Configuration model for correlation matrices preserving the node strength

Correlation matrices are a major type of multivariate data. To examine properties of a given correlation matrix, a common practice is to compare the same quantity between the original correlation matrix and reference correlation matrices, such as those derived from random matrix theory, that partially preserve properties of the original matrix. We propose a model to generate such reference correlation and covariance matrices for the given matrix. Correlation matrices are often analysed as networks, which are heterogeneous across nodes in terms of the total connectivity to other nodes for each node. Given this background, the present algorithm generates random networks that preserve the expectation of total connectivity of each node to other nodes, akin to configuration models for conventional networks. Our algorithm is derived from the maximum entropy principle. We will apply the proposed algorithm to measurement of clustering coefficients and community detection, both of which require a null model to assess the statistical significance of the obtained results.Comment: 8 figures, 4 table

Effective Theory of Dark Energy at Redshift Survey Scales

We explore the phenomenological consequences of general late-time modifications of gravity in the quasi-static approximation, in the case where cold dark matter is non-minimally coupled to the gravitational sector. Assuming spectroscopic and photometric surveys with configuration parameters similar to those of the Euclid mission, we derive constraints on our effective description from three observables: the galaxy power spectrum in redshift space, tomographic weak-lensing shear power spectrum and the correlation spectrum between the integrated Sachs-Wolfe effect and the galaxy distribution. In particular, with $\Lambda$CDM as fiducial model and a specific choice for the time dependence of our effective functions, we perform a Fisher matrix analysis and find that the unmarginalized $68\%$ CL errors on the parameters describing the modifications of gravity are of order $\sigma\sim10^{-2}$--$10^{-3}$. We also consider two other fiducial models. A nonminimal coupling of CDM enhances the effects of modified gravity and reduces the above statistical errors accordingly. In all cases, we find that the parameters are highly degenerate, which prevents the inversion of the Fisher matrices. Some of these degeneracies can be broken by combining all three observational probes.Comment: 41 pages, 5 figures, 2 tables, improved analysis of ISW-galaxy correlation, matches published version on JCA

Exact solution of the 1D Hubbard model with NN and NNN interactions in the narrow-band limit

We present the exact solution, obtained by means of the Transfer Matrix (TM) method, of the 1D Hubbard model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) Coulomb interactions in the atomic limit (t=0). The competition among the interactions ($U$, $V_1$, and $V_2$) generates a plethora of T=0 phases in the whole range of fillings. $U$, $V_1$, and $V_2$ are the intensities of the local, NN and NNN interactions, respectively. We report the T=0 phase diagram, in which the phases are classified according to the behavior of the principal correlation functions, and reconstruct a representative electronic configuration for each phase. In order to do that, we make an analytic limit $T\to 0$ in the transfer matrix, which allows us to obtain analytic expressions for the ground state energies even for extended transfer matrices. Such an extension of the standard TM technique can be easily applied to a wide class of 1D models with the interaction range beyond NN distance, allowing for a complete determination of the T=0 phase diagrams.Comment: 13 pages, 7 figures, to appear in European Physical Journal

Developing A New Storage Format And A Warp-Based Spmv Kernel For Configuration Interaction Sparse Matrices On The Gpu

Configuration interaction (CI) is a post Hartree–Fock method that is commonly used for solving the nonrelativistic Schrödinger equation for quantum many-electron systems of molecular scale. CI includes instantaneous electron correlation and it can deal with the ground state as well as multiple excited states. The CI matrix is a sparse matrix, and the bigger the CI matrix, the more electron correlation can be captured. However, due to the large size of the CI sparse matrix that is involved in CI computations, a good amount of the time spent on the eigenvalue computations is associated with the multiplication of the CI sparse matrix by numerous dense vectors, which is basically known as Sparse matrix-vector multiplication (SpMV). Sparse matrix-vector multiplication (SpMV) can be used to solve diverse-scaled linear systems and eigenvalue problems that exist in numerous and varying scientific applications. One of the scientific applications that SpMV is involved in is Configuration Interaction (CI). In this work, we have developed a new hybrid approach to deal with CI sparse matrices. The proposed model includes a newly-developed hybrid format for storing CI sparse matrices on the Graphics Processing Unit (GPU). In addition to the new developed format, the proposed model includes the SpMV kernel for multiplying the CI matrix (proposed format) by a vector using the C language and the CUDA platform. The proposed SpMV kernel is a vector kernel that uses the warp approach. We have gauged the newly developed model in terms of two primary factors, memory usage and performance. Our proposed kernel was compared to the cuSPARSE library and the CSR5 (Compressed Sparse Row 5) format and already outperformed both. Our proposed kernel outperformed the CSR5 format by 250.7% and the cuSPARSE library by 395.1% Keywords— CI, SpMV, Linear System, GPU, Kernel, CUDA

Conservation Laws and Integrability of a One-dimensional Model of Diffusing Dimers

We study a model of assisted diffusion of hard-core particles on a line. The model shows strongly ergodicity breaking : configuration space breaks up into an exponentially large number of disconnected sectors. We determine this sector-decomposion exactly. Within each sector the model is reducible to the simple exclusion process, and is thus equivalent to the Heisenberg model and is fully integrable. We discuss additional symmetries of the equivalent quantum Hamiltonian which relate observables in different sectors. In some sectors, the long-time decay of correlation functions is qualitatively different from that of the simple exclusion process. These decays in different sectors are deduced from an exact mapping to a model of the diffusion of hard-core random walkers with conserved spins, and are also verified numerically. We also discuss some implications of the existence of an infinity of conservation laws for a hydrodynamic description.Comment: 39 pages, with 5 eps figures, to appear in J. Stat. Phys. (March 1997