Fig. 3
Learning the parameters of the noisy channel. Each subgraph report results obtained on an artificial network constructed according to a synthetic model similar to the Girvan-Newman benchmark, where N=128 are divided into Q=4 communities of equal size. Nodes within the same group are connected with probability \(\hat {p}_{in}\), while pairs of nodes belonging to different groups are connected with probability \(\hat {p}_{out}\). We consider four different combinations \(\left (\hat {p}_{in}, \hat {p}_{out}\right)\) to generate four different instances of the model. The four different instances of the model are represented in panels a, b, c, and d. Ground-truth values of \(\hat {p}_{in}\) and \(\hat {p}_{out}\) are denoted by the green star symbol in the various panels. We apply the method for community detection introduced in this paper to the graph using the parameters values \(\tilde {p}_{in}\) and \(\tilde {p}_{out}\), randomly sampled in the regime of detectability. The value of the normalized mutual information (NMI) between retrieved and ground-truth community structure is represented by the color of the various points. The green line in the plot identifies combinations of pin and pout compatible with the observed average degree 〈k〉 of the graph. The blue line is y=x, and denotes the region where community structure is present. The orange line is the detectability threshold \(\frac {N(Q-1)}{Q}\left (p_{in} - p_{out}\right) = \sqrt {\frac {N(Q-1)}{Q}\left (p_{in} - p_{out}\right)}\)