# A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks

arXiv: https://arxiv.org/abs/1809.07955 · HTML: https://mlsanity.com/paper/1809.07955/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Sep 2018.

## Built on by (most cited first, 2 of 2)

- [Distributed Linear Solvers and Data Heterogeneity](https://mlsanity.com/paper/2304.10640/) (2023)
- [Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis](https://mlsanity.com/paper/2609.04329/) (2026)

## Abstract (arXiv)

In this paper, we consider the problem of solving linear algebraic equations of the form Ax=b among multi agents which seek a solution by using local information in presence of random communication topologies. The equation is solved by m agents where each agent only knows a subset of rows of the partitioned matrix [A,b]. We formulate the problem such that this formulation does not need the distribution of random interconnection graphs. Therefore, this framework includes asynchronous updates or unreliable communication protocols without B-connectivity assumption. We apply the random Krasnoselskii-Mann iterative algorithm which converges almost surely and in mean square to a solution of the problem for any matrices A and b and any initial conditions of agents' states. We demonestrate that the limit point to which the agents' states converge is determined by the unique solution of a convex optimization problem regardless of the distribution of random communication graphs. Eventually, we show by two numerical examples that the rate of convergence of the algorithm cannot be guaranteed.