Figure 1.

Left column: 3D heat maps with top-view scatter plots of posterior probabilities over block assignments, according to our belief-propagation algorithm on one of the sample networks. It shows that the ordinary and the degree-corrected block models converge to the same marginal distributions of G_u (first assumption). Furthermore, the concentration of the marginals suggests the strong dominance of the ground state (second assumption). The x- and y-axes in both parts are the marginal probabilities of being in block 1 according to ordinary and degree-corrected models, respectively. The Z-axes are logarithmic. Right column: QQ plots (100 samples) for comparing the distributions of the ground state energy difference to those of free energy difference under the null model. A point in a QQ plot corresponds to one of the quantiles of the first distribution (x-coordinate) plotted against the same quantile of the second distribution (y-coordinate). Observe that here the points approximately lie on the line y = x, which means that the two distributions are very similar. Top row: synthetic network with n = 10³, k = 2 equally-sized blocks (γ₁ = γ₂ = 1/2), average degree μ_r = 11, and associative structure with ω₁₂/ω₁₁ = ω₂₁/ω₂₂ = 1/11. Bottom row: bootstrapped networks with the assignments and parameters of the political blog network.