Column Sums Research Articles

Structural transition in interdependent networks with regular interconnections.

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.

Estimation and Inference in Spatial Models with Dominant Units

In spatial econometrics literature estimation and inference are carried out assuming that the matrix of spatial or network connections has uniformly bounded absolute column sums in the number of units, n, in the network. This paper relaxes this restriction and allows for one or more units to have pervasive effects in the network. The linear-quadratic central limit theorem of Kelejian and Prucha (2001) is generalized to allow for such dominant units, and the asymptotic properties of the GMM estimators are established in this more general setting. A new bias-corrected method of moments (BMM) estimator is also proposed that avoids the problem of weak instruments by self-instrumenting the spatially lagged dependent variable. Both cases of homoskedastic and heteroskedastic errors are considered and the associated estimators are shown to be consistent and asymptotically normal, depending on the rate at which the maximum column sum of the weights matrix rises with n. The small sample properties of GMM and BMM estimators are investigated by Monte Carlo experiments and shown to be satisfactory. An empirical application to sectoral price changes in the US over the pre- and post-2008 financial crisis is also provided. It is shown that the share of capital can be estimated reasonably well from the degree of sectoral interdependence using the input-output tables, despite the evidence of dominant sectors being present in the US economy.

Stuttering blocks of Ariki–Koike algebras

We study a shift action defined on multipartitions and on residue multisets of their Young diagrams. We prove that the minimal orbit cardinality among all multipartitions associated to a given multiset depends only on the orbit cardinality of the multiset. Using abaci, this problem reduces to a convex optimisation problem over the integers with linear constraints. We solve it by proving an existence theorem for binary matrices with prescribed row, column and block sums. Finally, we give some applications to the representation theory of the Hecke algebra of the complex reflection group $G(r,p,n)$.

Ideal flow of Markov Chain

Ideal flow network is a strongly connected network with flow, where the flows are in steady state and conserved. The matrix of ideal flow is premagic, where vector, the sum of rows, is equal to the transposed vector containing the sum of columns. The premagic property guarantees the flow conservation in all nodes. The scaling factor as the sum of node probabilities of all nodes is equal to the total flow of an ideal flow network. The same scaling factor can also be applied to create the identical ideal flow network, which has from the same transition probability matrix. Perturbation analysis of the elements of the stationary node probability vector shows an insight that the limiting distribution or the stationary distribution is also the flow-equilibrium distribution. The process is reversible that the Markov probability matrix can be obtained from the invariant state distribution through linear algebra of ideal flow matrix. Finally, we show that recursive transformation [Formula: see text] to represent [Formula: see text]-vertices path-tracing also preserved the properties of ideal flow, which is irreducible and premagic.

Partitions of the polytope of doubly substochastic matrices

In this paper, we provide three different ways to partition the polytope of doubly substochastic matrices into subpolytopes via the prescribed row and column sums, the sum of all elements and the sub-defect respectively. Then we characterize the extreme points of each type of convex subpolytopes. The relations of the extreme points of the subpolytopes in the three partitions are also given.

Fuzzy Analytic Hierarchy Process: A performance analysis of various algorithms

Analytical Hierarchical Process (AHP) along with fuzzy set theory has been used extensively in the Multi-Criteria Decision Making (MCDM) process in which fuzzy numbers are utilized to represent human judgments more realistically. Over the past couple of decades, numerous articles have been published proposing algorithms through which priority vector (or weight vector) can be calculated from fuzzy comparison matrices. The aim of this study is to conduct a comprehensive performance analysis of the most popular algorithms proposed in this domain in terms of accuracy of weights calculated from fuzzy comparison matrices. Such an analysis is much needed by the researchers and practitioners. However none is available. An experimental analysis is conducted and the performance of various algorithms are evaluated with varying three parameters i.e., the size of the comparison matrix, the level of fuzziness and the level of inconsistency. We found that modified Logarithmic Least Squares Method and Fuzzy Inverse of Column Sum Method (FICSM) generally outperformed other algorithms, while Fuzzy Extent Analysis (FEA) which is the most frequently used algorithm in the literature provides the least accurate results. Furthermore, it was observed that a modified version of FEA method significantly improved its performance.

On the norm of a random jointly exchangeable matrix

In this note, we show that the norm of an $n\times n$ random jointly exchangeable matrix with zero diagonal can be estimated in terms of the norm of its $n/2\times n/2$ submatrix located in the top right corner. As a consequence, we prove a relation between the second largest singular values of a random matrix with constant row and column sums and its top right $n/2\times n/2$ submatrix. The result has an application to estimating the spectral gap of random undirected $d$-regular graphs in terms of the second singular value of {\it directed} random graphs with predefined degree sequences.

Multiplying and Factoring Matrices

All of us learn and teach matrix multiplication using rows times columns. Those inner products are the entries of AB. But to go backward—to factor a matrix into triangular or orthogonal or diagonal matrices—outer products are much better. Now AB is the sum of columns of A times rows of B: rank one matrices. Our goal is to produce those columns and rows as simply as possible for A = LU (elimination) and A = CE (echelon form) and A = QR (Gram–Schmidt). Diagonalization by eigenvectors and by singular vectors is also expressed by columns times rows.

Quantifying multiple-site compositional turnover in an Afrotemperate forest, using zeta diversity

BackgroundSpecies turnover is typically measured by partitioning diversity components into alpha and pairwise beta diversity. However, alpha and beta components cannot express the full spectrum of multiple-site compositional turnover. To this end, zeta diversity has been proposed as an extended framework to allow complete biodiversity partitioning and to measure multiple-site species turnover. We use a zeta-diversity framework to explore the turnover and potential community assembly processes of an African Montane Forest.MethodsUsing a 20 m grid, we explore the species turnover in a 4.55 ha forest plot located in the Garden Route National Park of South Africa, with 47 and 27 canopy and sub-canopy tree species in the regional pool. We first calculate how zeta diversity declines and how the probability of retention of species with particular occupancies changes with increasing zeta orders (i.e. the number of sites [grid cells] involved in the calculation). Using null models with row sums and column sums constrained respectively, we explore whether species turnover is driven by mechanisms of ecological differences (species-specific occupancies) or habitat heterogeneity (site-specific alpha diversity and thus environmental filters).ResultsThe decline of zeta diversity with zeta order followed a power law; that is, the probability of retention increased with species occupancies, suggesting common species being more likely to be discovered in extra sites. The null model retaining row sums (species’ occupancy) of the species-by-site matrix recreated perfectly the decline of zeta diversity, while the null model of habitat heterogeneity (retaining column sums) was rejected. This suggests that mechanisms driving species-specific occupancies (i.e. ecological differences between species) dictate the multi-site species turnover in the community. The spatial patterns of zeta diversity revealed little spatial structuring forces, supporting a fine-grain structure in these southern Cape forests.ConclusionsThe framework of zeta diversity revealed mechanisms driving the large discrepancies in the occupancy among species that are behind the species turnover in the African Montane forest plot. Future studies could further link species turnover to spatial distance decay. Environmental filters and temporal turnover from landscape demography could bring a cohesive understanding of community assembly in these unique forest ecosystems.

Restricted lonesum matrices

Lonesum matrices are matrices that are uniquely reconstructible from their row and column sum vectors. These matrices are enumerated by the poly-Bernoulli numbers; a sequence related to the multiple zeta values with a rich literature in number theory. Lonesum matrices are in bijection with many other combinatorial objects: several permutation classes, matrix classes, acyclic orientations in complete bipartite graphs etc. Motivated of these facts, we study in this paper lonesum matrices with restriction on the number of columns and rows of the same type. Keywords: enumeration, poly-Bernoulli numbers, restricted Stirling numbers, lonesum matrices MSC: 05A05, 05A15, 05A18, 05A19

Cutset-type possibilistic c-means clustering algorithm

The possibilistic c-means (PCM) clustering algorithm was introduced to avoid the sensitivity of the fuzzy c-means (FCM) clustering algorithm to outliers by relaxing the column sum constraint of the membership matrix of the FCM, namely the between-class relationships. The membership value is then interpreted as the typicality value of a data point in the PCM which reflects the absolute distance of data point to one cluster well. However, the PCM has a significant defect of coincident clustering problem because of relaxing the relations of clusters. In this paper, a novel cutset-type possibilistic clustering (C-PCM) algorithm is proposed. The C-PCM firstly generates a cluster core resulting from a β-cutset for each cluster and searches for data points located inside the cluster core. Then, the typicalities of these points to other clusters are modified, thus introducing the between-class relationships and avoiding coincident clusters. Simultaneously an adaptive determination method is also given for the parameter β in the C-PCM. Moreover, a novel segmentation method for images corrupted by salt-and-pepper noise is proposed by taking advantage of the strong robustness of the C-PCM to outliers. Several experiments are done on synthetic data-sets, high dimensional data sets, noisy images, which demonstrate the good performance of the proposed algorithm.

A Multi-Objective Interpretation of Optimal Transport

This paper connects discrete optimal transport to a certain class of multi-objective optimization problems. In both settings, the decision variables can be organized into a matrix. In the multi-objective problem, the notion of Pareto efficiency is defined in terms of the objectives together with nonnegativity constraints and with equality constraints that are specified in terms of column sums. A second set of equality constraints, defined in terms of row sums, is used to single out particular points in the Pareto-efficient set which are referred to as “balanced solutions.” Examples from several fields are shown in which this solution concept appears naturally. Balanced solutions are shown to be in one-to-one correspondence with solutions of optimal transport problems. As an example of the use of alternative interpretations, the computation of solutions via regularization is discussed.

Dihedral transportation and (0, 1)-matrix classes

ABSTRACTLet R and S be two vectors of real numbers whose entries have the same sum. In the transportation problems, one wishes to find a matrix A with row sum vector R and column sum vector S. If, in addition, the two vectors only contain nonnegative integers then one wants the same to be true for A. This can always be done and the transportation algorithm gives a method for explicitly calculating A. We can restrict things even further and insist that A have only entries zero and one. In this case, the Gale-Ryser Theorem gives necessary and sufficient conditions for A to exist and this result can be proved constructively. One can let the dihedral group of the square act on matrices. Then a subgroup of defines a set of matrices invariant under the subgroup. So one can consider analogues of the transportation and (0, 1) problems for these sets of matrices. For every subgroup, we give conditions equivalent to the existence of the desired type of matrix.

Sampling contingency tables

Let Ω be the set of all m×n matrices, where ri and cj are the sums of entries in row i and column j, respectively. Sampling efficiently uniformly at random elements of Ω is a problem with interesting applications in Combinatorics and Statistics. To calibrate the statistic χ2 for testing independence, Diaconis and Gangolli proposed a Markov chain on Ω that samples uniformly at random contingency tables of fixed row and column sums. Although the scheme works well for practical purposes, no formal proof is available on its rate of convergence. By using a canonical path argument, we prove that this Markov chain is fast mixing and the mixing time τx(ϵ) is given by τx(ϵ)≤2cmax(mn)4ln(cmaxϵ−1),where cmax−1 is the maximal value in a cell.

Lonesum decomposable matrices

A lonesum matrix is a (0,1)-matrix that is uniquely determined by its row and column sum vectors. In this paper, we introduce lonesum decomposable matrices and study their properties. We provide a necessary and sufficient condition for a matrix A to be lonesum decomposable, and give a generating function for the number Dk(m,n) of m×n lonesum decomposable matrices of order k. Moreover, by using this generating function we prove some congruences for Dk(m,n) modulo a prime.

Constructing (0,1)-matrices with large minimal defining sets

If D is a partially filled-in (0,1)-matrix with a unique completion to a (0,1)-matrix M (with prescribed row and column sums), we say that D is a defining set for M. Let A2m,m be the set of (0,1)-matrices of dimensions 2m×2m with uniform row and column sum m. It is shown in Cavenagh (2013) [3] that the smallest possible size for a defining set of an element of A2m,m is precisely m2. In this note when m is a power of two we construct an element of A2m,m which has no defining set of size less than 2m2−o(m2). Given that it is easy to show any A2m,m has a defining set of size at most 2m2, this construction is asymptotically optimal. Our construction is based on a array, defined using linear algebra, in which any subarray has asymptotically the same number of 0's and 1's.

Aggregation may or may not eliminate reproductive uncertainty

In matrix population models, there cannot be any ‘reproductive uncertainty’ when the life cycle graph contains only one reproductive stage. Otherwise, it is logical to expect that aggregating all the reproductive stages into a single one would exclude the very basis of uncertainty. However, can the aggregation change principally the model characteristics such as the dominant eigenvalue λ1 of the projection matrix, thus signifying the aggregation failure? I demonstrate that it can with the data mined in a case study on the dynamics of a local stage-structured population of Eritrichium caucasicum, a short-lived perennial plant species inhabiting an alpine lichen heath. Frobenius Theorem for nonnegative matrices specifies the upper and lower bounds for λ1 via the row (or column) sums of matrix elements, and the lower bound, when it exceeds the maximal possible λ1 of the original, disaggregated matrix, does explain why the aggregation may fail to eliminate reproductive uncertainty.

Assessment of European Use tables at basic prices and valuation matrices in the absence of official data

ABSTRACTInput–Output modellers are often faced with the task of estimating missing Use tables at basic prices and also valuation matrices of the individual countries. This paper examines a selection of estimation methods applied to the European context where the analysts are not in possession of superior data. The estimation methods are restricted to the use of automated methods that would require more than just the row and column sums of the tables (as in projections) but less than a combination of various conflicting information (as in compilation). The results are assessed against the official Supply, Use and Input–Output tables of Belgium, Germany, Italy, Netherlands, Finland, Austria and Slovakia by using matrix difference metrics. The main conclusion is that using the structures of previous years usually performs better than any other approach.

Reconstructing Binary Matrices under Window Constraints from their Row and Column Sums

The present paper deals with the discrete inverse problem of reconstructing binary matrices from their row and column sums under additional constraints on the number and pattern of entries in specified minors. While the classical consistency and reconstruction problems for two directions in discrete tomography can be solved in polynomial time, it turns out that these window constraints cause various unexpected complexity jumps back and forth from polynomial-time solvability to NP-hardness.

Reduction of species coexistence through mixing in a spatial competition model

Many ecological systems exhibit self-organized spatial patterns due to local interactions. Such patterns can promote species diversity and therefore serve as an important mechanism for biodiversity maintenance. Previous work has shown that when species interactions occurred at local spatial scales, species diversity was greatest when robust mosaic spatial patterns formed. Also, intransitive interactions led to the emergence of spiral patterns, frequently resulting in multispecies coexistence. In some instances, intransitive interactions reduced species diversity as the consequence of competitive hierarchies. Here, we extend and broaden this line of investigation and examine the role of global competition along a continuum ranging from spatial mosaics to spiral patterns. While previous models have predicted that species diversity is reduced when interactions occur over larger spatial scales, our model considers the effects of various levels of mixing on species diversity, in the context of various network structures as measured by the covariance of row and column sums of the competition matrix. First, we compare local competition (unmixed system) versus global competition (mixed systems) and show that greater species diversity is maintained under a positive covariance. Second, we show that under various levels of mixing, species diversity declines more rapidly under a negative covariance. Lastly, we demonstrate that time to extinction in our model occurs much more rapidly under a negative covariance.