Earthquake Sequencing: Significance of Kuramoto Model on Directed Graphs

Abstract:

Earthquake sequencing studies allow us to investigate empirical relationships among spatio-temporal parameters describing the complexity of earthquake properties. We have recently studied the relevance of Markov chain models to draw information from global earthquake catalogues. In these studies, we have considered directed graphs as graph theoretic representation of the Markov chain model, and their properties. Here, we look at earthquake sequencing itself as a directed graph. In general, earthquakes are occurrences resulting from significant stress-interactions among faults. As a result, stress-field fluctuations evolve continuously. We propose that they are akin to the dynamics of the collective behaviour of weakly-coupled non-linear oscillators. Since mapping of global stress-field fluctuations in real time at all scales is an impossible task, we consider an earthquake zone as a proxy for a collection of weakly-coupled oscillators the dynamics of which would befit the ubiquitous Kuramoto model. In the present work, we apply the Kuramoto model to understand the non-linear dynamics on a directed graph of a sequence of earthquakes. For directed graphs with certain properties, the Kuramoto model yields synchronization and inclusion of non-local effects evokes the occurrence of chimera states or the co-existence of synchronous and asynchronous behaviour of oscillators. In this presentation, we show how we build the model for both synthetic networks honouring different orders and average degrees, differing asymmetries and network heterogeneity and directed graphs derived from global seismicity data. Then, we present conditions under which chimera states could occur and subsequently, point out the role of Kuramoto model in understanding the evolution of synchronous and asynchronous regions. We interpret our results using spectral properties of directed graphs.