heterEigenvectorCentrality.multiplex: Multiplex heterogeneous eigenvector centrality indices of the...
In Achab94/mplex: mplex

Description Usage Arguments Details Value References See Also Examples

The function calculates the multiplex heterogeneous eigenvector centrality indices proposed by Sola' et al.

heterEigenvectorCentrality.multiplex(obj,
                                    indexNode = 1:length(nodes.multiplex(obj)),
                                    W = matrix(1, length(layers.multiplex(obj)),
                                    length(layers.multiplex(obj))),
                                    rowStand = TRUE)

`obj`	An object of class `multiplex`.
`indexNode`	A vector of IDs (or labels) for the selected nodes on which to calculate the multiplex heterogeneous eigenvector centrality coefficients.
`W`	The influence matrix of the multiplex network. For further information, see References.
`rowStand`	Default is `TRUE`. If `FALSE`, the vectors given in output are not standardized (i.e. the sum of their components is != 1) and comparison with other (intra)layers or multiplex networks is more difficult.

This measure refeirs to the eigenvector of a special N*L x N*L matrix obtained from the (inter)layer adjacency matrices and the so-called influence matrix of the multiplex network, where N is the number of nodes and L the number of layers of the multiplex network taken into consideration. Its eigenvector has length N*L, thus it's breaked into L sub-vectors, each of them refers to a certain (intra)layer of the multiplex network.

Irreducibility is a required assumption to satisfy the Perron-Frobenius theorem, which ensures the positivity of the eigenvector assosicated to the maximum eigenvalue of the supra adjacency matrix of the multiplex network; nevertheless, results are usually good even if it is not strictly satisfied.

A list with the L vectors of the multiplex heterogeneous eigenvector centrality indices of the nodes (eventually selected with indexNode argument), where L is the number of layers of the multiplex network.

Sola' et al. (2013) Eigenvector centrality of nodes in multiplex networks. Chaos Volume 23, Issue 3.

create.multiplex

# Loading Aarhus CS Department dataset.

data(aarhus_mplex)

# Creating the multiplex object using the dataset loaded into aarhus_mplex object.

mplexObj <- create.multiplex(nodes = aarhus_mplex$nodes,
                            layersNames = aarhus_mplex$layerNames,
                            layer1 = aarhus_mplex$L1,
                            type1 = "undirected",
                            aarhus_mplex$L2,
                            aarhus_mplex$L3,
                            aarhus_mplex$L4,
                            aarhus_mplex$L5
                            )

# Calculating the multiplex heterogeneous eigenvector centrality indices for the multiplex network:

heterEigenvectorCentrality.multiplex(mplexObj)

# It could be useful to select a set of nodes on which to calculate the index. This can be done
# using the 'indexNode' argument, as it follows:

heterEigenvectorCentrality.multiplex(mplexObj,
                                      indexNode = sample(1:length(nodes.multiplex(mplexObj)), 10)
                                      )

# The particularity of this index is strictly linked to the possibility to include a so-called
# influence matrix in the argument 'W'. This matrix rapresents the weights of the links (i.e. the
# interlayer edges) of the multiplex network. If we set, as an example, a random influence
# matrix, we see that the values of the index changes:

W <- matrix(rbinom(25, 5, .5), 5, 5)
diag(W) <- 0
heterEigenvectorCentrality.multiplex(mplexObj, W = W)

# Another way to visualize the results is to consider the standardized measures. In this case,
# comparisons between indices on different layers can be done, because the sum of the indices
# for each layer are forced to be 1:

heterEigenvectorCentrality.multiplex(mplexObj, rowStand = TRUE)
apply(heterEigenvectorCentrality.multiplex(mplexObj, rowStand = TRUE), 1, sum)