Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvatal, Trotter and the two senior authors proved that for any Delta there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph K(N) with N >= Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Delta. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N(2-1/Delta)log(1/Delta)N) edges, with N = [B`n] for some constant B` that depends only on Delta. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Delta is O(n(2-1/Delta)log(1/Delta)n) Our approach is based on random graphs; in fact, we show that the classical Erdos-Renyi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Delta. The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed. (C) 2011 Elsevier Inc. All rights reserved.

UNIVERSALITY OF RANDOM GRAPHS

We prove that asymptotically (as n -> infinity) almost all graphs with n vertices and C(d)n(2-1/2d) log(1/d) n edges are universal with respect to the family of all graphs with maximum degree bounded by d. Moreover, we provide an efficient deterministic embedding algorithm for finding copies of bounded degree graphs in graphs satisfying certain pseudorandom properties. We also prove a counterpart result for random bipartite graphs, where the threshold number of edges is even smaller but the embedding is randomized.

Speech Graphs Provide a Quantitative Measure of Thought Disorder in Psychosis

Background: Psychosis has various causes, including mania and schizophrenia. Since the differential diagnosis of psychosis is exclusively based on subjective assessments of oral interviews with patients, an objective quantification of the speech disturbances that characterize mania and schizophrenia is in order. In principle, such quantification could be achieved by the analysis of speech graphs. A graph represents a network with nodes connected by edges; in speech graphs, nodes correspond to words and edges correspond to semantic and grammatical relationships. Methodology/Principal Findings: To quantify speech differences related to psychosis, interviews with schizophrenics, manics and normal subjects were recorded and represented as graphs. Manics scored significantly higher than schizophrenics in ten graph measures. Psychopathological symptoms such as logorrhea, poor speech, and flight of thoughts were grasped by the analysis even when verbosity differences were discounted. Binary classifiers based on speech graph measures sorted schizophrenics from manics with up to 93.8% of sensitivity and 93.7% of specificity. In contrast, sorting based on the scores of two standard psychiatric scales (BPRS and PANSS) reached only 62.5% of sensitivity and specificity. Conclusions/Significance: The results demonstrate that alterations of the thought process manifested in the speech of psychotic patients can be objectively measured using graph-theoretical tools, developed to capture specific features of the normal and dysfunctional flow of thought, such as divergence and recurrence. The quantitative analysis of speech graphs is not redundant with standard psychometric scales but rather complementary, as it yields a very accurate sorting of schizophrenics and manics. Overall, the results point to automated psychiatric diagnosis based not on what is said, but on how it is said.

Mapping Dynamics into Graphs. The Visibility Algorithm

Si no tenemos en cuenta posibles procesos subyacentes con significado físico, químico, económico, etc., podemos considerar una serie temporal como un mero conjunto ordenado de valores y jugar con él algún inocente juego matemático como transformar dicho conjunto en otro objeto con la ayuda de una operación matemática para ver qué sucede: qué propiedades del conjunto original se conservan, cuáles se transforman y cómo, qué podemos decir de alguna de las dos representaciones matemáticas del objeto con sólo atender a la otra... Este ejercicio sería de cierto interés matemático por sí solo. Ocurre, además, que las series temporales son un método universal de extraer información de sistemas dinámicos en cualquier campo de la ciencia. Esto hace ganar un inesperado interés práctico al juego matemático anteriormente descrito, ya que abre la posibilidad de analizar las series temporales (vistas ahora como evolución temporal de procesos dinámicos) desde una nueva perspectiva. Hemos para esto de asumir la hipótesis de que la información codificada en la serie original se conserva de algún modo en la transformación (al menos una parte de ella). El interés resulta completo cuando la nueva representación del objeto pertencece a un campo de la matemáticas relativamente maduro, en el cual la información codificada en dicha representación puede ser descodificada y procesada de manera efectiva. ABSTRACT Disregarding any underlying process (and therefore any physical, chemical, economical or whichever meaning of its mere numeric values), we can consider a time series just as an ordered set of values and play the naive mathematical game of turning this set into a different mathematical object with the aids of an abstract mapping, and see what happens: which properties of the original set are conserved, which are transformed and how, what can we say about one of the mathematical representations just by looking at the other... This exercise is of mathematical interest by itself. In addition, it turns out that time series or signals is a universal method of extracting information from dynamical systems in any field of science. Therefore, the preceding mathematical game gains some unexpected practical interest as it opens the possibility of analyzing a time series (i.e. the outcome of a dynamical process) from an alternative angle. Of course, the information stored in the original time series should be somehow conserved in the mapping. The motivation is completed when the new representation belongs to a relatively mature mathematical field, where information encoded in such a representation can be effectively disentangled and processed. This is, in a nutshell, a first motivation to map time series into networks.

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs G and H. Let H^k[G] be the lexicographic product of H^k and G, where H^k is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H^k[G]H and H^k when G and H are regular and the Laplacian spectrum of H^k[G] and H^k for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10^100 ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.

Is There Justification For Universal Leukoreduction?

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Padrão fundamental de movimento : uma analise universal ou contextual?

Universidade Estadual de Campinas . Faculdade de Educação Física

Numerical evidence against a conjecture on the cover time of planar graphs

We investigate a conjecture on the cover times of planar graphs by means of large Monte Carlo simulations. The conjecture states that the cover time tau (G(N)) of a planar graph G(N) of N vertices and maximal degree d is lower bounded by tau (G(N)) >= C(d)N(lnN)(2) with C(d) = (d/4 pi) tan(pi/d), with equality holding for some geometries. We tested this conjecture on the regular honeycomb (d = 3), regular square (d = 4), regular elongated triangular (d = 5), and regular triangular (d = 6) lattices, as well as on the nonregular Union Jack lattice (d(min) = 4, d(max) = 8). Indeed, the Monte Carlo data suggest that the rigorous lower bound may hold as an equality for most of these lattices, with an interesting issue in the case of the Union Jack lattice. The data for the honeycomb lattice, however, violate the bound with the conjectured constant. The empirical probability distribution function of the cover time for the square lattice is also briefly presented, since very little is known about cover time probability distribution functions in general.

Universal resistance capacitance crossover in current-voltage characteristics for unshunted array of overdamped Nb-AlOx-Nb Josephson junctions

We report on some unusual behavior of the measured current-voltage characteristics (CVC) in artificially prepared two-dimensional unshunted array of overdamped Nb-AlO(x)-Nb Josephson junctions. The obtained nonlinear CVC are found to exhibit a pronounced (and practically temperature independent) crossover at some current I(cr) = (1/2 beta(C)-1)I(C) from a resistance R dominated state with V(R)=R root I(2)-I(C)(2) below I(cr) to a capacitance C dominated state with V(C) = root(h) over bar /4eC root I-I(C) above I(cr). The origin of the observed behavior is discussed within a single-plaquette approximation assuming the conventional resistively shunted junction model with a finite capacitance and the Ambegaokar-Baratoff relation for the critical current of the single junction. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3407566]

Universal and nonuniversal contributions to block-block entanglement in many-fermion systems

We calculate the entanglement entropy of blocks of size x embedded in a larger system of size L, by means of a combination of analytical and numerical techniques. The complete entanglement entropy in this case is a sum of three terms. One is a universal x- and L-dependent term, first predicted by Calabrese and Cardy, the second is a nonuniversal term arising from the thermodynamic limit, and the third is a finite size correction. We give an explicit expression for the second, nonuniversal, term for the one-dimensional Hubbard model, and numerically assess the importance of all three contributions by comparing to the entropy obtained from fully numerical diagonalization of the many-body Hamiltonian. We find that finite-size corrections are very small. The universal Calabrese-Cardy term is equally small for small blocks, but becomes larger for x > 1. In all investigated situations, however, the by far dominating contribution is the nonuniversal term stemming from the thermodynamic limit.

Universal zero-bias conductance through a quantum wire side-coupled to a quantum dot

A numerical renormalization-group study of the conductance through a quantum wire containing noninteracting electrons side-coupled to a quantum dot is reported. The temperature and the dot-energy dependence of the conductance are examined in the light of a recently derived linear mapping between the temperature-dependent conductance and the universal function describing the conductance for the symmetric Anderson model of a quantum wire with an embedded quantum dot. Two conduction paths, one traversing the wire, the other a bypass through the quantum dot, are identified. A gate potential applied to the quantum wire is shown to control the current through the bypass. When the potential favors transport through the wire, the conductance in the Kondo regime rises from nearly zero at low temperatures to nearly ballistic at high temperatures. When it favors the dot, the pattern is reversed: the conductance decays from nearly ballistic to nearly zero. When comparable currents flow through the two channels, the conductance is nearly temperature independent in the Kondo regime, and Fano antiresonances in the fixed-temperature plots of the conductance as a function of the dot-energy signal interference between them. Throughout the Kondo regime and, at low temperatures, even in the mixed-valence regime, the numerical data are in excellent agreement with the universal mapping.

Universal zero-bias conductance for the single-electron transistor

The thermal dependence of the zero-bias conductance for the single electron transistor is the target of two independent renormalization-group approaches, both based on the spin-degenerate Anderson impurity model. The first approach, an analytical derivation, maps the Kondo-regime conductance onto the universal conductance function for the particle-hole symmetric model. Linear, the mapping is parametrized by the Kondo temperature and the charge in the Kondo cloud. The second approach, a numerical renormalization-group computation of the conductance as a function the temperature and applied gate voltages offers a comprehensive view of zero-bias charge transport through the device. The first approach is exact in the Kondo regime; the second, essentially exact throughout the parametric space of the model. For illustrative purposes, conductance curves resulting from the two approaches are compared.

On the k-restricted structure ratio in planar and outerplanar graphs

A planar k-restricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar k-restricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar k-restricted ratio is the infimum, over simple planar graphs H, of the ratio of the number of edges in a maximum k-restricted structure subgraph of H to the number edges of H. We prove that, as k tends to infinity, the planar k-restricted ratio tends to 1/2. The same result holds for the weighted version. Our results are based on analyzing the analogous ratios for outerplanar and weighted outerplanar graphs. Here both ratios tend to 1 as k goes to infinity, and we provide good estimates of the rates of convergence, showing that they differ in the weighted from the unweighted case.

On the resilience of long cycles in random graphs

In this paper we determine the local and global resilience of random graphs G(n,p) (p >> n(-1)) with respect to the property of containing a cycle of length at least (1 - alpha)n. Roughly speaking, given alpha > 0, we determine the smallest r(g) (G, alpha) with the property that almost surely every subgraph of G = G(n,p) having more than r(g) (G, alpha)vertical bar E(G)vertical bar edges contains a cycle of length at least (1 - alpha)n (global resilience). We also obtain, for alpha < 1/2, the smallest r(l) (G, alpha) such that any H subset of G having deg(H) (v) larger than r(l) (G, alpha) deg(G) (v) for all v is an element of V(G) contains a cycle of length at least (1 - alpha)n (local resilience). The results above are in fact proved in the more general setting of pseudorandom graphs.

Gibbs random graphs on point processes

Consider a discrete locally finite subset Gamma of R(d) and the cornplete graph (Gamma, E), with vertices Gamma and edges E. We consider Gibbs measures on the set of sub-graphs with vertices Gamma and edges E` subset of E. The Gibbs interaction acts between open edges having a vertex in common. We study percolation properties of the Gibbs distribution of the graph ensemble. The main results concern percolation properties of the open edges in two cases: (a) when Gamma is sampled from a homogeneous Poisson process; and (b) for a fixed Gamma with sufficiently sparse points. (c) 2010 American Institute of Physics. [doi:10.1063/1.3514605]