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On the Formalization of Network Topology Matrices in HOL

This paper presents a formalization of network topology matrices (adjacency, degree, Laplacian, and incidence) within the Isabelle/HOL proof assistant based on directed graphs, where classical properties and inter-matrix relationships are verified to support rigorous analysis of systems like electrical networks through examples such as Kron reduction and power dissipation verification.

Original authors: Kubra Aksoy, Adnan Rashid, Osman Hasan, Sofiene Tahar

Published 2026-03-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kubra Aksoy, Adnan Rashid, Osman Hasan, Sofiene Tahar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to design a massive, complex city. You have roads (edges) connecting buildings (nodes), and traffic flows between them. To understand how this city works—how electricity flows, how data moves, or how water travels—you need a map. But a simple drawing isn't enough for a city this big; you need a mathematical blueprint that a computer can check for errors.

This paper is about building that blueprint using a very strict, unyielding digital architect named Isabelle/HOL.

Here is the story of what they did, explained simply:

1. The Problem: "Paper-and-Pencil" Mistakes

For decades, engineers and scientists have used math to model networks (like power grids, the internet, or traffic systems). They use special grids of numbers called Matrices to represent these networks.

The Old Way: They would write proofs on paper or run computer simulations.
The Risk: Paper proofs can have hidden mistakes (like a typo in a blueprint). Computer simulations are like "guessing and checking"—they might work 99% of the time, but they might miss that one tiny, catastrophic failure that causes a blackout.

2. The Solution: The "Digital Lawyer"

The authors decided to use Isabelle/HOL, which is like a super-smart digital lawyer. It doesn't just "guess"; it demands absolute proof for every single step. If you say "A plus B equals C," the lawyer checks the laws of logic to ensure that is impossible to be false.

They built a massive library of these "legal proofs" for network matrices.

3. The Building Blocks: The Matrix Toolkit

To model a network, you need different types of "maps" (matrices). The authors created formal, unbreakable definitions for four main types:

The Adjacency Matrix (The "Who's Next Door?" Map):
Imagine a seating chart at a wedding. This matrix tells you exactly who is sitting next to whom. If Person A is connected to Person B, the chart gets a "1" (or a weight, like a distance). If they aren't connected, it's a "0."
- The Paper's Job: They proved that this map is built correctly and that if you flip it (transpose it), it still makes sense for certain types of networks.
The Degree Matrix (The "Popularity" List):
This is a list that counts how many connections each node has. In a weighted network (where connections have different strengths, like heavy traffic vs. light traffic), this list sums up the "weight" of all connections for each building.
- The Paper's Job: They proved that this list accurately reflects the connections found in the "Who's Next Door?" map.
The Laplacian Matrix (The "Big Picture" Engine):
This is the most important one. It combines the "Who's Next Door?" map and the "Popularity" list. Think of it as the engine of the network. It tells you how the whole system behaves as a unit. It's used to solve problems like "How much power is lost in this grid?" or "How does heat spread through this metal?"
- The Paper's Job: They proved that this engine works exactly as the math textbooks say it should, down to the smallest detail.
The Incidence Matrix (The "Connector" List):
This is a list that connects the buildings to the roads. It tracks which road starts at which building and ends at which building.
- The Paper's Job: They created two versions (one for roads going out, one for roads coming in) and proved how they fit together to build the other maps.

4. The Magic Trick: Connecting the Dots

The coolest part of the paper is showing how these maps talk to each other.

They proved that if you take the Connector List and the Popularity List, you can mathematically build the Big Picture Engine (Laplacian).
They proved that if you take the Big Picture Engine and remove a few buildings (a process called Kron Reduction), you get a smaller, simpler map that still acts exactly like the original giant city. This is huge for engineers who want to simplify complex power grids without losing accuracy.

5. Real-World Testing: The Power Grid

To show this wasn't just theory, they tested it on two real-world scenarios:

Kron Reduction: They took a complex power grid (like the IEEE 5-Bus system) and mathematically "pruned" it. The computer proved that the pruned version behaves exactly like the original.
Power Dissipation: They calculated how much energy is lost as heat in a resistor network. Using their "Big Picture Engine" (Laplacian), they proved the formula for total energy loss is 100% correct.

The Analogy Summary

Imagine you are building a skyscraper.

Traditional Math is like drawing the plans on a napkin. It usually works, but if you miss one beam, the building might fall.
Computer Simulation is like building a scale model out of clay. It looks good, but it's not the real thing.
This Paper is like hiring a team of robot inspectors who check every single bolt, beam, and wire against the laws of physics before you even pour the concrete. They don't just say "it looks right"; they prove it is impossible for it to be wrong.

Why Does This Matter?

In safety-critical fields like nuclear power plants, airplane navigation, or medical devices, a small math error can be fatal. By formalizing these network matrices in Isabelle/HOL, the authors have created a "gold standard" library. Now, engineers can build their systems on top of these proven, unbreakable foundations, knowing that the math behind their designs is 100% trustworthy.

1. Problem Statement

Network topology matrices (such as adjacency, degree, Laplacian, and incidence matrices) are fundamental algebraic tools used to model and analyze complex systems in electrical engineering, transportation, and biology. Traditionally, the analysis of these matrices relies on:

Paper-and-pencil proofs: These are prone to human error, especially for large-scale or complex systems.
Computer-based simulations: These methods lack sound mathematical guarantees, often rely on trial-and-error, and may miss safety-critical edge cases due to unverified underlying algorithms.

There is a critical need for a rigorous, logic-based framework to formally verify the properties and relationships of these matrices to ensure the reliability of safety-critical systems (e.g., power grids). While some graph theory formalizations exist in interactive theorem provers, comprehensive formalizations of network topology matrices specifically for weighted directed graphs are largely missing.

2. Methodology

The authors utilize Isabelle/HOL, a Higher-Order Logic (HOL) interactive theorem prover, to formalize network topology matrices. The methodology involves:

Modular System Modeling using Locales:
- The authors employ Isabelle's locale mechanism to create a hierarchical, modular system.
- They define a generic netw_sys locale and extend it to specialized contexts (e.g., wdg_sys for weighted directed graphs, sym_wdg_sys for symmetric/undirected graphs, partitioned_wdg_sys for subsystems).
- This approach allows definitions and theorems to be inherited and reused, ensuring consistency across different graph types.
Matrix Library Selection:
- The formalization is built upon the Jordan Normal Form (JNF) matrix library in Isabelle/HOL. The authors chose JNF over the HOL-Analysis library because JNF better supports block matrices and varying dimensions, which are crucial for operations like Kron reduction.
- They extended the JNF library to include definitions for diagonal, symmetric, and specific network matrices.
Formalization Strategy:
- Graph Representation: Networks are modeled as directed graphs where nodes are lists and edges are pairs of nodes. Weights are real-valued functions.
- Matrix Definitions: Adjacency, Degree (in/out), Laplacian, and Incidence matrices are defined functionally.
- Verification: The authors prove fundamental properties (e.g., indexing, row/column sums) and establish algebraic relationships between the matrices.
- Applications: The framework is applied to two specific case studies: Kron reduction and power dissipation analysis in resistive circuits.

3. Key Contributions

The paper makes several significant contributions to the field of formal methods and network theory:

Comprehensive Formalization of Topology Matrices:
- Formalized Adjacency, Degree (in-degree and out-degree), and Laplacian matrices for weighted directed graphs.
- Introduced and formalized In-Incidence and Out-Incidence matrices, distinguishing them from the standard incidence matrix to handle directed edges rigorously.
Verification of Fundamental Properties:
- Proved classical properties such as the relationship between matrix entries and node degrees.
- Verified that the sum of rows in a Laplacian matrix is zero ( $L \mathbf{1} = \mathbf{0}$ ).
- Established the condition for column sums to be zero (weight-balanced graphs).
Formal Relationships Between Matrices:
- Proved the algebraic connections between incidence, weight, degree, adjacency, and Laplacian matrices.
- Specifically verified the equation: $L = I_{out} W (I_{out} - I_{in})^T$ , where $W$ is the weight matrix.
Application to Kron Reduction:
- Provided the first known formalization of Kron reduction (an algebraic method to eliminate interior nodes while preserving network topology) for weighted directed graphs.
- Proved the closure property: The Kron-reduced Laplacian matrix is itself a valid Laplacian matrix.
Application to Power Systems:
- Formally verified Ohm's Law and Kirchhoff's Current Law (KCL) within the framework.
- Proved the formula for total power dissipation in a resistive network: $P_{dissipated} = \mathbf{v}^T L \mathbf{v}$ .

4. Results

Correctness: All definitions and theorems were mechanically verified by the Isabelle/HOL proof assistant, eliminating human error.
Generality: The formalization is generic, supporting any finite number of nodes and edges, and can be instantiated for real or complex numbers.
Code Availability: The complete Isabelle/HOL code is publicly available, serving as a reusable library for future research.
Proof Complexity: The authors noted that formal proofs are significantly more verbose than textbook proofs (e.g., a 5-line textbook proof required ~73 lines of code) but offer superior rigor by capturing every intermediate step.
Automation: The use of the Sledgehammer tool allowed for the automation of certain complex lemmas, reducing proof effort while maintaining soundness.

5. Significance

Safety-Critical Assurance: This work provides a mathematically sound foundation for analyzing safety-critical systems (like power grids) where errors in topological analysis can lead to catastrophic failures.
Bridging Theory and Practice: It bridges the gap between abstract graph theory and practical engineering applications by providing a verified framework for algebraic methods like Kron reduction.
Foundation for Future Work: The modular locale-based approach allows for easy extension to:
- Dynamical systems (stability and controllability analysis).
- Complex-valued network systems (essential for graph signal processing).
- Integration with other Isabelle/HOL graph libraries.
Benchmark for Formal Methods: The paper demonstrates the capability of interactive theorem proving to handle complex, real-world algebraic structures that are often too intricate for manual verification.

In conclusion, this paper successfully establishes a rigorous, machine-checked foundation for network topology matrices, enabling trustworthy analysis of complex interconnected systems.