gateway_coeff calculates the gateway coefficient of each vertex,
based on community membership.
part_coeff calculates the participation coefficient of each vertex,
based on community membership.
within_module_deg_z_score is a measure of the connectivity from a
given vertex to other vertices in its module/community.
gateway_coeff(g, memb, centr = c("btwn.cent", "degree", "strength"))part_coeff(g, memb)
within_module_deg_z_score(g, memb)
An igraph graph object
A numeric vector of membership indices of each vertex
Character string; the type of centrality to use in calculating
GC (default: btwn.cent)
A vector of the participation coefficients, within-module degree z-scores, or gateway coefficients for each vertex of the graph.
The gateway coefficient \(G_i\) of vertex i is: $$G_i = 1 - \sum_{S=1}^{N_M} \left ( \frac{\kappa_{iS}}{\kappa_i} \right )^2 (g_{iS})^2$$ where \(\kappa_{iS}\) is the number of edges from vertex i to vertices in module S, and \(\kappa_i\) is the degree of vertex i. \(N_M\) equals the number of modules. \(g_{ii}\) is a weight, defined as: $$g_{iS} = 1 - \bar{\kappa_{iS}} \bar{c_{iS}}$$ where $$\bar{\kappa_{iS}} = \frac{\kappa_{iS}}{\sum_j \kappa_{jS}}$$ for all nodes \(j\) in node \(i\)'s module, and $$\bar{c_{iS}} = c_{iS} / max(c_n)$$
The participation coefficient \(P_i\) of vertex i is: $$P_i = 1 - \sum_{s=1}^{N_M} \left ( \frac{\kappa_{is}}{\kappa_i} \right )^2$$ where \(\kappa_{is}\) is the number of edges from vertex i to vertices in module s, and \(\kappa_s\) is the degree of vertex i. \(N_M\) equals the number of modules.
As discussed in Guimera et al., \(P_i = 0\) if vertex i is connected only to vertices in the same module, and \(P_i = 1\) if vertex i is equally connected to all other modules.
The within-module degree z-score is: $$z_i = \frac{\kappa_i - \bar{\kappa}_{s_i}}{\sigma_{\kappa_{s_i}}}$$ where \(\kappa_i\) is the number of edges from vertex i to vertices in the same module \(s_i\), \(\bar{\kappa}_{s_i}\) is the average of \(\kappa\) over all vertices in \(s_i\), and \(\sigma_{\kappa_{s_i}}\) is the standard deviation.
Vargas E.R. & Wahl L.M. (2014) The gateway coefficient: a novel metric for identifying critical connections in modular networks. Eur Phys J B, 87:161-170.
Guimera, R. and Amaral, L.A.N. (2005) Cartography of complex networks: modules and universal roles, Journal of Statistical Mechanics: Theory and Experiment, 02, P02001.