Graph topology optimization for fast consensus based on the influence of new links to a quadratic criterion

This paper studies the problem of graph choice in order to maximize consensus convergence speed. Instead of algebraic connectivity, a quadratic criterion is used which is shown to depend on the whole spectrum of the consensus dynamics matrix. The influence of new links to this cost is investigated and a first-order approximation is given. This approximation can be used to compare the effect that different links will have and this is exploited to propose a simple algorithm that starts from a regular lattice and adds links until all nodes have a specific number of neighbours. The resulting graph is compared with random graphs known to have high connectivity because of the small-world effect. Numerical experimentation showed that, for specific parameters, this algorithm provides a graph outperforming a large number of realizations of random graphs both in terms of algebraic connectivity and of the proposed quadratic criterion; for these random graphs it was also shown that this quadratic criterion correlates better with the characteristic path length than algebraic connectivity does.