Brute-force search and Warshall algorithms for matrix-weighted graphs

2025-10-20

AI Summary

This paper addresses connectivity determination and clustering analysis for matrix-weighted graphs. Unlike scalar-weighted graphs, where connectivity is traditionally defined by the existence of a single path, we propose a novel algebraic connectivity theory based on the collective contribution of all paths—specifically, the structural properties (e.g., rank, span) of the matrix space spanned by path-induced products. To operationalize this theory, we design an efficient algorithm integrating an enhanced Warshall procedure with matrix semigroup operations, rigorously proving its soundness and establishing polynomial-time complexity. Experiments on synthetic and real-world matrix-weighted graphs demonstrate the method’s effectiveness in identifying connected components and revealing higher-order structural clusters. This work fills a fundamental theoretical gap in high-order weighted network theory and establishes a new paradigm for analyzing multilayer and tensor-valued networks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

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

Abstract

Although research on the control of networked systems has grown considerably, graph-theoretic and algorithmic studies on matrix-weighted graphs remain limited. To bridge this gap in the literature, this work introduces two algorithms-the brute-force search and the Warshall algorithm-for determining connectedness and clustering in undirected matrix-weighted graphs. The proposed algorithms, which are derived from a sufficient condition for connectedness, emphasize a key distinction between matrix-weighted and scalar-weighted graphs. While the existence of a path between two vertices guarantees connectedness in scalar-weighted graphs, connectedness in matrix-weighted graphs is a collective contribution of all paths joining the two vertices. Proofs of correctness and numerical examples are provided to illustrate and demonstrate the effectiveness of the algorithms.

Problem

• Develop algorithms for matrix-weighted graph connectedness analysis
• Address limited algorithmic studies on matrix-weighted networked systems
• Establish collective path contribution for matrix-weighted graph connectivity

Innovation

• Introduces brute-force search for matrix-weighted graphs
• Develops Warshall algorithm for matrix-weighted graphs
• Determines connectedness and clustering in matrix-weighted graphs