Mapping Sandpiles to Complex Networks

Abstract

In this paper, we address a longstanding challenge in self-organized criticality (SOC) systems: establishing a connection between sandpiles and complex networks. Our approach employs a similarity-based transfer function characterized by two parameters, $\mathcal{R}=(r_1, r_2)$. Here, $r_1$ quantifies the similarity of local activities, while $r_2$ governs the filtration process used to convert a weighted network into a binary one. We reveal that the degree centrality distribution of the resulting network follows a generalized Gamma distribution (GGD), which transitions to a power-law distribution under specific conditions. The GGD exponents, estimated numerically, exhibit a dependency on $\mathcal{R}$. Notably, while both decreasing $r_1$ and $r_2$ lead to denser networks, $r_2$ plays a more significant role in influencing network density. Furthermore, the Shannon entropy is observed to decrease linearly with increasing $r_2$, whereas its variation with $r_1$ is more gradual. An analytical expression for the Shannon entropy is proposed. To characterize the network structure, we investigate the clustering coefficient ($cc$), eigenvalue centrality ($e$), closeness centrality ($c$), and betweenness centrality ($b$). The distributions of $cc$, $e$, and $c$ exhibit peaked profiles, while $b$ displays a power-law distribution over a finite interval of $k$. Additionally, we explore correlations between the exponents and identify a specific parameter regime of $\mathcal{R}$ and $k$ where the $e-k$, $c-k$, and $b-k$ correlations become negative.