Unified Perturbation Research Articles

Unified robust network embedding framework for community detection via extreme adversarial attacks

Graph data are widely available in complex network systems. Numerous community detection algorithms have been investigated to study graph problems wherein the network topology offers abundant behavioral and functional information. In real scenarios, networks are extremely susceptible to external perturbations, primarily because of sparse topological information. Consequently, maintaining robust community detection performance when confronted with intricate network attacks or perturbations is challenging. Previous anti-graph perturbation methods have typically been adjusted to specific perturbation types, leading to the degradation or failure of these defences when confronted with other types of attack. Accordingly, a unified robust framework that leverages extreme adversarial attacks is proposed in this paper. Specifically, a novel graph perturbation module is introduced into the model to generate extreme undirected or directed perturbations through dynamic updating; this ensures that the model can fit the attacked network well. The proposed framework derives a unified perturbation formulation capable of simultaneously attacking symmetric and asymmetric networks. Furthermore, this framework can be applied to many existing non-negative matrix factorization-based community detection methods. Extensive experiments on artificial and real-world networks demonstrate that the proposed framework significantly improves the robustness of detection tasks, particularly in noisy networks.

Impact of Model Bias Correction on a Hybrid Data Assimilation System

Hybrid data assimilation combines a conventional 3-D or 4-D variational system with background error covariance (BEC) generated from ensemble forecast systems. In order to achieve better BEC, three perturbation schemes, namely, the random combination of multiple physical paramterization schemes (referred to as MP), the MP plus stochastical perturbation on physical process tendencies (MP-SPPT), and the unified perturbation of stochastic physics with bias correction (UPSB, proposed by the authors of this paper in a previous work), were first used in a regional ensemble model, i.e., the Global and Regional Assimilation and Prediction System-Regional Ensemble Prediction System (GRAPES-REPS), and the BECs thus obtained were compared for 7-day ensemble forecasts. The results show that UPSB, which is in fact an MP-SPPT but with the systematic model bias removed, has a better consistency, i.e., the ratio between root-mean-square error (RMSE) and ensemble spread is much closer to 1, especially at low model levels, compared to the other two schemes. Moreover, the BEC derived from UPSB captured more reasonable distributions of forecast errors. Second, performance of a hybrid data assimilation system (the GRAPES-MESO hybrid En-3DVar) was evaluated by using the BECs from the three perturbation schemes for 7-day hybrid data assimilation forecasts, and thus disclosing the effect of the model bias correction (assuming that the random stocastical features are in general offset in the three perturbation schemes) on the hybrid system forecasts. A covariance weight of 0.8 was prescribed, and this value was determined through sensitivity experiments. The forecast results from the hybrid data assimilation system show that UPSB reduced the false correlation between distant points. The quality of analysis fields of the UPSB scheme shows visible improvement, i.e., the analysis fields produced by UPSB have much smaller RMSEs than those of the other two schemes, at all vertical model levels. The quality of the hybrid data assimilation forecast fields was also improved by this scheme. Furthermore, the improvement was much greater in the early stage of the assimilation cycle than in the late stage. Generally, the quality of the hybrid data assimilation of GRAPES-MESO hybrid En-3DVar could be efficiently improved by the model bias correction in the UPSB scheme.

A unified perturbation analysis framework for countable Markov chains

In this paper, we are devoted to singular perturbation analysis for discrete-time or continuous-time Markov chains. We modify and extend the drift condition method, well known for regular perturbation, to develop a new framework for singular perturbation analysis. Our results extend and improve the corresponding ones in [2] for singularly perturbed Markov chains by allowing a general perturbation form, less restrictive conditions, and more computable bounds. Our analysis covers the regular perturbation analysis, and hence unifies singular and regular perturbation analysis. Furthermore, our results are illustrated by two two-dimensional Markov chains, including a discrete-time queue and a continuous-time level dependent quasi-birth-death process.

Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach

A novel computational method, namely the unified perturbation mathematical programming (UPMP) approach, for hybrid uncertainty analysis of engineering structures is proposed in this paper. The presented study considers a mixture of random and interval system parameters which are frequently encountered in engineering applications. Within the UPMP approach, matrix perturbation theory is adopted in combination with the mathematical programming approach. The proposed computational method provides a non-simulative hybrid uncertainty analysis framework, which is competent to offer the extreme bounds of the statistical characteristics (i.e., mean and variance) of any concerned structural responses in computationally tractable fashion. In order to thoroughly explore various intricate aspects of the engineering system involving hybrid uncertainties, systematic numerical experiments have also been conducted. Diverse statistical analyses are implemented to identify the bounded probability profile of the uncertain structural responses. Both academic and practical engineering structures are investigated to justify the applicability, accuracy and efficiency of the proposed UPMP approach.

Constructing explicit homoclinic solution of oscillators: An improvement for perturbation procedure based on nonlinear time transformation

An improved procedure for perturbation method is presented for constructing homoclinic solutions of strongly nonlinear self-excited oscillators. Compared with current perturbation methods based on nonlinear time transformations, the preference of the present method is that the explicit solutions, in respect to the original time variable, can be derived. In the paper, the equivalence and unified perturbation procedure with nonlinear time transformations, by which implicit solutions can be derived at nonlinear time scales, are firstly presented. Then an explicit generating homoclinic solution for power-law strongly nonlinear oscillator is derived with proposed hyperbolic function balance procedure. An approximation scheme is presented to improve the perturbation procedure and the explicit expression for nonlinear time transformation can be achieved. Applications and comparisons with other methods are performed to assess the advantage of the present method.

Effective interaction in unified perturbation theory

We present a unified description of the Bloch and Rayleigh–Schrödinger perturbation theories of the effective interaction in both algebraic and graphic representations.

A unified perturbation approach to static/dynamic coupled instabilities of nonlinear structures

A unified perturbation approach, based on the multiple scale method, able to analyse interactions among static and/or dynamic bifurcations of nonlinear structures, is illustrated. The method appears as the natural extension to the dynamical case of the static perturbation method, well-known in buckling of elastic structures. To illustrate the algorithm, a general, multiparameter, finite-dimensional autonomous system is considered, simultaneously undergoing a static (divergence) and two dynamic (Hopf) bifurcations, of nonresonant or resonant type. The method furnishes bifurcation equations governing the time-evolution of amplitudes and phases of the critical interacting modes. Results relevant to sample structures are discussed.

Acoustic field in a quasi-spherical resonator: Unified perturbation model

Gas-filled quasi-spherical resonators are excellent tools for the measurement of thermophysical properties of gas and have also been retained for the determination of the Boltzmann constant with a low uncertainty, which can be derived from measurements of both the speed of sound in a noble gas and the volume of the resonator. To achieve this, a detailed modeling of the acoustic field in quasi-spherical resonators is of importance. Several phenomena and perturbations must be taken into account, including, among inertia and compressibility, heat conduction, viscosity, the shape of the resonator, small irregularities on the wall, and so on. The aim of this paper is to provide improvements to the current models of the acoustic field in such resonator. Namely, the model given here takes into account all the different perturbing elements together in a unique formalism, including the coupling between the different perturbing elements and the resulting modal coupling in a consistent manner. The first results obtained from this analytical model on a simple configuration show that the effect of modal coupling is small but should not be neglected regarding the accuracy required here, even if several improvements could still be provided to this new unified model.

On solitary waves. Part 2 A unified perturbation theory for higher-order waves

A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on (i) the accuracy of the theoretically predicted wave properties and (ii) the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property (e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy), on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ(n) (the relative mean-square difference between successive orders n of wave elevations) reaching a minimum, δ m , at a specific order, n=n m , both depending on the base adopted, e.g. n m , α =11-12 based on parameter α (wave amplitude), n m , β =15 on β (amplitude-speed square ratio), and n m , ∈ =17 on ∈ ( wave number squared). The asymptotic range is brought to completion by the highest order of n=18 reached in this work.

An improvement on perturbation bounds for the Drazin inverse

AbstractThe Drazin inverse of a square matrix occurs in a number of applications. It is of importance to analyse the perturbation bounds for the Drazin inverse of a matrix. Let B=A+E. Under the assumption of rank(Bj) =rank(Ak), where j and k are the indices of B and A, respectively, upper bounds of ∥BD‐AD∥/∥AD∥ and ∥BBD‐AAD∥/∥AAD∥ have been recently studied. However, these upper bounds do not cover the perturbation bounds of the group inverse recently given by the authors as a special case.Moreover, these perturbation bounds for the Drazin inverse are too large to be practical. In this paper, we present sharper unified perturbation bounds for the Drazin inverse, which are the extensions of the recent result in the case of group inverse. It solves the problem posed by Campbell and Meyer in 1975. A numerical example is given to illustrate the sharpness of the new general bounds. Copyright © 2003 John Wiley & Sons, Ltd.

A Unified Theoryof Conditioning for Linear Least Squares and Tikhonov Regularization Solutions

We develop a unified perturbation theory for the unconstrained linear least squares problem, least squares with linear equality constraints, and least squares with quadratic inequality constraint and Tikhonov regularization solution. The computable condition numbers are exact with respect to the Frobenius norm. For the 2-norm, the computed bounds may differ from the exact condition numbers only by a factor less than $\sqrt{2}$.

Canonical averaging of the Schrödinger equation

The representation of the Schrodinger equation in the form of a classical Hamiltonian system makes it possible to construct a unified perturbation theory that is based on the theory of canonical transformations and covers both classical and quantum mechanics. Also, the closeness of the exact and approximate solutions of the Schrodinger equation can be approximately estimated with such a representation.

Magnetic field induced alignment–orientation conversion: Nonlinear energy shift and predissociation in Te2 B1u state

The paper analyzes magnetic field induced alignment–orientation conversion (AOC) phenomenon caused by simultaneous effect of quadratic terms in Zeeman energy shift and magnetic predissociation (PD), producing asymmetry either in energy splitting ωMM±1≠ω−M∓1−M or in relaxation of coherence ΓMM±1≠Γ−M∓1−M between coherently excited M, M±1 magnetic sublevels. The AOC is registered via the appearance of circular polarization (C) of fluorescence under linearly polarized excitation. The unified perturbation treatment of a molecule in external magnetic field B is presented, accounting for magnetic and intramolecular perturbations via interaction with bonded or continuum states, considering Hund’s (c)-case coupling and dividing the intramolecular perturbation operator into homogeneous (ΔΩ=0) and heterogeneous (ΔΩ=±1) parts. Explicit expressions up to B2 terms are given for energy shift and PD rate, adapted to 1u state in conditions relevant to the B 3Σ−u complex of Te2 molecule. Numeric simulation revealed that nonlinear magnetic energy shift and heterogeneous magnetic PD produce dispersion type fluorescence circularity signals C(B) of different sign. Fitting of experimental data on B1−u, v(J)=2(96) state of 130Te2 molecule allowed to determine the electronic matrix element of paramagnetic Hamiltonian (Ω=0|Ĥpm|Ω=1)≡G±=2.7, as well as the natural Cvhet=±6 s−1/2 and the magnetic αvhet=∓9×103 s−1/2 T−1 rate constants of heterogeneous PD, supposing that the B1−u state PD takes place through 0−u state continuum. As a result, magnetic AOC represents a sensitive method to investigate molecular structure and intramolecular interaction between both bonded and continuum states. Additionally, it has been shown that the magnetic PD effect leads to strong amplification of nonzero field level crossing signals caused by B2 terms in Zeeman energy shift.

A unified perturbation approach to volume and roughness scattering

A unified approach is developed to study scattering from a fluid medium with irregularities of two different types: volume inhomogeneities (spatial fluctuations of the compressibility and density) and roughness of the interfaces. The approach treats roughness as a perturbation of the volume acoustic properties in the vicinity of the flat (unperturbed) surface. This permits a description of the scattering problem in terms of a unique integral equation for both types of irregularities. In the case of small roughness, the first iteration of this equation gives results corresponding to the extended Born approximation but provides a new approach for the case of interfaces with large, smooth roughness. As an application, acoustic scattering from a seabed consisting of an arbitrary number of irregular layers with volume inhomogeneities and rough interfaces is considered. [Work partially supported by ONR.]

A numerical assessment of rough surface scattering theories: Horizontal polarization

We present a numerical evaluation of the regime of validity for various rough surface scattering theories against numerical results obtained by using the method of moments. The theories are the small perturbation method, the Kirchhoff approximation, the momentum transfer expansion, the two‐scale expansion, and a new theory, which we call the unified perturbation method. Both the field and phase perturbation expansions for these theories are examined. It is outlined how all these theories can be derived in a unified manner starting with the extinction theorem. We consider the contribution of each theory up to second order in the perturbation expansion for the surface current. The rough surfaces considered are oceanlike surfaces, which exhibit height power law spectra.

A numerical assessment of rough surface scattering theories: Vertical polarization

In a previous paper, we presented a numerical evaluation of the regime of validity for various rough surface scattering theories. The evaluation was done for horizontal polarization by comparing these theories against numerical results obtained by the method of moments. In this paper we address the same issues for vertical polarization. The theories considered here are the small perturbation method, the Kirchhoff approximation, the momentum transfer expansion, the two‐scale expansion, and the unified perturbation method (UPM). All these theories can be derived in a unified manner starting with the extinction theorem. The rough surfaces of interest are oceanlike surfaces, which exhibit height power law spectra. Considering both horizontal and vertical polarizations, the UPM provides best results among all theories considered.

Comparison of the unified perturbation method with the two-scale expansion

The authors consider whether the unified perturbation method (UPM), which converges faster over a wider domain of surface roughness than other perturbations and intrinsically possesses characteristics similar to the two-scale expansion without requiring a free parameter, can be improved by applying the two-scale concept. To do so, they expand the unknown source current in a two-scale approximation. It is shown that the UPM performs best without two-scale expansions. The optimum spectral split of the various two-scale theories considered is presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A unified perturbation expansion for surface scattering

Starting with the extinction theorem, we present a perturbation expansion which, to first and second orders, converges over a wider domain than the small perturbation expansion and the momentum transfer expansion. We show that, in the appropriate limits, both of these theories, as well as the two‐scale expansion, are recovered. There is no adjustable parameter, such as a spectral split, in the theory. We apply this theory to random rough surfaces and derive analytic expressions for the coherent field and the bistatic cross section. Finally, we present a numerical test of the theory against method of moments results for Gaussian random rough surfaces with a power law spectrum. These results show that the expansion is remarkably accurate over a large range of surface heights and slopes for both horizontal and vertical polarization.

A self-consistent theory of wave propagation and scattering through weakly inhomogeneous media

A unified perturbation approach to wave propagation and scattering in weakly inhomogeneous media is presented. By separating the total wave field into an averaged coherent propagation part and a random scattering part, a set of coupled wave equations is found. The problem of a wave propagating in a random medium is changed to a problem of a pair of coupled waves propagating in a deterministic medium with randomly distributed sources. This method not only improves upon the conventional Born approximation for the scattered field, but also gives the attenuation behavior of the coherent propagating field related to the statistics of the random media. Examples of underwater acoustic waves in an inhomogeneous water column and an inhomogeneous bottom are given.

Unified Perturbation Treatment for Phonons in Metallic Covalent and Ionic Crystals

AbstractAn interpolation formula which interpolates the exact Hartree expression for the electronic part of the dynamical matrix between the limits of strong and weak crystal potentials and which becomes rigorous in the zero wave vector limit is used to avoid to the problems connected with the inversion of the screening tensor. In the ionic part of the dynamical matrix the Gaussian charge densities of the usual Ewald technique are replaced by the true charge densities around each ion in the unperturbed crystal. The arising expressions are transformed in such a way that the compensation with the corresponding terms in the electronic part can be done explicitely. As a result one gets an alternative formula for the dynamical matrix of simple metals, a practicable formula for transition and noble metals and covalent crystals, and a rather simple expression for ionic crystals which has some link to the physical picture of a shell model. It is shown that the given theory describes the qualitatively correct pressure dependence of the phonon frequencies.