Patterns and anomalies in k -cores of real-world graphs with applications

How do the k -core structures of real-world graphs look like? What are the common patterns and the anomalies? How can we exploit them for applications? A k -core is the maximal subgraph in which all vertices have degree at least k . This concept has been applied to such diverse areas as hierarchical structure analysis, graph visualization, and graph clustering. Here, we explore pervasive patterns related to k -cores and emerging in graphs from diverse domains. Our discoveries are: (1) Mirror Pattern : coreness (i.e., maximum k such that each vertex belongs to the k -core) is strongly correlated with degree. (2) Core-Triangle Pattern : degeneracy (i.e., maximum k such that the k -core exists) obeys a 3-to-1 power-law with respect to the count of triangles. (3) Structured Core Pattern : degeneracy–cores are not cliques but have non-trivial structures such as core–periphery and communities. Our algorithmic contributions show the usefulness of these patterns. (1) Core-A , which measures the deviation from Mirror Pattern , successfully spots anomalies in real-world graphs, (2) Core-D , a single-pass streaming algorithm based on Core-Triangle Pattern , accurately estimates degeneracy up to 12 × faster than its competitor. (3) Core-S , inspired by Structured Core Pattern , identifies influential spreaders up to 17 × faster than its competitors with comparable accuracy.