Beyond Neighbors: Distance-Generalized Graphlets for Enhanced Graph Characterization

Graphs are widely used to model complex systems across various domains, including social networks and biological systems. A key task in graph analysis is identifying recurring structural patterns, known as graphlets, which capture connectivity among a fixed-size subset of nodes. While graphlets have been extensively applied in tasks such as measuring graph similarity and identifying communities, conventional graphlets focus only on direct connections between nodes. This limitation overlooks potential insights from more distant relationships within the graph structure. In this paper, we introduce (d,s)-graphlets, a generalization of size-s graphlets that incorporates indirect connections between nodes up to distance d. This new formulation provides a more fine-grained and comprehensive understanding of local graph structures. To efficiently count (d,s)-graphlets in a graph, we present EDGE, an exact counting algorithm that employs optimized combinatorial techniques to significantly reduce computational complexity compared to naive enumeration. Our empirical analysis across diverse real-world datasets demonstrates that (d,s)-graphlets provide superior graph characterization, outperforming conventional graphlets in a graph clustering task. Moreover, our case studies show that (d,s)-graphlets uncover non-trivial insights that would remain undiscovered when using conventional graphlets.