get_distance_matrix_from_T {diffudist}

Diffusion distance matrix from a custom transition matrix

Description

Returns a matrix where each entry encodes the diffusion distance between two nodes of a network, given a transition matrix on the network and a diffusion time.

The diffusion distance at time $\tau$ between nodes $i, j \in G$ is defined as

$D_{\tau}(i, j) = \vert \mathbf{p}(t|i) - \mathbf{p}(t|j) \vert_2$

with $\mathbf{p}(t|i) = (e^{- \tau L})_{i\cdot} = \mathbf{e}_i e^{- \tau L}$ indicating the i-th row of the stochastic matrix $e^{- \tau L}$ and representing the probability (row) vector of a random walk dynamics corresponding to the initial condition $\mathbf{e}_i$, i.e. the random walker is in node $i$ at time $\tau = 0$ with probability 1.

The Laplacian $L$ is the normalised laplacian corresponding to the given transition matrix, i.e. $L = I - Pi$.

Usage

get_distance_matrix_from_T(Pi, tau, verbose = TRUE)

get_DDM_from_T(Pi, tau, verbose = TRUE)

get_distance_matrix_from_Pi(Pi, tau, verbose = TRUE)

get_DDM_from_Pi(Pi, tau, verbose = TRUE)

Arguments

Pi

a transition matrix (it should be a stochastic matrix)

tau

diffusion time

verbose

default TRUE

Value

The diffusion distance matrix $D_t$, a square numeric matrix of the $L^2$-norm distances between posterior probability vectors, i.e. Euclidean distances between the rows of the stochastic matrix $P(t) = e^{-\tau L}$, where $-L = -(I - T)$ is the generator of the continuous-time random walk (Markov chain) corresponding to the discrete-time transition matrix $T=$Pi.

References

De Domenico, M. (2017). Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena. Physical Review Letters. doi:10.1103/PhysRevLett.118.168301

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_distance_matrix get_diffusion_probability_matrix, get_diffusion_probability_matrix_from_T

Examples

g <- igraph::sample_pa(10, directed = FALSE)
dm <- get_distance_matrix(g, tau = 1)