Mean degree of a neighbor of a node of degree , as a function of , for the different ensembles considered, with and the same set of imposed degrees sampled from Eq. () with . (The error bars on the values are smaller than the symbols used.)

Degree assortativity as a function of the Zipf exponent , for networks with nodes sampled from the models indicated in the legend.

Poisson multigraph reconstruction accuracy as measured via the similarity of Eq. () for simple graphs sampled from the erased Poisson model with imposed degrees sampled from a Zipf distribution with exponent , as a function of the number of nodes . Each point was averaged over 100 realizations.

Mean degree of a neighbor of a node of degree , as a function of , for a network with nodes sampled from the maximum-entropy ensemble with imposed degrees sampled from Eq. () with , and its inferred Poisson multigraph, using the algorithm described in the text.

Degree assortativity as function of the Zipf exponent , for networks with nodes sampled from the maximum entropy ensemble with Zipf-distributed imposed degrees, and their corresponding inferred Poisson multigraphs.

Degree assortativity for original simple graph , and for the reconstructed multigraph for 816 empirical networks, obtained from the CommunityFitNet [] and Konect [] databases.

Mean degree of a neighbor of a node of degree as a function of , for the internet at the autonomous systems level, both for the original network and reconstructed multigraph, as shown in the legend, which includes the degree assortativity coefficient of each case.

Mean degree of a neighbor of node of degree as a function of , for (a) the metabolic network of  [], (b) the network of leaked emails of the Democratic National Committee, (c) the class dependency graph of a large software project [], and (d) the online social network Flixter [], both for the original network and reconstructed multigraph, as shown in the legend, which includes the degree assortativity coefficient of each case.

Mean degree of a neighbor of a node of degree as a function of , for a political blog network generated form the maximum-entropy DC-SBM, and the corresponding inferred latent multigraph.

Posterior distribution of effective number of groups for the network in Fig. , obtained with (a) the Poisson DC-SBM, (b) the nested Poisson DC-SBM, (c) the erased Poisson DC-SBM, and (d) the nested erased Poisson DC-SBM, the latter showing a distribution highly concentrated on .

Probability of observing a sample from the Poisson and erased Poisson models, as a function of the parameter , as shown in the legend. The vertical and horizontal lines show the maximum of the Poisson at and , and the asymptotic value of for the erased Poisson model as .