Chaos in Nonlinear Random Walks with Non-Monotonic Transition Probabilities

TL;DR

This paper investigates how nonlinear random walks with non-monotonic transition probabilities can lead to chaotic dynamics across various network structures, revealing complex behaviors in nonlinear transport systems.

Contribution

It introduces the concept that non-monotonic transition probabilities in nonlinear random walks induce chaos, a phenomenon previously underexplored in network dynamics.

Findings

Chaotic behavior emerges in nonlinear random walks with non-monotonic transition probabilities.

The phenomenon is shown to be generic across different network types.

Non-monotonic transition functions lead to complex, unpredictable dynamics.

Abstract

Random walks serve as important tools for studying complex network structures, yet their dynamics in cases where transition probabilities are not static remain under explored and poorly understood. Here we study nonlinear random walks that occur when transition probabilities depend on the state of the system. We show that when these transition probabilities are non-monotonic, i.e., are not uniformly biased towards the most densely or sparsely populated nodes, but rather direct random walkers with more nuance, chaotic dynamics emerge. Using multiple transition probability functions and a range of networks with different connectivity properties, we demonstrate that this phenomenon is generic. Thus, when such non-monotonic properties are key ingredients in nonlinear transport applications complicated and unpredictable behaviors may result.