Table 1.

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

MeasureDefinition for Undirected GraphDefinition for Directed Graph
(Out)-degree (kijNaij jNaij
In-degree ($kiin$jNaji
Shortest path length (dijauv, auvgij auv, auvgij
Number of triangles (ti$12$j,haijaihajh $12$j,h (aij + aji)(aih + ahi)(ajh + ahj
Global efficiency (E$1N$iNEi = $1N$iN$∑j∈N,j≠idij−1N−1$
Local efficiency (Eloc$1N$iN$∑i,j∈N,j≠haijaihdjh−1Nikiki−1$ $12N$iN$∑i,j∈N,j≠haij+ajiaih+ahidjh−1Ni+dhj−1Nikiout+kiinkiout+kiin−1−2∑jaijaji$
Clustering coefficient (C$1N$iN$2tikiki−1$ $1N$iN$tikiout+kiinkiout+kiin−1−2∑jaijaji$
Transitivity (T$∑i∈N2ti∑i∈Nkiki−1$ $∑i∈Ntikiout+kiinkiout+kiin−1−2∑jaijaji$
Modularity (QuM [euu − (∑vMeuv)]2
MeasureDefinition for Undirected GraphDefinition for Directed Graph
(Out)-degree (kijNaij jNaij
In-degree ($kiin$jNaji
Shortest path length (dijauv, auvgij auv, auvgij
Number of triangles (ti$12$j,haijaihajh $12$j,h (aij + aji)(aih + ahi)(ajh + ahj
Global efficiency (E$1N$iNEi = $1N$iN$∑j∈N,j≠idij−1N−1$
Local efficiency (Eloc$1N$iN$∑i,j∈N,j≠haijaihdjh−1Nikiki−1$ $12N$iN$∑i,j∈N,j≠haij+ajiaih+ahidjh−1Ni+dhj−1Nikiout+kiinkiout+kiin−1−2∑jaijaji$
Clustering coefficient (C$1N$iN$2tikiki−1$ $1N$iN$tikiout+kiinkiout+kiin−1−2∑jaijaji$
Transitivity (T$∑i∈N2ti∑i∈Nkiki−1$ $∑i∈Ntikiout+kiinkiout+kiin−1−2∑jaijaji$
Modularity (QuM [euu − (∑vMeuv)]2
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