Optimizing doubly stochastic matrices for average consensus through swarm and evolutionary algorithms

Doubly-stochastic matrices play a vital role in modern applications of complex networks such as tracking and decentralized state estimation, coordination and control of autonomous agents. A central theme in all of the above is consensus, that is, nodes reaching agreement about the value of an underlying variable (e.g. the state of the environment). Despite the fact that complex networks have been studied thoroughly, the communication graphs are usually described by symmetric matrices due to their advantageous theoretical properties. We do not yet have methods for optimizing generic doubly-stochastic matrices. In this paper, we propose a novel formulation and framework, EvoDSM, for achieving fast linear distributed averaging by: (a) optimizing the weights of a fixed graph topology, and (b) optimizing for the topology itself. We are concerned with graphs that can be described by positive doubly-stochastic matrices. Our method relies on swarm and evolutionary optimization algorithms and our experimental results and analysis showcase that our method (1) achieves comparable performance with traditional methods for symmetric graphs, (2) is applicable to non-symmetric network structures and edge weights, and (3) is scalable and can operate effectively with moderately large graphs without engineering overhead.