  We analyze the properties of order parameters measuring synchronization and
phase locking in complex oscillator networks. First, we review network order
parameters previously introduced and reveal several shortcomings: none of the
introduced order parameters capture all transitions from incoherence over phase
locking to full synchrony for arbitrary, finite networks. We then introduce an
alternative, universal order parameter that accurately tracks the degree of
partial phase locking and synchronization, adapting the traditional definition
to account for the network topology and its influence on the phase coherence of
the oscillators. We rigorously proof that this order parameter is strictly
monotonously increasing with the coupling strength in the phase locked state,
directly reflecting the dynamic stability of the network. Furthermore, it
indicates the onset of full phase locking by a diverging slope at the critical
coupling strength. The order parameter may find applications across systems
where different types of synchrony are possible, including biological networks
and power grids.
