Algorithm 1 VGAECD-OPT

Input: Features $\mathbf{X}$, Adjacency Matrix $\mathbf{A}$, no. of comm. K, filter size $\mathcal{D}$, number of epochs L,              NEM steps R.      Output: Community Assignment Probability $\mathbf{\gamma }$ and Reconstructed Adjacency matrix $\tilde{\mathbf{A}}$ 1:$\mathbf{\pi }\sim \mathcal{U}(0,1)$ 2:for$l=1,\dots ,L$do 3:    for $i=1,\dots ,N$ do 4:        ${\mathbf{\mu }}_i={\mathrm{SGC}}_{\mathbf{\mu }}({\mathbf{x}}_i,{\mathbf{a}}_i)$ 5:        ${\mathbf{\sigma }}_i={\mathrm{SGC}}_{\mathbf{\sigma }}({\mathbf{x}}_i,{\mathbf{a}}_i)$ 6:        Sample ${\mathbf{z}}_i\sim \mathcal{N}({\mathbf{\mu }}_x{|}_i,\mathrm{diag}\left({\mathbf{\sigma }}_x^2{|}_i\right))$ 7:        Obtain ${\tilde{\mathbf{a}}}_i=\sigma \left({\mathbf{z}}_i^{\top }{\mathbf{z}}_j\right)$ 8:        Compute loss, ${\mathcal{L}}_{\mathrm{ELBO}}$            ▹ From Equation (12) 9:           and backpropagate gradients. 10:    end for 11:    for $r=1,\dots ,R$ do 12:        Compute E-Step: $\mathbf{\gamma }$, $\{{\mathbf{\mu }}_c$, ${\mathbf{\sigma }}_c\}$        ▹ From Equation (14) 13:        Compute M-Step: ${\mathbf{\mu }}_c$, ${\mathbf{\sigma }}_c$, $\left\{\mathbf{\gamma }\right\}$        ▹ From Equation (16) 14:        Compute loss, ${\mathcal{L}}_{\mathrm{comm}}$ 15:           and backpropagate gradients. 16:    end for 17:end for 18:Extract community assignment ${arg\; max}_k\mathbf{\gamma }$ 19:Return $\tilde{\mathbf{A}}=\{{\tilde{\mathbf{a}}}_1,\dots ,{\tilde{\mathbf{a}}}_N\}$