Distributed consensus-based estimation of the leading eigenvalue of a non-negative irreducible matrix

This paper presents an algorithm to solve the problem of estimating the largest eigenvalue and its corresponding eigenvector for irreducible matrices in a distributed manner. The proposed algorithm utilizes a network of computational nodes that interact with each other, forming a strongly connected digraph where each node handles one row of the matrix, without the need for centralized storage or knowledge of the entire matrix. Each node possesses a solution space, and the intersection of all these solution spaces contains the leading eigenvector of the matrix. Initially, each node selects a random vector from its solution space, and then, while interacting with its neighbors, updates the vector at each step by solving a quadratically constrained linear program (QCLP). The updates are done so that the nodes reach a consensus on the leading eigenvector of the matrix. The numerical outcomes demonstrate the effectiveness of our proposed method.