Recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices

Abstract

We analyze a random walk strategy on undirected regular networks involving power matrix functions of the type L α 2 where L indicates a 'simple' Laplacian matrix. We refer such walks to as 'Fractional Random Walks' with admissible interval 0 < α ≤ 2. We deduce for the Fractional Random Walk probability generating functions (network Green's functions). From these analytical results we establish a generalization of Polya's recurrence theorem for Fractional Random Walks on d-dimensional infinite lattices: The Fractional Random Walk is transient for dimensions d > α (recurrent for d ≤ α) of the lattice. As a consequence for 0 < α < 1 the Fractional Random Walk is transient for all lattice dimensions d = 1, 2, .. and in the range 1 ≤ α < 2 for dimensions d ≥ 2. Finally, for α = 2 Polya's classical recurrence theorem is recovered, namely the walk is transient only for lattice dimensions d ≥ 3. The generalization of Polya's recurrence theorem remains valid for the class of random walks with Lévy flight asymptotics for long-range steps. We also analyze for the Fractional Random Walk mean first passage probabilities, mean residence times, mean first passage times, and global mean first passage times (Kemeny constant). For the infinite 1D lattice (infinite ring) we obtain for the transient regime 0 < α < 1 closed form expressions for the fractional lattice Green's function matrix containing the escape and ever passage probabilities. The ever passage probabilities fulfill Riesz potential power law decay asymptotic behavior for nodes far from the departure node. The non-locality of the Fractional Random Walk is generated by the non-diagonality of the fractional Laplacian matrix with Lévy type heavy tailed inverse power law decay for the probability of long-range moves. This non-local and asymptotic behavior of the Fractional Random Walk introduces small world properties with emergence of Lévy flights on large (infinite) lattices.

Continuous time random walk and diffusion with generalized fractional Poisson process

“…We refer the resulting diffusion process to as 'generalized fractional diffusion'. The eigenvalues of the one-step transition matrix (55) for the d-dimensional infinite lattice are given by the Fourier transforms [33,34]…”

Generalized Space–Time Fractional Dynamics in Networks and Lattices

“…, N . The topology of the network is described by the positive-semidefinite N × N Laplacian matrix which is defined by [8,12,36,37,38,39,40] L…”

“…For the random walk on the network we allow long-range jumps which can be described when we replace the Laplacian matrix by its fractional power in the one-step transition matrix (45). In this way, the walker cannot only jump to connected neighbor nodes, but also to far distant nodes in the network [5,6,8,34,36,37,38,39,42,43]. The model to be developed in this section involves both space-and time-fractional calculus.…”

“…, k d ) denotes the wave-vector with −π ≤ k j ≤ π. One can show that the fractional index µ is restricted to the interval 0 < µ ≤ 2 as a requirement for stochasticity of the one-step transition matrix [8,34,38]. In (50) the constant K (µ) can be conceived as a fractional generalization of the degree and is given by the trace of the fractional power of Laplacian matrix, namely [8,34]…”

“…In order to derive the 'diffusive limit' which corresponds to the space-time representation of this equation, it appears instructive to consider the long-wave contribution of some kernels such as the fractional power of the Laplacian matrix in Z d . The fractional Laplacian matrix in Z d has the canonic form [8,38] [L…”