Ordered Avalanches on the Bethe Lattice

Abstract

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