Distance-generalized Core Decomposition

Francesco Bonchi, Arijit Khan, Lorenzo Severini

Abstract: The k-core of a graph is defined as the maximal subgraph in which every vertex is connected to at least k other vertices within that subgraph. In this work we introduce a distance-based generalization of the notion of k-core, which we refer to as the $(k,h)$-core, i.e., the maximal subgraph in which every vertex has at least k other vertices at distance $łeq h$ within that subgraph. We study the properties of the $(k,h)$-core showing that it preserves many of the nice features of the classic core decomposition (e.g., its connection with the notion of distance-generalized chromatic number ) and it preserves its usefulness to speed-up or approximate distance-generalized notions of dense structures, such as h-club. Computing the distance-generalized core decomposition over large networks is intrinsically complex. However, by exploiting clever upper and lower bounds we can partition the computation in a set of totally independent subcomputations, opening the door to top-down exploration and to multithreading, and thus achieving an efficient algorithm.