FIG. 1. Multi-subject modularity, communities, and areal entropy.

(a) Single-subject networks are represented as layers in a multi-layer network ensemble. Each node is linked to itself across layers, here illustrated by interlayer connections. Note that community labels are indicated by node color. (b) Maximizing a multi-layer modularity function returns a set of single-subject partitions. Importantly, community labels are preserved across layers; thus, if the label C₁ appears in layers r and s, we assume that the same community has recurred. This property allows us to make several useful measurements. We can calculate, for each node, the mode of its community assignment across subjects to generate a consensus partition. We can also calculate the entropy of each node’s community assignments, which measures the variability of communities across subjects. (c) The preservation of community labels also allows for a direct comparison of any one subject to any other subject. Given partitions of subjects (or layers), denoted here with variables r and s, we can generate a bit vector whose values are {0, 1} depending on whether a given node has the same/different community assignment. Doing so for all pairs of subjects generates a three-dimensional entropy tensor. When averaged over nodes, this tensor generates a T ×T matrix whose elements indicate, in total, the number of non-identical community assignments between pairs of subjects. When averaged over either of its other dimensions, the result is an N × T matrix, whose elements indicate, in total, the similarity of a node’s community assignment within a given subject to that of the remaining T − 1 subjects.