Alfredo Blanco Serrano

arXiv2019

Markov chain approach to anomalous diffusion on Newman-Watts networks

A Markov chain (MC) formalism is used to investigate the mean-square displacement (MSD) of a random walker on Newman-Watts (NW) networks. It leads to a precise analysis of the conditions for the emergence of anomalous sub- or super-diffusive regimes in such random media. Whereas results provided by most numerical approaches used so far base their results on the computation of a large number of independent runs over many equivalent substrates, the MC framework is applied only once to each equivalent sample. Starting from the simple cycle graph with $2k$ nearest neighbor connections, for which exact MSD expressions within the MC formalism can be derived, the randomness and complexity of the substrate is easily controlled by the number $x$ of added links. Results for different values of $k$, $x$, and the number $N$ of nodes make it possible to distinguish actual anomalous regimes from transient behavior and finite size effects. Albeit the high computing cost restricts the size of our networks to $N\leq1500$ nodes, our very precise results justify a new and more comprehensive scaling ansatz for walker dynamics, from which the behavior for very large networks can be derived.

physics.soc-ph↗

arXiv2018

Superdiffusion on complex networks: the role of shortcuts and long-range interactions

This work addresses the superdiffusive motion of a discrete time random walker on ordered discrete substrates and complex networks with the presence of long-range interactions (LRIs). In ordered regular lattices, where LRIs have a clear geometrical meaning, their presence allow for hoppings between more distant sites, yet with a smaller probability. In such cases, it is found that LRIs do not affect the dependency of the mean square displacement (MSD) traveled by the walker: exact analytical results for the the cycle graph within the Markov chain framework shows that MSD follows the same linearly increasing behavior with time when LRIs are absent, independently of the strength of LRI. This contrasts with the superdiffusive scenario in complex networks. When they have very short diameter ($\sim \log N$), the analysis of the time dependency of MSD becomes quite difficult, as it saturates very quickly even when LRIs are absent. The presence of a faster than linearly increasing growth phase can be noticed, but it can hardly be measured with precision. This effect is sidestepped on small-world Newman-Watts (NW) networks, where the network diameter can be controlled by the number of new links (shortcuts) that are added to the cycle graph. The time duration $t_f$ of the superdiffusive regime and the power law exponent can be adequately evaluated by numerical methods. They depend on the number of nodes and shortcuts, as well as the strength of LRIs. Although the later causes a strong reduction in $t_f$ when shortcuts are present, their presence by itself is not sufficient to trigger a superdiffusive behavior.

physics.soc-ph↗