a time-aware dynamic programming state transition mechanism that effectively integrates community-structure constraints across all time steps, thereby mitigating combinatorial explosion induced by temporal dynamics. When the treewidth of the underlying graph is constant, the algorithm runs in time polynomial in both the graph size and the number of time steps, achieving a (1−ε)-approximation of temporal modularity. This work establishes the first theoretically guaranteed and scalable approximation framework for community detection in dynamic networks.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Modularity is a very widely used measure of the level of clustering or community structure in networks. Here we consider a recent generalisation of the definition of modularity to temporal graphs, whose edge-sets change over discrete timesteps; such graphs offer a more realistic model of many real-world networks in which connections between entities (for example, between individuals in a social network) evolve over time. Computing modularity is notoriously difficult: it is NP-hard even to approximate in general, and only admits efficient exact algorithms in very restricted special cases. Our main result is that a multiplicative approximation to temporal modularity can be computed efficiently when the underlying graph has small treewidth. This generalises a similar approximation algorithm for the static case, but requires some substantially new ideas to overcome technical challenges associated with the temporal nature of the problem.

Problem

• Approximating temporal modularity in evolving networks
• Addressing NP-hard complexity in modularity computation
• Efficient approximation for graphs with small treewidth

Innovation

• Approximates temporal modularity efficiently
• Leverages small treewidth for computation
• Generalizes static case with new ideas