On The Limits of Finite-Time Distributed Consensus Through Successive Local Linear Operations

In this work, we explore the limits of finite-time distributed consensus through the intersection of graph filters and matrix function theory. We focus on algorithms capable to compute the consensus exactly through filtering operations over a graph, and that have been proven to converge in finite time. In this context, we show that there exists an algebraic algorithm that can minimize the minimum polynomial of a matrix whose support is known. Different from previous works, we leverage the structure of matrices that share the same support and are diagonalizable by the eigenbasis of the graph shift operator to prove a theoretical result with respect to the minimum number of diffusion steps required to reach consensus. We show that the previously known bound on the number of consensus iterations can be further reduced in accordance to the algebraic properties of the matrix representation of the network. Finally, insights with respect to the relation between the graph topology and the algebraic properties of such matrices are provided in order to encourage further discussion on the role of eigenvalues and eigenvectors in the network topology.