Weight balance for directed networks: Conditions and algorithms

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 for directed communication networks, this work explores the conditions and algorithms for the digraph balancing problem. It has been proved that a directed graph can be balanced if and only if the null space of its incidence matrix contains positive vectors. Then two weight balance algorithms have been proposed, and the conditions for obtaining a unique balanced solution have been investigated. This work has also pointed 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.