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Hyperbolic Property of Earthquake Networks
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Advanced statistical research on robust quantification of seismicity patterns is required for mitigating the devastating impacts of earthquakes. The need for such research is highlighted by increasing population density in large urban areas near major active faults (e.g., Tokyo, Istanbul, Los Angeles, San Francisco), the catastrophic earthquakes in Japan, Haiti and Indonesia (life loss over 500,000, economical damage over $US 100 billion), and recent earthquakes in the Midwestern US and other areas with hydrocarbon and geothermal production. A particularly important aspect of seismicity that might benefit from better statistical and physical understanding is earthquake clustering in time and space. A nearest-neighbor analysis has been shown recently to be an effective tool for identifying earthquake clusters, quantifying regional cluster styles in relation to the physical properties of lithosphere, discriminating natural and human-induced seismicity, and developing earthquake declustering techniques. The key technical component of this analysis is a network (graph) of earthquakes connected according to their space-time-energy proximity. Large-scale geometric analysis and, particularly, hyperbolic embedding of the examined networks has facilitated network analysis in various applied fields during the recent decade. This thesis is focused on exploring hyperbolic properties of the earthquake networks. We examine the geometry of earthquakes in time-space-magnitude domain using the Gromov hyperbolic property of metric spaces. Gromov delta-hyperbolicity quantifies the curvature of a metric space via four point condition, which is a computationally convenient analog of the famous slim triangle property. We estimate the delta-hyperbolicity for observed events from several different earthquake catalogs. A set of earthquakes is represented by a point field in space-time-magnitude domain D. The separation between earthquakes is quantified by the Baiesi-Paczuski proximity n that has been shown efficient in applied cluster analyses of natural and human-induced seismicity and acoustic emission experiments. The Gromov delta is estimated in the earthquake space (D,n) and in proximity graphs obtained by connecting pairs of earthquakes within proximity n0. All experiments result in the values of delta that are bounded from above and do not tend to increase as the examined region expands. This suggests that the earthquake field has hyperbolic geometry. We discuss the properties naturally associated with hyperbolicity in terms of the examined earthquake field. The results improve the understanding of the dynamics of seismicity and further expand the list of natural processes characterized by underlying hyperbolic geometry.