Abstract
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 with strong coupling and sufficiently fast rewiring the network reaches partial synchronization even in the vanishing connectivity limit. In particular, we provide an approximate 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 beyond the formal fast rewiring limit. We interpret our results as a qualitative mechanism for emergence of consensus in social communities. In particular, our result suggest that groups of individuals are capable of aligning their opinions under extremely sparse exchanges of views, which is reminiscent of fast communications that take place in the modern social media. Our results may also be relevant to characterize the onset of collective behavior in engineered systems of mobile units with limited wireless capabilities.
Highlights
The emergence of collective phenomena in complex systems is related to the interplay between interaction topology and local dynamics[1,2,3,4]
Here we wish to go beyond the strict fast switching limit asking under which conditions macroscopic synchronization may emerge in Erdös-Rény networks with random rewiring and, possibly, arbitrarily small instantaneous connectivity
Numerical simulations and analytical arguments show that this system is able to achieve statistically stable macroscopic synchronization even for arbitrarily small network connectivity, provided sufficiently fast switching and strong couplings are considered
Summary
Characteristic time scales and the onset of synchronization. We seek to understand the physical mechanism leading to synchronisation in the low connectivity region via switching. Once the link is removed in a rewiring event, nodes can be left without any link, so that the phase of previously connected and synchronized oscillators will start to drift away one from each other due to their natural frequencies difference δω, losing any information regarding their previous mutual synchronization when their phase difference approaches π/2. This allows one to define the typical local desynchronization timescale τLD such that τLD〈δω〉.
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