  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. We present analytical results for the statistical
properties of APs in configuration model networks. In order to quantify their
abundance, we calculate the probability $P(i \in {\rm AP})$, that a random
node, i, in a configuration model network with P(K=k), is an AP. We also obtain
the conditional probability $P(i \in {\rm AP}|k)$ 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' theorem, we obtain the conditional degree
distribution, $P(K=k|{\rm AP})$, over the set of APs and compare it to P(K=k).
We propose a new 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 $r \ge 1$. We obtain
a closed form expression for the distribution of articulation ranks, P(R=r).
Configuration model networks often exhibit a coexistence between a giant
component and finite components. 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 a Poisson,
exponential and power-law degree distributions. The implications of these
results are discussed in the context of common attack scenarios and network
dismantling processes.
