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

Minh Hoang Trinh, Hyo-Sung Ahn

TL;DR

This work addresses determining connectedness and clustering in undirected matrix-weighted graphs, where edge weights are PSD matrices and vertex states are $d$-dimensional vectors. It introduces two heuristic algorithms, brute-force search and Warshall, rooted in a sufficient kernel-based connectivity condition involving the matrix-weighted Laplacian $\mathbf{L}$ and per-path kernels, to decide connectivity and partition vertices into clusters. The authors prove correctness, analyze worst-case complexities, and demonstrate through numerical examples that the Warshall-based method can efficiently reveal clusters while the brute-force approach remains exhaustive; examples also reveal limitations and heuristic characteristics. Overall, the paper extends classical graph connectivity concepts to matrix-weighted networks, enabling practical analysis of multidimensional and multi-layered systems in applications such as multiagent coordination and clustering dynamics.

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.

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

TL;DR

This work addresses determining connectedness and clustering in undirected matrix-weighted graphs, where edge weights are PSD matrices and vertex states are -dimensional vectors. It introduces two heuristic algorithms, brute-force search and Warshall, rooted in a sufficient kernel-based connectivity condition involving the matrix-weighted Laplacian and per-path kernels, to decide connectivity and partition vertices into clusters. The authors prove correctness, analyze worst-case complexities, and demonstrate through numerical examples that the Warshall-based method can efficiently reveal clusters while the brute-force approach remains exhaustive; examples also reveal limitations and heuristic characteristics. Overall, the paper extends classical graph connectivity concepts to matrix-weighted networks, enabling practical analysis of multidimensional and multi-layered systems in applications such as multiagent coordination and clustering dynamics.

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.
Paper Structure (17 sections, 7 theorems, 28 equations, 6 figures, 2 algorithms)

This paper contains 17 sections, 7 theorems, 28 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parallel and series additions of symmetric positive semidefinite matrices
  4. Matrix-weighted graphs
  5. Algorithms
  6. Brute-force search algorithm
  7. Warshall algorithm
  8. Numerical examples
  9. Example 1: Connected matrix-weighted graph
  10. Example 2: A graph which is topologically connected but not connected
  11. Example 3: Zero submatrix in M(G,n-1) does not imply topological disconnectedness
  12. Example 4: The algorithms are heuristic
  13. Conclusion
  14. Proofs
  15. Proof of Lemma \ref{['lem:distribution']}
  16. ...and 2 more sections

Key Result

Lemma 1

The "$\vee$" operator of $d\times d$ symmetric positive semidefinite matrices satisfies the following properties:

Figures (6)

  • Figure 1: The path $\mathcal{P}_k=v_{i_1}^kv_{i_2}^k\ldots v_{i_{|\mathcal{P}_k|+1}}^k$ joins two vertices $v_i$ and $v_j$.
  • Figure 2: The matrix-weighted graph describing a four-country network: (a) each day of week corresponds to a small black node in each country and two nodes from two countries are connected by a line if there is a flight between two countries in the corresponding day; (b) Abstraction of the four-country network by a matrix-weighted graph.
  • Figure 3: The matrix-weighted graph considered in Example 1.
  • Figure 4: The matrix-weighted graph considered in Example 2
  • Figure 5: The matrix-weighted graph considered in Example 3
  • ...and 1 more figures

Theorems & Definitions (16)

  • Lemma 1: The "$\vee$" operator
  • Lemma 2: The "$\wedge$" operator
  • Lemma 3: Distributivity of "$\wedge$" and "$\vee$" operators
  • Definition 1: Connectedness/Clustering
  • Definition 2
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3: Computational complexity analysis
  • ...and 6 more