Periodic patterns of synchrony are lattice networks whose cells are coloured according to a local rule, or balanced colouring, and such that the overall system has spatial periodicity. These patterns depict the finite-dimensional flow-invariant subspaces for all the lattice dynamical systems, in the given lattice network, that exhibit those periods. Previous results relate the existence of peri...

We propose a new algorithm that simultaneously estimates the intrinsic dimension and intrinsic entropy of random data sets lying on smooth manifolds. The method is based on asymptotic properties of entropic graph constructions. In particular, we compute the Euclidean -nearest neighbors ( NN) graph over the sample points and use its overall total edge length to estimate intrinsic dimension and e...

Two-sample tests for multivariate and especially for non-Euclidean data are not fully explored. This paper presents a novel test based on a similarity graph constructed on the pooled observations from the two samples. It can be applied to multivariate data and non-Euclidean data as long as a dissimilarity measure on the sample space can be defined, which can usually be provided by domain expert...

In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian eigen-system over time. A Young–Householder decomposition is performed on the heat kernel to obtain the matrix of the em...

Extremal eigenvalues and eigenvectors of the Laplace matrix of a graph form the core of many bounds on graph parameters and graph optimization problems. In order to advance the understanding of connections between structural properties of the graph and these eigenvectors and eigenvalues we study the problem minimizing the difference between maximum and second smallest eigenvalue over edge weigh...

Abstract The objective of the traveling salesperson problem (TSP) is to find a shortest tour through all nodes in graph. Euclidean TSPs are traditionally represented with “cities” placed on 2D plane. When straight line obstacles added plane, has visit cities while going around obstacles. resulting remains metric, but not because paths no longer lines. We first revise previous version multiresol...

A graph G(V, E) is a threshold graph if there exist non-negative reals wv, v ∈ V and t such that for every U ⊆ V , P v∈U wv ≤ t if and only if U is a stable set. The threshold dimension of a graph G(V,E), denoted as t(G), is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. A permutation graph is a graph that can be represented as the intersection graph o...

Convolutional neural networks (CNNs), in a few decades, have outperformed the existing state of art methods classification context. However, way they were formalised, CNNs are bound to operate on euclidean spaces. Indeed, convolution is signal operation that defined This has restricted deep learning main use euclidean-defined data such as sound or image. And yet, numerous computer application f...

A graph is called (generically) rigid in Rd if, for any choice of sufficiently generic edge lengths, it can be embedded in Rd in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining the maximum number of planar Euclidean embeddings of minimally rigid graphs with 8 vertices, because this is the smallest unknown case in the plane. Until now...

We consider the cubic nonlinear Schr\"{o}dinger equation on star graph with Kirchhoff boundary condition. prove modified scattering for final state problem and initial value problem. Moreover, we also failure of power-type long-range nonlinearities. These results are extension NLS one dimensional Euclidean space.