A Consistent Histogram Estimator for Exchangeable Graph Models

Exchangeable graph models (ExGM) subsume a number of popular network models. The mathematical object that characterizes an ExGM is termed a graphon. Finding scalable estimators of graphons, provably consistent, remains an open issue. In this paper, we propose a histogram estimator of a graphon that is provably consistent and numerically efficient. The proposed estimator is based on a sorting-and-smoothing (SAS) algorithm, which first sorts the empirical degree of a graph, then smooths the sorted graph using total variation minimization. The consistency of the SAS algorithm is proved by leveraging sparsity concepts from compressed sensing.Comment: 28 pages, 5 figure

Hydrogen-Bonded Liquids: Effects of Correlations of Orientational Degrees of Freedom

We improve a lattice model of water introduced by Sastry, Debenedetti, Sciortino, and Stanley to give insight on experimental thermodynamic anomalies in supercooled phase, taking into account the correlations between intra-molecular orientational degrees of freedom. The original Sastry et al. model including energetic, entropic and volumic effect of the orientation-dependent hydrogen bonds (HBs), captures qualitatively the experimental water behavior, but it ignores the geometrical correlation between HBs. Our mean-field calculation shows that adding these correlations gives a more water-like phase diagram than previously shown, with the appearance of a solid phase and first-order liquid-solid and gas-solid phase transitions. Further investigation is necessary to be able to use this model to characterize the thermodynamic properties of the supercooled region.Comment: 7 pages latex, 3 figures EP

Novel Six-Quark Hidden-Color Dibaryon States in QCD

The recent observation of a hadronic resonance $d^*$ in the proton-neutron system with isospin $I = 0$ and spin-parity $J^P = 3^+$ raises the possibility of producing other novel six-quark dibaryon configurations allowed by QCD. A dramatic example of an exotic six-quark color-singlet system is the charge $Q=+4$, isospin I=3, $I^z=+3$ $|uuuuuu>$ state which couples strongly to $\Delta^{++}$ + $\Delta^{++} .$ The width and decay properties of such six-quark resonances could be regarded as manifestations of "hidden-color" six-quark configurations, a first-principle prediction of QCD -- SU(3)-color gauge theory for the deuteron distribution amplitude. Other implications and possible future experiments are discussed

Scaling behavior in economics: I. Empirical results for company growth

We address the question of the growth of firm size. To this end, we analyze the Compustat data base comprising all publicly-traded United States manufacturing firms within the years 1974-1993. We find that the distribution of firm sizes remains stable for the 20 years we study, i.e., the mean value and standard deviation remain approximately constant. We study the distribution of sizes of the ``new'' companies in each year and find it to be well approximated by a log-normal. We find (i) the distribution of the logarithm of the growth rates, for a fixed growth period of one year, and for companies with approximately the same size $S$ displays an exponential form, and (ii) the fluctuations in the growth rates -- measured by the width of this distribution $\sigma_1$ -- scale as a power law with $S$, $\sigma_1\sim S^{-\beta}$. We find that the exponent $\beta$ takes the same value, within the error bars, for several measures of the size of a company. In particular, we obtain: $\beta=0.20\pm0.03$ for sales, $\beta=0.18\pm0.03$ for number of employees, $\beta=0.18\pm0.03$ for assets, $\beta=0.18\pm0.03$ for cost of goods sold, and $\beta=0.20\pm0.03$ for property, plant, & equipment.Comment: 16 pages LateX, RevTeX 3, 10 figures, to appear J. Phys. I France (April 1997

Scaling behavior in economics: II. Modeling of company growth

In the preceding paper we presented empirical results describing the growth of publicly-traded United States manufacturing firms within the years 1974--1993. Our results suggest that the data can be described by a scaling approach. Here, we propose models that may lead to some insight into these phenomena. First, we study a model in which the growth rate of a company is affected by a tendency to retain an ``optimal'' size. That model leads to an exponential distribution of the logarithm of the growth rate in agreement with the empirical results. Then, we study a hierarchical tree-like model of a company that enables us to relate the two parameters of the model to the exponent $\beta$, which describes the dependence of the standard deviation of the distribution of growth rates on size. We find that $\beta = -\ln \Pi / \ln z$, where $z$ defines the mean branching ratio of the hierarchical tree and $\Pi$ is the probability that the lower levels follow the policy of higher levels in the hierarchy. We also study the distribution of growth rates of this hierarchical model. We find that the distribution is consistent with the exponential form found empirically.Comment: 19 pages LateX, RevTeX 3, 6 figures, to appear J. Phys. I France (April 1997

Stochastic blockmodel approximation of a graphon: Theory and consistent estimation

Non-parametric approaches for analyzing network data based on exchangeable graph models (ExGM) have recently gained interest. The key object that defines an ExGM is often referred to as a graphon. This non-parametric perspective on network modeling poses challenging questions on how to make inference on the graphon underlying observed network data. In this paper, we propose a computationally efficient procedure to estimate a graphon from a set of observed networks generated from it. This procedure is based on a stochastic blockmodel approximation (SBA) of the graphon. We show that, by approximating the graphon with a stochastic block model, the graphon can be consistently estimated, that is, the estimation error vanishes as the size of the graph approaches infinity.Comment: 20 pages, 4 figures, 2 algorithms. Neural Information Processing Systems (NIPS), 201

Bigger Bursts From Merging Neutron Stars

GRB 990123 may have radiated more than one solar mass equivalent in just its gamma emissions. Though this may be within the upper limit of the binding energy available from neutron stars in the Schwarzschild metric, it is difficult to imagine a process with the required efficiency of conversion to gamma rays. Neutron stars of ~10 solar mass are permitted in the Yilmaz metric. A merger of two neutron stars of maximum mass could release approximately 10 solar mass equivalent binding energy.Comment: 5 pages, 1 figure, submitted to ApJ Letter