# Distributed estimation and control of node centrality in undirected asymmetric networks

## TL;DR

This paper develops distributed algorithms for estimating and controlling node alpha-centrality in undirected networks with asymmetric weights, enabling network protection and influence balancing.

## Contribution

It introduces a novel distributed estimation and control framework for alpha-centrality, extending eigenvector centrality to asymmetric undirected graphs.

## Key findings

- Convergence of the distributed alpha-centrality estimation protocol.
- Effective consensus algorithm weighted by node influence.
- Successful application to protect key nodes against targeted attacks.

## Abstract

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.