Density-friendly Graph Decomposition

Abstract

Decomposing a graph into a hierarchical structure via k-core analysis is a standard operation in any modern graph-mining toolkit. k-core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere de-gree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connected-ness, and it allows to reveal the structural organization of the graph. Despite the fact that k-core analysis relies on vertex de-grees, k-cores do not satisfy a certain, rather natural, density property. Simply put, the most central k-core is not nec-essarily the densest subgraph. This inconsistency between k-cores and graph density provides the basis of our study. We start by defining what it means for a subgraph to be locally-dense, and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by k-cores, but in this case the components are ar-ranged in order of increasing density. We show that such a locally-dense decomposition for a graph G = (V,E) can be computed in polynomial time. The running time of the exact decomposition algorithm is O(|V |2|E|) but is signifi-cantly faster in practice. In addition, we develop a linear-time algorithm that provides a factor-2 approximation to the optimal locally-dense decomposition. Furthermore, we show that the k-core decomposition is also a factor-2 ap-proximation, however, as demonstrated by our experimental evaluation, in practice k-cores have different structure than locally-dense subgraphs, and as predicted by the theory, k-cores are not always well-aligned with graph density