for each $l\in\Omega_{n}(t)$ as follows: [$$\begin{aligned}
            \label{eq:cens_2}
            \mathbf{e}_{j}^{\top}\mathbf{x}_{l,n}^{s}(t)=
            \begin{cases}
            \mathbf{e}_{j}^{\top}\mathbf{x}_{n}(t) & \mathcal{I}_{n}(j)\in\mathcal{I}_{l}\\
            0 & \mbox{otherwise}.
            \end{cases}
            \end{aligned}$$]{} where $j\in\{1,\cdots,\left|\mathcal{I}_{n}\right|\}$. The agent $n$ also incorporates the latest sensed information $\mathbf{y}_{n}(t)$ while updating the parameter estimate at each sampling epoch and only retains the components of interest, i.e., those in $\mathcal{I}_n$. For a given vector $\mathbf{z}\in\mathbb{R}^{\left|\mathcal{I}_{n}\right|}$, let $\mathbf{z}^{\mathcal{P}_{\mathcal{I}_n}}\in\mathbb{R}^{N}$ be the vector whose $j$-th component is given by [$$\begin{aligned}
            \label{eq:cens_3}
            \mathbf{e}_{j}^{\top}\mathbf{z}^{\mathcal{P}_{\mathcal{I}_n}}=
            \begin{cases}
            \mathbf{e}_{\mathcal{I}_{n}^{-1}(j)}^{\top}\mathbf{z} & j\in\mathcal{I}_{n}\\
            0 & \mbox{otherwise}.
            \end{cases}
            \end{aligned}$$]{} Finally, for a given vector $\mathbf{z}\in\mathbb{R}^{N}$, $\mathbf{z}_{\mathcal{I}_n}$ denotes the vector in $\mathbb{R}^{\left|\mathcal{I}_{n}\right|}$, where $\mathbf{e}_{j}^{\top}\mathbf{z}_{\mathcal{I}_n}=\mathbf{e}_{\mathcal{I}_{n}(j)}^{\top}\mathbf{z}$.

We now introduce the algorithm $\mathcal{CIRFE}$ for distributed parameter estimation: [$$\begin{aligned}
            \label{eq:cirfe_1}
            &\mathbf{x}_{n}(t+1)=\mathbf{x}_{n}(t)-\underbrace{\sum_{l\in\Omega_{n}(t)}\beta_{t}\left(\mathbf{x}_{l,n}^{s}(t)-\mathbf{x}_{l,n}^{r}(t)\right)}_{\text{Neighborhood Consensus}}\nonumber\\&+\underbrace{\alpha_{t}\mathbf{H}_{n}^{\top}\mathbf{R}_{n}^{-1}\left(\mathbf{y}_{n}(t)-\mathbf{H}_{n}\mathbf{x}_{n}^{\mathcal{P}_{\mathcal{I}_n}}(t)\right)_{\mathcal{I}_{n}}}_{\text{Local Innovation}},
            \end{aligned}$$]{} where $\Omega_{n}(t)$ represents the neighborhood of agent $n$ at time $t$; and $\left\{\beta_{t}\right\}$ and $\left\{\alpha_{t}\right\}$ are the consensus and innovation weight sequences given by [$$\begin{aligned}
            \label{eq:ab}
            \beta_{t} = \frac{\beta_{0}}{(t+1)^{\delta_{1}}}, \alpha_{t}=\frac{a}{t+1},
            \end{aligned}$$]{} where $a, b > 0$ and $0<\delta_{1}<1/2-1/(2+\epsilon_{1})$ and $\epsilon_{1}$ was as defined in Assumption \[as:1\]. It is to be noted that with the interest set of each agent being $\mathcal{I}_{n}=\{1,2,\cdots,N\}$, we have that the update in reduces to the classical consensus+innovations update for linear parameter estimation schemes (see, [@kar2013distributed] for example). Thus, the classical consensus+innovations parameter estimation scheme, is strictly a special 