Fig. 2: Construction diagram of the v-control network Gv.
From: Identifying key players in complex networks via network entanglement
The process of constructing Gv is somewhat analogous to shrinking v and its neighbors into one super vertex and redistributing the original weights evenly within the super vertex. Specifically, the transition from G to Gv occurs merely in v and its neighbors, being restructured into a probabilistic complete graph. The weight of each link in the probabilistic complete graph is \({\omega }_{v}=\frac{{{{{{{{\rm{sum}}}}}}}}\,{{{{{{{\rm{of}}}}}}}}\,{{{{{{{\rm{link}}}}}}}}\,{{{{{{{\rm{weights}}}}}}}}\,{{{{{{{\rm{among}}}}}}}}\,v\,{{{{{{{\rm{and}}}}}}}}\,{{{{{{{\rm{its}}}}}}}}\,{{{{{{{\rm{neighbors}}}}}}}}}{{{{{{{{\rm{possible}}}}}}}}\,{{{{{{{\rm{links}}}}}}}}\,{{{{{{{\rm{number}}}}}}}}}=\frac{{k}_{v}+0.5{k}_{v}\left({k}_{v}-1\right){c}_{v}}{{k}_{v}\left({k}_{v}+1\right)}\)\(=\frac{{c}_{v}}{2}\) \(-\, \frac{{c}_{v}+1}{{k}_{v}+1}\), with cv presenting the clustering coefficient of Gv. For example, in the above network, \({\omega }_{v}=\frac{4}{6}=\frac{2}{3}\).
