R/vertex_roles.R

In brainGraph: Graph Theory Analysis of Brain MRI Data

#' Gateway coefficient, participation coefficient, and within-mod degree z-score
#'
#' \code{gateway_coeff} calculates the gateway coefficient of each vertex,
#' based on community membership.
#'
#' The gateway coefficient \eqn{G_i} of vertex \emph{i} is:
#' \deqn{G_i = 1 - \sum_{S=1}^{N_M} \left ( \frac{\kappa_{iS}}{\kappa_i} \right
#' )^2 (g_{iS})^2}
#' where \eqn{\kappa_{iS}} is the number of edges from vertex \emph{i} to
#' vertices in module \emph{S}, and \eqn{\kappa_i} is the degree of vertex
#' \emph{i}. \eqn{N_M} equals the number of modules. \eqn{g_{ii}} is a weight,
#' defined as:
#' \deqn{g_{iS} = 1 - \bar{\kappa_{iS}} \bar{c_{iS}}}
#' where
#' \deqn{\bar{\kappa_{iS}} = \frac{\kappa_{iS}}{\sum_j \kappa_{jS}}}
#' for all nodes \eqn{j} in node \eqn{i}'s module, and
#' \deqn{\bar{c_{iS}} = c_{iS} / max(c_n)}
#'
#' @inheritParams efficiency
#' @param memb A numeric vector of membership indices of each vertex
#' @param centr Character string; the type of centrality to use in calculating
#'   GC. Default: \code{btwn.cent}
#' @param weighted Logical indicating whether to calculate metrics using edge
#' weights. Default: \code{FALSE}
#' @export
#' @return A vector of the participation coefficients, within-module degree
#'   z-scores, or gateway coefficients for each vertex of the graph.
#'
#' @name Vertex Roles
#' @rdname vertex_roles
#'
#' @author Christopher G. Watson, \email{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.
#'   \emph{Eur Phys J B}, \bold{87}, 161--170.
#'   \doi{10.1140/epjb/e2014-40800-7}

gateway_coeff <- function(g, memb, centr=c('btwn.cent', 'degree', 'strength'),
                          A=NULL, weighted=FALSE) {
  stopifnot(is_igraph(g))
  if (is.null(A)) {
    if (isTRUE(weighted)) {
      A <- as_adj(g, sparse=FALSE, names=FALSE, attr='weight')
    } else {
      A <- as_adj(g, sparse=FALSE, names=FALSE)
    }
  }
  Ki <- colSums(A)
  centr <- match.arg(centr)
  if (centr == 'btwn.cent') {
    if ('btwn.cent' %in% vertex_attr_names(g)) {
      cent <- V(g)$btwn.cent
    } else {
      cent <- centr_betw(g)$res
    }
  } else {
    cent <- Ki
  }
  N <- max(memb)
  if (N == 1L) return(rep.int(0, length(memb)))
  Cn <- max(vapply(seq_len(N), function(x) sum(cent[memb == x]), numeric(1L)))

  Kis <- t(rowsum(A, memb))
  Kjs <- rowsum(Kis, memb)
  barKis <- Cis <- matrix(0, nrow=length(Ki), ncol=N)

  for (i in which(Ki > 0)) {
    barKis[i, ] <- Kis[i, ] / Kjs[, memb[i]]
    Vi <- which(A[, i] != 0)
    Cis[i, ] <- vapply(seq_len(N), function(x) sum(cent[Vi[memb[Vi] == x]]), numeric(1L))
  }

  barCis <- Cis / Cn
  gis <- 1 - barKis * barCis
  1 - ((1 / Ki^2) * rowSums(Kis^2 * gis^2, na.rm=TRUE))
}

#' Participation coefficient
#'
#' \code{part_coeff} calculates the participation coefficient of each vertex,
#' based on community membership.
#'
#' The participation coefficient \eqn{P_i} of vertex \emph{i} is:
#' \deqn{P_i = 1 - \sum_{s=1}^{N_M} \left ( \frac{\kappa_{is}}{\kappa_i} \right )^2}
#' where \eqn{\kappa_{is}} is the number of edges from vertex \emph{i} to
#' vertices in module \emph{s}, and \eqn{\kappa_s} is the degree of vertex
#' \emph{i}. \eqn{N_M} equals the number of modules.
#'
#' As discussed in Guimera et al., \eqn{P_i = 0} if vertex \emph{i} is connected
#' only to vertices in the same module, and \eqn{P_i = 1} if vertex \emph{i} is
#' equally connected to all other modules.
#'
#' @export
#' @rdname vertex_roles
#'
#' @references Guimera, R. and Amaral, L.A.N. (2005) Cartography of complex
#'   networks: modules and universal roles. \emph{Journal of Statistical
#'   Mechanics: Theory and Experiment}, \bold{02}, P02001.
#'   \doi{10.1088/1742-5468/2005/02/P02001}

part_coeff <- function(g, memb, A=NULL, weighted=FALSE) {
  stopifnot(is_igraph(g))
  if (is.null(A)) {
    if (isTRUE(weighted)) {
      A <- as_adj(g, sparse=FALSE, names=FALSE, attr='weight')
    } else {
      A <- as_adj(g, sparse=FALSE, names=FALSE)
    }
  }
  Ki <- colSums(A)
  N <- max(memb)
  Kis <- t(rowsum(A, memb))
  1 - ((1 / Ki^2) * rowSums(Kis^2))
}

#' Calculate vertex within-module degree z-score
#'
#' \code{within_module_deg_z_score} is a measure of the connectivity from a
#' given vertex to other vertices in its module/community.
#'
#' The within-module degree z-score is:
#' \deqn{z_i = \frac{\kappa_i - \bar{\kappa}_{s_i}}{\sigma_{\kappa_{s_i}}}}
#' where \eqn{\kappa_i} is the number of edges from vertex \emph{i} to vertices
#' in the same module \eqn{s_i}, \eqn{\bar{\kappa}_{s_i}} is the average of
#' \eqn{\kappa} over all vertices in \eqn{s_i}, and \eqn{\sigma_{\kappa_{s_i}}}
#' is the standard deviation.
#'
#' @export
#' @rdname vertex_roles

within_module_deg_z_score <- function(g, memb, A=NULL, weighted=FALSE) {
  stopifnot(is_igraph(g))
  if (is.null(A)) {
    if (isTRUE(weighted)) {
      A <- as_adj(g, sparse=FALSE, names=FALSE, attr='weight')
    } else {
      A <- as_adj(g, sparse=FALSE, names=FALSE)
    }
  }
  N <- max(memb)
  nS <- tabulate(memb)
  z <- Ki <- rep.int(0, dim(A)[1L])
  Ksi <- sigKsi <- rep.int(0, N)

  for (S in seq_len(N)) {
    x <- rowSums(A[memb == S, memb == S, drop=FALSE])
    Ki[memb == S] <- x
    Ksi[S] <- sum(x) / nS[S]
    sigKsi[S] <- sqrt(sum((x - Ksi[S])^2) / (nS[S] - 1L))
  }
  z <- (Ki - Ksi[memb]) / sigKsi[memb]
  z[is.infinite(z)] <- 0
  return(z)
}