Communicability Centrality

Total communicability (Estrada and Hatano 2008; Benzi and Klymko 2013) sums the walks of every length that start at a node, a walk of length \(k\) weighted by \(1/k!\): $$TC(v) = \sum_{w} \left[ e^{A} \right]_{vw} = \sum_{w} \sum_{k = 0}^{\infty} \frac{(A^k)_{vw}}{k!}.$$

Usage

centrality_communicability(x, ...)

Arguments

x

Network input accepted by centrality.

...

Further arguments to centrality, such as directed and normalized.

Value

A named numeric vector with one score per node, in input node order.

Details

\(A\) is the binary adjacency matrix, so edge weights are ignored. On an undirected network the matrix exponential is formed from the eigendecomposition of the symmetric \(A\). On a directed network it is computed by scaling and squaring with a Pade approximation (Moler and Van Loan 2003), and the scores are the row sums of \(e^{A}\), the walks that leave each node. directed = FALSE gives the undirected reading.

References

Estrada, E., & Hatano, N. (2008). Communicability in complex networks. Physical Review E, 77(3), 036111. doi:10.1103/PhysRevE.77.036111 .

Benzi, M., & Klymko, C. (2013). Total communicability as a centrality measure. Journal of Complex Networks, 1(2), 124-149. doi:10.1093/comnet/cnt007 .

Moler, C., & Van Loan, C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review, 45(1), 3-49. doi:10.1137/S00361445024180 .

See also

centrality_subgraph, centrality_communicability_betweenness, centrality.

Examples

centrality_communicability(regulation_net, directed = FALSE)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>   218.0164   264.8024   302.7023   255.7658   212.7336   222.1101   188.5025 
#>   Evaluate     Create      Share 
#>   237.1442   269.3303   238.0564