  A sufficiently connected topology linking the constituent units of a complex
system is usually seen as a prerequisite for the emergence of collective
phenomena such as synchronization. We present a random network of heterogeneous
phase oscillators in which the links mediating the interactions are constantly
rearranged with a characteristic timescale and, possibly, an extremely low
instantaneous connectivity. We show that, provided strong coupling and fast
enough rewiring are considered, the network is able to reach partial
synchronization even in the vanishing connectivity limit. We also provide an
intuitive analytical argument, based on the comparison between the different
characteristic timescales of our system in the low connectivity regime, which
is able to predict the transition to synchronization threshold with
satisfactory precision. In the formal fast switching limit, finally, we argue
that the onset of collective synchronization is captured by the time-averaged
connectivity network. Our results may be relevant to qualitatively describe the
emergence of consensus in social communities with time-varying interactions and
to study the onset of collective behavior in engineered systems of mobile units
with limited wireless capabilities.
