Random Self-Similar Trees: Emergence of Scaling Laws

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, power-law 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.