Enhance the Efficiency of Heuristic Algorithm for Maximizing Modularity Q

Modularity Q is an important function for identifying community structure in complex networks. In this paper, we prove that the modularity maximization problem is equivalent to a nonconvex quadratic programming problem. This result provide us a simple way to improve the efficiency of heuristic algorithms for maximizing modularity Q. Many numerical results demonstrate that it is very effective.Comment: 9 pages, 3 figure

Size reduction of complex networks preserving modularity

The ubiquity of modular structure in real-world complex networks is being the focus of attention in many trials to understand the interplay between network topology and functionality. The best approaches to the identification of modular structure are based on the optimization of a quality function known as modularity. However this optimization is a hard task provided that the computational complexity of the problem is in the NP-hard class. Here we propose an exact method for reducing the size of weighted (directed and undirected) complex networks while maintaining invariant its modularity. This size reduction allows the heuristic algorithms that optimize modularity for a better exploration of the modularity landscape. We compare the modularity obtained in several real complex-networks by using the Extremal Optimization algorithm, before and after the size reduction, showing the improvement obtained. We speculate that the proposed analytical size reduction could be extended to an exact coarse graining of the network in the scope of real-space renormalization.Comment: 14 pages, 2 figure

Comparing community structure identification

We compare recent approaches to community structure identification in terms of sensitivity and computational cost. The recently proposed modularity measure is revisited and the performance of the methods as applied to ad hoc networks with known community structure, is compared. We find that the most accurate methods tend to be more computationally expensive, and that both aspects need to be considered when choosing a method for practical purposes. The work is intended as an introduction as well as a proposal for a standard benchmark test of community detection methods.Comment: 10 pages, 3 figures, 1 table. v2: condensed, updated version as appears in JSTA

Deciphering Network Community Structure by Surprise

The analysis of complex networks permeates all sciences, from biology to sociology. A fundamental, unsolved problem is how to characterize the community structure of a network. Here, using both standard and novel benchmarks, we show that maximization of a simple global parameter, which we call Surprise (S), leads to a very efficient characterization of the community structure of complex synthetic networks. Particularly, S qualitatively outperforms the most commonly used criterion to define communities, Newman and Girvan's modularity (Q). Applying S maximization to real networks often provides natural, well-supported partitions, but also sometimes counterintuitive solutions that expose the limitations of our previous knowledge. These results indicate that it is possible to define an effective global criterion for community structure and open new routes for the understanding of complex networks.Comment: 7 pages, 5 figure

Nanoscale imaging of buried topological defects with quantitative X-ray magnetic microscopy

This work is licensed under a Creative Commons Attribution 4.0 International License.-- et al.Advances in nanoscale magnetism increasingly require characterization tools providing detailed descriptions of magnetic configurations. Magnetic transmission X-ray microscopy produces element specific magnetic domain images with nanometric lateral resolution in films up to ∼100 nm thick. Here we present an imaging method using the angular dependence of magnetic contrast in a series of high resolution transmission X-ray microscopy images to obtain quantitative descriptions of the magnetization (canting angles relative to surface normal and sense). This method is applied to 55-120 nm thick ferromagnetic NdCo 5 layers (canting angles between 65° and 22°), and to a NdCo 5 film covered with permalloy. Interestingly, permalloy induces a 43° rotation of Co magnetization towards surface normal. Our method allows identifying complex topological defects (merons or 1/2 skyrmions) in a NdCo 5 film that are only partially replicated by the permalloy overlayer. These results open possibilities for the characterization of deeply buried magnetic topological defects, nanostructures and devices.Work supported by Spanish MINECO under grant FIS2013-45469. A. Hierro-Rodriguez acknowledges support from FCT of Portugal (Grant SFRH/BPD/90471/2012). C. Blanco-Roldán thanks support from CSIC JAE Predoc Program.Peer Reviewe

Hot Streaks in Artistic, Cultural, and Scientific Careers

The hot streak, loosely defined as winning begets more winnings, highlights a specific period during which an individual's performance is substantially higher than her typical performance. While widely debated in sports, gambling, and financial markets over the past several decades, little is known if hot streaks apply to individual careers. Here, building on rich literature on lifecycle of creativity, we collected large-scale career histories of individual artists, movie directors and scientists, tracing the artworks, movies, and scientific publications they produced. We find that, across all three domains, hit works within a career show a high degree of temporal regularity, each career being characterized by bursts of high-impact works occurring in sequence. We demonstrate that these observations can be explained by a simple hot-streak model we developed, allowing us to probe quantitatively the hot streak phenomenon governing individual careers, which we find to be remarkably universal across diverse domains we analyzed: The hot streaks are ubiquitous yet unique across different careers. While the vast majority of individuals have at least one hot streak, hot streaks are most likely to occur only once. The hot streak emerges randomly within an individual's sequence of works, is temporally localized, and is unassociated with any detectable change in productivity. We show that, since works produced during hot streaks garner significantly more impact, the uncovered hot streaks fundamentally drives the collective impact of an individual, ignoring which leads us to systematically over- or under-estimate the future impact of a career. These results not only deepen our quantitative understanding of patterns governing individual ingenuity and success, they may also have implications for decisions and policies involving predicting and nurturing individuals with lasting impact