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, particularly suitable for graphs with asymmetric interactions. In this setting, our contribution is twofold: first we derive a distributed protocol where agents can locally compute their $\alpha$-centrality index by means of local interactions; then, we focus our attention on the problem of controlling the weight matrix of the graph in such a way that the network reaches a desired $\alpha$-centrality vector. The interest of our solution lies in obtaining the control objective with minimum changes in the original influence matrix. Moreover, with our algorithm every agent is able to reach the desired value locally and in finite-time. The two algorithms are then applied to two problems of interest in real life. The estimation method is used together with a consensus algorithm to achieve a consensus value weighted by the influence of each node in the network. The control algorithm is exploited 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.