Internal stability of linear consensus processes

In a network of n agents, consensus means that all n agents reach an agreement on a specific value of some quantity via local interactions. A linear consensus process can typically be modeled by a discrete-time linear recursion equation or a continuous-time linear differential equation, whose equilibria include nonzero states of the form a1 where a is a constant and 1 is a column vector in ℝⁿ whose entries all equal 1. Using a suitably defined semi-norm, this paper extends the standard notions of uniform asymptotic stability and exponential stability from linear systems to linear recursions and differential equations of this type. It is shown that these notions are equivalent just as they are for conventional linear systems. The main contributions of this paper are first to provide a simple, direct proof of the necessary graph-theoretic condition given in [1] for a discrete-time linear consensus process to be exponentially stable, and second to derive a necessary graph-theoretic condition for a piecewise time-invariant continuous-time linear consensus process to be exponentially stable.