Returns a specific Laplacian matrix corresponding to the chosen dynamics type and network. The available types are:
for the classical combinatorial Laplacian matrix; it governs the diffusion dynamics on the network
for the Laplacian matrix normalized by degree matrix, the so-called classical random walk normalized Laplacian; it governs stochastic walks on the network
for the Laplacian matrix normalized to be symmetric; it governs quantum walks on the network
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)a network
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.
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).
default TRUE. If information on the type of Laplacian
or on edge weights should be printed.
the (`type`) Laplacian matrix of network `g`
[1] Burda, Z., et al. (2009). Phys Rev. Lett. 102 160602(April), 1–4. doi:10.1103/PhysRevLett.102.160602