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Table 3 Preference rule table: rule for the new node addition in the subgraph based on the threshold \(\nu\) value and preference value

From: Theory of preference modelling for communities in scale-free networks

Preference value \({\bf{P_{\nu }^{(\alpha )}{(x,y)}}}\) Fitness function rule Decision of a new node addition
\(0< P_{\nu }^{(\alpha )}{(x,y)} < 0.5 \quad P_{\nu }^{(\alpha )}{(x,y)} > \delta\) \(f_{\mathcal {G}} > f_{\mathcal {G'}}\) Not desirable for a community membership but it can have a very weak overlapping membership
\(\qquad P_{\nu }^{(\alpha )}{(x,y)} = 0.5 \quad P_{\nu }^{(\alpha )}{(x,y)} > \delta\) \(f_{\mathcal {G}} = f_{\mathcal {G'}}\) Desirable for Community membership and it can have a weak overlapping membership
\(1>P_{\nu }^{(\alpha )}{(x,y)}> 0.5 \quad P_{\nu }^{(\alpha )}{(x,y)} > \delta\) \(f_{\mathcal {G}} < f_{\mathcal {G'}}\) Strong community membership and it can have a strong overlapping membership