Abstract
We consider a fairly wide class of nonlinear diffusion equations on networks, and derive several common and basic behaviors of solutions to them. Further, we demonstrate that the Legendre structure can be naturally introduced for such a class of dynamical systems, and discuss their information geometric aspects.
Supported by JSPS Grant278 in-Aid (C) 19K03633.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Barrat, A., Barthelemy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008)
Thurner, S., Hanel, R., Klimek, P.: Introduction to the Theory of Complex Systems. Oxford University Press, Oxford (2018)
Kumagai, T.: Random Walks on Disordered Media and Their Scaling Limits. Lecture Notes in Mathematics, vol. 2101. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-03152-1
Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: Araki, H. (ed.) International Symposium on Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol. 39, pp. 420–422. Springer, Heidelberg (1975). https://doi.org/10.1007/BFb0013365
Arenas, A., DÃaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Phys. Rep. 469, 93–153 (2008)
Zhang, X., Senyo, T., Sakai, H., Ohara, A.: Behaviors of solutions to network diffusion equation with power-nonlinearity. Eur. Phys. J. Spec. Top. 229(5), 729–741 (2020). https://doi.org/10.1140/epjst/e2020-900202-5
Chung, F.R.K.: Spectral Graph Theory. AMS, Providence (1997)
Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197–198, 143–176 (1994)
Newman, M.: Networks: An Introduction. Oxford University Press Inc., New York (2018)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Amari, S., Nagaoka, H.: Methods of Information Geometry. AMS and Oxford (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Ohara, A., Zhang, X. (2021). Properties of Nonlinear Diffusion Equations on Networks and Their Geometric Aspects. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_79
Download citation
DOI: https://doi.org/10.1007/978-3-030-80209-7_79
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-80208-0
Online ISBN: 978-3-030-80209-7
eBook Packages: Computer ScienceComputer Science (R0)