MIT Press

Table 1.

Mathematical definition of network topological measure for (un)directed graphs

Measure	Definition for Undirected Graph	Definition for Directed Graph
(Out)-degree (k_i)	∑_j∈Na_ij	∑_j∈Na_ij
In-degree (⁠ $k_{i}^{in}$ ⁠)	/	∑_j∈Na_ji
Shortest path length (d_ij)	∑ a_uv, a_uv ∈ g_i↔j	∑ a_uv, a_uv ∈ g_i→j
Number of triangles (t_i)	$\frac{1}{2}$ ∑_j,ha_ija_iha_jh	$\frac{1}{2}$ ∑_j,h (a_ij + a_ji)(a_ih + a_hi)(a_jh + a_hj)
Global efficiency (E)	$\frac{1}{N}$ ∑_i∈NE_i = $\frac{1}{N}$ ∑_i∈N $\frac{\sum_{j \in N, j \neq i} d_{i j}^{- 1}}{N - 1}$
Local efficiency (E_loc)	$\frac{1}{N}$ ∑_i∈N $\frac{\sum_{i, j \in N, j \neq h} a_{i j} a_{i h} d_{j h}^{- 1} (N_{i})}{k_{i} (k_{i} - 1)}$	$\frac{1}{2 N}$ ∑_i∈N $\frac{\sum_{i, j \in N, j \neq h} (a_{i j} + a_{j i}) (a_{i h} + a_{h i}) (d_{j h}^{- 1} (N_{i}) + d_{h j}^{- 1} (N_{i}))}{(k_{i}^{out} + k_{i}^{in}) (k_{i}^{out} + k_{i}^{in} - 1) - 2 \sum_{j} a_{i j} a_{j i}}$
Clustering coefficient (C)	$\frac{1}{N}$ ∑_i∈N $\frac{2 t_{i}}{k_{i} (k_{i} - 1)}$	$\frac{1}{N}$ ∑_i∈N $\frac{t_{i}}{(k_{i}^{out} + k_{i}^{in}) (k_{i}^{out} + k_{i}^{in} - 1) - 2 \sum_{j} a_{i j} a_{j i}}$
Transitivity (T)	$\frac{\sum_{i \in N} 2 t_{i}}{\sum_{i \in N} k_{i} (k_{i} - 1)}$	$\frac{\sum_{i \in N} t_{i}}{(k_{i}^{out} + k_{i}^{in}) (k_{i}^{out} + k_{i}^{in} - 1) - 2 \sum_{j} a_{i j} a_{j i}}$
Modularity (Q)	∑_u∈M [e_uu − (∑_v∈Me_uv)]²

Measure	Definition for Undirected Graph	Definition for Directed Graph
(Out)-degree (k_i)	∑_j∈Na_ij	∑_j∈Na_ij
In-degree (⁠ $k_{i}^{in}$ ⁠)	/	∑_j∈Na_ji
Shortest path length (d_ij)	∑ a_uv, a_uv ∈ g_i↔j	∑ a_uv, a_uv ∈ g_i→j
Number of triangles (t_i)	$\frac{1}{2}$ ∑_j,ha_ija_iha_jh	$\frac{1}{2}$ ∑_j,h (a_ij + a_ji)(a_ih + a_hi)(a_jh + a_hj)
Global efficiency (E)	$\frac{1}{N}$ ∑_i∈NE_i = $\frac{1}{N}$ ∑_i∈N $\frac{\sum_{j \in N, j \neq i} d_{i j}^{- 1}}{N - 1}$
Local efficiency (E_loc)	$\frac{1}{N}$ ∑_i∈N $\frac{\sum_{i, j \in N, j \neq h} a_{i j} a_{i h} d_{j h}^{- 1} (N_{i})}{k_{i} (k_{i} - 1)}$	$\frac{1}{2 N}$ ∑_i∈N $\frac{\sum_{i, j \in N, j \neq h} (a_{i j} + a_{j i}) (a_{i h} + a_{h i}) (d_{j h}^{- 1} (N_{i}) + d_{h j}^{- 1} (N_{i}))}{(k_{i}^{out} + k_{i}^{in}) (k_{i}^{out} + k_{i}^{in} - 1) - 2 \sum_{j} a_{i j} a_{j i}}$
Clustering coefficient (C)	$\frac{1}{N}$ ∑_i∈N $\frac{2 t_{i}}{k_{i} (k_{i} - 1)}$	$\frac{1}{N}$ ∑_i∈N $\frac{t_{i}}{(k_{i}^{out} + k_{i}^{in}) (k_{i}^{out} + k_{i}^{in} - 1) - 2 \sum_{j} a_{i j} a_{j i}}$
Transitivity (T)	$\frac{\sum_{i \in N} 2 t_{i}}{\sum_{i \in N} k_{i} (k_{i} - 1)}$	$\frac{\sum_{i \in N} t_{i}}{(k_{i}^{out} + k_{i}^{in}) (k_{i}^{out} + k_{i}^{in} - 1) - 2 \sum_{j} a_{i j} a_{j i}}$
Modularity (Q)	∑_u∈M [e_uu − (∑_v∈Me_uv)]²