Evolving power grids with self-organized intermittent strain releases: An analogy with sandpile models and earthquakes

Ordered Avalanches on the Bethe Lattice

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 ≈ log(b)/log(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 αmax in the bifractal spectrum is found to be 1/τ.