Poisson Form Research Articles

Degree distributions under general node removal: Power-law or Poisson?

Perturbations made to networked systems may result in partial structural loss, such as a blackout in a power-grid system. Investigating the resulting disturbance in network properties is quintessential to understand real networks in action. The removal of nodes is a representative disturbance, but previous studies are seemingly contrasting about its effect on arguably the most fundamental network statistic, the degree distribution. The key question is about the functional form of the degree distributions that can be altered during node removal or sampling. The functional form is decisive in the remaining subnetwork's static and dynamical properties. In this work, we clarify the situation by utilizing the relative entropies with respect to the reference distributions in the Poisson and power-law form, to quantify the distance between the subnetwork's degree distribution and either of the reference distributions. Introducing general sequential node removal processes with continuously different levels of hub protection to encompass a series of scenarios including uniform random removal and preferred or protective (i.e., biased random) removal of the hub, we classify the altered degree distributions starting from various power-law forms by comparing two relative entropy values. From the extensive investigation in various scenarios based on direct node-removal simulations and by solving the rate equationof degree distributions, we discover in the parameter space two distinct regimes, one where the degree distribution is closer to the power-law reference distribution and the other closer to the Poisson distribution.

Bayesian inference for pseudo-Poisson data

The bivariate pseudo-Poisson model discussed in this paper is one in which one marginal density and the conditional density of the other variable are both of the Poisson form. Such models find application for modelling bivariate count data with usually positive correlation. In the present note, we examine Bayesian estimation of the unknown parameters of bivariate pseudo-Poisson models utilizing independent gamma priors for the parameters as well as pseudo-gamma priors. Possible conjugacy is investigated in both full and submodels. In some special subcases, conjugate priors can be identified. A simulation study is included to illustrate the performance of Bayesian parameter estimates using a variety of priors, both informative and non-informative. The techniques are illustrated by re-analysing two well-known bivariate count data sets.

Fresnel diffraction from sector apertures

The derivation of the analytic representation of the diffraction field from the sector aperture in the Fresnel zone is presented. The new solution is expressed in the form of a convergent series on incomplete cylindrical functions in the Poisson form that makes it possible to calculate and obtain three-dimensional patterns of the field amplitude in the whole image plane. The obtained patterns correspond to the Fresnel approximation and are true for arbitrary values of the opening angle and for the radius of sectoral aperture.

Statistical inference for distributions with one Poisson conditional

It will be recalled that the classical bivariate normal distributions have normal marginals and normal conditionals. It is natural to ask whether a similar phenomenon can be encountered involving Poisson marginals and conditionals. However, it is known, from research on conditionally specified models, that Poisson marginals will be encountered, together with both conditionals being of the Poisson form, only in the case in which the variables are independent. In order to have a flexible dependent bivariate model with some Poisson components, in the present article, we will be focusing on bivariate distributions with one marginal and the other family of conditionals being of the Poisson form. Such distributions are called Pseudo-Poisson distributions. We discuss distributional features of such models, explore inferential aspects and include an example of applications of the Pseudo-Poisson model to sets of over-dispersed data.

Multiplicity distribution of gluons in pQCD

The multiplicity distribution of the gluons produced at the high energy is evaluated in BFKL approach. The distribution has Poisson form that can explain experimentally observed KNO scaling.

Distortions in Macroeconomics

Distortions in Macroeconomics

Prolongation of Poisson $2$-form on Weil bundles

In this paper, $M$ denotes a smooth manifold of dimension $n$, $A$ a Weil algebra and $M^{A}$ the associated Weil bundle. When $(M,\omega _{M})$ is a Poisson manifold with $2$-form $\omega _{M}$, we construct the $2$-Poisson form $\omega _{M^{A}}^{A}$, prolongation on $M^{A}$ of the $2$-Poisson form $\omega _{M}$. We give a necessary and sufficient condition for that $M^{A}$ be an $A$-Poisson manifold.

Universal Poisson Statistics of mRNAs with Complex Decay Pathways

Messenger RNA (mRNA) dynamics in single cells are often modeled as a memoryless birth-death process with a constant probability per unit time that an mRNA molecule is synthesized or degraded. This predicts a Poisson steady-state distribution of mRNA number, in close agreement with experiments. This is surprising, since mRNA decay is known to be a complex process. The paradox is resolved by realizing that the Poisson steady state generalizes to arbitrary mRNA lifetime distributions. A mapping between mRNA dynamics and queueing theory highlights an identifiability problem: a measured Poisson steady state is consistent with a large variety of microscopic models. Here, I provide a rigorous and intuitive explanation for the universality of the Poisson steady state. I show that the mRNA birth-death process and its complex decay variants all take the form of the familiar Poisson law of rare events, under a nonlinear rescaling of time. As a corollary, not only steady-states but also transients are Poisson distributed. Deviations from the Poisson form occur only under two conditions, promoter fluctuations leading to transcriptional bursts or nonindependent degradation of mRNA molecules. These results place severe limits on the power of single-cell experiments to probe microscopic mechanisms, and they highlight the need for single-molecule measurements.

Orbital Mechanics near a Rotating Asteroid

This study investigates the different novel forms of the dynamical equations of a particle orbiting a rotating asteroid and the effective potential, the Jacobi integral, etc. on different manifolds. Nine new forms of the dynamical equations of a particle orbiting a rotating asteroid are presented, and the classical form of the dynamical equations has also been found. The dynamical equations with the potential and the effective potential in scalar form in the arbitrary body-fixed frame and the special body-fixed frame are presented and discussed. Moreover, the simplified forms of the effective potential and the Jacobi integral have been derived. The dynamical equation in coefficient-matrix form has been derived. Other forms of the dynamical equations near the asteroid are presented and discussed, including the Lagrange form, the Hamilton form, the symplectic form, the Poisson form, the Poisson-bracket form, the cohomology form, and the dynamical equations on the K\"ahler manifold and another complex manifold. Novel forms of the effective potential and the Jacobi integral are also presented. The dynamical equations in scalar form and coefficient-matrix form can aid in the study of the dynamical system, the bifurcation, and the chaotic motion of the orbital dynamics of a particle near a rotating asteroid. The dynamical equations of a particle near a rotating asteroid are presented on several manifolds, including the symplectic manifold, the Poisson manifold, and complex manifolds, which may lead to novel methods of studying the motion of a particle in the potential field of a rotating asteroid.

Onset of chaos in finite interacting Boson systems with F-spin

For m bosons, carrying spin (f = 1 2) degree of freedom, in Ω single particle levels which are doubly degenerate, one-plus two-body Embedded Gaussian Orthogonal random matrix Ensemble which conserves spin (F) [BEGOE(1 + 2)-F] is defined and spectral properties are examined. Using mean field defined by random single particle energies, for fixed-(m,F) BEGOE(1 + 2)-F, density of states being close to Gaussian, we analyze spectral fluctuation properties, like Nearest Neighbour Spacing Distribution (NNSD) and Dyson-Mehta (Δ3) statistic, as a function of λ- the two-body interaction strength. Numerical calculations are used to demonstrate that for very small value of λ, NNSD and Δ3 are found close to Poisson form and they move steadily to GOE form as λ increases. Moreover, spin dependence of the transition point λ C is obtained. It is also verified using periodogram analysis that BEGOE(1 + 2)-F spectra shows 1/f–noise behaviour.

Complete Closed-Form Expression of Dyadic Green's Function and Its Far- and Near-Field Approximations for an Impedance Half-Plane

A closed-form expression of the dyadic Green's function is formulated for an impedance half-plane, which is written in terms of the incomplete cylindrical function of Poisson form. Due to the branch-cut of the logarithm function that is required to calculate the input argument of the incomplete cylindrical function, the closed-form representation consists of two formulations. Since the closed-form expression contains a singularity at ρ = 0, the small argument expansion of the expression is also derived to rigorously characterize the behavior of the function at ρ ≈ 0. The previously-reported complete asymptotic expansion for the Sommerfeld integral for an impedance half-plane is not accurate for practically important cases such as near-earth propagation and/or when the surface is highly conductive. Hence, in this paper, a new complete asymptotic series of the Sommerfeld integral are derived for the case that the existing asymptotic series is not accurate. The two asymptotic series not only allow efficient numerical computation but also provide more accurate results for virtually all propagation scenarios. Based on the two asymptotic series, the complete asymptotic series of the dyadic Green's function is derived. All derived formulations are numerically verified, and their accuracies are investigated.

Comments on “Scale-free networks without growth”

In [Y.-B. Xie, T. Zhou, B.-H. Wang, Scale-free networks without growth, Physica A 387 (2008) 1683–1688], a nongrowing scale-free network model has been introduced, which shows that the degree distribution of the model varies from the power-law form to the Poisson form as the free parameter α increases, and indicates that the growth may not be necessary for a scale-free network structure to emerge. However, the model implicitly assumes that self-loops and multiple-links are allowed in the model and counted in the degree distribution. In many real-life networks, such an assumption may not be reasonable. We showed here that the degree distribution of the emergent network does not obey a power-law form if self-loops and multiple-links are allowed in the model but not counted in the degree distribution. We also observed the same result when self-loops and multiple-links are not allowed in the model. Furthermore, we showed that the effect of self-loops and multiple-links on the degree distribution weakens as α increases and even becomes negligible when α is sufficiently large.

Algebraic construction of a Nambu bracket for the two-dimensional vorticity equation

So far fluid mechanical Nambu brackets have mainly been given on an intuitive basis. Alternatively an algorithmic construction of such a bracket for the two-dimensional vorticity equation is presented here. Starting from the Lie–Poisson form and its algebraic properties it is shown how the Nambu representation can be explicitly constructed as the continuum limit from the structure preserving Zeitlin discretization.

A recursive method to calculate the expected molecule numbers for a polymerization network with a small number of subunits

We compare the deterministic method and the stochastic method for a polymerization network when the number of available subunits is small. For the stochastic method, we prove there is a recursive method to compute the expected molecule numbers of various components in the reaction network, using the stationary probability distribution of molecule numbers which we illustrate to have a multivariate Poisson form. For the deterministic method, ordinary differential equations for the component concentrations are built following the mass action law. The steady state of the system is extracted to estimate the corresponding molecule numbers. Identities involving the propensity function parameters for the stochastic method and the reaction rate constants in the deterministic method are used to connect the two methods. Computations are conducted for a group of combinations of total number of subunits and reaction rate constant ratios, and the results are compared.

Information filtering via weighted heat conduction algorithm

In this paper, by taking into account effects of the user and object correlations on a heat conduction (HC) algorithm, a weighted heat conduction (WHC) algorithm is presented. We argue that the edge weight of the user–object bipartite network should be embedded into the HC algorithm to measure the object similarity. The numerical results indicate that both the accuracy and diversity could be improved greatly compared with the standard HC algorithm and the optimal values reached simultaneously. On the Movielens and Netflix datasets, the algorithmic accuracy, measured by the average ranking score, can be improved by 39.7% and 56.1% in the optimal case, respectively, and the diversity could reach 0.9587 and 0.9317 when the recommendation list equals to 5. Further statistical analysis indicates that, in the optimal case, the distributions of the edge weight are changed to the Poisson form, which may be the reason why HC algorithm performance could be improved. This work highlights the effect of edge weight on a personalized recommendation study, which maybe an important factor affecting personalized recommendation performance.

Distributional analysis of a generalization of the Polya process

A nonhomogeneous birth process generalizing the Polya process is analyzed, and the distribution of the transition probabilities is shown to be the convolution of a negative binomial distribution and a compound Poisson distribution, whose secondary distribution is a mixture of zero-truncated geometric distributions. A simplified form of the secondary distribution is obtained when the transition intensities have a particular structure, and may sometimes be expressed in terms of Stirling numbers and special functions such as the incomplete gamma function, the incomplete beta function, and the exponential integral. Conditions under which the compound Poisson form of the marginal distributions may be improved to a geometric mixture are also given.

Global existence and asymptotic behavior of solutions to the nonisentropic bipolar hydrodynamic models

In this paper, we consider a one-dimensional bipolar nonisentropic hydrodynamical model from semiconductor devices. This system takes the nonisentropic Euler–Poisson form with electric field and frictional damping added to the momentum equations. First, we prove global existence of smooth solutions to the Cauchy problem. Next, we also discuss the asymptotic behavior of the smooth solutions. We find that in large time, the densities of electron and hole tend to the same nonlinear diffusive wave, the momentums tend to the Darcy's law, and the temperatures tend to the ambient device temperature. Finally, we can obtain the algebraic decay rate of the densities to the same nonlinear diffusive wave, the momentums to the Darcy's law and the temperatures to the ambient device temperature, and the exponential decay of their difference and the electric field to zero. We can show our results by precise energy methods.

Weighted bipartite network and personalized recommendation

In this paper, the degree distributions of a bipartite network, namely Movielens, are investigated. The statistical analysis shows that the distribution of the degree product, ku ko, has an exponential from, where ku and ko denote the user and object degrees respectively. By introducing the edge weight effect on the recommendation performance, an improved recommendation algorithm based on mass diffusion (MD) process is presented. We argue that the edges weight of the user-object bipartite network should be taken into account to measure the object similarity. By taking into account the user and object degree correlations, the weighted bipartite network is constructed. The numerical results of the MD algorithms on the weighted network indicate that both of the accuracy and diversity could be increased at the optimal case. More importantly, we find that, at the optimal case, the edge weight distribution would change from the exponential form to the poisson form. This work may shed some light on how to improve the recommendation algorithm performance by considering the statistical properties.

The Hamiltonian description of incompressible fluid ellipsoids

We construct the noncanonical Poisson bracket associated with the phase space of first order moments of the velocity field and quadratic moments of the density of a fluid with a free-boundary, constrained by the condition of incompressibility. Two methods are used to obtain the bracket, both based on Dirac’s procedure for incorporating constraints. First, the Poisson bracket of moments of the unconstrained Euler equations is used to construct a Dirac bracket, with Casimir invariants corresponding to volume preservation and incompressibility. Second, the Dirac procedure is applied directly to the continuum, noncanonical Poisson bracket that describes the compressible Euler equations, and the moment reduction is applied to this bracket. When the Hamiltonian can be expressed exactly in terms of these moments, a closure is achieved and the resulting finite-dimensional Hamiltonian system provides exact solutions of Euler’s equations. This is shown to be the case for the classical, incompressible Riemann ellipsoids, which have velocities that vary linearly with position and have constant density within an ellipsoidal boundary. The incompressible, noncanonical Poisson bracket differs from its counterpart for the compressible case in that it is not of Lie–Poisson form.

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.