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Consensus conditions and convergence speeds are crucial for distributedconsensus algorithms of networked systems. Based on a basic first-orderaverage-consensus protocol with time-varying topologies and additive noises,this paper first investigates its critical consensus condition on networktopology by stochastic approximation frameworks. A new joint-connectivitycondition called extensible joint-connectivity that contains a parameter$\delta$ (termed the extensible exponent) is proposed. With this and a balancedtopology condition, we show that a critical value of $\delta$ for consensus is$1/2$. Optimization on convergence rate of this protocol is furtherinvestigated. It is proved that the fastest convergence rate, which is thetheoretic optimal rate among all controls, is of the order $1/t$ for the besttopologies, and is of the order $1/t^{1-2\delta}$ for the worst topologieswhich are balanced and satisfy the extensible joint-connectivity condition. Forpractical implementation, certain open-loop control strategies are introducedto achieve consensus with a convergence rate of the same order as the fastestconvergence rate. Furthermore, a consensus condition is derived for nonstationary and strongly correlated random topologies. The algorithms andconsensus conditions are applied to distributed consensus computation of mobilead-hoc networks; and their related critical exponents are derived from relativevelocities of mobile agents for guaranteeing consensus.

Critical Connectivity and Fastest Convergence Rates of Distributed Consensus with Switching Topologies and Additive Noises