On the Von Neumann Entropy of Graphs

The von Neumann entropy of a graph is a spectral complexity measure that has recently found applications in complex networks analysis and pattern recognition. Two variants of the von Neumann entropy exist based on the graph Laplacian and normalized graph Laplacian, respectively. Due to its computational complexity, previous works have proposed to approximate the von Neumann entropy, effectively reducing it to the computation of simple node degree statistics. Unfortunately, a number of issues surrounding the von Neumann entropy remain unsolved to date, including the interpretation of this spectral measure in terms of structural patterns, understanding the relation between its two variants, and evaluating the quality of the corresponding approximations. In this paper we aim to answer these questions by first analysing and comparing the quadratic approximations of the two variants and then performing an extensive set of experiments on both synthetic and real-world graphs. We find that 1) the two entropies lead to the emergence of similar structures, but with some significant differences; 2) the correlation between them ranges from weakly positive to strongly negative, depending on the topology of the underlying graph; 3) the quadratic approximations fail to capture the presence of non-trivial structural patterns that seem to influence the value of the exact entropies; 4) the quality of the approximations, as well as which variant of the von Neumann entropy is better approximated, depends on the topology of the underlying graph

Polynomials constant on a hyperplane and CR maps of spheres

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of nonnegative distinct monomials. This bound was conjectured by John P. D'Angelo, proved in two dimensions by D'Angelo, Kos and Riehl and in three dimensions by the authors. The current work builds upon these results to settle the conjecture in all dimensions. We also give a complete description of all polynomials in dimensions 4 and higher for which the sharp bound is obtained. The results prove the sharp degree bounds for monomial CR mappings of spheres in all dimensions.Comment: 17 pages, 10 figures; accepted to Illinois J. Math., added 3 figures and improved expositio

Asymmetric First-Price Menu Auctions under Intricate Uncertainty

This paper studies asymmetric first-price menu auctions in the procurement environment where the buyer does not commit to a decision rule and asymmetric sellers have interdependent costs and statistically affiliated signals. Sellers compete in bidding a menu of contracts, where a contract specifies a vector of characteristics and a payment required from the buyer for delivering these characteristics. The buyer does not commit ex-ante to a decision rule but rather upon observing all the menus offered by sellers chooses the best contract. This paper establishes the existence of a continuum of separating monotone equilibria in this game bounded above by the jointly ex-post efficient outcome and below by the jointly interim efficient outcome. It shows that the jointly ex-post efficient equilibrium outcome is the only ex-post renegotiation proof outcome and it is also ex-ante robust to all continuation equilibriafirst-price menu auction, interdependent values, monotone equilibria, joint ex-post renegotiation-proofness, ex-ante robustness

On Take It or Leave It Offers in Common Agency

If the agent's preference relation satisfies a strict monotonicity condition in common agency under the asymmetric information, the set of all equilibrium allocations in the menu game where menus of contracts are allowed coincides with the set of all equilibrium allocations in the single contract game where only single contracts are allowed.take it or leave it offers, menus, common agency, robust equilibrium allocations, mixed-strategy equilibrium

Is there a third order phase transition for supercritical fluids?

We prove that according to Molecular Dynamics (MD) simulations of liquid mixtures of Lennard-Jones (L-J) particles, there is no third order phase transition in the supercritical regime beyond Andrew's critical point. This result is in open contrast with recent theoretical studies and experiments which instead suggest not only its existence but also its universality regarding the chemical nature of the fluid. We argue that our results are solid enough to go beyond the limitations of MD and the generic character of L-J models, thus suggesting a rather smooth liquid-vapor thermodynamic behavior of fluids in supercritical regime.Comment: 13 pages, 6 figure

HAN

Tuesday, September 07, 2004, 10:30 EDT (10:30 AM EDT)CDCHAN-00212-2004-09-07-UPD-NThe 38-year old man who died of Lassa fever in a New Jersey hospital traveled to the United States from Liberia via London. Further progress has been made in identifying passengers at risk of exposure to Lassa fever during travel on Continental Flight 29, which departed London Gatwick Airport on August 24 and arrived in Newark, N.J., at 3:20 pm on August 24.The plane was a Boeing 777-200, which has seats configured in groups of 3 with 9 seats and two aisles in each row. The case patient was seated in seat 21L, which is a window seat in economy class. Passengers seated within a six-foot radius of the case patient (i.e., passengers seated in seats F, J, K, L in rows 19-23) are considered at low risk for potential exposure, based on guidance outlined in the CDC HAN issued September 2, 2004.The airline has provided CDC the names of passengers who occupied seats 19 F-L, 20 F-L, 21 F-L, 22 F-L, and 23 F-L. CDC\u2019s Division of Global Migration and Quarantine is distributing this information to appropriate local and state health departments and national public health authorities, who in turn will advise passengers of their potential for exposure. In addition, these passengers will be advised to monitor their health for 21 days (i.e., the maximum incubation period) after their travel on Continental Flight 29. For airline passengers, this 21-day period ends September 14, 2004 and NOT September 18 as was incorrectly stated in the previous HAN. During this time, passengers should report a fever 101oF or greater to their local or state health department.Based upon currently available information and the configuration of the aircraft, only passengers seated in these 20 seat locations are considered to be at low risk of exposure. While no specific evaluation or monitoring is currently advised for persons seated in rows other than those listed above, any passenger who traveled on the implicated flight and who has concerns or develops a fever of 101oF or greater should seek medical evaluation and contact public health authorities.2004Lassa Fever, 2004VirusLassa viru

Lagrangian-based Hydrodynamic Model: Freeway Traffic Estimation

This paper is concerned with highway traffic estimation using traffic sensing data, in a Lagrangian-based modeling framework. We consider the Lighthill-Whitham-Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956) in Lagrangian-coordinates, and provide rigorous mathematical results regarding the equivalence of viscosity solutions to the Hamilton-Jacobi equations in Eulerian and Lagrangian coordinates. We derive closed-form solutions to the Lagrangian-based Hamilton-Jacobi equation using the Lax-Hopf formula (Daganzo, 2005; Aubin et al., 2008), and discuss issues of fusing traffic data of various types into the Lagrangian-based H-J equation. A numerical study of the Mobile Century field experiment (Herrera et al., 2009) demonstrates the unique modeling features and insights provided by the Lagrangian-based approach.Comment: 17 pages, 7 figures, current version submitted to Transportation Research Part