A Gauss--Newton iteration for Total Least Squares problems

The Total Least Squares solution of an overdetermined, approximate linear equation $Ax \approx b$ minimizes a nonlinear function which characterizes the backward error. We show that a globally convergent variant of the Gauss--Newton iteration can be tailored to compute that solution. At each iteration, the proposed method requires the solution of an ordinary least squares problem where the matrix $A$ is perturbed by a rank-one term.Comment: 14 pages, no figure

An algebraic analysis of the graph modularity

One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix $M$, defined in terms of the adjacency matrix and a rank one null model matrix. That matrix could be posed inside the set of relevant matrices involved in graph theory, alongside adjacency, incidence and Laplacian matrices. This is the reason we propose a graph analysis based on the algebraic and spectral properties of such matrix. In particular, we propose a nodal domain theorem for the eigenvectors of $M$; we point out several relations occurring between graph's communities and nonnegative eigenvalues of $M$; and we derive a Cheeger-type inequality for the graph optimal modularity

Modularity bounds for clusters located by leading eigenvectors of the normalized modularity matrix

Nodal theorems for generalized modularity matrices ensure that the cluster located by the positive entries of the leading eigenvector of various modularity matrices induces a connected subgraph. In this paper we obtain lower bounds for the modularity of that set of nodes showing that, under certain conditions, the nodal domains induced by eigenvectors corresponding to highly positive eigenvalues of the normalized modularity matrix have indeed positive modularity, that is they can be recognized as modules inside the network. Moreover we establish Cheeger-type inequalities for the cut-modularity of the graph, providing a theoretical support to the common understanding that highly positive eigenvalues of modularity matrices are related with the possibility of subdividing a network into communities

Orthogonal Cauchy-like matrices

Cauchy-like matrices arise often as building blocks in decomposition formulas and fast algorithms for various displacement-structured matrices. A complete characterization for orthogonal Cauchy-like matrices is given here. In particular, we show that orthogonal Cauchy-like matrices correspond to eigenvector matrices of certain symmetric matrices related to the solution of secular equations. Moreover, the construction of orthogonal Cauchy-like matrices is related to that of orthogonal rational functions with variable poles

Fundamental relations in simple and 0-simple semihypergroups of small size

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The expected adjacency and modularity matrices in the degree corrected stochastic block model

We provide explicit expressions for the eigenvalues andeigenvectors of matrices that can be written as the Hadamard product of a blockpartitioned matrix with constant blocks and a rank one matrix. Such matricesarise as the expected adjacency or modularity matrices in certain random graphmodels that are widely used as benchmarks for community detection algorithms

Generalized modularity matrices

Various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory. Their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems. In this paper we put in evidence certain common traits of various modularity matrices and shed light on their spectral properties that are at the basis of various theoretical results and practical spectral-type algorithms for community detection

Isomorphism classes of the hypergroups of type U on the right of size five

AbstractBy means of a blend of theoretical arguments and computer algebra techniques, we prove that the number of isomorphism classes of hypergroups of type U on the right of order five, having a scalar (bilateral) identity, is 14751. In this way, we complete the classification of hypergroups of type U on the right of order five, started in our preceding papers [M. De Salvo, D. Freni, G. Lo Faro, A new family of hypergroups and hypergroups of type U on the right of size five, Far East J. Math. Sci. 26(2) (2007) 393–418; M. De Salvo, D. Freni, G. Lo Faro, A new family of hypergroups and hypergroups of type U on the right of size five Part two, Mathematicki Vesnik 60 (2008) 23–45; M. De Salvo, D. Freni, G. Lo Faro, On the hypergroups of type U on the right of size five, with scalar identity (submitted for publication)]. In particular, we obtain that the number of isomorphism classes of such hypergroups is 14865