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Journal of Nonlinear Science

, Volume 29, Issue 4, pp 1419–1444 | Cite as

Dynamics of Nonlinear Random Walks on Complex Networks

  • Per Sebastian SkardalEmail author
  • Sabina Adhikari
Article
  • 149 Downloads

Abstract

In this paper, we study the dynamics of nonlinear random walks. While typical random walks on networks consist of standard Markov chains whose static transition probabilities dictate the flow of random walkers through the network, nonlinear random walks consist of nonlinear Markov chains whose transition probabilities change in time depending on the current state of the system. This framework allows us to model more complex flows through networks that may depend on the current system state. For instance, under humanitarian or capitalistic direction, resource flow between institutions may be diverted preferentially to poorer or wealthier institutions, respectively. Importantly, the nonlinearity in this framework gives rise to richer dynamical behavior than occurs in linear random walks. Here we study these dynamics that arise in weakly and strongly nonlinear regimes in a family of nonlinear random walks where random walkers are biased either toward (positive bias) or away from (negative bias) nodes that currently have more random walkers. In the weakly nonlinear regime, we prove the existence and uniqueness of a stable stationary state fixed point provided that the network structure is primitive that is analogous to the stationary distribution of a typical (linear) random walk. We also present an asymptotic analysis that allows us to approximate the stationary state fixed point in the weakly nonlinear regime. We then turn our attention to the strongly nonlinear regime. For negative bias, we characterize a period-doubling bifurcation where the stationary state fixed point loses stability and gives rise to a periodic orbit below a critical value. For positive bias, we investigate the emergence of multistability of several stable stationary state fixed points.

Keywords

Random walks Complex networks Nonlinear Markov chains Bifurcations 

Mathematics Subject Classification

05C81 05C21 39A28 60J10 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTrinity CollegeHartfordUSA

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