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Ordered Avalanches on the Bethe Lattice

Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
Complex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków, Poland
Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, ul. Warszawska 24, 31-155 Kraków, Poland
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Entropy 2019, 21(10), 968;
Received: 5 September 2019 / Revised: 29 September 2019 / Accepted: 30 September 2019 / Published: 3 October 2019
(This article belongs to the Special Issue Disordered Systems, Fractals and Chaos)
We discuss deterministic sequences of avalanches on a directed Bethe lattice. The approach is motivated by the phenomenon of self-organized criticality. Grains are added only at one node of the network. When the number of grains at any node exceeds a threshold b, each of k out-neighbors gets one grain. The probability of an avalanche of size s is proportional to s τ . When the avalanche mass is conserved ( k = b ), we get τ = 1 . For an application of the model to social phenomena, the conservation condition can be released. Then, the exponent τ is found to depend on the model parameters; τ     l o g ( b ) / l o g ( k ) . The distribution of the time duration of avalanches is exponential. Multifractal analysis of the avalanche sequences reveals their strongly non-uniform fractal organization. Maximal value of the singularity strength α m a x in the bifractal spectrum is found to be 1 / τ .
Keywords: self-organized criticality; multifractals; mean field; social processes self-organized criticality; multifractals; mean field; social processes
MDPI and ACS Style

Krawczyk, M.J.; Oświęcimka, P.; Kułakowski, K.; Drożdż, S. Ordered Avalanches on the Bethe Lattice. Entropy 2019, 21, 968.

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