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Pathways of diffusion observed in real-world systems often require stochasticprocesses going beyond first-order Markov models, as implicitly assumed innetwork theory. In this work, we focus on second-order Markov models, andderive an analytical expression for the effect of memory on the spectral gapand thus, equivalently, on the characteristic time needed for the stochasticprocess to asymptotically reach equilibrium. Perturbation analysis shows thatstandard first-order Markov models can either overestimate or underestimate thediffusion rate of flows across the modular structure of a system captured by asecond-order Markov network. We test the theoretical predictions on a toyexample and on numerical data, and discuss their implications for networktheory, in particular in the case of temporal or multiplex networks.

Effect of Memory on the Dynamics of Random Walks on Networks