Title: Phase Synchronization on Spacially Embeded Duplex Networks with Total Cost Constraint,
Abstract: Synchronization on multiplex networks have attracted increasing attention in
the past few years. We investigate collective behaviors of Kuramoto oscillators
on single layer and duplex spacial networks with total cost restriction, which
was introduced by Li et. al [Li G., Reis S. D., Moreira A. A., Havlin S.,
Stanley H. E. and Jr A. J., {\it Phys. Rev. Lett.} 104, 018701 (2010)] and
termed as the Li network afterwards. In the Li network model, with the increase
of its spacial exponent, the network's structure will vary from the random type
to the small-world one, and finally to the regular lattice.We first explore how
the spacial exponent influences the synchronizability of Kuramoto oscillators
on single layer Li networks and find that the closer the Li network is to a
regular lattice, the more difficult for it to evolve into synchronization. Then
we investigate synchronizability of duplex Li networks and find that the
existence of inter-layer interaction can greatly enhance inter-layer and global
synchronizability. When the inter-layer coupling strength is larger than a
certain critical value, whatever the intra-layer coupling strength is, the
inter-layer synchronization will always occur. Furthermore, on single layer Li
networks, nodes with larger degrees more easily reach global synchronization,
while on duplex Li networks, this phenomenon becomes much less obvious.
Finally, we study the impact of inter-link density on global synchronization
and obtain that sparse inter-links can lead to the emergence of global
synchronization for duplex Li networks just as dense inter-links do. In a word,
inter-layer interaction plays a vital role in determining synchronizability for
duplex spacial networks with total cost constraint.