Table 1 The notations and their meaning used in the preference relation approach

Input graph on which communities are to be detected

\(G=(V,E)\)

Out-degree: number of links outside graph G

\(K_{out}^{\mathcal {G}}\)

In-degree: number of links inside graph G

\(K_{in}^{\mathcal {G}}\)

Internal-links: for node i, links shared with neighboring nodes belonging to community c

\(k_i^{int} (c)\)

External-links: for node i, links shared with neighboring nodes not belonging to community c

\(k_i^{ext} (c)\)

Preference implication for comparing x and y

\(P_{\nu }^{(\alpha )}{(x,y)}\)