# Distributed Averaging With Random Network Graphs and Noises

## TL;DR

This paper analyzes distributed averaging algorithms over random, noisy, time-varying networks, providing conditions for convergence and quantifying steady-state errors.

## Contribution

It introduces new convergence conditions for stochastic averaging algorithms on random graphs with measurement noise, extending existing theories.

## Key findings

- Agents converge to a common state in mean square and almost surely.
- Convergence requires the graph flow to be conditionally balanced and jointly connected.
- Steady-state error bounds depend on noise energy and algorithm parameters.

## Abstract

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.