Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Degree Matrix Comparison for Graph Alignment

Ashley Wang, Peter Chin

TL;DR

The paper addresses unsupervised graph alignment for unattributed graphs by introducing Degree Matrix Comparison (DMC), a simple degree-based method that encodes local and global structure in an $(N,m)$ degree matrix and solves the alignment via the Hungarian algorithm ($O(N^3)$ in general, $O(N^2m)$ when $m\ll N$). It extends DMC with Greedy DMC (auction pre-processing) and Weighted DMC for weighted graphs, providing theoretical motivation including invariance under node relabeling and analysis of cost contrasts that favor distinguishing high-degree nodes in heterogeneous networks. Empirically, DMC and its variations outperform several baselines on real-world and synthetic networks, with DMC excelling on heterogeneous graphs and Weighted DMC performing strongly on weighted datasets such as protein–protein interaction networks. The work suggests a simple, scalable, and robust route for graph alignment across diverse network types and lays groundwork for future extensions and broader validations.

Abstract

The graph alignment problem, which considers the optimal node correspondence across networks, has recently gained significant attention due to its wide applications. There are graph alignment methods suited for various network types, but we focus on the unsupervised geometric alignment algorithms. We propose Degree Matrix Comparison (DMC), a very simple degree-based method that has shown to be effective for heterogeneous networks. Through extensive experiments and mathematical proofs, we demonstrate the potential of this method. Remarkably, DMC achieves up to 99% correct node alignment for 90%-overlap networks and 100% accuracy for isomorphic graphs. Additionally, we propose a reduced Greedy DMC with lower time complexity and Weighted DMC that has demonstrated potential for aligning weighted graphs. Positive results from applying Greedy DMC and the Weighted DMC furthermore speaks to the validity and potential of the DMC. The sequence of DMC methods could significantly impact graph alignment, offering reliable solutions for the task.

Degree Matrix Comparison for Graph Alignment

TL;DR

The paper addresses unsupervised graph alignment for unattributed graphs by introducing Degree Matrix Comparison (DMC), a simple degree-based method that encodes local and global structure in an degree matrix and solves the alignment via the Hungarian algorithm ( in general, when ). It extends DMC with Greedy DMC (auction pre-processing) and Weighted DMC for weighted graphs, providing theoretical motivation including invariance under node relabeling and analysis of cost contrasts that favor distinguishing high-degree nodes in heterogeneous networks. Empirically, DMC and its variations outperform several baselines on real-world and synthetic networks, with DMC excelling on heterogeneous graphs and Weighted DMC performing strongly on weighted datasets such as protein–protein interaction networks. The work suggests a simple, scalable, and robust route for graph alignment across diverse network types and lays groundwork for future extensions and broader validations.

Abstract

The graph alignment problem, which considers the optimal node correspondence across networks, has recently gained significant attention due to its wide applications. There are graph alignment methods suited for various network types, but we focus on the unsupervised geometric alignment algorithms. We propose Degree Matrix Comparison (DMC), a very simple degree-based method that has shown to be effective for heterogeneous networks. Through extensive experiments and mathematical proofs, we demonstrate the potential of this method. Remarkably, DMC achieves up to 99% correct node alignment for 90%-overlap networks and 100% accuracy for isomorphic graphs. Additionally, we propose a reduced Greedy DMC with lower time complexity and Weighted DMC that has demonstrated potential for aligning weighted graphs. Positive results from applying Greedy DMC and the Weighted DMC furthermore speaks to the validity and potential of the DMC. The sequence of DMC methods could significantly impact graph alignment, offering reliable solutions for the task.

Paper Structure

This paper contains 17 sections, 1 theorem, 11 equations, 15 figures, 6 tables.

Table of Contents

  1. Problem Set Up
  2. Random Walk Graph Sampling
  3. Edge Deletion Graph Sampling
  4. Proposed Graph Alignment Methods
  5. Degree Matrix Comparison
  6. Variation 1: Greedy DMC
  7. Variation 2: Weighted DMC
  8. Theoretical Motivations
  9. Invariance
  10. Performance Analysis
  11. More General Theoretical Motivations
  12. Experimental Results
  13. Unweighted Graphs: DMC and Greedy DMC
  14. Real World Networks
  15. Artificial Networks
  16. ...and 2 more sections

Key Result

Theorem 1

We define a binary relation between two degree matrices: degree matrices $A,B$ are said to be related (i.e. $A \sim B$) if we can transform $A$ to become $B$ only through row swap operations. Moreover, we claim that this is an equivalence relation.

Figures (15)

  • Figure 1: Graph Sampling with Random Walk
  • Figure 2: Graph Sampling with Edge Deletion
  • Figure 3: Example Graph to Demonstrate Degree Matrix Formulation
  • Figure 4: $F_{1}$
  • Figure 5: $F_{2}$
  • ...and 10 more figures

Theorems & Definitions (7)

  • Definition 1: Graph
  • Definition 2: Graph Alignment
  • Definition 3: Random Walk (on graphs)
  • Definition 4: Degree Matrices
  • Theorem 1: Equivalence Relation on Degree Matrices.
  • proof
  • Definition 5: k-partite Graph