Table 2.

Summary of the Network Measures for Weighted Networks

Type	Measure	Definition	Clinical meaning	Equation
Macroscopic	Density	Average of all the edge weights in the network	To what extent symptoms in the network are interconnected	$D=\frac{2\text{*}{{\displaystyle \sum }}_{i,j\in V}{w}_{ij}}{N(N-1)}$
	Harmonic mean shortest path length	Average shortest path length between all nodes	Level of information efficiency in the network	$L=\frac{N(N-1)}{{{\displaystyle \sum }}_{i,j\in V }\frac{1}{d(i,j)}}$
	Average clustering coefficient	Overall clustering in the network	To what extent symptoms tend to cluster together	$Avg CC= \frac{{{\displaystyle \sum }}_i{C}_i}{N}$ ${\text{C}}_i=\frac{1}{k_i(k_i-1)}{\displaystyle \sum }_{u,v\in V}{(W_{iu}{W}_{iv}{W}_{uv})}^{\frac{1}{3}}$
	Modularity	Partitioning networks into a collection of modules (groups)	To what extent symptoms can be separated into distinct groups	$Q=\frac{1}{2m}{\displaystyle \sum }_{i,j\in V}(w_{ij}-\frac{k_i{k}_j}{2m})\delta \left(c_i,c_j\right)$
Microscopic	Degree centrality	Sum of the edge weights connected to a node	Level of connectivity of a symptom in the network	$k_i=\frac{{{\displaystyle \sum }}_{j\in V}{w}_{ij}}{N-1}$
	Betweenness centrality	# of shortest paths between pairs of nodes that pass-through node i	How frequently a symptom emerges as part of interactions among other symptoms	$B_i={\displaystyle \sum }_{j,k\in V}\frac{g(j,u,i)}{g(j,u)}$
	Clustering coefficient	How well the neighborhoods of a node connect to each other	To what extent symptoms tend to cluster together	${\text{C}}_i=\frac{1}{k_i(k_i-1)}{\displaystyle \sum }_{u,v\in V}{(W_{iu}{W}_{iv}{W}_{uv})}^{\frac{1}{3}}$

Note: $i, j, u,v$ = Node (Brief Negative Symptom Scale symptom) Index; N = total number of nodes; V is the set of all nodes in the network; $w_{ij}$, $w_{iu}$, $w_{iv}$= weight between nodes i and j, i and u,i and v; $d_{i,j }$ = 1/$w_{ij}$= distance between nodes i and j; $k_i$ and $k_j$ = degrees of nodes i and j; $g(j,u,i)$ = the number of shortest paths from node j to node u that go through node i; and $g\left(j,u\right)$= the total number of shortest paths from node j to node u.