Vertex Roles {brainGraph}

Gateway coefficient, participation coefficient, and within-mod degree z-score

Description

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.

Usage

gateway_coeff(g, memb, centr = c("btwn.cent", "degree", "strength"),
  A = NULL, weighted = FALSE)

part_coeff(g, memb, A = NULL, weighted = FALSE)

within_module_deg_z_score(g, memb, A = NULL, weighted = FALSE)

Arguments

g

An igraph graph object

memb

A numeric vector of membership indices of each vertex

centr

Character string; the type of centrality to use in calculating GC. Default: btwn.cent

A

Numeric matrix; the adjacency matrix of the input graph. Default: NULL

weighted

Logical indicating whether to calculate metrics using edge weights. Default: FALSE

Details

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.

Value

A vector of the participation coefficients, within-module degree z-scores, or gateway coefficients for each vertex of the graph.

Author(s)

Christopher G. Watson, [e-mail]

References

Vargas, E.R. and Wahl, L.M. (2014) The gateway coefficient: a novel metric for identifying critical connections in modular networks. Eur Phys J B, 87, 161–170. doi: 10.1140/epjb/e2014-40800-7

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. doi: 10.1088/1742-5468/2005/02/P02001