# Connectivity measures

## Graph (ROI-level) measures

All ROI-level graph measures below are based on user-defined nondirectional graphs with nodes = ROIs, and edges = supra-threshold connections. For each subject (and condition) a graph adjacency matrix *A(i,j)* is computed by thresholding an ROI-to-ROI Correlation (RRC) matrix *r(i,j)*. Then from the resulting graphs, the following measures can be computed locally for each node / ROI and globally aggregated across all nodes / ROIs

**Degree & Cost:**Number (degree) or proportion (cost) of edges for each node. Degree and Cost represent both measures of network centrality at each node/ROI, characterizing the degree of local connectedness of each ROI within the graph. Network cost represents the proportion of edges among all possible node pairs, and this is typically fixed to allow meaningful between-network comparisons

where Aij = adjacency matrix (Aij=1 if there is an edge between nodes i and j, Aij=0 otherwise; note: Aii=0)

N = number of nodes in graph

**d_i = Degree of i-th node / ROI**

**c_i = Cost of i-th node / ROI**

** d = Degree of graph**

** c = Cost of graph**

**Average path distance:**Average minimum path distance between each node and all other nodes in the graph. Average path distance represents a measure of node centrality within a network, characterizing the degree of global connectedness of each ROI within the graph. Network average path distance represents a measure of graph inter-connectedness/radius (e.g. random graphs have comparatively low/compact average path distances)

where Dij = distance matrix (minimum number of edges transversed in a path from node i to node j; Dij=inf if no such path exists)

Omega_i = sub-graph of nodes that can be reached by paths starting from the i-th node, including the i-th node itself

Ni = number of nodes in Omega_i (equal to N for fully connected graphs)

**L_i = Average path distance of i-th node / ROI**

** L = Average path distance of graph**

**Clustering Coefficient:**Fraction of edges among all possible edges in the local neighboring sub-graph for each node / ROI. Clustering coefficient represents a measure of local integration, characterizing the degree of inter-connectedness among all nodes within a node neighboring sub-graph. Network clustering coefficient represents a measure of network locality (e.g. grid topologies have comparatively high clustering coefficients)

where Gamma_i = neighboring sub-graph of i-th node (graph formed only by those nodes with edges between the i-th node and them)

A* = adjacency matrix of Gamma_i graph

di = number of nodes in Gamma_i graph (degree of i-th node)

** CCi = Clustering coefficient of i-th node / ROI**

** CC = Clustering coefficient of graph**

**Global Efficiency:**average of inverse-distances between each node and all other nodes in the graph. Global efficiency represents a measure of node centrality within a network, characterizing the degree of global connectedness of each ROI within the graph. Network global efficiency represents a measure of graph inter-connectedness/radius (e.g. random graphs have comparatively high/compact global efficiency)

where Dij = distance matrix (minimum number of edges transversed in a path from node i to node j; Dij=inf if no such path exists)

N = number of nodes in graph

**GE_i = Global efficiency of i-th node / ROI**

** GE = Global efficiency of graph**

**Local Efficiency:**Global efficiency of neighboring sub-graph. Local efficiency represents a measure of local integration, characterizing the degree of inter-connectedness among all nodes within a node neighboring sub-graph. Network local efficiency represents a measure of network locality (e.g. grid topologies have comparatively high local efficiency)

where Gamma_i = neighboring sub-graph of i-th node (graph formed only by those nodes with edges between the i-th node and them)

D* = distance matrix of Gamma_i graph

di = number of nodes in Gamma_i graph (degree of i-th node)

** LEi = Local efficiency of i-th node / ROI**

** LE = Local efficiency of graph**

**Betweenness Centrality:**Proportion of times that a node is part of an optimal minimum-distance path between any two pairs of nodes within the graph. Betweenness centrality represents a measure of node centrality within the graph

where P_jk = set of nodes in minimum-distance path between j-th and k-th nodes

N = number of nodes in graph

** BCi = Betweenness Centrality of i-th node / ROI**

** BC = Betweenness Centrality of graph**