Community Landscapes: An Integrative Approach to Determine Overlapping Network Module Hierarchy, Identify Key Nodes and Predict Network Dynamics
The figure shows the decreasing integrity of the USA Western Power Grid network  as a function of the number of nodes removed. Nodes were removed in the order of their decreasing degree (black alternating dashes and dots) betweenness centrality  (red dashed lines) or “bridgeness” (solid blue lines), where “bridgeness” measures the overlap of the given node between different modules as described in detail in Section V.6.d. in the Electronic Supplementary Material S1. Network integrity has been calculated after Latora and Marchiori . Bridgeness was calculated from the modular structure of the lowest hierarchical level as determined by the LinkLand influence function calculation method and the TotalHill module membership assignment method. During the post-processing of the module assignment, we merged the modules with ProportionalHill module membership assignment-based correlation higher than 0.9 (see Section VI. in the Electronic Supplementary Material S1, we received similar results without this merging process; data not shown). On the vertical axis of the insets the betweenness centrality (left, color-coded from red to yellow) and bridgeness (right, color-coded from blue to green) of the nodes of the USA Western Power Grid network are shown. Networks on the insets were laid out using the Kamada-Kawai algorithm and visualized with a custom Blender script.