Statistical analysis of articulation points in configuration model networks

An articulation point (AP) in a network is a node whose deletion would split the network component on which it resides into two or more components. APs are vulnerable spots that play an important role in network collapse processes, which may result from node failures, attacks, or epidemics. Therefore, the abundance and properties of APs affect the resilience of the network to these collapse scenarios. Here we present analytical results for the statistical properties of APs in configuration model networks. In order to quantify the abundance of APs, we calculate the probability P(iAP) that a random node, i, in a configuration model network with a given degree distribution, P(K=k), is an AP. We also obtain the conditional probability P(iAP|k) that a random node of degree k is an AP and find that high-degree nodes are more likely to be APs than low-degree nodes. Using Bayes's theorem, we obtain the conditional degree distribution, P(K=k|AP), over the set of APs and compare it to the overall degree distribution P(K=k). We propose a centrality measure based on APs: Each node can be characterized by its articulation rank, r, which is the number of components that would be added to the network upon deletion of that node. For nodes which are not APs, the articulation rank is r=0, while for APs it satisfies r≥1. We obtain a closed-form analytical expression for the distribution of articulation ranks, P(R=r). Configuration model networks often exhibit a coexistence between a giant component and finite components. While the giant component is extensive in the network size and exhibits cycles, the finite components are nonextensive tree structures. To examine the distinct properties of APs on the giant and on the finite components, we calculate the probabilities presented above separately for the giant and the finite components. We apply these results to ensembles of configuration model networks with degree distributions that follow a Poisson distribution (Erdos-Rényi networks), an exponential distribution of the form P(K=k)∼e-αk, and a power-law distribution of the form P(K=k)∼k-γ (scale-free networks), where k≥kmin=1. The implications of these results are discussed in the context of common attack scenarios and network dismantling processes.