Doubly stochastic matrix equations

Networks are often made up of several layers that exhibit diverse degrees of interdependencies. An interdependent network consists of a set of graphs G that are interconnected through a weighted interconnection matrix B, where the weight of each intergraph link is a non-negative real number p. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix Q characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix B=pI, I being the identity matrix, it has been shown that there exists a structural transition at some critical coupling p^{*}. This transition is such that dynamical processes are separated into two regimes: if p>p^{*}, the network acts as a whole; whereas when p<p^{*}, the network operates as if the graphs encoding the layers were isolated. In this paper, we extend and generalize the structural transition threshold p^{*} to a regular interconnection matrix B (constant row and column sum). Specifically, we provide upper and lower bounds for the transition threshold p^{*} in interdependent networks with a regular interconnection matrix B and derive the exact transition threshold for special scenarios using the formalism of quotient graphs. Additionally, we discuss the physical meaning of the transition threshold p^{*} in terms of the minimum cut and show, through a counterexample, that the structural transition does not always exist. Our results are one step forward on the characterization of more realistic multilayer networks and might be relevant for systems that deviate from the topological constraints imposed by multiplex networks.

The spectral radius r(A) of matrix A is the maximum modulus of the Eigen values. In this paper, the studies about the lower and upper bounds for the spectral radius and the lower bounds for the minimum eigen value of appositive and nonnegative matrices are investigate.The matrix norm, the spectral radius norm,and the column (row) sums of nonnegative and positive matrices are widely used to establish some inequalities for matrices. Then several existing results are improved for these inequalities for nonnegative and positive matrix. Furthermore, the lower and upper bounds of the Perron roots for nonnegative matrices are examined, and some upper bounds are computed.

We consider the problem of counting the number of contingency tables with given row and column sums. This problem is known to be #P-complete, even when there are only two rows [7]. In this paper we present the first fully-polynomial randomized approximation scheme for counting contingency tables when the number of rows is constant. A novel feature of our algorithm is that it is a hybrid of an exact counting technique with an approximation algorithm, giving two distinct phases. In the first, the columns are partitioned into "small" and "large". We show that the number of contingency tables can be expressed as the weighted sum of a polynomial number of new instances of the problem, where each instance consists of some new row sums and the original large column sums. In the second phase, we show how to approximately count contingency tables when all the column sums are large. In this case, we show that the solution lies in approximating the volume of a single convex body, a problem which is known to be solvable in polynomial time [5].

The maximum absolute value of the determinant of n x n nonsinguiar (0,1) matrices that have constant line sums (i.e., row sums and column sums) k is investigated. For n≠4k=2, this maximum determinant is determined to be 2t if n=3t or 3t+2, and 2t−1 if n=3t+1. Restriction to a subset of these matrices, namely those that are symmetric and have zero trace (their graphs are regular of degree k), leaves the maximum unchanged for k=2. For this restricted class, when n≥7k=n−3, the maximum absolute value of the determinant is (n−3)3[n/4]−-1, This maximum gives a lower bound for the maximum absolute value of the determinant for the larger class, but in general this bound is not tight. Other determinantal values and bounds for specific n and k are given.

Asymptotic enumeration of sparse 0–1 matrices with irregular row and column sums

where the biological activity is expressed as the reciprocal of the concentration that elicit a given biological, pharmacological or toxicological endpoint such as ED 50 (effective dose 50%), IC 50 (inhibitory concentration 50%) or LD 50 (lethal dose 50%), and where x i is the i predictor (from a set of n predictors) raised to the powers from 1 to m and weighted by its coefficients a ij obtained from a multivariate statistical fitting [4,5]. Usually, most of the coefficients a ij can be zero, leading to much simpler looking QSAR models than the general form in Equation 1, as in the example below. Often, other functions of x i are used in the QSAR model such as the logarithmic or exponential function instead of a simple power expansion as in Equation 1. As a simple illustration of the operation of Equation 1 is the following example of a QSAR model to predict the inhibitory effect of 3-nitroflavones (3NF) based on a single predictor, x (the energy of the highest occupied molecular orbital, E HOMO ) raised to the power 1 (see also [6]). Invasive tumor growth and metastasis are sustained by angiogenesis (the formation of new blood vessels [7,8]), which led Folkman to suggest anticancer therapies by the use of antiangiogenic agents [9,10]. Lichtenberg et al. have shown that the antiangiogenicity of six 3NF is correlated with the energies of the highest occupied molecular orbital, E HOMO, the lower the oxidation potential the greater the inhibitory effects on several biomarkers of angiogenesis [11,12]. These authors report the following correlation between the E HOMO and the IC 50 of six 3NF for inhibition of proliferation of microvascular endothelial HMEC-1 cells after 72 h of incubation:

A matrix of the form A = BBT where B is nonnegative is called completely positive (CP). Berman and Xu (2005) investigated a subclass of CP-matrices, called f0, 1g-completely positive matrices. We introduce a related concept and show connections between the two notions. An important relation to the so-called cut cone is established. Some results are shown for f0, 1g-completely positive matrices with given graphs, and for {0,1}-completely positive matrices constructed from the classes of (0, 1)-matrices with fixed row and column sums.

Using the Rudin-Shapiro sequence, the existence of a regular but not strongly regular positive matrix that sums {exp(2irikt)} to 0 for all f E (0,1) is established.As a corollary it is shown that there exist matrices that sum all almost periodic sequences without possessing the Borel property and vice versa.1.A matrix A -(ank) of real or complex numbers is called strongly regular if it is regular and 00 (1) lim 2 K.fc -an.k+l\ = 0.Such matrices possess many interesting properties.Thus a matrix is translative if and only if it is strongly regular (cf.Petersen [5, p. 21]).Lorentz [3] has shown that a matrix sums every almost convergent sequence to its almost convergence limit if and only if it is strongly regular.If A = (an k) is strongly regular, then it necessarily satisfies the conditions CO (2) lim 2 an.k exp(27Tz/ci) = 0 and 00 (3) Hm 2 |fl.*lexp(27rifc) = 0 fl- 00 k=0for all / e (0, 1) as a simple Abel's transformation shows.The question naturally arises whether there exist regular matrices satisfying (2) and (3) that are not strongly regular.A positive answer to this question becomes much more significant if we observe that the Fourier-effective matrices

Let denote the set of Generalized Doubly Stochastic n × n real matrices, and Ωn denote the set of Doubly Stochastic n × n nonnegative matrices ; that is, matrices whose row and column sums are 1. The research in this paper concerns finding a theoretical algorithm for the closest matrix C* in Ωn to a given matrix A in Mm , the space of n × n real matrices. That is, we prove that it is necessary for an entry of the closest matrix C* in Ωn to A to be 0 whenever the corresponding entry in the closest matrix B* in to A is negative. In addition, if C* is the closest matrix in Ωn to A in Mn to B* is the closest matrix in to A , then C* is the closest matrix in Ωn to B*.

On (0, 1)-matrices with prescribed row and column sum vectors

We prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorthmic and combinatorial applications. Our main application is the first poly-time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor. This new algorithm is a particular instance of new polynomial time deterministic algorithms related to the multiple partial differentiation of polynomials given by evaluation oracles.

The network flows approach for matrices with given row and column sums

General matrices can be split uniquely into Frobenius-orthogonal components: a constant row and column sum (type S) part, a vertex cross sum (type V) part and a weight part. We show that for square matrices, the type S part can be expressed as a sum of squares of type V matrices. We investigate the properties of such decomposition under matrix multiplication, in particular how the pseudoinverses of a matrix relates to the pseudoinverses of its component parts. For invertible matrices, this yields an expression for the inverse where only the type S part needs to be (pseudo)inverted; in the example of the Wilson matrix, this component is considerably better conditioned than the whole matrix. We also show a relation between matrix determinants and the weight of their matrix inverses and give a simple proof for Frobenius-optimal approximations with the constant row and column sum and the vertex cross sum properties, respectively, to a given matrix.

This paper is concerned with a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri (i = 1, … , m) and let the sum of column i of A be denoted by si (i = 1, … , n).

Let be real n-tuplesn>3, satisfying It is shown that a necessary and sufficient condition for the existence of a non-negative matrix with main diagonal with row sums and column sums is that ft Equality can hold if and only if all the off-diagonal positive entries of the matrix are restricted to the kth row and the fcth column, for some k,1<k<n.