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The mutual information of a class of graphical channels

This paper studies a class of channels obtained by combining a graph and a memoryless channel. A particular example is a memoryless channel pre-coded with a graph-based codes. Other examples are planted constrained satisfaction problems, and probabilistic network models with communities. In an earlier work by the authors, concentration results are obtained for such channels when the graphs is a sparse Erdos-Renyi graph. We overview in this note the approach in an information-theoretic framework, describe how it establishes a conjecture on parity-check codes, and connect the coding and community detection problems.