# vertex_roles: Gateway coefficient, participation coefficient, and... In brainGraph: Graph Theory Analysis of Brain MRI Data

## 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

 1 2 3 4 5 6 gateway_coeff(g, memb, centr = c("btwn.cent", "degree", "strength"), A = NULL) part_coeff(g, memb, A = NULL) within_module_deg_z_score(g, memb, A = NULL) 

## 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

## Details

The gateway coefficient G_i of vertex i is:

G_i = 1 - ∑_{S=1}^{N_M} ≤ft ( \frac{κ_{iS}}{κ_i} \right )^2 (g_{iS})^2

where κ_{iS} is the number of edges from vertex i to vertices in module S, and κ_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{κ_{iS}} \bar{c_{iS}}

where

\bar{κ_{iS}} = \frac{κ_{iS}}{∑_j κ_{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 - ∑_{s=1}^{N_M} ≤ft ( \frac{κ_{is}}{κ_i} \right )^2

where κ_{is} is the number of edges from vertex i to vertices in module s, and κ_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{κ_i - \bar{κ}_{s_i}}{σ_{κ_{s_i}}}

where κ_i is the number of edges from vertex i to vertices in the same module s_i, \bar{κ}_{s_i} is the average of κ over all vertices in s_i, and σ_{κ_{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, cgwatson@bu.edu

## 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. https://dx.doi.org/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. https://dx.doi.org/10.1088/1742-5468/2005/02/P02001

brainGraph documentation built on Oct. 23, 2020, 6:37 p.m.