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


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.


Random walks Complex networks Nonlinear Markov chains Bifurcations 

Mathematics Subject Classification

05C81 05C21 39A28 60J10 


  1. Atkinson, K.E.: An Introduction to Numerical Analysis. Wiley, London (2008)Google Scholar
  2. Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. In: Computer Networks and ISDN Systems. Proceedings of the Seventh International World Wide Web Conference, 30, pp. 107–117 (1998)Google Scholar
  3. Butkovsky, O.: On ergodic properties of nonlinear Markov chains and stochastic Mckean–Vlasov equations. Theory Probab. Appl. 58, 661–674 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Durrett, R., Durrett, R.: Essentials of Stochastic Processes. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar
  5. Frank, T.: Markov chains of nonlinear Markov processes and an application to a winner-takes-all model for social conformity. J. Phys. A Math. Theor. 41, 282001 (2008a)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Frank, T.: Nonlinear Markov processes: deterministic case. Phys. Lett. A 372, 6235–6239 (2008b)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Frank, T.: Deterministic and stochastic components of nonlinear Markov models with an application to decision making during the bailout votes 2008 (USA). Eur. Phys. J. B 70, 249–255 (2009)CrossRefGoogle Scholar
  8. Frank, T.: Stochastic processes and mean field systems defined by nonlinear Markov chains: an illustration for a model of evolutionary population dynamics. Braz. J. Phys. 41, 129 (2011)CrossRefGoogle Scholar
  9. Frank, T.: Strongly nonlinear stochastic processes in physics and the life sciences. ISRN Math. Phys. 2013, 149169 (2013)CrossRefzbMATHGoogle Scholar
  10. Gleich, D.F.: Pagerank beyond the web. SIAM Rev. 57, 321–363 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Gómez-Gardeñes, J., Latora, V.: Entropy rate of diffusion processes on complex networks. Phys. Rev. E 78, 065102 (2008)CrossRefGoogle Scholar
  12. Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space–time fractional diffusion. Chem. Phys. 284, 521–541 (2002)CrossRefzbMATHGoogle Scholar
  13. Hunter, J.K., Nachtergaele, B.: Applied Analysis. World Scientific, Singapore (2001)CrossRefzbMATHGoogle Scholar
  14. Kolokoltsov, V.N.: Nonlinear Markov Processes and Kinetic Equations, vol. 182. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  15. Kolokoltsov, V.N.: Nonlinear Markov games on a finite state space (mean-field and binary interactions). Int. J. Stat. Probab. 1, 77 (2012)CrossRefGoogle Scholar
  16. MacCluer, C.R.: The many proofs and applications of Perron’s theorem. SIAM Rev. 42, 487–498 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  18. Nicosia, V., Skardal, P.S., Arenas, A., Latora, V.: Collective phenomena emerging from the interactions between dynamical processes in multiplex networks. Phys. Rev. Lett. 118, 138302 (2017)CrossRefGoogle Scholar
  19. Noh, J.D., Rieger, H.: Random walks on complex networks. Phys. Rev. Lett. 92, 118701 (2004)CrossRefGoogle Scholar
  20. Page, L., Brin, S., Motwani, R., Winograd, T.: The pagerank citation ranking: bringing order to the web. Technical Report 1999-66, Stanford InfoLab (1999)Google Scholar
  21. Rosvall, M., Bergstrom, C.T.: Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. 105, 1118–1123 (2008)CrossRefGoogle Scholar
  22. Saburov, M.: Ergodicity of nonlinear Markov operators on the finite dimensional space. Nonlinear Anal. Theory Methods Appl. 143, 105–119 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Schultz, P., Menck, P.J., Heitzig, J., Kurths, J.: Potentials and limits to basin stability estimation. New J. Phys. 19, 023005 (2017)CrossRefGoogle Scholar
  24. Sinatra, R., Gómez-Gardeñes, J., Lambiotte, R., Nicosia, V., Latora, V.: Maximal-entropy random walks in complex networks with limited information. Phys. Rev. E 83, 030103 (2011)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsTrinity CollegeHartfordUSA

Personalised recommendations