  We invoke a metric to quantify the correlation between any two earthquakes.
This provides a simple and straightforward alternative to using space-time
windows to detect aftershock sequences and obviates the need to distinguish
main shocks from aftershocks. Directed networks of earthquakes are constructed
by placing a link, directed from the past to the future, between pairs of
events that are strongly correlated. Each link has a weight giving the relative
strength of correlation such that the sum over the incoming links to any node
equals unity for aftershocks, or zero if the event had no correlated
predecessors. A correlation threshold is set to drastically reduce the size of
the data set without losing significant information. Events can be aftershocks
of many previous events, and also generate many aftershocks. The probability
distribution for the number of incoming and outgoing links are both scale free,
and the networks are highly clustered. The Omori law holds for aftershock rates
up to a decorrelation time that scales with the magnitude, $m$, of the
initiating shock as $t_{\rm cutoff} \sim 10^{\beta m}$ with $\beta \simeq 3/4$.
Another scaling law relates distances between earthquakes and their aftershocks
to the magnitude of the initiating shock. Our results are inconsistent with the
hypothesis of finite aftershock zones. We also find evidence that seismicity is
dominantly triggered by small earthquakes. Our approach, using concepts from
the modern theory of complex networks, together with a metric to estimate
correlations, opens up new avenues of research, as well as new tools to
understand seismicity.
