Algorithm 1: The seminal diffusion distributed Kalman filter.

For the node $i\in V$ and node $j\in N_i$,   Initialize with:     ${\widehat{x}}_{i,0\mid 0}=\mathbb{E}{x}_0,$     $P_{i,0\mid 0}=\mathbb{E}\left[(x_0-\mathbb{E}{x}_0){(x_0-\mathbb{E}{x}_0)}^{\mathrm{T}}\right];$   Local update:     ${\widehat{x}}_{i,t\mid t-1}=A{\widehat{x}}_{i,t-1\mid t-1}$,     $P_{i,t\mid t-1}=A{P}_{i,t-1\mid t-1}{A}^{\mathrm{T}}+\mathsf{\Gamma }Q{\mathsf{\Gamma }}^{\mathrm{T}}$,     ${\left(P_{i,t\mid t}\right)}^{-1}={\left(P_{i,t\mid t-1}\right)}^{-1}+H_i^{\mathrm{T}}{R}_i^{-1}{H}_i$,     ${\widehat{x}}_{i,t\mid t}^{\mathrm{loc}}={\widehat{x}}_{i,t\mid t-1}+P_{i,t\mid t}{H}_i^{\mathrm{T}}{R}_i^{-1}(y_{i,t}-H_i{\widehat{x}}_{i,t\mid t-1})$;   Communication and fusion update:     Send ${\widehat{x}}_{j,t\mid t}^{\mathrm{loc}}$ to adjacent node i,     ${\widehat{x}}_{i,t\mid t}={\sum }_{k\in N_i+i}{w}_k{\widehat{x}}_{k,t\mid t}^{\mathrm{loc}},$   where     ${\sum }_{k\in N_i+i}{w}_k=1, 0\le w_k\le 1$.