we consider a distributed estimation setup over time-varying networks connected only on average and provide asymptotic characterization of the estimator as time goes to $\infty$. Furthermore, we specifically characterize the scaling of the asymptotic variance of each entry of the parameter in terms of the number of agents interested in reconstructing the particular entry in question. We also characterize the fundamental condition so as to generate consistent estimates of each entry of the parameter and show that connectivity of the network and global observability is not enough to ensure consistency of the estimates. We direct the reader to assumption A5 and the discussion after assumption A6 for a detailed illustration. In particular, we establish that connectivity of the subgraphs induced by the interest sets is a sufficient condition to enforce assumption A5. It is an open question as to what is a necessary condition (in terms of the network structure, sensing structure, and the interest sets’ structure) so as to enforce assumption A5.

$\mathcal{CIRFE}$: DISTRIBUTED RANDOM FIELDS ESTIMATION {#sec:cirfe}
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In this section, we develop the algorithm $\mathcal{CIRFE}$. The parameter to be reconstructed which is the vector of states accumulated over the entire network is $\mathbf{\btheta}^{\ast}\in\mathbb{R}^{N}$. The sparsifying nature of 