Network topology measures (definitions)

Note: These definitions are for simple graphs (undirected, unweighted, no self-loops, no multiple edges). For non-simple graphs definitions may be different.
m: Total number of edges
n: Total number of nodes
A: Adjacency matrix. A_ij is 1 if node i and j are connected. Otherwise 0.
k_i: degree of node i

Assortativity (degree assortativity coefficient, r) [Ref]

e_ij is the fraction of edges in a network that connect a node of degree i to one of degree j.
$\sum_{i,j}e_{ij} = 1, \, \sum_{j}e_{ij} = a_{i}, \, \sum_{i}e_{ij} = b_{j} $

$r = \frac{\sum_{i}e_{ii} - \sum_{i}a_{i}b_{i}}{1 - \sum_{i}a_{i}b_{i}}$

Modularity, Q [Ref]

$Q = \frac{1}{2m}\sum_{vw}{[A_{vw} - \frac{k_{v}k_{w}}{2m}]}\delta (c_v,c_w)$
where 𝛿(c_v,c_w) becomes 1 when node v and w are in the same community, and 0 otherwise.

Normalized Degree Variance [Ref]

$\frac{n-1}{nm(1-d)}var(\sum_{i}A_{ij})$
where d is density, 2m/(n(n-1)).

Average Clustering Coefficient, C

Clustering coefficient of node i, c_i, is
$c_i = \frac{2t_i}{k_i (k_i - 1)}$
where t_i denotes the number of triangles that contain node i.
Taking the average of these, average clustering coefficient becomes
$C = \sum_{i}c_{i} \, .$

Efficiency, E

$E = \frac{1}{n(n-1)}\sum_{i,j}\frac{1}{d(i,j)}$
where d(i,j) is the shortest path length between node i and j.