communicability {brainGraph}

Calculate communicability

Description

communicability calculates the communicability of a network, a measure which takes into account all possible paths (including non-shortest paths) between vertex pairs.

Usage

communicability(g, weights = NULL)

Arguments

g

An igraph graph object

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.

Details

The communicability G_{pq} is a weighted sum of the number of walks from vertex p to q and is calculated by taking the exponential of the adjacency matrix A:

G_{pq} = \sum_{k=0}^{\infty} \frac{(\mathbf{A}^k)_{pq}}{k!} = (e^{\mathbf{A}})_{pq}

where k is walk length.

For weighted graphs with D = diag(d_i) a diagonal matrix of vertex strength,

G_{pq} = (e^{\mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}})_{pq}

Value

A numeric matrix of the communicability

Author(s)

Christopher G. Watson, [e-mail]

References

Estrada, E. and Hatano, N. (2008) Communicability in complex networks. Physical Review E. 77, 036111. doi: 10.1103/PhysRevE.77.036111

Crofts, J.J. and Higham, D.J. (2009) A weighted communicability measure applied to complex brain networks. J. R. Soc. Interface. 6, 411–414. doi: 10.1098/rsif.2008.0484