Random Self-Similar Trees: Emergence of Scaling Laws

Abstract

The hierarchical organization and emergence of scaling laws in complex systems-geophysical, biological, technological, and socioeconomic-have been the topic of extensive research at the turn of the twentieth century. Although significant progress has been achieved, the mathematical origin of and relation among scaling laws for different system attributes remain unsettled. Paradigmatic examples are the Gutenberg-Richter law of seismology and Horton's laws of geomorphology. We review the results that clarify the appearance, parameterization, and implications of scaling laws in hierarchical systems conceptualized by tree graphs. A recently formulated theory of random self-similar trees yields a suite of results on scaling laws for branch attributes, tree fractal dimension, powerlaw distributions of link attributes, and power-law relations between distinct attributes. Given the relevance of power laws to extreme events and hazards, our review informs related theoretical and modeling efforts and provides a framework for unified analysis in hierarchical complex systems.

The Multiscale Principle in Nature (Principium luxuriæ): Linking Multiscale Thermodynamics to Living and Non-Living Complex Systems

“…The multiscale properties of a system refer to the existence of multiple scales of organization within the system [1][2][3][4], usually during phase transition, where each scale could be characterized by different mathematical physics [5][6][7], biochemical [8,9], or biological [10][11][12][13][14][15] properties. A wide range of varied systems, such as cosmology [16], zoology [17], networks [18], cities [19], ecology [20], computational imaging [21], interface physics [22,23], geophysics [24], emergent processes [25], genetic [26], or complex systems [27] often exhibit hierarchical organization [28][29][30], where smaller-scale components or subsystems interact to create emergent behavior at larger scales. These emergent behaviors are typically described as a type of dynamic that cannot be described as the sum of its parts [31][32][33].…”

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“…The dominance of fluvial erosion leads to the emergence of self-similarity in natural landscapes, as the interlocked network of ridges and valleys grows in complexity. Statistical self-similarity in such landscapes reveals invariant statistical structures across different observation scales ( 1 – 6 ), giving rise to scaling laws for several key properties like contributing area, stream length, and drainage density, as the drainage networks become fractal ( 4 , 7 , 8 ). Such universal scaling laws suggest that landscape dynamics eventually become independent of the precise fluvial erosion intensity, attaining complete self-similarity with respect to the fluvial erosion process ( 9 , 10 ).…”

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“…The Special Issue highlights the state of the art in complex dynamic systems, quantitative seismology, geodynamics, earthquake forecasting, and hazard assessments by overviewing past developments and looking at future possibilities and perspectives. Kovchegov et al (2022) point up the importance of a hierarchical organisation and the emergence of scaling laws in complex systems, such as the Earth's lithosphere (e.g. Keilis-Borok 1990).…”