Ebook: Fractional Dynamics on Networks and Lattices

Random walks are among the most fundamental stochastic processes that occur
ubiquitously in various interdisciplinary contexts such as in biological networks, the
foraging of animals, the spread of diseases, in finance, human mobility in cities,
friendship networks, among many other “complex systems”. Generally, random
walks are microscopic models for diffusion processes and play a crucial role in the
development of “random search” strategies. One of the main goals of this field is to
find a strategy that a given set of targets distributed among a larger set is most
quickly visited by the walker. This and similar questions constitute one of the major
motivations and driving forces for the subjects of this book. The analysis of new
random walk strategies, especially those which allow faster exploration of a network
or a subset of it, is highly desirable for many interdisciplinary applications and
interesting from a theoretical point of view. Last but not least, the recent upswing of
online networks such as Google and social networks has launched a huge interest in
stochastic motions on networks. In many of these problems of real life such as in
population dynamics, chaotic motions, the time evolution of stock-market prices and
many others, it is impossible and even meaningless to describe time evolutions using
deterministic equations. Instead, one is interested in extracting a maximum of
“statistical information” from these processes. As a result, many different kinds of
random walk models have been proposed and extensively studied.