that extends the basic Erdős-Rényi model [@ER-seminal], which often turns out to be too simplistic since all vertices have the same expected degree and no cluster structure appears. One possibility to obtain cluster structure is precisely to attach latent variables to the vertices and assume an edge distribution that depends on these variables. There are various models with latent variables, such as the exchangeable, inhomogeneous or stochastic block models [@airoldi; @sbm1; @sbm2; @dyer; @newman2; @sbm-book]. The general model in can be used for this purpose, as explained above in the special case of . The vertex-variables represent the community assignment, the edge-variables the connectivity, and the graph $G$ encodes where the information is available. The model is related to the stochastic block model through the following [**censored block model**]{}, introduced in [@random] in a different context. Given a base-graph $G=(V,E(G))$ and a community assignment $X \in \{0,1\}^V$, the following random graph is generated on the vertex set $V$ with ternary edge labels $E_{ij} \in \{*,0,1\}$ drawn independently with the following probability distribution:

    \[bl:filBlMod\] $$\begin{aligned}
    &\mathbb{P}\{ E_{ij} = * | E(G)_{ij}=0 \} =1\\
    &\mathbb{P}\{ E_{ij} = 1 | X_i=X_j , E(G)_{ij}=1\} = q_1,\\
    &\mathbb{P}\{ E_{ij} = 1 | X_i \neq X_j 