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Table 4 Statistics of network configurations of bipartite networks

From: Local interactions and homophily effects in actor collaboration networks for urban resilience governance

Network configurations Network statistics Interpretation
Edges: L \(\mathop \sum \nolimits_{R = 1}^{R} \mathop \sum \nolimits_{P = 1}^{P} M_{RP}\) Number of edges in the bipartite network
Two stars:\(S_{R2}\) \(\mathop \sum \nolimits_{R = 1}^{R} \mathop \sum \nolimits_{{P^{\prime } > P}}^{P} M_{RP} M_{{RP^{\prime } }}\) Correspondent to an edge between node set P in the 1-mode network
Two stars:\(S_{P2}\) \(\mathop \sum \nolimits_{P = 1}^{P} \mathop \sum \nolimits_{{R^{\prime } > R}}^{R} M_{PR} M_{{PR^{\prime } }}\) Correspondent to an edge between node set R in the 1-mode network
Three stars:\(S_{R3}\) \(\mathop \sum \nolimits_{R = 1}^{R} \mathop \sum \nolimits_{{P^{\prime \prime } > P^{\prime } > P}}^{P} M_{RP} M_{{RP^{\prime } }} M_{{RP^{\prime \prime } }}\) Correspondent to a triangle between node set P in the 1-mode network
Three stars:\(S_{P3}\) \(\mathop \sum \nolimits_{P = 1}^{P} \mathop \sum \nolimits_{{R^{\prime \prime } > R^{\prime } > R}}^{R} M_{PR} M_{{PR^{\prime } }} M_{{PR^{\prime \prime } }}\) Correspondent to a triangle between node set R in the 1-mode network
Three trails:\(L_{3}\) \(\mathop \sum \nolimits_{{P^{\prime } > P}}^{P} \mathop \sum \nolimits_{{R^{\prime } > R}}^{R} M_{PR} M_{{PR^{\prime } }} M_{{P^{\prime } R}} \left( {1 - M_{{P^{\prime } R^{\prime } }} } \right)\) Reflect global connectivity in bipartite networks
Cycle:\(C_{4}\) \(\mathop \sum \nolimits_{{P^{\prime } > P}}^{P} \mathop \sum \nolimits_{{R^{\prime } > R}}^{R} M_{PR} M_{{PR^{\prime } }} M_{{P^{\prime } R}} M_{{P^{\prime } R^{\prime } }}\) Local closures in bipartite networks
  1. \({M}_{RP}\) represents the value of the elements in the bi-adjacent matrix of the bipartite network. If node R and P are linked, \({M}_{RP}=1\). Otherwise \({M}_{RP}=0\)