Hybrid Quantum-Classical Walks for Graph Representation Learning in Community Detection

TL;DR

The paper addresses the challenge of learning informative node representations on graphs with complex community structure by combining quantum and classical walk dynamics. It introduces Hybrid Quantum-Classical Walks (HQCWs) governed by a Lindblad master equation, enabling a tunable blend of quantum and classical propagation, and generates node embeddings via a skip-gram objective from walk trajectories. Empirical results on a four-cluster Erdős-Rényi graph show that HQCWs with $\\alpha \\\approx 0.8$ achieve clearer community separation and higher ARI than classical walks, highlighting the potential of quantum-inspired dynamics for GRL. This approach offers a scalable path to improved community detection and node-ranking tasks in networks with nontrivial topology, with future work aimed at discrete-time implementations and broader graph topologies.

Abstract

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.

Figures (3)

• Figure 1: Graph analyzed in this work. Random graph of $100$ nodes with Erdös-Renyi subclusters of different sizes with high intraconnectivity ($p_{\mathrm{intra}} = 0.25$) inside the clusters and small interconnectivity $p_{\mathrm{inter}} = 0.0015$ between clusters.
• Figure 2: t-SNE representations of node embeddings ($d=32$) for CRWs (top): 1st order (left), 2nd order (right) and HQCWs (bottom): $\alpha=0.3$ (left), $\alpha=0.8$ (right).
• Figure 3: t-SNE representations for the 2nd order CRW (top row) and the HQCW with $\alpha=0.8$ (bottom row), for different embedding sizes $d$.