Shared Nodes of Overlapping Communities in Complex Networks

TL;DR

The paper presents a threshold-based framework to identify overlapping nodes within overlapping communities in complex networks, separating genuine shared nodes from noise and internal substructures. It builds on a quality-based community detection approach using an influence-spreading matrix to obtain building blocks (intersections) and applies a threshold rule to classify overlapping nodes. Through five real-world network datasets, the method demonstrates robustness: increasing the threshold systematically reduces overlaps while preserving core bridging nodes. The framework is designed as a flexible post-processing tool compatible with various overlapping community detection methods and highlights implications for understanding information flow and network resilience. Overall, it offers a practical, tunable approach to clarify complex overlapping structures in social, communication, and ecological networks.

Abstract

Overlapping communities are key characteristics of the structure and function analysis of complex networks. Shared or overlapping nodes within overlapping communities can form either subcommunities or act as intersections between larger communities. Nodes at the intersections that do not form subcommunities can be identified as overlapping nodes or as part of an internal structure of nested communities. To identify overlapping nodes, we apply a threshold rule based on the number of nodes in the nested structure. As the threshold value increases, the number of selected overlapping nodes decreases. This approach allows us to analyse the roles of nodes considered overlapping according to selection criteria, for example to reduce the effect of noise. We illustrate our method by using three small and two larger real-world network structures. In larger networks, minor disturbances can produce a multitude of slightly different solutions, but the core communities remain robust, allowing other variations to be treated as noise. While this study employs our own method for community detection, other approaches can also be applied. Exploring the properties of shared nodes in overlapping communities of complex networks is a novel area of research with diverse applications in social network analysis, cybersecurity, and other fields in network science.

Figures (8)

• Figure 1: Diagram showing the process by which the building blocks of overlapping communities are generated and shared nodes are identified using a threshold. The purple boxes indicate the algorithms in Appendices A and B that describe each phase.
• Figure 2: An illustrative example of two divisions of the network into two communities $M_1$ and $M_2$ (left) and $K_1$ and $K_2$ (middle). Nodes in $K_2$ are almost the same as in $M_2$. The interception of these communities is marked as $M_2\cap K_2$ (right).
• Figure 3: Zachary's Karate Club network overlapping nodes identified for two threshold parameter values ($thr=0.0$ and $0.5$).
• Figure 4: Les Misérables network overlapping nodes identified for three increasing threshold parameter values ($thr=0.0,1.0$, and $3.0$).
• Figure 5: Building blocks identified by Riolo and Newman using the information-theoretic method in their study riolo. These building blocks include both communities and their intersections.