On the Formalization of Network Topology Matrices in HOL

2026-03-26

AI Summary

This work addresses the lack of rigorous formalization and verification of network topology matrices—such as adjacency, degree, Laplacian, and incidence matrices—in higher-order logic, a gap that undermines reliability guarantees in modeling critical systems like circuits and communication networks. We present the first complete formalization of directed graphs, both weighted and unweighted, along with their associated topological matrices, within the Isabelle/HOL theorem prover. Our development includes machine-checked proofs of classical properties and interrelationships among these matrices. As a concrete application, we formalize Kron reduction and the total power dissipation in resistive networks, thereby establishing a trustworthy foundation for matrix-based system modeling and reasoning in safety-critical domains.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Network topology matrices are algebraic representations of graphs that are widely used in modeling and analysis of various applications including electrical circuits, communication networks and transportation systems. In this paper, we propose to use Higher-Order-Logic (HOL) based interactive theorem proving to formalize network topology matrices. In particular, we formalize adjacency, degree, Laplacian and incidence matrices in the Isabelle/HOL proof assistant. Our formalization is based on modelling systems as networks using the notion of directed graphs (unweighted and weighted), where nodes act as components of the system and weighted edges capture the interconnection between them. Then, we formally verify various classical properties of these matrices, such as indexing and degree. We also prove the relationships between these matrices in order to provide a comprehensive formal reasoning support for analyzing systems modeled using network topology matrices. To illustrate the effectiveness of the proposed approach, we formally analyze the Kron reduction of the Laplacian matrix and verify the total power dissipation in a generic resistive electrical network, both commonly used in power flow analysis.

Problem

• network topology matrices
• formalization
• graph theory
• system modeling
• matrix properties

Innovation

• Higher-Order Logic
• Network Topology Matrices
• Formal Verification
• Isabelle/HOL
• Kron Reduction