Topological data analysis of the visibility graphs of the BTW avalanche-size sequences

Abstract In this paper, we use topological data analysis to explore both pairwise and higher-order connectivity aspects of the visibility graph (VG) representation of sandpile models. By transforming the time series of avalanche sizes into VGs, we investigate the structural characteristics of the Bak–Tang–Wiesenfeld model. Through an analysis of degree and betweenness centralities, we find that the resulting graphs exhibit scale-free behavior, with degree and betweenness exponents <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2.50</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.02</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1.585</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.008</mml:mn> </mml:mrow> </mml:math> , respectively. We further apply persistent homology to extract topological features, revealing that the distributions of one-, two-, and three-dimensional simplexes follow power-law forms with exponents <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>γ</mml:mi> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.795</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.006</mml:mn> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>γ</mml:mi> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.602</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.009</mml:mn> </mml:mrow> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>γ</mml:mi> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.422</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.008</mml:mn> </mml:mrow> </mml:math> , respectively. Similar scaling behavior is observed in the Betti number distributions derived from homology group analysis. In addition, the persistent entropy associated with the lifetimes of d -dimensional holes increases logarithmically with the network size N for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Although alternative methods of mapping avalanche-size sequences into VGs may yield different sets of exponents, this work contribute to a broader understanding of topological and structural patterns in complex networks derived from dynamical systems, with potential applications in modeling phenomena such as earthquakes, forest fires, and financial market dynamics.