  Consensus conditions and convergence speeds are crucial for distributed
consensus algorithms of networked systems. Based on a basic first-order
average-consensus protocol with time-varying topologies and additive noises,
this paper first investigates its critical consensus condition on network
topology by stochastic approximation frameworks. A new joint-connectivity
condition called extensible joint-connectivity that contains a parameter
$\delta$ (termed the extensible exponent) is proposed. With this and a balanced
topology condition, we show that a critical value of $\delta$ for consensus is
$1/2$. Optimization on convergence rate of this protocol is further
investigated. It is proved that the fastest convergence rate, which is the
theoretic optimal rate among all controls, is of the order $1/t$ for the best
topologies, and is of the order $1/t^{1-2\delta}$ for the worst topologies
which are balanced and satisfy the extensible joint-connectivity condition. For
practical implementation, certain open-loop control strategies are introduced
to achieve consensus with a convergence rate of the same order as the fastest
convergence rate. Furthermore, a consensus condition is derived for non
stationary and strongly correlated random topologies. The algorithms and
consensus conditions are applied to distributed consensus computation of mobile
ad-hoc networks; and their related critical exponents are derived from relative
velocities of mobile agents for guaranteeing consensus.
