This function calculates the global efficiency of a graph or the local or nodal efficiency of each vertex of a graph.
efficiency(g, type = c("local", "nodal", "global"), weights = NULL,
xfm = FALSE, xfm.type = NULL, use.parallel = TRUE, A = NULL,
D = NULL)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).
An igraph graph object
Character string; either local, nodal, or
global. Default: local
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.
Logical indicating whether to transform the edge weights. Default:
FALSE
Character string specifying how to transform the weights.
Default: 1/w
Logical indicating whether or not to use foreach.
Default: TRUE
Numeric matrix; the adjacency matrix of the input graph. Default:
NULL
Numeric matrix; the graph's “distance matrix”
Christopher G. Watson, cgwatson@bu.edu
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.
Latora, V. and Marchiori, M. (2001) Efficient behavior of small-world networks. Phys Rev Lett, 87.19, 198701. 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. tools:::Rd_expr_doi("10.1140/epjb/e2003-00095-5")