The Emergence of Informative Higher Scales in Complex Networks

Figure 4

Macronodes. (a) The original network, along with its adjacency matrix (left). The shaded oval indicates that subgraph member nodes and will be grouped together, forming a macronode, . All macronodes are some transformation of the original adjacency matrix via recasting it as a new adjacency matrix (right). The manner of this recasting depends on the type of macronode. (b) The simplest form of a macronode is when is an average of the of each node in the subgraph. (c) A macronode that represents some path-dependency, such as input from . Here, in averaging to create the , the out-weights of nodes and are weighted by their input from . (d) A macronode that represents the subgraph’s output over the network’s stationary dynamics. Each node has some associated , which is the probability of in the stationary distribution of the network. The of a macronode is created by weighting each of the micronodes in the subgraph by . (e) A macronode with a single timestep delay between input and its output , each constructed using the same techniques as its components. However, always deterministically outputs to . See SM V A for the full equations governing the creation of the of each of the different HOMs shown.

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