Graph-balancing algorithms for average consensus over directed networks

Consensus strategies find extensive applications in coordination of robot groups and decision-making of agents. Since balanced graph plays an important role in the average consensus problem and many other coordination problems for directed communication networks, this work explores the conditions and algorithms for the digraph balancing problem. Based on the analysis of graph cycles, we prove that a digraph can be balanced if and only if the null space of its incidence matrix contains positive vectors. Then, based on this result and the corresponding analysis, two weight balance algorithms have been proposed, and the conditions for obtaining a unique balanced solution and a set of analytical results on weight balance problems have been introduced. Then, we point out the relationship between the weight balance problem and the features of the corresponding underlying Markov chain. Finally, two numerical examples are presented to verify the proposed algorithms.