We investigate collective synchronization in a system of coupled oscillators on small-world networks. The order parameters that measure synchronization of phases and frequencies are introduced and analyzed by means of dynamic simulations and finite-size scaling. Phase synchronization is observed to emerge in the presence of even a tiny fraction P of shortcuts and to display saturated behavior for $P\gtrsim \mathrm{0.5.}$ This indicates that the same synchronizability as the random network $\mathrm{}(P=1\mathrm{})\mathrm{}$ can be achieved with relatively small number of shortcuts. The transient behavior of the synchronization, obtained from the measurement of the relaxation time, is also discussed.

Phase transitions in water are normally classified as first or second order. But in confined quasi-one-dimensional films of water, simulations show that the solid–liquid transition can take place by means of a first-order transition or a continuous one without a distinction between solid and liquid. Bulk water has three phases: solid, liquid and vapour. In addition to undergoing a phase transition (of the first order) between them, liquid and vapour can deform continuously into each other without crossing a transition line—in other words, there is no intrinsic distinction between the two phases. Hence, the first-order line of the liquid–vapour phase transition should terminate at a critical point. In contrast, the first-order transition line between solid and liquid is believed to persist indefinitely without terminating at a critical point1. In recent years, however, it was reported that inside carbon nanotubes, freezing of water may occur continuously as well as discontinuously through a first-order phase transition2. Here we present simulation results for water in a quasi-two-dimensional hydrophobic nanopore slit, which are consistent with the idea that water may freeze by means of both first-order and continuous phase transitions. Our results lead us to hypothesize the existence of a connection point at which first-order and continuous transition lines meet3,4.

While shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network, the suppressing tendency of the heterogeneity of the degree distribution, even for shorter characteristic path length, has also been reported. To see this, we investigate the effects of various factors such as the degree, characteristic path length, heterogeneity, and betweenness centrality on synchronization, and find a consistent trend between the synchronization and the betweenness centrality. The betweenness centrality is thus proposed as a good indicator for synchronizability.

A two-dimensional small-world-type network, subject to spatial prisoners’ dilemma dynamics and containing an influential node defined as a special node, with a finite density of directed random links to the other nodes in the network, is numerically investigated. It is shown that the degree of cooperation does not remain at a steady state level but displays a punctuated equilibrium-type behavior manifested by the existence of sudden breakdowns of cooperation. The breakdown of cooperation is linked to an imitation of a successful selfish strategy of the influential node. It is also found that while the breakdown of cooperation occurs suddenly, its recovery requires longer time. This recovery time may, depending on the degree of steady state cooperation, either increase or decrease with an increasing number of long-range connections.

We develop the Hubbard-Stratanovich transform for uniformly frustrated XY models both on a square lattice and on a triangular lattice, and construct the Landau-Ginzburg-Wilson Hamiltonians, which reflect the formation of various superlattices according to values of the frustration f. Near the critical point the system f=(1/4) on a triangular lattice is shown to belong to the same universality class as the fully frustrated (f=(1/2)) system on a square lattice. By decomposing two-mode systems into two coupled XY models and by applying the Migdal-Kadanoff approximation, we show the possibility of Ising-like or three-state Potts-like transitions in addition to the Kosterlitz-Thouless–like ones.

Concentrations of cations and ATP affect the binding activities of synaptotagmin-1, to trigger synaptic vesicle exocytosis. In physiological ionic conditions, synaptotagmin-1's interaction with SNARE proteins is prevented, and its binding is confined to PIP2-containing membrane. The Ca2+ sensor synaptotagmin-1 is thought to trigger membrane fusion by binding to acidic membrane lipids and SNARE proteins. Previous work has shown that binding is mediated by electrostatic interactions that are sensitive to the ionic environment. However, the influence of divalent or polyvalent ions, at physiological concentrations, on synaptotagmin's binding to membranes or SNAREs has not been explored. Here we show that binding of rat synaptotagmin-1 to membranes containing phosphatidylinositol 4,5-bisphosphate (PIP2) is regulated by charge shielding caused by the presence of divalent cations. Surprisingly, polyvalent ions such as ATP and Mg2+ completely abrogate synaptotagmin-1 binding to SNAREs regardless of the presence of Ca2+. Altogether, our data indicate that at physiological ion concentrations Ca2+-dependent synaptotagmin-1 binding is confined to PIP2-containing membrane patches in the plasma membrane, suggesting that membrane interaction of synaptotagmin-1 rather than SNARE binding triggers exocytosis of vesicles.