Figure 3

(A) Sketches showing the k-core structure of a toy network and heat maps with a decentralized and centralized network organization. The patterns exemplified in the heat maps A1 to A4 represent scenarios in which the connectivity is involving predominantly a single k-shell. In A1 most connections are between nodes of the central core (\(K=4\)) and other central shells (as in assortative networks). In A2 most connections are between the central core and nodes of external shells (as in dis-assortative networks). In A3 and A4 most connections involve the most external shell (\(K=1\)). This can also happen in an assortative way (A3), where most communication will be between isolated couples of nodes of the external shells, or a dis-assortative way (A4) where most communication is between a large external shell of leaves and the central shells (core-periphery structure). The patterns of heat maps A5 and A6 describe instead networks where connections are distributed between multiple shells in such a way that nodes in a k-shell are mostly connected to the same shell or to shells below and above in the hierarchy. They can be obtained by overlapping single shell effects similar to that of A1. In A5, connections are homogeneously distributed across shells, while in A6 connections are more abundant in the central shells. In (B) and (C), k-shell decomposition and associated heat map for Twitter and Gab, respectively. The distribution of nodes and their degrees across the different k-shells can be appreciated by representing the different nodes in concentric circumferences (plotted above using LaNet-vi⁵²). In the heat maps we focus instead on the connectivity. In the light of the pattern exemplified above, we observe how the Twitter reply and mention and the Gab reply networks can be seen as an overlap of A1 and A2 (with the A3 patterns also playing a minor role in Twitter). The Gab mention network differs notably, and can be seen as an overlap of the A2, A4 and A6 patterns: more communication is thus seen in the intermediate shells (A6) and between marginal and central nodes (A4).