Optimal k-centers of a graph: a control-theoretic approach

Abstract: In a network consisting of n nodes, our goal is to identify the most central k nodes with respect to the proposed definitions of centrality. Depending on the specific application, there exist several metrics for quantifying k-centrality, and the subset of the best k nodes naturally varies based on the chosen metric. In this paper, we propose two metrics and establish connections to a well-studied metric from the literature (specifically for stochastic matrices). We prove these three notions match for path graphs. We then list a few more control-theoretic notions and compare these various notions for a general randomly generated graph. Our first metric involves maximizing the shift in the smallest eigenvalue of the Laplacian matrix. This shift can be interpreted as an improvement in the time constant when the RC circuit experiences leakage at certain k capacitors. The second metric focuses on minimizing the Perron root of a principal sub-matrix of a stochastic matrix, an idea proposed and interpreted in the literature as manufacturing consent. The third one explores minimizing the Perron root of a perturbed (now super-stochastic) matrix, which can be seen as minimizing the impact of added stubbornness. It is important to emphasize that we consider applications (for example, facility location) when the notions of central ports are such that the set of the best k ports does not necessarily contain the set of the best k-1 ports. We apply our k-port selection metric to various network structures. Notably, we prove the equivalence of three definitions for a path graph and extend the concept of central port linkage beyond Fiedler vectors to other eigenvectors associated with path graphs.