twork
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