Hard optimization problems have soft edges.

Finding a Maximum Clique is a classic property test from graph theory; find any one of the largest complete subgraphs in an Erdös-Rényi G(N, p) random graph. We use Maximum Clique to explore the structure of the problem as a function of N, the graph size, and K, the clique size sought. It displays a complex phase boundary, a staircase of steps at each of which [Formula: see text] and [Formula: see text], the maximum size of a clique that can be found, increases by 1. Each of its boundaries has a finite width, and these widths allow local algorithms to find cliques beyond the limits defined by the study of infinite systems. We explore the performance of a number of extensions of traditional fast local algorithms, and find that much of the "hard" space remains accessible at finite N. The "hidden clique" problem embeds a clique somewhat larger than those which occur naturally in a G(N, p) random graph. Since such a clique is unique, we find that local searches which stop early, once evidence for the hidden clique is found, may outperform the best message passing or spectral algorithms.

Mining the Thin Air-for Understanding of Urban Society.

We explore the potential of crowd-sourced information on human mobility and activities in an urban population drawn from a significant fraction of smartphones in the Los Angeles basin during February-May 2015. The raw dataset was collected by WeFi, a smartphone app provider. The dataset is noisy, irregular, and lean; however, it is large scale (over a billion events), cheap to collect, and arguably unbiased. We employ the state-of-the-art Big Data techniques to turn this structurally thin dataset into semantically rich insights on commuting, overworking, recreational traveling, shopping, and fast food consumption of the Greater LA population. For example, we reveal that Greater LA residents commute substantially longer than what is reported in the US census data. Also, we show that younger individuals dine at McDonald's significantly more than the older population does. Our results have implications for public health, inequality, urban traffic, and other research areas in social sciences. The large number of phones participating in our "crowd" makes it possible to obtain those results without the risk of compromising individual privacy.

Once the Internet can measure itself.

In communications, the obstacle to high bandwidth and reliable transmission is usually the interconnections, not the links. Nowhere is this more evident than on the Internet, where broadband connections to homes, offices and now mobile smart phones are a frequent source of frustration, and the interconnections between the roughly 50,000 subnetworks (autonomous systems or ASes) from which it is formed, even more so. The structure of the AS graph that is formed by these interconnections is unspecified, undocumented and only guessed-at through measurement, but it shows surprising efficiencies. Under recent pressures for network neutrality and openness or 'transparency', operators, several classes of users and regulatory bodies have a good chance of realizing these efficiencies, but they need improved measurement technology to manage this under continued growth. A long-standing vision, an Internet that measures itself, in which every intelligent port takes a part in monitoring, can make this possible and may now be within reach.

White light lateral shear interferometer with holographic shear lenses and spatial Fourier transform.

Lateral shear interferometer is a simple yet powerful method for testing wavefronts or measuring refractive index changes. Previously, we have reported a method of using two holographic lenses to obtain shear and to generate a spatial frequency carrier, which was used for quantitative analysis. This technique has some advantages such as stability and instantaneous measurements. In this Letter, we report a method of using white light with holographic gratings to obtain shear and also to perform wavefront analysis using the spatial carrier fringes generated at a specific selected wavelength from the white light spectrum. We show that the sensitivity of the setup can be changed by selecting different wavelengths from the spectrum.

A model of Internet topology using k-shell decomposition.

We study a map of the Internet (at the autonomous systems level), by introducing and using the method of k-shell decomposition and the methods of percolation theory and fractal geometry, to find a model for the structure of the Internet. In particular, our analysis uses information on the connectivity of the network shells to separate, in a unique (no parameters) way, the Internet into three subcomponents: (i) a nucleus that is a small ( approximately 100 nodes), very well connected globally distributed subgraph; (ii) a fractal subcomponent that is able to connect the bulk of the Internet without congesting the nucleus, with self-similar properties and critical exponents predicted from percolation theory; and (iii) dendrite-like structures, usually isolated nodes that are connected to the rest of the network through the nucleus only. We show that our method of decomposition is robust and provides insight into the underlying structure of the Internet and its functional consequences. Our approach of decomposing the network is general and also useful when studying other complex networks.