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Social networks involve both positive and negative relationships, which can be captured in signed graphs. The {\em edge sign prediction problem} aims to predict whether an interaction between a pair of nodes will be positive or negative. We provide theoretical results for this problem that motivate natural improvements to recent heuristics. The edge sign prediction problem is related to correlation clustering; a positive relationship means being in the same cluster. We consider the following model for two clusters: we are allowed to query any pair of nodes whether they belong to the same cluster or not, but the answer to the query is corrupted with some probability $0

The recent conjecture of a topologically-protected surface state in SmB$_6$ and the verification of robust surface conduction below 4 K have prompted a large effort to understand the surface states. Conventional Hall transport measurements allow current to flow on all surfaces of a topological insulator, so such measurements are influenced by contributions from multiple surfaces of varying transport character. Instead, we study magnetotransport of SmB$_6$ using a Corbino geometry, which can directly measure the conductivity of a single, independent surface. Both (011) and (001) crystal surfaces show a strong negative magnetoresistance at all magnetic field angles measured. The (011) surface has a carrier mobility of $122\text{ cm}^2/\text{V}\cdot\text{sec}$ with a carrier density of $2.5\times10^{13} \text{ cm}^{-2}$, which are significantly smaller than indicated by Hall transport studies. This mobility value can explain a failure so far to observe Shubnikov-de Haas oscillations. Analysis of the angle-dependence of conductivity on the (011) surface suggests a combination of a field-dependent enhancement of the carrier density and a suppression of Kondo scattering from native oxide layer magnetic moments as the likely origin of the negative magnetoresistance. Our results also reveal a hysteretic behavior whose magnitude depends on the magnetic field sweep rate and temperature. Although this feature becomes smaller when the field sweep is slower, does not disappear or saturate during our slowest sweep-rate measurements, which is much slower than a typical magnetotransport trace. These observations cannot be explained by quantum interference corrections such as weak anti-localization, but are more likely due to an extrinsic magnetic effect such as the magnetocaloric effect or glassy ordering., Comment: 29 pages, 12 figures, PDFLaTeX. This new version is a combination of the prior version and arXiv:1410.7430

Quantum oscillations are generally studied to resolve the electronic structure of topological insulators. In Cu$_{0.25}$Bi$_2$Se$_3$, the prime candidate of topological superconductors, quantum oscillations are still not observed in magnetotransport measurement. However, using torque magnetometry, quantum oscillations (the de Hass - van Alphen effect) were observed in Cu$_{0.25}$Bi$_2$Se$_3$ . The doping of Cu in Bi$_2$Se$_3$ increases the carrier density and the effective mass without increasing the scattering rate or decreasing the mean free path. In addition, the Fermi velocity remains the same in Cu$_{0.25}$Bi$_2$Se$_3$ as that in Bi$_2$Se$_3$. Our results imply that the insertion of Cu does not change the band structure of Bi$_2$Se$_3$., Comment: 5 pages, 4 figures