  Components of complex systems are often classified according to the way they
interact with each other. In graph theory such groups are known as clusters or
communities. Many different techniques have been recently proposed to detect
them, some of which involve inference methods using either Bayesian or Maximum
Likelihood approaches. In this article, we study a statistical model designed
for detecting clusters based on connection similarity. The basic assumption of
the model is that the graph was generated by a certain grouping of the nodes
and an Expectation Maximization algorithm is employed to infer that grouping.
We show that the method admits further development to yield a stability
analysis of the groupings that quantifies the extent to which each node
influences its neighbors group membership. Our approach naturally allows for
the identification of the key elements responsible for the grouping and their
resilience to changes in the network. Given the generality of the assumptions
underlying the statistical model, such nodes are likely to play special roles
in the original system. We illustrate this point by analyzing several empirical
networks for which further information about the properties of the nodes is
available. The search and identification of stabilizing nodes constitutes thus
a novel technique to characterize the relevance of nodes in complex networks.
