On the distance and algorithms of strong product digraphs

Strong product is an efficient way to construct a larger digraph through some specific small digraphs. The large digraph constructed by the strong product method contains the factor digraphs as its subgraphs, and can retain some good properties of the factor digraphs. The distance of digraphs is one of the most basic structural parameters in graph theory, and it plays an important role in analyzing the effectiveness of interconnection networks. In particular, it provides a basis for measuring the transmission delay of networks. When the topological structure of an interconnection network is represented by a digraph, the average distance of the directed graph is a good measure of the communication performance of the network. In this paper, we mainly investigate the distance and average distance of strong product digraphs, and give a formula for the distance of strong product digraphs and an algorithm for solving the average distance of strong product digraphs

Landau Damping of Baryon Structure Formation in the Post Reionization Epoch

It has been suggested by Chen and Lai that the proper description of the large scale structure formation of the universe in the post-reionization era, which is conventionally characterized via gas hydrodynamics, should include the plasma collective effects in the formulation. Specifically, it is the combined pressure from the baryon thermal motions and the residual long-range electrostatic potentials resulted from the imperfect Debye shielding, that fights against the gravitational collapse. As a result, at small-scales the baryons would oscillate at the ion-acoustic, instead of the conventional neutral acoustic, frequency. In this paper we extend and improve the Chen-Lai formulation with the attention to the Landau damping of the ion-acoustic oscillations. Since T_e \sim T_i in the post-reionization era, the ion acoustic oscillations would inevitably suffer the Landau damping which severely suppresses the baryon density spectrum in the regimes of intermediate and high wavenumber k. To describe this Landau-damping phenomenon more appropriately, we find it necessary to modify the filtering wavenumber k_f in our analysis. It would be interesting if our predicted Landau damping of the ion-acoustic oscillations can be observed at high redshifts.Comment: 5 page

Acceleration of SVRG and Katyusha X by Inexact Preconditioning

Empirical risk minimization is an important class of optimization problems with many popular machine learning applications, and stochastic variance reduction methods are popular choices for solving them. Among these methods, SVRG and Katyusha X (a Nesterov accelerated SVRG) achieve fast convergence without substantial memory requirement. In this paper, we propose to accelerate these two algorithms by \textit{inexact preconditioning}, the proposed methods employ \textit{fixed} preconditioners, although the subproblem in each epoch becomes harder, it suffices to apply \textit{fixed} number of simple subroutines to solve it inexactly, without losing the overall convergence. As a result, this inexact preconditioning strategy gives provably better iteration complexity and gradient complexity over SVRG and Katyusha X. We also allow each function in the finite sum to be nonconvex while the sum is strongly convex. In our numerical experiments, we observe an on average $8\times$ speedup on the number of iterations and $7\times$ speedup on runtime

A2BCD: An Asynchronous Accelerated Block Coordinate Descent Algorithm With Optimal Complexity

In this paper, we propose the Asynchronous Accelerated Nonuniform Randomized Block Coordinate Descent algorithm (A2BCD), the first asynchronous Nesterov-accelerated algorithm that achieves optimal complexity. This parallel algorithm solves the unconstrained convex minimization problem, using p computing nodes which compute updates to shared solution vectors, in an asynchronous fashion with no central coordination. Nodes in asynchronous algorithms do not wait for updates from other nodes before starting a new iteration, but simply compute updates using the most recent solution information available. This allows them to complete iterations much faster than traditional ones, especially at scale, by eliminating the costly synchronization penalty of traditional algorithms. We first prove that A2BCD converges linearly to a solution with a fast accelerated rate that matches the recently proposed NU_ACDM, so long as the maximum delay is not too large. Somewhat surprisingly, A2BCD pays no complexity penalty for using outdated information. We then prove lower complexity bounds for randomized coordinate descent methods, which show that A2BCD (and hence NU_ACDM) has optimal complexity to within a constant factor. We confirm with numerical experiments that A2BCD outperforms NU_ACDM, which is the current fastest coordinate descent algorithm, even at small scale. We also derive and analyze a second-order ordinary differential equation, which is the continuous-time limit of our algorithm, and prove it converges linearly to a solution with a similar accelerated rate.Comment: 33 pages, 6 figure

Theory of Driven Nonequilibrium Critical Phenomena

A system driven in the vicinity of its critical point by varying a relevant field in an arbitrary function of time is a generic system that possesses a long relaxation time compared with the driving time scale and thus represents a large class of nonequilibrium systems. For such a manifestly nonlinear nonequilibrium strongly fluctuating system, we show that there exists universal nonequilibrium critical behavior that is well described incredibly by its equilibrium critical properties. A dynamic renormalization-group theory is developed to account for the behavior. The weak driving may give rise to several time scales depending on its form and thus rich nonequilibrium phenomena of various regimes and their crossovers, negative susceptibilities, as well as violation of fluctuation-dissipation theorem. An initial condition that can be in either equilibrium or nonequilibrium but has longer correlations than the driving scales also results in a unique regime and complicates the situation. Implication of the results on measurement is also discussed. The theory may shed light on study of other nonequilibrium systems and even nonlinear science.Comment: 15 pages, 11 figure

Two Time-dependent Solutions of Magnetic field Annihilation in Two Dimensions

In this paper, two classes of exact analytic time-dependent soultion of magnetic annihilation for incompressible magnetic fluid, have been obtained by solving the magnetohydrodynamic (MHD) equations directly. The solutions derived here possess scaling property with time $t$ as the scale factor. Based on these two solutions, we find that, for some given inflow fields, the evolution of the annihilating magnetic field can be described by the solutions of certain ordinary differential equations whose variables are dilated simply by time $t$. The relevant evolution characteristics in the process of magnetic annihilation are also revealed.Comment: 16 pages, Latex file, 7 EPS figure

Distributed Metropolis Sampler with Optimal Parallelism

The Metropolis-Hastings algorithm is a fundamental Markov chain Monte Carlo (MCMC) method for sampling and inference. With the advent of Big Data, distributed and parallel variants of MCMC methods are attracting increased attention. In this paper, we give a distributed algorithm that can correctly simulate sequential single-site Metropolis chains without any bias in a fully asynchronous message-passing model. Furthermore, if a natural Lipschitz condition is satisfied by the Metropolis filters, our algorithm can simulate $N$-step Metropolis chains within $O(N/n+\log n)$ rounds of asynchronous communications, where $n$ is the number of variables. For sequential single-site dynamics, whose mixing requires $\Omega(n\log n)$ steps, this achieves an optimal linear speedup. For several well-studied important graphical models, including proper graph coloring, hardcore model, and Ising model, our condition for linear speedup is weaker than the respective uniqueness (mixing) conditions. The novel idea in our algorithm is to resolve updates in advance: the local Metropolis filters can often be executed correctly before the full information about neighboring spins is available. This achieves optimal parallelism without introducing any bias

Dynamic Sampling from Graphical Models

In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update, on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or anti-ferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, change of functions), both efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new "equilibrium" property that is shown to be satisfied by our algorithm at each step, and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest

Useful vacancies in Single Wall Carbon Nanotubes

The electronic and structural properties of zigzag and armchair single-wall carbon nanotubes (SWCNT) with a single vacancy or two vacancies located at various distances have been obtained within the frame of the Density Function Theory (DFT) and a Molecular Dynamics method. It is found that the vacancy defects interact at long ranges in armchair SWCNTs unlike the short-range interaction in zigzag SWCNTs. The density of states for different vacancy densities shows that the local energy gap shrinks with the vacancy density increase. This and other results of the investigation provide insight into understanding the relation between the local deformation of a defective nanotube and its measurable electronic properties.Comment: 7 pages, 5 figures, 213 ECS, 18-22 May 2008, Phoenix, A

Z2Z_{2} fractionalized Chern/topological insulators in an exactly soluble correlated model

In this paper we propose an exactly soluble model in two-dimensional honeycomb lattice, from which two phases are found. One is the usual Chern/topological insulating state and the other is an interesting $Z_2$ fractionalized Chern/topological insulator. While their bulk properties are similar, the edge-states of physical electrons are quite different. The single electron excitation of the former shows a free particle-like behavior while the latter one is gapped, which provides a definite signature to identify the fractionalized states. The transition between these two phases is found to fall into the 3D Ising universal class. Significantly, near the quantum transition point the physical electron in the edge-states shows strong Luttinger liquid behavior. An extension to the interesting case of the square lattice is also made. In addition, we also discuss some relationship between our exactly soluble model and various Hubbard-like models existing in the literature. The essential difference between the proposed $Z_{2}$ fractionalized Chern insulator and the hotly pursued fractional Chern insulator is also pointed out. The present work may be helpful for further study on the fractionalized insulating phase and related novel correlated quantum phases.Comment: 13pages,no figures, some physics clarified and acknowledgement update