Fast Consensus over Almost Regular Directed Graphs^*

This paper studies an open consensus network design problem: identifying the optimal simple directed graphs, given a fixed number of vertices and arcs, that maximize the second smallest real part of all Laplacian eigenvalues, referred to as algebraic connectivity. For sparse and dense graphs, the class of all optimal directed graphs that maximize algebraic connectivity is theoretically identified, leading to the fastest consensus. For general graphs, a computationally efficient sequence of almost regular directed graphs is proposed to achieve fast consensus, with algebraic connectivity close to the optimal value.