On the steady-state range of averaging dynamics

The steady-state range, or simply the range, of a network of multiple agents with fixed initial conditions refers to the intersection of all convex subsets of the state space, each containing all agent states as time grows. The notion of range is similarly defined for a subset of agents. Given the update rule of a network with unknown initial conditions, a subset of agents is said to be range-defining if, for any initial conditions, its range equals the range of the whole network. If, in addition, the subset remains range-defining regardless of its members' update rules-in the sense that the agents in that subset may or may not abide by their own update rules- the subset is said to be range-deciding. In this paper, minimal range-defining/-deciding sets given a general linear averaging update rule, which is uniquely characterized by a sequence of row-stochastic matrices, are investigated. More specifically, convergence properties of the sequence are employed to obtain a minimal range-defining/-deciding set and its size.