Power-law distributions in empirical data

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution -- the part of the distribution representing large but rare events -- and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data while in others the power law is ruled out.Comment: 43 pages, 11 figures, 7 tables, 4 appendices; code available at http://www.santafe.edu/~aaronc/powerlaws

Crystallographic orientation inhomogeneity and crystal splitting in biogenic calcite

The calcitic prismatic units forming the outer shell of the bivalve Pinctada margaritifera have been analysed using scanning electron microscopy–electron back-scatter diffraction, transmission electron microscopy and atomic force microscopy. In the initial stages of growth, the individual prismatic units are single crystals. Their crystalline orientation is not consistent but rather changes gradually during growth. The gradients in crystallographic orientation occur mainly in a direction parallel to the long axis of the prism, i.e. perpendicular to the shell surface and do not show preferential tilting along any of the calcite lattice axes. At a certain growth stage, gradients begin to spread and diverge, implying that the prismatic units split into several crystalline domains. In this way, a branched crystal, in which the ends of the branches are independent crystalline domains, is formed. At the nanometre scale, the material is composed of slightly misoriented domains, which are separated by planes approximately perpendicular to the c-axis. Orientational gradients and splitting processes are described in biocrystals for the first time and are undoubtedly related to the high content of intracrystalline organic molecules, although the way in which these act to induce the observed crystalline patterns is a matter of future research

Highly optimized tolerance and power laws in dense and sparse resource regimes

Power law cumulative frequency $(P)$ vs. event size $(l)$ distributions $P(\geq l)\sim l^{-\alpha}$ are frequently cited as evidence for complexity and serve as a starting point for linking theoretical models and mechanisms with observed data. Systems exhibiting this behavior present fundamental mathematical challenges in probability and statistics. The broad span of length and time scales associated with heavy tailed processes often require special sensitivity to distinctions between discrete and continuous phenomena. A discrete Highly Optimized Tolerance (HOT) model, referred to as the Probability, Loss, Resource (PLR) model, gives the exponent $\alpha=1/d$ as a function of the dimension $d$ of the underlying substrate in the sparse resource regime. This agrees well with data for wildfires, web file sizes, and electric power outages. However, another HOT model, based on a continuous (dense) distribution of resources, predicts $\alpha= 1+ 1/d$. In this paper we describe and analyze a third model, the cuts model, which exhibits both behaviors but in different regimes. We use the cuts model to show all three models agree in the dense resource limit. In the sparse resource regime, the continuum model breaks down, but in this case, the cuts and PLR models are described by the same exponent.Comment: 19 pages, 13 figure

Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous both and space.Comment: To appear in Czech. Math.

Fluctuation-driven capacity distribution in complex networks

Maximizing robustness and minimizing cost are common objectives in the design of infrastructure networks. However, most infrastructure networks evolve and operate in a highly decentralized fashion, which may significantly impact the allocation of resources across the system. Here, we investigate this question by focusing on the relation between capacity and load in different types of real-world communication and transportation networks. We find strong empirical evidence that the actual capacity of the network elements tends to be similar to the maximum available capacity, if the cost is not strongly constraining. As more weight is given to the cost, however, the capacity approaches the load nonlinearly. In particular, all systems analyzed show larger unoccupied portions of the capacities on network elements subjected to smaller loads, which is in sharp contrast with the assumptions involved in (linear) models proposed in previous theoretical studies. We describe the observed behavior of the capacity-load relation as a function of the relative importance of the cost by using a model that optimizes capacities to cope with network traffic fluctuations. These results suggest that infrastructure systems have evolved under pressure to minimize local failures, but not necessarily global failures that can be caused by the spread of local damage through cascading processes

Multicenter evaluation of a lateral-flow device test for diagnosing invasive pulmonary aspergillosis in ICU patients.

Published onlineClinical TrialJournal ArticleMulticenter StudyResearch Support, Non-U.S. Gov'tINTRODUCTION: The incidence of invasive pulmonary aspergillosis (IPA) in intensive care unit (ICU) patients is increasing, and early diagnosis of the disease and treatment with antifungal drugs is critical for patient survival. Serum biomarker tests for IPA typically give false-negative results in non-neutropenic patients, and galactomannan (GM) detection, the preferred diagnostic test for IPA using bronchoalveolar lavage (BAL), is often not readily available. Novel approaches to IPA detection in ICU patients are needed. In this multicenter study, we evaluated the performance of an Aspergillus lateral-flow device (LFD) test for BAL IPA detection in critically ill patients. METHODS: A total of 149 BAL samples from 133 ICU patients were included in this semiprospective study. Participating centers were the medical university hospitals of Graz, Vienna and Innsbruck in Austria and the University Hospital of Mannheim, Germany. Fungal infections were classified according to modified European Organization for Research and Treatment of Cancer/Mycoses Study Group criteria. RESULTS: Two patients (four BALs) had proven IPA, fourteen patients (sixteen BALs) had probable IPA, twenty patients (twenty-one BALs) had possible IPA and ninety-seven patients (one hundred eight BALs) did not fulfill IPA criteria. Sensitivity, specificity, negative predictive value, positive predictive value and diagnostic odds ratios for diagnosing proven and probable IPA using LFD tests of BAL were 80%, 81%, 96%, 44% and 17.6, respectively. Fungal BAL culture exhibited a sensitivity of 50% and a specificity of 85%. CONCLUSION: LFD tests of BAL showed promising results for IPA diagnosis in ICU patients. Furthermore, the LFD test can be performed easily and provides rapid results. Therefore, it may be a reliable alternative for IPA diagnosis in ICU patients if GM results are not rapidly available. TRIAL REGISTRATION: ClinicalTrials.gov NCT02058316. Registered 20 January 2014.PfizerOesterreichische Nationalbank (Anniversary Fund, project number 15346)

Modeling the variability of shapes of a human placenta

While it is well-understood what a normal human placenta should look like, a deviation from the norm can take many possible shapes. In this paper we propose a mechanism for this variability based on the change in the structure of the vascular tree

Large Scale Cross-Correlations in Internet Traffic

The Internet is a complex network of interconnected routers and the existence of collective behavior such as congestion suggests that the correlations between different connections play a crucial role. It is thus critical to measure and quantify these correlations. We use methods of random matrix theory (RMT) to analyze the cross-correlation matrix C of information flow changes of 650 connections between 26 routers of the French scientific network `Renater'. We find that C has the universal properties of the Gaussian orthogonal ensemble of random matrices: The distribution of eigenvalues--up to a rescaling which exhibits a typical correlation time of the order 10 minutes--and the spacing distribution follow the predictions of RMT. There are some deviations for large eigenvalues which contain network-specific information and which identify genuine correlations between connections. The study of the most correlated connections reveals the existence of `active centers' which are exchanging information with a large number of routers thereby inducing correlations between the corresponding connections. These strong correlations could be a reason for the observed self-similarity in the WWW traffic.Comment: 7 pages, 6 figures, final versio

Statistical mechanics of complex networks

Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks is governed by robust organizing principles. Here we review the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, we discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, as well as the interplay between topology and the network's robustness against failures and attacks.Comment: 54 pages, submitted to Reviews of Modern Physic

Multicenter evaluation of a lateral-flow device test for diagnosing invasive pulmonary aspergillosis in ICU patients

Introduction: The incidence of invasive pulmonary aspergillosis (IPA) in intensive care unit (ICU) patients is increasing, and early diagnosis of the disease and treatment with antifungal drugs is critical for patient survival. Serum biomarker tests for IPA typically give false-negative results in non-neutropenic patients, and galactomannan (GM) detection, the preferred diagnostic test for IPA using bronchoalveolar lavage (BAL), is often not readily available. Novel approaches to IPA detection in ICU patients are needed. In this multicenter study, we evaluated the performance of an Aspergillus lateral-flow device (LFD) test for BAL IPA detection in critically ill patients. Methods: A total of 149 BAL samples from 133 ICU patients were included in this semiprospective study. Participating centers were the medical university hospitals of Graz, Vienna and Innsbruck in Austria and the University Hospital of Mannheim, Germany. Fungal infections were classified according to modified European Organization for Research and Treatment of Cancer/Mycoses Study Group criteria. Results: Two patients (four BALs) had proven IPA, fourteen patients (sixteen BALs) had probable IPA, twenty patients (twenty-one BALs) had possible IPA and ninety-seven patients (one hundred eight BALs) did not fulfill IPA criteria. Sensitivity, specificity, negative predictive value, positive predictive value and diagnostic odds ratios for diagnosing proven and probable IPA using LFD tests of BAL were 80%, 81%, 96%, 44% and 17.6, respectively. Fungal BAL culture exhibited a sensitivity of 50% and a specificity of 85%. Conclusion: LFD tests of BAL showed promising results for IPA diagnosis in ICU patients. Furthermore, the LFD test can be performed easily and provides rapid results. Therefore, it may be a reliable alternative for IPA diagnosis in ICU patients if GM results are not rapidly available. Trial registration: ClinicalTrials.gov NCT02058316. Registered 20 January 2014