communicability calculates the communicability of a network, a measure
which takes into account all possible paths (including non-shortest paths)
between vertex pairs.
communicability(g, weights = NULL)An igraph graph object
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.
A numeric matrix of the communicability
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, $$G_{pq} = (e^{\mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}})_{pq}$$
Estrada E. & Hatano N. (2008) Communicability in complex networks. Physical Review E, 77:036111.
Crofts J.J. & Higham D.J. (2009) A weighted communicability measure applied to complex brain networks. J. R. Soc. Interface, 6:411-414.