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  Criticality in conserved dynamical systems: Experimental observation vs. exact properties

Markovic, D., Gros, C., & Schuelein, A. (2013). Criticality in conserved dynamical systems: Experimental observation vs. exact properties. Chaos: an interdisciplinary journal of nonlinear science, 23: 013106. doi:10.1063/1.4773003.

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Markovic_Criticality.pdf (Publisher version), 575KB
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Markovic, Dimitrije1, Author           
Gros, Claudius1, Author
Schuelein, André1, Author
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1Institute for Theoretical Physics, Goethe University, Frankfurt, Germany, ou_persistent22              

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Free keywords: Power laws; Dynamical systems; Information flow; Cyclic attractors; Networks
 Abstract: Conserved dynamical systems are generally considered to be critical. We study a class of critical routing models, equivalent to random maps, which can be solved rigorously in the thermodynamic limit. The information flow is conserved for these routing models and governed by cyclic attractors. We consider two classes of information flow, Markovian routing without memory and vertex routing involving a one-step routing memory. Investigating the respective cycle length distributions for complete graphs, we find log corrections to power-law scaling for the mean cycle length, as a function of the number of vertices, and a sub-polynomial growth for the overall number of cycles. When observing experimentally a real-world dynamical system one normally samples stochastically its phase space. The number and the length of the attractors are then weighted by the size of their respective basins of attraction. This situation is equivalent, for theory studies, to “on the fly” generation of the dynamical transition probabilities. For the case of vertex routing models, we find in this case power law scaling for the weighted average length of attractors, for both conserved routing models. These results show that the critical dynamical systems are generically not scale-invariant but may show power-law scaling when sampled stochastically. It is hence important to distinguish between intrinsic properties of a critical dynamical system and its behavior that one would observe when randomly probing its phase space.

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Language(s): eng - English
 Dates: 2012-05-302012-12-062013-01-092013
 Publication Status: Published in print
 Pages: -
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 Table of Contents: -
 Rev. Type: Peer
 Identifiers: DOI: 10.1063/1.4773003
PMID: 23556943
 Degree: -

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Title: Chaos : an interdisciplinary journal of nonlinear science
Source Genre: Journal
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Publ. Info: Woodbury, NY : American Institute of Physics
Pages: - Volume / Issue: 23 Sequence Number: 013106 Start / End Page: - Identifier: ISSN: 1054-1500
CoNE: https://pure.mpg.de/cone/journals/resource/954922836228