Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)

Although the ``scale-free'' literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet Mathematics (2005

Strong blocking sets and minimal codes from expander graphs

A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the $(k-1)$-dimensional projective space over $\mathbb{F}_q$ that have size $O( q k )$. Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of $\mathbb{F}_q$-linear minimal codes of length $n$ and dimension $k$, for every prime power $q$, for which $n = O (q k)$. This solves one of the main open problems on minimal codes.Comment: 20 page

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that are triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media and their application to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact in cardiac arrhythmias.Peer ReviewedPreprin

Cinema and Transcendence. Xavier Beauvois and Terrence Malick: Two Attempts at Revelation

In Des Hommes et des Dieuxand The Tree of Life, both Xavier Beauvois and Terrence Malick are concerned with values that from the beginning of time to our day have set the human heart wondering and speculating. These issues are brought to their respective films in a distinctive style, which, however, may be closely associated with a common trait in both directors –creative imagination and zest to make the most out of the art of cinema. Their skillful exploration of filmic devices, their invitation to other artistic expressions, namely music and painting, to figure in their films in a signifying role, is tentatively accounted for in this essay, in order to show how a visual narrative, in the case of Beauvois, and an ever-flowing succession of images, in the case of Malick, may contribute to illustrate unsuspected structural and thematic affinities, their remarkable differences notwithstanding.Em Des Hommes et des Dieuxand The Tree of Life, tanto Xavier Beauvois como Terrence Malick se ocupam de valores que, desde o princípio dos tempos aos dias de hoje, trouxeram aos nossos corações o assombro e a especulação. Estes assuntos assumem um estilo específico nos respectivos filmes, que, contudo, pode ser ligado a um traço comum a ambos os realizadores – a imaginação criadora e o gosto de tirar o melhor partido da arte cinematográfica. A artificiosa exploração dos instrumentos fílmicos, o convite a que outras expressões artísticas, nomeadamente a música e a pintura, participem substancialmente a níveis de sentido, são experimentalmente ensaiados neste ensaio, de modo a pôr em evidência o que numa narrativa visual, no caso de Beauvois, e no ininterrupto fluir de imagens, no caso de Malick, pode, apesar das notáveis diferenças entre eles, ilustrar insuspeitadas afinidades temáticas e estruturais

Diversity of Arkansas Water Resources Research

In order to understand, protect, and manage our water resources effectively knowledge is required from many diverse areas of science, engineering, economics, and sociology. These proceedings of the conference on the Diversity of Arkansas Water Resources Research reflect this need and demonstrate how researchers in the state are responding to water issues and problems in Arkansas. The papers in these proceedings are representative of the research in Arkansas, but are only a sample of the work being conducted by universities and government agencies in Arkansas. We are grateful that Arkansas has the expertise available to provide the information necessary to maintain our water resources

Unsupervised Discovery of Extreme Weather Events Using Universal Representations of Emergent Organization

Spontaneous self-organization is ubiquitous in systems far from thermodynamic equilibrium. While organized structures that emerge dominate transport properties, universal representations that identify and describe these key objects remain elusive. Here, we introduce a theoretically-grounded framework for describing emergent organization that, via data-driven algorithms, is constructive in practice. Its building blocks are spacetime lightcones that embody how information propagates across a system through local interactions. We show that predictive equivalence classes of lightcones -- local causal states -- capture organized behaviors and coherent structures in complex spatiotemporal systems. Employing an unsupervised physics-informed machine learning algorithm and a high-performance computing implementation, we demonstrate automatically discovering coherent structures in two real world domain science problems. We show that local causal states identify vortices and track their power-law decay behavior in two-dimensional fluid turbulence. We then show how to detect and track familiar extreme weather events -- hurricanes and atmospheric rivers -- and discover other novel coherent structures associated with precipitation extremes in high-resolution climate data at the grid-cell level