efficiency {brainGraph} | R Documentation |
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 |
type |
Character string; either |
weights |
Numeric vector of edge weights; if |
xfm |
Logical indicating whether to transform the edge weights. Default:
|
xfm.type |
Character string specifying how to transform the weights.
Default: |
use.parallel |
Logical indicating whether or not to use |
A |
Numeric matrix; the adjacency matrix of the input graph. Default:
|
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, cgwatson@bu.edu
References
Latora, V. and Marchiori, M. (2001) Efficient behavior of small-world networks. Phys Rev Lett, 87.19, 198701. 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. doi: 10.1140/epjb/e2003-00095-5