The Emergence of Informative Higher Scales in Complex Networks
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.
Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.