Clustering Spectrum of scale-free networks

Real-world networks often have power-law degrees and scale-free properties such as ultra-small distances and ultra-fast information spreading. In this paper, we study a third universal property: three-point correlations that suppress the creation of triangles and signal the presence of hierarchy. We quantify this property in terms of $\bar c(k)$, the probability that two neighbors of a degree-$k$ node are neighbors themselves. We investigate how the clustering spectrum $k\mapsto\bar c(k)$ scales with $k$ in the hidden variable model and show that $c(k)$ follows a {\it universal curve} that consists of three $k$-ranges where $\bar c(k)$ remains flat, starts declining, and eventually settles on a power law $\bar c(k)\sim k^{-\alpha}$ with $\alpha$ depending on the power law of the degree distribution. We test these results against ten contemporary real-world networks and explain analytically why the universal curve properties only reveal themselves in large networks

Magnetic-Field Induced Gap in One-Dimensional Antiferromagnet KCuGaF6_6

Magnetic susceptibility and specific heat measurements in magnetic fields were performed on an $S=1/2$ one-dimensional antiferromagnet KCuGaF$_6$. Exchange interaction was evaluated as $J/k_{\rm B}\simeq 100$ K. However, no magnetic ordering was observed down to 0.46 K. It was found that an applied magnetic field induces a staggered magnetic susceptibility obeying the Curie law and an excitation gap, both of which should be attributed to the antisymmetric interaction of the Dzyaloshinsky-Moriya type and/or the staggered $g$-tensor. With increasing magnetic field $H$, the gap increases almost in proportion to $H^{2/3}$.Comment: Submitted to Proceedings of Research in High Magnetic Fiel