Measuring node similarity using minimum cycles in networks

TL;DR

This work introduces a cycle-based node similarity measure that uses minimal cycles up to a bound $L_{max}^c$ to compute $s_{xy}$, enabling analysis of higher-order structural relationships in networks. It extends this measure to two tasks—link prediction and community detection—by proposing an edge-addition correction and a Cycle-based Hierarchical Clustering (CHC) algorithm, with evaluations on eight real networks. The findings show that the edge-correction can bridge to the Common Neighbors baseline and improve AUC in several datasets, while CHC achieves competitive hierarchical community discovery, particularly when networks exhibit strong cycle signals. Overall, the approach highlights the importance of cyclic motifs (captured by $Z_{cycle}^k$) and the presence of pendant or cut vertices in determining method performance, offering a principled framework for incorporating higher-order topology into network analysis.

Abstract

Cycles are ubiquitous in various networks such as social, biological, and technological systems, where they play a significant functional and dynamical role. This paper proposes a node similarity measure based on minimal simple cycles, referred to as cycle similarity. Specifically, the metric quantifies the similarity between two nodes by considering the minimal cycles that connect them through their neighboring nodes, with an upper bound imposed on the cycle size to ensure computational feasibility. We then systematically examine the effectiveness and applicability of this similarity measure through two fundamental tasks: link prediction and community detection. To address the scarcity of cycles in link prediction, an edge-addition correction strategy is introduced, whereby the existence of a candidate edge is hypothetically assumed before computing node similarity. Experimental results demonstrate that this correction leads to improved performance on datasets including karate, INT, PPI, and Grid. In hierarchical community detection using cycle similarity, we find that the significance of cyclic structures (reflected by Z-scores), the presence of pendant nodes with degree one, and the existence of cut vertices are the primary factors influencing the algorithm's performance.

Figures (3)

• Figure 1: Schematic representation of the example network used to demonstrate the pairwise cycle similarity computation.
• Figure 2: Hierarchical clustering dendrogram and modularity progression for the Karate Club network.
• Figure 3: Visualization of the detected community structure in the Karate Club network.