The paper addresses the problem of distributed average consensus with least-mean-square deviation, a stochastic model used in applications such as load balancing, coordination of mobile agents, and network synchronization. Each node updates its variable using a weighted average of its neighbors' values corrupted by additive noise. The quality of consensus is measured by the total mean-square deviation of the individual variables from their average, which converges to a steady-state value. The paper aims to find the edge weights that minimize this steady-state mean-square deviation, formulating the problem as a convex optimization task. It provides computational methods for solving this problem, including exploiting sparsity in the graph structure. The paper also discusses special cases, such as constant edge weights and edge-transitive graphs, and compares the solutions with other weight design methods, including those that minimize the asymptotic convergence factor. Numerical examples are provided to illustrate the performance of different weight designs.The paper addresses the problem of distributed average consensus with least-mean-square deviation, a stochastic model used in applications such as load balancing, coordination of mobile agents, and network synchronization. Each node updates its variable using a weighted average of its neighbors' values corrupted by additive noise. The quality of consensus is measured by the total mean-square deviation of the individual variables from their average, which converges to a steady-state value. The paper aims to find the edge weights that minimize this steady-state mean-square deviation, formulating the problem as a convex optimization task. It provides computational methods for solving this problem, including exploiting sparsity in the graph structure. The paper also discusses special cases, such as constant edge weights and edge-transitive graphs, and compares the solutions with other weight design methods, including those that minimize the asymptotic convergence factor. Numerical examples are provided to illustrate the performance of different weight designs.