efficiency: Calculate graph global, local, or nodal efficiency

In brainGraph: Graph Theory Analysis of Brain MRI Data

Calculate graph global, local, or nodal efficiency

Description

This function calculates the global efficiency of a graph or the local or nodal efficiency of each vertex of a graph.

Usage

efficiency(g, type = c("local", "nodal", "global"), weights = NULL,
  xfm = FALSE, xfm.type = NULL, use.parallel = TRUE, A = NULL,
  D = NULL)

Arguments

g	An igraph graph object
type	Character string; either local, nodal, or global. Default: local
weights	Numeric vector of edge weights; if NULL (the default), and if the graph has edge attribute weight, then that will be used. To avoid using weights, this should be NA.
xfm	Logical indicating whether to transform the edge weights. Default: FALSE
xfm.type	Character string specifying how to transform the weights. Default: 1/w
use.parallel	Logical indicating whether or not to use foreach. Default: TRUE
A	Numeric matrix; the adjacency matrix of the input graph. Default: NULL
D	Numeric matrix; the graph's “distance matrix”

Details

Local efficiency for vertex i is:

E_{local}(i) = \frac{1}{N} \sum_{i \in G} E_{global}(G_i)

where G_i is the subgraph of neighbors of i, and N is the number of vertices in that subgraph.

Nodal efficiency for vertex i is:

E_{nodal}(i) = \frac{1}{N-1} \sum_{j \in G} \frac{1}{d_{ij}}

Global efficiency for graph G with N vertices is:

E_{global}(G) = \frac{1}{N(N-1)} \sum_{i \ne j \in G} \frac{1}{d_{ij}}

where d_{ij} is the shortest path length between vertices i and j. Alternatively, global efficiency is equal to the mean of all nodal efficiencies.

Value

A numeric vector of the efficiencies for each vertex of the graph (if type is local|nodal) or a single number (if type is global).

Author(s)

Christopher G. Watson, [e-mail]

References

Latora, V. and Marchiori, M. (2001) Efficient behavior of small-world networks. Phys Rev Lett, 87.19, 198701. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1103/PhysRevLett.87.198701")}

Latora, V. and Marchiori, M. (2003) Economic small-world behavior in weighted networks. Eur Phys J B, 32, 249–263. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1140/epjb/e2003-00095-5")}

brainGraph documentation built on June 22, 2024, 7:36 p.m.