  Reconstructing network connectivity from the collective dynamics of a system
typically requires access to its complete continuous-time evolution although
these are often experimentally inaccessible. Here we propose a theory for
revealing physical connectivity of networked systems only from the event time
series their intrinsic collective dynamics generate. Representing the patterns
of event timings in an event space spanned by inter-event and cross-event
intervals, we reveal which other units directly influence the inter-event times
of any given unit. For illustration, we linearize an event space mapping
constructed from the spiking patterns in model neural circuits to reveal the
presence or absence of synapses between any pair of neurons as well as whether
the coupling acts in an inhibiting or activating (excitatory) manner. The
proposed model-independent reconstruction theory is scalable to larger networks
and may thus play an important role in the reconstruction of networks from
biology to social science and engineering.
