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Community detection in network using Szegedy quantum walk

Md Samsur Rahaman, Supriyo Dutta

TL;DR

This work develops a quantum-graph clustering framework based on Szegedy's discrete-time quantum walk to detect communities in undirected networks. It constructs an edge-centered limiting distribution $\Pi$ from time-averaged quantum walk probabilities, driven by an initial state focused on vertices of maximum degree, and then uses a Main/Refinement procedure with path-weight criteria $PW(u,v)$ and a threshold $q$ to assign vertices into communities. Applied to Barbell, relaxed caveman, planted $l$-partition, karate club, Les Misérables, dolphins, and Euroroad networks, the approach reveals coherent communities and, in some cases, partitions that differ from classical methods, with performance depending on graph structure and $q$. The results establish a quantum graph clustering methodology that leverages the limiting behavior of quantum walks and initial-state design to uncover modular structure in complex networks, offering a new tool for quantum-assisted community detection.

Abstract

In a network, the vertices with similar characteristics construct communities. The vertices in a community are well-connected. Detecting the communities in a network is a challenging and important problem in the theory of complex networks. One approach to solve this problem uses the classical random walks on the graphs. In quantum computing, quantum walks are the quantum mechanical counterparts of classical random walks. In this article, we employ a variant of Szegedy's quantum walk to develop a procedure for discovering the communities in networks. The limiting probability distribution of quantum walks assists us in determining the inclusion of a vertex in a community. We apply our procedure of community detection on a number of graphs and social networks, such as the relaxed caveman graph, $l$-partition graph, Karate club graph, dolphin's social network, etc.

Community detection in network using Szegedy quantum walk

TL;DR

This work develops a quantum-graph clustering framework based on Szegedy's discrete-time quantum walk to detect communities in undirected networks. It constructs an edge-centered limiting distribution from time-averaged quantum walk probabilities, driven by an initial state focused on vertices of maximum degree, and then uses a Main/Refinement procedure with path-weight criteria and a threshold to assign vertices into communities. Applied to Barbell, relaxed caveman, planted -partition, karate club, Les Misérables, dolphins, and Euroroad networks, the approach reveals coherent communities and, in some cases, partitions that differ from classical methods, with performance depending on graph structure and . The results establish a quantum graph clustering methodology that leverages the limiting behavior of quantum walks and initial-state design to uncover modular structure in complex networks, offering a new tool for quantum-assisted community detection.

Abstract

In a network, the vertices with similar characteristics construct communities. The vertices in a community are well-connected. Detecting the communities in a network is a challenging and important problem in the theory of complex networks. One approach to solve this problem uses the classical random walks on the graphs. In quantum computing, quantum walks are the quantum mechanical counterparts of classical random walks. In this article, we employ a variant of Szegedy's quantum walk to develop a procedure for discovering the communities in networks. The limiting probability distribution of quantum walks assists us in determining the inclusion of a vertex in a community. We apply our procedure of community detection on a number of graphs and social networks, such as the relaxed caveman graph, -partition graph, Karate club graph, dolphin's social network, etc.
Paper Structure (16 sections, 16 equations, 10 figures)

This paper contains 16 sections, 16 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. A brief Overview of graph theory
  4. An introduction to quantum walks
  5. Community detection on graphs
  6. Numerical experiments
  7. Barbell graphs
  8. Relaxed caveman graph
  9. Planted $l$-partition graph
  10. Zachary's karate club graph
  11. Les Misérable graph
  12. Dolphin's social network
  13. Euroroad network
  14. Conclusion
  15. Dependency on the initial state
  16. ...and 1 more sections

Figures (10)

  • Figure 1: Different steps for generating communities on the graph $G$, depicted in Sub-figure \ref{['Random_graph']}, are presented.
  • Figure 2: Community detection on a barbell graph
  • Figure 3: Community detection in a relaxed caveman graph.
  • Figure 4: Community detection in a planted partition graph.
  • Figure 5: Karate club graph are considered to generate communities.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Definition 3