get_distance_matrix_from_T: Diffusion distance matrix from a custom transition matrix

In diffudist: Diffusion Distance for Complex Networks

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 τ between nodes i, j \in G is defined as

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

with \mathbf{p}(t|i) = (e^{- τ L})_{i\cdot} = \mathbf{e}_i e^{- τ L} indicating the i-th row of the stochastic matrix e^{- τ 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 τ = 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^{-τ 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)

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