Growth, collapse, and self-organized criticality in complex networks

To understand how certain dynamical behaviors can or cannot persist as the underlying network grows is a problem of increasing importance in complex dynamical systems as well as sustainability science and engineering. We address the question of whether a complex network of nonlinear oscillators can maintain its synchronization stability as it expands or grows. A network in the real world can never be completely synchronized due to noise and/or external disturbances. This is especially the case when, mathematically, the transient synchronous state during the growth process becomes marginally stable, as a local perturbation can trigger a rapid deviation of the system from the vicinity of the synchronous state. In terms of the nodal dynamics, a large scale avalanche over the entire network can be triggered in the sense that the individual nodal dynamics diverge from the synchronous state in a cascading manner within a short time period. Because of the high dimensionality of the networked system, the transient process for the system to recover to the synchronous state can be extremely long. Introducing a tolerance threshold to identify the desynchronized nodes, we find that, after an initial stage of linear growth, the network typically evolves into a critical state where the addition of a single new node can cause a group of nodes to lose synchronization, leading to synchronization collapse for the entire network. A statistical analysis indicates that, the distribution of the size of the collapse is approximately algebraic (power law), regardless of the fluctuations in the system parameters. This is indication of the emergence of self-organized criticality. We demonstrate the generality of the phenomenon of synchronization collapse using a variety of complex network models, and uncover the underlying dynamical mechanism through an eigenvector analysis.Comment: 10pages, 6 figure

Supersymmetry Constraints and String Theory on K3

We study supervertices in six dimensional (2,0) supergravity theories, and derive supersymmetry non-renormalization conditions on the 4- and 6-derivative four-point couplings of tensor multiplets. As an application, we obtain exact non-perturbative results of such effective couplings in type IIB string theory compactified on K3 surface, extending previous work on type II/heterotic duality. The weak coupling limit thereof, in particular, gives certain integrated four-point functions of half-BPS operators in the nonlinear sigma model on K3 surface, that depend nontrivially on the moduli, and capture worldsheet instanton contributions.Comment: 47 pages, 4 figure

Romans Supergravity from Five-Dimensional Holograms

We study five-dimensional superconformal field theories and their holographic dual, matter-coupled Romans supergravity. On the one hand, some recently derived formulae allow us to extract the central charges from deformations of the supersymmetric five-sphere partition function, whose large N expansion can be computed using matrix model techniques. On the other hand, the conformal and flavor central charges can be extracted from the six-dimensional supergravity action, by carefully analyzing its embedding into type I' string theory. The results match on the two sides of the holographic duality. Our results also provide analytic evidence for the symmetry enhancement in five-dimensional superconformal field theories.Comment: 57 pages, 4 figures, 6 tables; v2: references adde