get_ddm_from_eigendec: Distance Matrix from Laplacian spectral decomposition

In diffudist: Diffusion Distance for Complex Networks

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 I - 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 \mathcal{L} = D^{-\frac{1}{2}} L D^{-\frac{1}{2}} = D^{-\frac{1}{2}} (D - A) D^{-\frac{1}{2}}. More specifically, \bar{L} = I - D^{-1} A = D^{-\frac{1}{2}} \mathcal{L} D^{\frac{1}{2}} and, since \mathcal{L} is symmetric it can be decomposed into \mathcal{L} = ∑_{l = 1}^N λ_l u_l u_l^T, hence

\bar{L} = ∑_{l = 1}^N λ_l u^R_l u^L_l

where u^L_l = u_l^T D^{\frac{1}{2}} and u^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 D_t, a square numeric matrix of the Euclidean distances between the rows of the stochastic matrix P(t) = e^{-τ L}, where -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 e^{-τ L} = Q e^{-τ Λ} 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

diffudist documentation built on March 7, 2023, 6:12 p.m.