get_ddm_from_eigendec {diffudist}R Documentation

Distance Matrix from Laplacian spectral decomposition

Description

Returns the diffusion distance matrix when the spectrum (more precisely, the eigendecomposition) of the Laplacian is provided as input (useful to speed up batch calculations).

For instance, the random walk normalised Laplacian ID1AI - D^{-1}A, which generates the classical continuous-time random walk over a network, can be easily and obtained from the spectral decomposition of the symmetric normalised Laplacian L=D12LD12=D12(DA)D12\mathcal{L} = D^{-\frac{1}{2}} L D^{-\frac{1}{2}} = D^{-\frac{1}{2}} (D - A) D^{-\frac{1}{2}}. More specifically, Lˉ=ID1A=D12LD12\bar{L} = I - D^{-1} A = D^{-\frac{1}{2}} \mathcal{L} D^{\frac{1}{2}} and, since L\mathcal{L} is symmetric it can be decomposed into L=l=1NλlululT\mathcal{L} = \sum_{l = 1}^N \lambda_l u_l u_l^T, hence

Lˉ=l=1NλlulRulL\bar{L} = \sum_{l = 1}^N \lambda_l u^R_l u^L_l

where ulL=ulTD12u^L_l = u_l^T D^{\frac{1}{2}} and ulR=ulD12u^R_l = u_l D^{-\frac{1}{2}}.

Usage

get_ddm_from_eigendec(tau, Q, Q_inv, lambdas, verbose = FALSE)

Arguments

tau

diffusion time (scalar)

Q

eigenvector matrix

Q_inv

inverse of the eigenvector matrix

lambdas

eigenvalues (vector)

verbose

whether warnings have to be printed or not

Value

The diffusion distance matrix DtD_t, a square numeric matrix of the Euclidean distances between the rows of the stochastic matrix P(t)=eτLP(t) = e^{-\tau L}, where L-L is the Laplacian generating a continuous-time random walk (Markov chain) over the network. The matrix exponential is here computed using the given eigendecomposition of the Laplacian matrix eτL=QeτΛQ1e^{-\tau L} = Q e^{-\tau \Lambda} Q^{-1}.

References

Bertagnolli, G., & De Domenico, M. (2021). Diffusion geometry of multiplex and interdependent systems. Physical Review E, 103(4), 042301. doi:10.1103/PhysRevE.103.042301 arXiv: 2006.13032

See Also

get_spectral_decomp


[Package diffudist version 1.0.1 Index]