Sand Model On The Apollo Network

Complex networks and self-organized criticality are two important branches in the research of complex system, the former based on the topology, the latter focusing on the dynamics. In order to deepen the understanding of complex network and understand-ing of self-organized criticality, this paper will study the directed sandpile model on the Apollonian network, trying to combine the two together and to study their relationship and interaction.We make the deterministic small world network and the deterministic Apollonian network included under the unifying framework to generate the Apollonian network with an iteration algorithm. On this basis, making the use of local self-similarity struc-ture of the one-dimensional Apollonian network and the method of renormalization group, spanning trees of the one-dimensional Apollonian network will be researched and the exact number of spanning trees can be obtained, avoiding using the matrix tree theory to catch the number of spanning trees and greatly reducing computational com-plexity of this.The analysis of the directed sandpile model on the Apollonian network is the focus of this paper. We will introduce only one parameter q to limit the number of neighbors who can receive sands from the collapsed node. Due to the special geometry of the Apollonian network, we can obtain the accurate expression of the two-and three-point correlation functions, leading to analytical results for avalanche distributions, mean-square flux and the critical exponent in the limit of an infinite system for q=1,2. Finally, we will discover that, for finite values of q, the avalanche distributions follow asymptotic power-law scaling forms with typical mean-field exponents. In other words, our results also show the emergence of large oscillatory deviations, suggesting that, in the qâ†'>∞limit, an exponential dependence of the average flux with the layer index is obtained. It also confirms us that the special topology of complex network has a great impact on the dynamic behavior of complex systems.