Connectivity of Random Annulus Graphs and the Geometric Block Model

We provide new connectivity results for {\em vertex-random graphs} or {\em random annulus graphs} which are significant generalizations of random geometric graphs. Random geometric graphs (RGG) are one of the most basic models of random graphs for spatial networks proposed by Gilbert in 1961, shortly after the introduction of the Erd\H{o}s-R\'{en}yi random graphs. They resemble social networks in many ways (e.g. by spontaneously creating cluster of nodes with high modularity). The connectivity properties of RGG have been studied since its introduction, and analyzing them has been significantly harder than their Erd\H{o}s-R\'{en}yi counterparts due to correlated edge formation. Our next contribution is in using the connectivity of random annulus graphs to provide necessary and sufficient conditions for efficient recovery of communities for {\em the geometric block model} (GBM). The GBM is a probabilistic model for community detection defined over an RGG in a similar spirit as the popular {\em stochastic block model}, which is defined over an Erd\H{o}s-R\'{en}yi random graph. The geometric block model inherits the transitivity properties of RGGs and thus models communities better than a stochastic block model. However, analyzing them requires fresh perspectives as all prior tools fail due to correlation in edge formation. We provide a simple and efficient algorithm that can recover communities in GBM exactly with high probability in the regime of connectivity

Investigation of upwind, multigrid, multiblock numerical schemes for three dimensional flows. Volume 1: Runge-Kutta methods for a thin layer Navier-Stokes solver

A state-of-the-art computer code has been developed that incorporates a modified Runge-Kutta time integration scheme, upwind numerical techniques, multigrid acceleration, and multi-block capabilities (RUMM). A three-dimensional thin-layer formulation of the Navier-Stokes equations is employed. For turbulent flow cases, the Baldwin-Lomax algebraic turbulence model is used. Two different upwind techniques are available: van Leer's flux-vector splitting and Roe's flux-difference splitting. Full approximation multi-grid plus implicit residual and corrector smoothing were implemented to enhance the rate of convergence. Multi-block capabilities were developed to provide geometric flexibility. This feature allows the developed computer code to accommodate any grid topology or grid configuration with multiple topologies. The results shown in this dissertation were chosen to validate the computer code and display its geometric flexibility, which is provided by the multi-block structure

Functional Tetrahedron Equation

We describe a scheme of constructing classical integrable models in 2+1-dimensional discrete space-time, based on the functional tetrahedron equation - equation that makes manifest the symmetries of a model in local form. We construct a very general "block-matrix model" together with its algebro-geometric solutions, study its various particular cases, and also present a remarkably simple scheme of quantization for one of those cases.Comment: LaTeX, 16 page

Consistent Estimation of Mixed Memberships with Successive Projections

This paper considers the parameter estimation problem in Mixed Membership Stochastic Block Model (MMSB), which is a quite general instance of random graph model allowing for overlapping community structure. We present the new algorithm successive projection overlapping clustering (SPOC) which combines the ideas of spectral clustering and geometric approach for separable non-negative matrix factorization. The proposed algorithm is provably consistent under MMSB with general conditions on the parameters of the model. SPOC is also shown to perform well experimentally in comparison to other algorithms

General entanglement scaling laws from time evolution

We establish a general scaling law for the entanglement of a large class of ground states and dynamically evolving states of quantum spin chains: we show that the geometric entropy of a distinguished block saturates, and hence follows an entanglement-boundary law. These results apply to any ground state of a gapped model resulting from dynamics generated by a local hamiltonian, as well as, dually, to states that are generated via a sudden quench of an interaction as recently studied in the case of dynamics of quantum phase transitions. We achieve these results by exploiting ideas from quantum information theory and making use of the powerful tools provided by Lieb-Robinson bounds. We also show that there exist noncritical fermionic systems and equivalent spin chains with rapidly decaying interactions whose geometric entropy scales logarithmically with block length. Implications for the classical simulatability are outlined.Comment: 4 pages, 1 figure (see also related work by S. Bravyi, M. Hastings, and F. Verstraete, quant-ph/0603121); replaced with final versio

Geometric Entanglement and Quantum Phase Transition in Generalized Cluster-XY models

In this work, we investigate quantum phase transition (QPT) in a generic family of spin chains using the ground-state energy, the energy gap, and the geometric measure of entanglement (GE). In many of prior works, GE per site was used. Here, we also consider GE per block with each block size being two. This can be regarded as a coarse grain of GE per site. We introduce a useful parameterization for the family of spin chains that includes the XY models with n-site interaction, the GHZ-cluster model and a cluster-antiferromagnetic model, the last of which exhibits QPT between a symmetry-protected topological (SPT) phase and a symmetry-breaking antiferromagnetic phase. As the models are exactly solvable, their ground-state wavefunctions can be obtained and thus their GE can be studied. It turns out that the overlap of the ground states with translationally invariant product states can be exactly calculated and hence the GE can be obtained via further parameter optimization. The QPTs exhibited in these models are detected by the energy gap and singular behavior of geometric entanglement. In particular, the XzY model exhibits transitions from the nontrivial SPT phase to a trivial paramagnetic phase. Moreover, the halfway XY model exhibits a first-order transition across the Barouch-McCoy circle, on which it was only a crossover in the standard XY model.Comment: 29 pages, 12 figure