Skeletal Cores and Graph Resilience

In network analysis, one of the most important structures is the k-core: the maximal set of nodes such that each node in the k-core has at least k neighbors within the core. Recently, the notion of the skeletal k-core- a minimal subgraph that preserves the core structure of the graph- has attracted attention. However, the literature to date has contained only a biased greedy heuristic for sampling skeletal cores, which resulted in a skewed analysis of the network. In this work, we introduce a novel MCMC algorithm for sampling skeletal cores uniformly at random, as well as a novel algorithm for estimating the size of the space of skeletal k-cores, which, as we show, is important for understanding the core resilience of the network. With these algorithms, we demonstrate the relationship between resilience of the network and the core structure of the graph and suggest fast heuristics for evaluating graph structure from a skeletal cores perspective. We show that the normalized number of skeletal cores in the graph correlates with the resilience of k-core towards edge deletion attacks.