Diffusion distance matrix from a custom transition matrix

Source: R/get_distance_matrix.R

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. \(L = I - Pi\).

get_distance_matrix_from_T(Pi, tau, as_dist = FALSE, verbose = TRUE)

get_DDM_from_T(Pi, tau, as_dist = FALSE, verbose = TRUE)

get_distance_matrix_from_Pi(Pi, tau, as_dist = FALSE, verbose = TRUE)

get_DDM_from_Pi(Pi, tau, as_dist = FALSE, 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