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arXiv:2601.21152

Community detection in network using Szegedy quantum walk

Md Samsur Rahaman, Supriyo Dutta

quant-phcs.SImath.COphysics.soc-phSubmitted Jan 29, 2026• Announced Jan 30, 2026• 0 citations• Revised Feb 9, 2026 (v2)
v1 · Jan 29, 2026v2 · Feb 9, 2026
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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 solving this problem uses the classical random walks on 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 community detection procedure to a variety of graphs and social networks, including the relaxed caveman graph, $l$-partition graph, Karate club graph, and the dolphin's social network, among others.

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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 solving this problem uses the classical random walks on 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 community detection procedure to a variety of graphs and social networks, including the relaxed caveman graph, lll-partition graph, Karate club graph, and the dolphin's social network, among others.
Many systems such as social networks and biological networks take the form of complex networks, which have a community structure.Community detection in complex networks is of great interest to many researchers in statistical physics and mathematical physics.There have been studies on community detection that use the classical random walk.The present study utilizes the discrete-time quantum walk instead.The quantum walk plays an important role in various fields, especially in research on quantum computers, and attracts much attention from mathematical physics too.The discrete-time quantum walk has two properties: it linearly spreads on a flat space, and it localizes in some cases because of quantum coherence.We demonstrate that these properties of the quantum walk are useful for community detection on complex networks.We define the discrete-time quantum walk on complex networks and utilize it for community detection.We numerically show that the quantum walk with a Fourier coin is localized in a community to which the initial node belongs.Meanwhile, the quantum walk with a Grover coin tends to be localized around the initial node, not over a community.The probability of a classical random walk on the same network converges to a uniform distribution with a relaxation time generally unknown a priori.We thus claim that the time average of the probability of a Fourier-coin quantum walk on complex networks reveals the community structure more explicitly than that of a Grover-coin quantum walk and a snapshot of the classical random walk.We first demonstrate our method of community detection for a prototypical three-community network, producing the correct grouping.We then apply our method to two real-world networks, namely, Zachary's karate club and the U.S. Airport network.We successfully reveal the community structure, the two communities of the instructor and the administrator in the former and major airline companies in the latter.
The dynamics and underlying structure of complex social networks (SNs) can be discovered using community detection. Considering how quantum computing might improve community detection techniques is becoming more and more popular in light of recent developments in computing technology. In this article, the rapidly developing topic of quantum-SN analysis for community discovery is thoroughly reviewed. Community detection is studied in the context of several quantum-inspired techniques, including quantum annealing and quantum-inspired optimization. To make use of the strengths of both conventional SN research methods and quantum computing techniques, hybrid quantum-classical approaches are being investigated. Case studies and applications that have made use of quantum-SN analysis methodologies are reviewed to highlight the practical consequences and potential advantages over conventional methods. The article also highlights the challenges and limitations of using quantum computing for SN analysis, including technical constraints and ethical issues. Finally, prospects and future research objectives in the area of quantum-SN analysis are highlighted. This covers possible developments in quantum algorithms for community detection, the incorporation of quantum computing with other SN research tasks, and the significance of multidisciplinary collaborations.
This paper investigates the application of quantum walk algorithms to address the computational challenges inherent in analyzing large and complex networks. Traditional graph algorithms often suffer from exponential scaling, rendering them impractical for networks with a substantial number of nodes and edges. We propose a framework leveraging quantum walks to perform network exploration and analysis tasks, aiming to achieve significant speedups compared to their classical counterparts. The core of our approach lies in simulating a quantum walk on the graph, where the walker's movement is governed by quantum mechanical principles, promoting parallel exploration of the graph's connectivity. We detail the theoretical foundations of this approach, focusing on the mathematical formulation of quantum walk operators and their application to graph traversal and algorithm design. Furthermore, we explore specific network analysis problems, such as shortest path finding and community detection, demonstrating the potential of this methodology. The results presented suggest that quantum walk-based algorithms offer a promising avenue for scalable network analysis, particularly for large networks where classical algorithms face limitations.
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Graph Representation Learning (GRL) has emerged as a cornerstone technique for analysing complex, networked data across diverse domains, including biological systems, social networks, and data analysis. Traditional GRL methods often struggle to capture intricate relationships within complex graphs, particularly those exhibiting non-trivial structural properties such as power-law distributions or hierarchical structures. This paper introduces a novel quantum-inspired algorithm for GRL, utilizing hybrid Quantum-Classical Walks to overcome these limitations. Our approach combines the benefits of both quantum and classical dynamics, allowing the walker to simultaneously explore both highly local and far-reaching connections within the graph. Preliminary results for a case study in network community detection shows that this hybrid dynamic enables the algorithm to adapt effectively to complex graph topologies, offering a robust and versatile solution for GRL tasks.