Convergence of type-symmetric and cut-balanced consensus

We consider continuous-time consensus seeking systems whose time-dependent interactions are cut-balanced, in the following sense: if a group of agents influences the remaining ones, the former group is also influenced by the remaining ones by at least a proportional amount. Models involving symmetric interconnections and models in which a weighted average of the agent values is conserved are special cases. We prove that such systems converge unconditionally. We give a sufficient condition on the evolving interaction topology for the limit values of two agents to be the same. Conversely, we show that if our condition is not satisfied, then these limits are generically different. Using the fact that our convergence result is unconditional, we show that it also applies to systems where the agent connectivity and interactions are random, or endogenous, that is, determined by the agent values. We also derive corresponding results for discrete-time systems.