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## Zero-one law for random distance graphs with vertices in {-1,0, 1}^n

We study zero-one laws for random distance graph with vertices in {−1, 0, 1} *^n* depending on a set of parameters. We give some conditions under which sequences of random distance graphs obey or do not obey the zero-one law.

The paper discusses the problems of preventing harmful information spreading in social Networks. Social networks are widespread nowadays and are used not only for managers and marketers propagation of advertising, promotion of goods, but also by attackers to spread harmful information. Thus, there is a need to counter the attackers. This paper presents simulation tools and several features that contribute to the successful application for modeling social networks and examine different strategies preventing rumors and harmful information spreading. The authors cite an example of a simulation model for identifying intruders in a social network, software tools and the results of simulation experiments.

We study the asymptotic behavior of the number of paths of length N on several classes of infinite graphs with a single special vertex. This vertex can work as an ‘entropic trap’ for the path, i.e. under certain conditions the dominant part of long paths becomes localized in the vicinity of the special point instead of spreading to infinity. We study the conditions for such localization on decorated star graphs, regular trees and regular hyperbolic graphs as a function of the functionality of the special vertex. In all cases the localization occurs for large enough functionality. The particular value of the transition point depends on the large-scale topology of the graph. The emergence of localization is supported by analysis of the spectra of the adjacency matrices of corresponding finite graphs.

Dynamical and spatial correlations of eigenfunctions as well as energy level correlations in the Anderson model on random regular graphs (RRG) are studied. We consider the critical point of the Anderson transition and the delocalized phase. In the delocalized phase near the transition point, the observables show a broad critical regime for system sizes N below the correlation volume Nξ and then cross over to the ergodic behavior. Eigenstate correlations allow us to visualize the correlation length ξ ∼ ln Nξ that controls the finite-size scaling near the transition. The critical-to-ergodic crossover is very peculiar, since the critical point is similar to the localized phase, whereas the ergodic regime is characterized by very fast “diffusion,” which is similar to the ballistic transport. In particular, the return probability crosses over from a logarithmically slow variation with time in the critical regime to an exponentially fast decay in the ergodic regime. We find a perfect agreement between results of exact diagonalization and those resulting from the solution of the self-consistency equation obtained within the saddle-point analysis of the effective supersymmetric action. We show that the RRG model can be viewed as an intricate d → ∞ limit of the Anderson model in d spatial dimensions.

Random matrix theory (RMT) is applied to investigate the cross-correlation matrix of a financial time series in four different stock markets: Russian, American, German, and Chinese. The deviations of distribution of eigenvalues of market correlation matrix from RMT global regime are investigated. Specific properties of each market are observed and discussed.

This book constitutes the proceedings of the 17th International Workshop on Algorithms and Models for the Web Graph, WAW 2020, held in Warsaw, Poland, in September 2020. The 12 full papers presented in this volume were carefully reviewed and selected from 19 submissions. The aim of the workshop was to further the understanding of graphs that arise from the Web and various user activities on the Web, and stimulate the development of high-performance algorithms and applications that exploit these graphs.

This book constitutes the proceedings of the 16th International Workshop on Algorithms and Models for the Web Graph, WAW 2019, held in Brisbane, QLD, Australia, in July 2019. The 9 full papers presented in this volume were carefully reviewed and selected from 13 submissions. The papers cover topics of all aspects of algorithmic and mathematical research in the areas pertaining to the World Wide Web, espousing the view of complex data as networks.

A numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder is performed. The problem can be described as a tight-binding model on a lattice with N sites that is locally a tree with constant connectivity. In a certain sense, the RRG ensemble can be seen as an infinite-dimensional (d→∞) cousin of the Anderson model in d dimensions. We focus on the delocalized side of the transition and stress the importance of finite-size effects. We show that the data can be interpreted in terms of the finite-size crossover from a small (N >> Nc) to a large (N >> Nc) system, where Nc is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. Our results support an analytical prediction that states in the delocalized phase (and at N >> Nc) are ergodic in the sense that their inverse participation ratio scales as 1/N

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.