Analytical approach to directed sandpile models on the Apollonian network

Abstract

We investigate a set of directed sandpile models on the Apollonian network, which are inspired on the work by Dhar and Ramaswamy (PRL \textbf{63}, 1659 (1989)) for Euclidian lattices. They are characterized by a single parameter $q$, that restricts the number of neighbors receiving grains from a toppling node. Due to the geometry of the network, two and three point correlation functions are amenable to exact treatment, leading to analytical results for the avalanche distributions in the limit of an infinite system, for $q=1,2$. The exact recurrence expressions for the correlation functions are numerically iterated to obtain results for finite size systems, when larger values of $q$ are considered. Finally, a detailed description of the local flux properties is provided by a multifractal scaling analysis.

7 pages in two-column format, 10 illustrations, 5 figures

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• Controlling self-organized criticality in complex networks
• Infinitely Robust Order and Local Order-Parameter Tulips in Apollonian Networks with Quenched Disorder
• Contact graphs of disk packings as a model of spatial planar networks
• Effect of rewiring for a sandpile model on a directed network