## 1. Introduction

[2] Fault systems as the San Andreas fault in California are prime examples of self-organizing systems in nature which are characterized by an internal dynamics that increases the inherent order of the system [*Rundle et al.*, 2002]. Very often they consist of interacting elements, each of which stays quiescent until its internal state variable reaches a trigger threshold leading to a rapid discharge or “firing”. Since the internal state variables evolve in time in response to external driving sources and inputs from other elements, the firing of an element may in turn trigger a discharge of other elements. In the context of fault systems, this corresponds to stress discharge in the form of earthquakes, or the deformation and sudden rupture of parts of the earth's crust driven by convective motion in the mantle.

[3] Self-organizing systems very often exhibit dynamics that is strongly correlated in space and time over many scales. In particular, the complex spatiotemporal dynamics of fault systems manifests itself in a number of generic, empirical features of earthquake occurrence including clustering, fault traces and epicenter locations with fractal statistics, as well as scaling laws like the Omori and Gutenberg-Richter laws [*Turcotte*, 1997; *Rundle et al.*, 2003], giving rise to a worldwide debate about their explanation. Resolving this dispute could conceivably require measuring the internal state variables — the stress and strain everywhere within the earth along active faults — and their exact dynamics. This is (currently) impossible. Yet, the associated earthquake patterns are readily observable making a statistical approach based on the concept of spatiotemporal point processes feasible, where the description of each earthquake is reduced to its size or magnitude, its epicenter and its time of occurrence. Describing the patterns of seismicity may shed light on the fundamental physics since these patterns are emergent processes of the underlying many-body nonlinear system.

[4] Recently, such an approach has brought to light new properties of the clustering of seismicity in space and time [*Bak et al.*, 2002; *Corral*, 2003, 2004; *Davidsen and Goltz*, 2004; *Davidsen and Paczuski*, 2005; *Baiesi and Paczuski*, 2005], which can potentially be exploited for earthquake prediction [*Goltz*, 2001; *Tiampo et al.*, 2002; *Baiesi*, 2006]. One aim has been to evaluate distances between subsequent events, including temporal and spatial measures. The observed spatiotemporal clustering of seismicity suggests that subsequent events are to a certain extent causally related. It further suggests that the usual mainshock/aftershock scenario — where each event has at most one correlated predecessor — is too simplistic and that the causal structure of seismicity could extend beyond immediately subsequent events, especially since the determination of the sequence is largely arbitrary depending on the size of the region considered and the completeness of the record of events.

[5] In this work we quantify the spatiotemporal clustering of seismicity in terms of a sparse, directed network, where each earthquake is a node in the graph and links connect events with their recurrences. This general network picture allows us to characterize clustering by using only the spatiotemporal structure of seismicity, without any additional assumptions.