Topology of the Visibility Graph of Sandpiles

Abstract

In this paper, we explore the higher- and lower-order connectivity aspects of the visibility graph representation of sandpile models, focusing on the Bak-Tang-Wiesenfeld (BTW) model. By converting the time series of avalanche events into visibility graphs, we investigate the structural properties that emerge at different scales of connectivity. Specifically, we utilize persistent homology and simplicial complex theory to identify higher-order topological features. Analyzing the statistics of degree and betweenness centralities reveal that the resulting graph is a scale-free network with the degree exponent of ${\gamma }_k=2.50\pm 0.02$ and betweenness exponent of ${\gamma }_b=1.585\pm 0.008$. The analysis of the simplexes show that the distribution function of one, two and three dimensional simplexes are power-law with the exponents of ${\gamma }_{{\sigma }_1}=0.795\pm 0.006$, ${\gamma }_{{\sigma }_2}=0.602\pm 0.009$ and ${\gamma }_{{\sigma }_3}=0.422\pm 0.008$ respectively. This power-law behavior is also seen in the homology group analysis, for the Betti numbers, the exponents of which are reported in the paper. The persistent entropy of life time of $d$ dimensional holes is analyzed indicating that it increases logarithmically in terms of the network size $N$ for $d=0,1$ and $2$.

References in corpus (17)

• From time series to complex networks: the visibility graph
• Networks beyond pairwise interactions: structure and dynamics
• Dynamical synapses causing self-organized criticality in neural networks
• Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems
• Persistent Homology of Complex Networks
• Small-world structure of earthquake network
• Self-organization without conservation: Are neuronal avalanches generically critical?
• Scaling fields in the two-dimensional abelian sandpile model
• Visibility graph analysis of wall turbulence time-series
• Dynamical evolution of clustering in complex network of earthquakes
• Avalanche frontiers in dissipative abelian sandpile model as off-critical SLE(2)
• Dynamical evolution of the community structure of complex earthquake network
• Dynamically Driven Renormalization Group Applied to Sandpile Models
• Renormalization group approach to an Abelian sandpile model on planar lattices
• Bak-Tang-Wiesenfeld Model in the Finite Range Random Link Lattice
• Some Properties of Sandpile Models as Prototype of Self-Organized Critical Systems
• Self-organized criticality in cumulus clouds