Iterative Maximum Likelihood on Networks

We consider $n$ agents located on the vertices of a connected graph. Each agent $v$ receives a signal $X_v(0)\sim N(\mu,1)$ where $\mu$ is an unknown quantity. A natural iterative way of estimating $\mu$ is to perform the following procedure. At iteration $t+1$ let $X_v(t+1)$ be the average of $X_v(t)$ and of $X_w(t)$ among all the neighbors $w$ of $v$. It is well known that this procedure converges to $X(\infty) = 1/2|E|^{-1} \sum d_v X_v$ where $d_v$ is the degree of $v$. In this paper we consider a variant of simple iterative averaging, which models "greedy" behavior of the agents. At iteration $t$, each agent $v$ declares the value of its estimator $X_v(t)$ to all of its neighbors. Then, it updates $X_v(t+1)$ by taking the maximum likelihood (or minimum variance) estimator of $\mu$, given $X_v(t)$ and $X_w(t)$ for all neighbors $w$ of $v$, and the structure of the graph. We give an explicit efficient procedure for calculating $X_v(t)$, study the convergence of the process as $t \to \infty$ and show that if the limit exists then $X_v(\infty) = X_w(\infty)$ for all $v$ and $w$. or graphs that are symmetric under actions of transitive groups, we show that the process is efficient. Finally, we show that the greedy process is in some cases more efficient than simple averaging, while in other cases the converse is true, so that, in this model, "greed" of the individual agents may or may not have an adverse affect on the outcome.