Distributed estimation and control of node centrality in undirected asymmetric networks

Measures of node centrality that describe the importance of a node within a network are crucial for understanding the behavior of social networks and graphs. In this paper, we address the problems of distributed estimation and control of node centrality in undirected graphs with asymmetric weight values. In particular, we focus our attention on $\alpha$-centrality, which can be seen as a generalization of eigenvector centrality. In this setting, we first consider a distributed protocol where agents compute their $\alpha$-centrality, focusing on the convergence properties of the method; then, we combine the estimation method with a consensus algorithm to achieve a consensus value weighted by the influence of each node in the network. Finally, we formulate an $\alpha$-centrality control problem which is naturally decoupled and, thus, suitable for a distributed setting and we apply this formulation to protect the most valuable nodes in a network against a targeted attack, by making every node in the network equally important in terms of {\alpha}-centrality. Simulations results are provided to corroborate the theoretical findings.