This book analyzes stochastic processes on networks and regular
structures such as lattices by employing the Markovian random walk
approach. Part 1 is devoted to the study of local and non-local
random walks. It shows how non-local random walk strategies can be
defined by functions of the Laplacian matrix that maintain the
stochasticity of the transition probabilities. A major result is
that only two types of functions are admissible: type (i) functions
generate asymptotically local walks with the emergence of Brownian
motion, whereas type (ii) functions generate asymptotically
scale-free non-local "fractional" walks with the emergence of Levy
flights. In Part 2, fractional dynamics and Levy flight behavior
are analyzed thoroughly, and a generalization of Polya's classical
recurrence theorem is developed for fractional walks. The authors
analyze primary fractional walk characteristics such as the mean
occupation time, the mean first passage time, the fractal scaling
of the set of distinct nodes visited, etc. The results show the
improved search capacities of fractional dynamics on networks.
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