Random walks

Modular hierarchical and power-law small-world networks bear structural optima for minimal first passage times and cover time

Networks that are organized as a hierarchy of modules have been the subject of much research, mainly focusing on algorithms that can extract this community structure from data. The question of why modular hierarchical (MH) organizations are so …

Cover time for random walks on arbitrary complex networks

We present an analytical method for computing the mean cover time of a discrete-time random walk process on arbitrary, complex networks. The cover time is defined as the time a random walker requires to visit every node in the network at least once. …

Complexity Explorables

This project is designed for people interested in complex systems and complex dynamical processes. Complexity Explorables hosts different collections of interactive illustrations of models for complex systems in physics, mathematics, biology, chemistry, social sciences, neuroscience, epidemiology, network science and ecology. Topics include pattern formation, synchronization, critical phenomena, chaotic dynamics, evolutionary dynamics, fractals, collective behavior, reaction-diffusion systems and more.

Human mobility and spatial disease dynamics

Lévy flights in inhomogeneous media

We investigate the impact of external periodic potentials on superdiffusive random walks known as Lévy flights and show that even strongly superdiffusive transport is substantially affected by the external field. Unlike ordinary random walks, Lévy …

Lévy flights in external force fields: from models to equations

We consider different generalizations of the Fokker–Planck equation (FPE) devised to describe Lévy processes in potential force fields. We show that such generalizations can proceed along different lines. On one hand, Lévy statistics can emerge from …