  Synchronization of non-identical oscillators coupled through complex networks
is an important example of collective behavior. It is interesting to ask how
the structural organization of network interactions influences this process.
Several studies have uncovered optimal topologies for synchronization by making
purposeful alterations to a network. Yet, the connectivity patterns of many
natural systems are often not static, but are rather modulated over time
according to their dynamics. This co-evolution - and the extent to which the
dynamics of the individual units can shape the organization of the network
itself - is not well understood. Here, we study initially randomly connected
but locally adaptive networks of Kuramoto oscillators. The system employs a
co-evolutionary rewiring strategy that depends only on instantaneous, pairwise
phase differences of neighboring oscillators, and that conserves the total
number of edges, allowing the effects of local reorganization to be isolated.
We find that a simple regulatory rule - which preserves connections between
more out-of-phase oscillators while rewiring connections between more in-phase
oscillators - can cause initially disordered networks to organize into more
structured topologies that support enhanced synchronization dynamics. We
examine how this process unfolds over time, finding both a dependence on the
intrinsic frequencies of the oscillators and the global coupling. For large
enough coupling and after sufficient adaptation, the resulting networks exhibit
degree - frequency and frequency - neighbor frequency correlations. These
properties have previously been associated with optimal synchronization or
explosive transitions. By considering a time-dependent interplay between
structure and dynamics, this work offers a mechanism through which emergent
phenomena can arise in complex systems utilizing local rules.
