TABLE 1.

Network and graph theory properties, definitions, and measures

Property	Definition	Example graph measure
Functional connectivity	Temporal correlation between time‐series of neurophysiological events	The average of the weight of node edges, e.g., BOLD correlation or PET tracer uptake
Path length	Distance between nodes in terms of edges traversed	Average minimum path length is the shortest mean distance from a particular node to all other nodes in the network (in terms of edges traversed), and is a measure of integration
Local efficiency	Efficiency of information transfer between nodes within a network	Inverse of the average shortest path length of each node to its neighbors
Global efficiency	Efficiency of information transfer across the entire network system	Inverse of the average shortest path length between networks in the system
Modularity	Degree to which the network is divided into distinct communities. Dense within relative to sparse between network connections. A modular network exhibits a ‘clustering’ of its nodes into multiple distinct subnetworks	Fraction of connections (edges) within the communities of a network compared to if the connections were distributed at random across the network, e.g., ICA or thresholded correlations aim to reflect modularity
Segregation	The amount of partitioning between distinct regions of a network. Reflects the ability for specialized processing to occur within densely interconnected groups of brain regions. Similar to modularity, except that it directly quantifies within versus between network connections	Dense or strong connectivity among nodes in the same network (high within‐network connectivity) coupled with sparse or weak connections between nodes belonging to different networks (low between‐network connectivity), e.g., clustering coefficient
Clustering	Nodes in the same network are more similar to each other than to those in other networks	Clustering coefficient is a measure of the proportion of a node's neighbors that are themselves neighbors and is a measure of segregation
Integration	Reflects the degree of integration among multiple brain regions. The ability to rapidly combine specialized information from distributed brain regions	The average number of edges of the shortest paths that connect nodes over the whole network. A smaller path length represents greater integration. Measures include shortest path length and node degree. Also related to global efficiency, i.e., as the inverse of the average shortest path length
Hubs	Nodes that have highly centralized connections and strong relationships with each other via long‐distance connection. A hub can be defined as a node that has many important connections running through it (is a member of the shortest path between many nodes), or a node that associates with (correlates to) many other networks	Measures of centrality, including how many (degree) and weight (strength) of the connections, together with the shortest paths within the network pass through a given node (betweenness). For example, participation coefficient measure the connections that are not within a node's own network or that are across networks. When the between node connections are denser than expected by chance, they are form “rich clubs”, serving as “way stations” for high volume transfer
“Small World” network	High level of local and global efficiency in information processing, with economical “sparse” wiring costs. High segregation and integration	In graph theoretical terms, a graph is “small world” if the path length is similar to that of a random graph, and the clustering coefficient is much greater than a random graph