  We consider discrete-time distributed averaging algorithms over multi-agent
networks with measurement noises and time-varying random graph flows. Each
agent updates its state by relative states between neighbours with both
additive and multiplicative measurement noises. The network structure is
modeled by time-varying random digraphs, which may be spatially and temporally
dependent. By developing difference inequalities of proper stochastic Lyapunov
function, the algebraic graph theory and martingale convergence theory, we
obtain sufficient conditions for stochastic approximation type algorithms to
achieve mean square and almost sure average consensus. We prove that all states
of agents converge to a common variable in mean square and almost surely if the
graph flow is conditionally balanced and uniformly conditionally jointly
connected. The mathematical expectation of the common variable is right the
average of initial values, and the upper bound of the mean square steady-state
error is given quantitatively related to the weights, the algorithm gain and
the energy level of the noises.
