Centrality Measure Based on the Laplacian Matrix Spectral Radius Eigenvector Application to the Identification of a Leader

Abstract

Centrality measures evaluate the importance of vertices in a graph. In a multi-agent framework i.e. robots swarm, computing these indicators may be useful in identifying a leader. As a first step, the paper deals with a centrality measure based on the analysis of the eigenvector associated with the largest eigenvalue of the Laplacian matrix. For the studied class of graphs, this centrality measure highlights the most connected vertex which is also associated with the largest binding energy. The results are established for complement of trees. For these highly connected graphs, the most popular centrality measures make hard to distinguish an important vertex. On the contrary, the proposed indicator distinctly highlights this vertex. As a second step, for a network of autonomous agents, the leader identification problem based on this centrality measure is solved in a decentralized way. All the agents cooperate to appoint their leader. The calculations are conducted in the frequency domain and are based on maximum-consensus functions.