Figure 2. Illustration of the power of our unified representation theory on the Zachary karate club network⁵².

The optimal layout (η = 2.1%, see Eq. (3)) in terms of d = 2 dimensional Gaussians is shown by a density plot in (a) and by circles of radiuses σ_i in (b). (c) the best layout is obtained in d = 3 (η = 1.7%), where the radiuses of the spheres are chosen to be proportional to σ_i. (d) the original data matrix of the network with an arbitrary ordering. (e) the d = 1 layout (η = 4.5%) yields an ordering of the original data matrix of the network. (f) the optimal coarse-gaining of the data matrix yields a tool to zoom out from the network in accordance with the underlying community structure. The colors indicate our results at the level of two clusters, being equivalent to the ones given by popular community detection techniques, such as the modularity optimization⁵ or the degree-corrected stochastic block model²⁵. We note, that the coarse-graining itself does not yield a unique ordering of the nodes, therefore an arbitrarily chosen compatible ordering is shown in this panel.