Community Landscapes: An Integrative Approach to Determine Overlapping Network Module Hierarchy, Identify Key Nodes and Predict Network Dynamics
For this illustrative example we used the network science co-authorship network  without link weights using the LinkLand influence function calculation method with the TotalHill module membership assignment method. The network was laid out using the Kamada-Kawai algorithm and was visualized with a custom Blender script. On the vertical axes influence function values (panel A), or community landscape values (panels B, C and D) of the links are shown. Influence functions of panels A1, or A2 belong to the Barabási-Vicsek, or Girvan-Newman author-pairs, respectively. Panel A3 shows the merged influence function of the Arenas-Pastor-Satorras and Guimera-Amaral co-authorship links. Links and nodes of panels C and D are colored in proportion of the colors of the modules they belong. Panel A: influence function calculation. First, the influence function of each link (or node) of the network were identified. If a link is in the ‘middle’ of a module, it is affected by many influence functions (all the three widely collaborating author-pairs, whose influence functions are shown by the arrows, are from this category). On the contrary, links at module ‘edges’ are affected by few influence functions only. At the bottom of the panel the names of the three algorithms we described in details are shown. Panel B: community landscape construction. Next, the community landscape is constructed by summing up the influence function values for all nodes or links. The hills of the community landscape correspond to the modules of the network. Panel C: determination of overlapping modules. Last, modular centers are identified as the links at the local maxima of the community landscapes, and memberships of links in all network modules are determined. At the top of the panel the names of the three methods we described in details are shown. Panel D: determination of network hierarchy. Optionally, a higher level hierarchical representation of the network can be created, where nodes of the higher level correspond to modules of the original network, and links of the higher level correspond to overlaps between the respective modules. Sizes of higher level nodes correspond to the log size of the respective lower level modules, where the module size is the sum of the membership assignment strengths of all nodes to that module.