Evaluate a Laplacian Matrix

Source: R/get_diff_prob_matrix.R

Returns a specific Laplacian matrix corresponding to the chosen dynamics type and network. The available types are:

Laplacian

for the classical combinatorial Laplacian matrix; it governs the diffusion dynamics on the network

Normalized Laplacian

for the Laplacian matrix normalized by degree matrix, the so-called classical random walk normalized Laplacian; it governs stochastic walks on the network

Quantum Laplacian

for the Laplacian matrix normalized to be symmetric; it governs quantum walks on the network

MERW normalized Laplacian

the maximal-entropy random walk (RW) normalized Laplacian; it governs stochastic walks on the network, in which the random walker moves according to a maximal-entropy RW [1].

The maximum entropy random walk (MERW) chooses the stochastic matrix which maximizes \(H(S)\), so that the walker can explore every walk of the same length with equal probability. Let \(\lambda_N, \phi\) be the leading eigenvalue and corresponding right eigenvector of the adjacency matrix \(A\). Then the transition matrix corresponding to the discrete-time random walk is \(\Pi_{ij} = \frac{A_{ij}}{\lambda_N}\frac{\phi_j}{\phi_i}.\) The MERW (normalized) Laplacian is then given by \(I - \Pi\). Note that we use the notation \(\Pi\) and Pi to avoid confusion with the abbreviation T for the logical TRUE.

get_laplacian(g, type = "Laplacian", weights = NULL, verbose = TRUE)

getLaplacianMatrix(g, type = "Laplacian", weights = NULL, verbose = TRUE)

Arguments

g

a network
type

the type of Laplacian matrix. default "Laplacian", the combinatorial Laplacian. Other types: c("Normalized Laplacian", "Quantum Laplacian", "MERW Normalized Laplacian"). Note that you can type abbreviations, e.g. "L", "N", "Q", "M" for the respective types (case is ignored). The argument match is done through match_arg.
weights

edge weights, representing the strength/intensity (not the cost!) of each link. if weights is NULL (the default) and g has an edge attribute called weight, then it will be used automatically. If this is NA then no weights are used (even if the graph has a weight attribute).
verbose

default TRUE. If information on the type of Laplacian or on edge weights should be printed.

Value

the (`type`) Laplacian matrix of network `g`

References

[1] Burda, Z., et al. (2009). Phys Rev. Lett. 102 160602(April), 1–4. doi: 10.1103/PhysRevLett.102.160602