Algorithm 1. DHIWCF implemented by node $i$ at time instant $k$.

1. Obtain the local measurement ${\mathit{z}}_{i,k}$ with covariance matrix ${\mathit{R}}_{i,k}^{}$.

2. Compute the measurement contribution vector and contribution matrix. $$\{\begin{array}{l}{\mathit{u}}_i^{}={\mathit{H}}_{i,k}^{\mathrm{T}}{\mathit{R}}_{i,k}^{-1}{\mathit{z}}_{i,k}\hfill \\ {\mathit{U}}_i^{}={\mathit{H}}_{i,k}^{\mathrm{T}}{\mathit{R}}_{i,k}^{-1}{\mathit{H}}_{i,k}^{}\hfill \end{array}$$ (35)

3. Broadcast state message $\left\{{\mathit{y}}_{i,k|k-1}^{},{\mathit{Y}}_{i,k|k-1}^{},{\mathit{u}}_i^{},{\mathit{U}}_i^{}\right\}$ to its neighboring nodes $j\in {\mathcal{N}}_i$.

4. Receive state message $\left\{{\mathit{y}}_{j,k|k-1}^{},{\mathit{Y}}_{j,k|k-1}^{},{\mathit{u}}_j^{},{\mathit{U}}_j^{}\right\}$ from its neighboring nodes $j\in {\mathcal{N}}_i$.

5. Compute the initial values. $$\{\begin{array}{l}{\widehat{\mathit{y}}}_{i,k|k}^0=\frac{1}{1+d_i}{\displaystyle \sum _{j\in {\mathcal{J}}_i}{\widehat{\mathit{y}}}_{j,k|k-1}^{}}+{\displaystyle \sum _{j\in {\mathcal{J}}_i}{\left({\mathit{H}}_{j,k}\right)}^{\mathrm{T}}{\mathit{R}}_{j,k}^{-1}{\mathit{z}}_{j,k}}\hfill \\ {\mathit{Y}}_{i,k|k}^0=\frac{1}{1+d_i}{\displaystyle \sum _{j\in {\mathcal{J}}_i}{\mathit{Y}}_{j,k|k-1}^{}}+{\displaystyle \sum _{j\in {\mathcal{J}}_i}{\left({\mathit{H}}_{j,k}\right)}^{\mathrm{T}}{\mathit{R}}_{j,k}^{-1}{\mathit{H}}_{j,k}}\hfill \end{array}$$ (36)

6. Perform consensus.   for $l=1:L$ do (1)Send ${\widehat{\mathit{y}}}_{i,k|k-1}^{l-1}$ and ${\mathit{Y}}_{i,k|k-1}^{l-1}$ to its neighbors $j\in {\mathcal{N}}_i$; (2)Receive ${\widehat{\mathit{y}}}_{j,k|k-1}^{l-1}$ and ${\mathit{Y}}_{j,k|k-1}^{l-1}$ from its neighbors $j\in {\mathcal{N}}_i$; (3) Update its consensus state. $$\{\begin{array}{l}{\widehat{\mathit{y}}}_{i,k|k}^l={\widehat{\mathit{y}}}_{i,k|k}^{l-1}+{\displaystyle \sum _{j\in {\mathcal{N}}_i}{\pi }_{i,j}\left({\widehat{\mathit{y}}}_{j,k|k}^{l-1}-{\widehat{\mathit{y}}}_{i,k|k}^{l-1}\right)}\hfill \\ {\mathit{Y}}_{i,k|k}^l={\mathit{Y}}_{i,k|k}^{l-1}+{\displaystyle \sum _{j\in {\mathcal{N}}_i}{\pi }_{i,j}\left({\mathit{Y}}_{j,k|k}^{l-1}-{\mathit{Y}}_{i,k|k}^{l-1}\right)}\hfill \end{array}$$ (37)   end for

7. Compute the posterior estimate.

${\widehat{\mathit{x}}}_{i,k|k}^{}={\left({\mathit{Y}}_{i,k|k}^L\right)}^{-1}{\widehat{\mathit{y}}}_{i,k|k}^L$, ${\mathit{Y}}_{i,k|k}^{}={\mathit{Y}}_{i,k|k}^L$, ${\widehat{\mathit{y}}}_{i,k|k}^{}={\widehat{\mathit{y}}}_{i,k|k}^L$

8. Prediction at time instant $k+1$

${\widehat{\mathit{x}}}_{i,k+1|k}^{}={\mathit{F}}_k{\widehat{\mathit{x}}}_{i,k|k}^{}$, ${\mathit{Y}}_{i,k+1|k}^{}={\Psi }_k\left({\mathit{Y}}_{i,k|k}^{}\right)$, ${\widehat{\mathit{y}}}_{i,k+1|k}^{}={\mathit{Y}}_{i,k+1|k}^{}{\widehat{\mathit{x}}}_{i,k+1|k}^{}$