, E(G)_{ij}=1\} = q_2.\end{aligned}$$

    Put differently, is a graph model where information is only available on the base-graph $G$, the $*$-variable encodes the absence of information, and when information is available, two vertices are connected with probability $q_1$ if they are in the same community and with probability $q_2$ if they are in different communities. When $G=K_n$ is the complete graph and $X$ is uniformly distributed, this is the standard stochastic block model with two communities, and $q_1=a/n$, $q_2=b/n$ gives the sparse regime of [@decelle; @mossel-sbm]. In the case of , the linear structure implies $q_1=1-q_2=\varepsilon$, which may be both of order 1, whereas the base-graph may be sparse. This raises an important distinction: in the sparse stochastic block model, it is assumed that most node pairs are unlikely to be connected, whereas in the model of this paper, it is assumed that information is not available for most node pairs. These are not the same, and the latter may help preventing false-alarm type of errors. However, we restrict ourselves in this paper to the symmetric case $q_1=1-q_2=\varepsilon$, which simplifies the computations.

-   [**Correlation clustering:**]{} [@corr_cluster] considers the problem of clustering a complete graph with edges labeled in $\{-,+\}$ in 