case of the update in .

We now illustrate the introduced setup and algorithm with a $5$ agents network example in Fig. \[5node-RLU\].

![A network example emphasizing the notion of structural observability.[]{data-label="5node-RLU"}](net_example.pdf){height="1in" width="1in"}

Each node $n$ corresponds to a physical component $\theta_{n}^{\ast}$. Thus, $\btheta^{\ast}\in\mathbb{R}^{5}$. The solid lines connecting the nodes correspond to the inter-node communication pattern. Each node observes a noisy scalar functional. In particular, we assume [$$\begin{aligned}
                \label{RLU-dis2}
                &y_{3}(t)=\frac{1}{3}\left(\theta_{2}^{\ast}+\theta_{3}^{\ast}+\theta_{4}^{\ast}\right)+\gamma_{3}(t)\nonumber\\
                &y_{n}(t)=\theta_{n}^{\ast}+\gamma_{n}(t), n=1,2,4,5.
                \end{aligned}$$]{} Note that then the noise covariance matrix $\mathbf{R}_n$ is a positive scalar, $n=1,2,...,5$. Also, for $n \neq 5$, $\mathbf{H}_n$ is a 5-dimensional (row) vector with all entries equal to zero except the $n$-th entry which equals one. On the other hand, $\mathbf{H}_3 = [\,0,\,1/3,\,1/3,\,1/3,\,0\,]$. we have that $\widetilde{\mathcal{I}_{n}}= \{n\}$ for $n=1,2,4,5$, and $\widetilde{\mathcal{I}_{3}}= \{2,3,4\}$. Let us also assume that the agents’ interest sets are given by ${\mathcal{I}_{n}} = \widetilde{\mathcal{I}_{n}}$, for each $n=1,2,...,5$. For notational simplicity, we omit time index $t$ when writing the agents’ estimates; that is, we write $\mathbf{x}_n$ in place of $\mathbf{x}_n(t)$. Also, we denote by $[\mathbf{x}_n]_i$ the $i$-th entry of $\mathbf{x}_n$. Then, agent 3’s estimate $\mathbf{x}_3$ is a $3 \times 1$ vector, with $[\mathbf{x}_3]_1$ being an estimate of $\theta_{2}^{\ast}$, $[\mathbf{x}_3]_2$ being an estimate of 