Random walks with long-range memory on networks

We study an exactly solvable random walk model with long-range memory on arbitrary networks. The walker performs unbiased random steps to nearest-neighbor nodes and intermittently resets to previously visited nodes in a preferential way such that the most visited nodes have proportionally a higher probability to be chosen for revisit. The occupation probability can be expressed as a sum over the eigenmodes of the standard random walk matrix of the network, where the amplitudes slowly decay as power-laws at large times, instead of exponentially. The stationary state is the same as in the absence of memory, and detailed balance is fulfilled. However, the relaxation of the transient part becomes critically self-organized at late times, as it is dominated by a single power-law whose exponent depends on the second largest eigenvalue and on the resetting probability. We apply our findings to finite networks, such as rings, complete graphs, Watts-Strogatz, and Barabási-Albert networks, and to Barbell and comb-like graphs. Our study could be of interest for modeling complex transport phenomena, such as human mobility, epidemic spreading, or animal foraging.