Abstract
We propose a method to search for signs of causal structure in
spatiotemporal data making minimal a priori assumptions
about the underlying dynamics. To this end, we generalize the
elementary concept of recurrence for a point process in time to
recurrent events in space and time. An event is defined to be a
recurrence of any previous event if it is closer to it in space
than all the intervening events. As such, each sequence of
recurrences for a given event is a record breaking process. This
definition provides a strictly data driven technique to search for
structure. Defining events to be nodes, and linking each event to
its recurrences, generates a network of recurrent events.
Significant deviations in properties of that network compared to
networks arising from random processes allows one to
infer attributes of the causal dynamics that generate observable
correlations in the patterns. We derive analytically a number of
properties for the network of recurrent events composed by a
random process in space and time. We extend the theory of records
to treat not only the variable where records happen, but also time
as continuous. In this way, we construct a fully symmetric theory
of records leading to a number of new results. Those analytic
results are compared in detail to the properties of a network
synthesized from time series of epicenter locations for
earthquakes in Southern California. Significant disparities from
the ensemble of random networks that can be plausibly attributed
to the causal structure of seismicity are: (1) Invariance of
network statistics with the time span of the events considered,
(2) Appearance of a fundamental length scale for recurrences,
independent of the time span of the catalog, which is consistent
with observations of the rupture length, (3) Hierarchy in the
distances and times of subsequent recurrences.
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