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 nonmonotonic, 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 nonmonotonic properties are key ingredients in nonlinear transport applications complicated and unpredictable behaviors may result.
Highlights
Random walks have long served as a powerful tool for studying and understanding the structural properties of complex networks due to the wealth of information they provide with remarkably simple dynamics [1,2]
We focus our attention on the case of discrete-time nonlinear random walks on complex networks
We explore the effect that the complicated dynamics that emerge in nonlinear random walks with nonmonotonic transition functions have on characteristic return times, i.e., the typical time it takes for a random walker to return to a particular node
Summary
Random walks have long served as a powerful tool for studying and understanding the structural properties of complex networks due to the wealth of information they provide with remarkably simple dynamics [1,2]. Random walkers might be biased more strongly towards nodes where a moderate number of random walkers are present, whereas being biased away from nodes that are both heavily and sparsely populated Such nonmonotonic transition probabilities may arise in nonlinear transport applications where the decision-making process that informs movement is nuanced. We explore the dynamics that emerge from nonlinear random walks with nonmonotonic transition probabilities and show that they give rise to chaotic dynamics that are not present when transition probabilities are monotonic We show that this phenomenon is widespread by exploring multiple choices for functions that define the transition probabilities of a system as well as a range of different network topologies.
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