Weak Multiplex Percolation

In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially generalised to multiple layers. This Element describes a generalisation of percolation to multilayer networks: weak multiplex percolation. A node belongs to a connected component if at least one of its neighbours in each layer is in this component. The authors fully describe the critical phenomena of this process. In two layers with finite second moments of the degree distributions the authors observe an unusual continuous transition with quadratic growth above the threshold. When the second moments diverge, the singularity is determined by the asymptotics of the degree distributions, creating a rich set of critical behaviours. In three or more layers the authors find a discontinuous hybrid transition which persists even in highly heterogeneous degree distributions, becoming continuous only when the powerlaw exponent reaches $1+1/(M-1)$ for $M$ layers.

The second moment of Sn(t) on the Riemann hypothesis

Let [Formula: see text] be the argument of the Riemann zeta-function at the point [Formula: see text]. For [Formula: see text] and [Formula: see text] define its antiderivatives as [Formula: see text] where [Formula: see text] is a specific constant depending on [Formula: see text] and [Formula: see text]. In 1925, Littlewood proved, under the Riemann Hypothesis (RH), that [Formula: see text] for [Formula: see text]. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of [Formula: see text] and [Formula: see text]. This was extended by Fujii for [Formula: see text], when [Formula: see text]. Assuming the RH, we give the explicit asymptotic formula for the second moment of [Formula: see text] up to the second-order term, for [Formula: see text]. Our result conditionally refines Selberg’s and Fujii’s formulas and extends previous work by Goldston in [Formula: see text], where the case [Formula: see text] was considered.

Convex Optimization for Bundle Size Pricing Problem

We study the bundle size pricing (BSP) problem in which a monopolist sells bundles of products to customers and the price of each bundle depends only on the size (number of items) of the bundle. Although this pricing mechanism is attractive in practice, finding optimal bundle prices is difficult because it involves characterizing distributions of the maximum partial sums of order statistics. In this paper, we propose to solve the BSP problem under a discrete choice model using only the first and second moments of customer valuations. Correlations between valuations of bundles are captured by the covariance matrix. We show that the BSP problem under this model is convex and can be efficiently solved using off-the-shelf solvers. Our approach is flexible in optimizing prices for any given bundle size. Numerical results show that it performs very well compared with state-of-the-art heuristics. This provides a unified and efficient approach to solve the BSP problem under various distributions and dimensions. This paper was accepted by David Simchi-Levi, revenue management and market analytics.

Equity Market Description under High and Low Volatility Regimes Using Maximum Entropy Pairwise Distribution

The financial market is a complex system in which the assets influence each other, causing, among other factors, price interactions and co-movement of returns. Using the Maximum Entropy Principle approach, we analyze the interactions between a selected set of stock assets and equity indices under different high and low return volatility episodes at the 2008 Subprime Crisis and the 2020 Covid-19 outbreak. We carry out an inference process to identify the interactions, in which we implement the a pairwise Ising distribution model describing the first and second moments of the distribution of the discretized returns of each asset. Our results indicate that second-order interactions explain more than 80% of the entropy in the system during the Subprime Crisis and slightly higher than 50% during the Covid-19 outbreak independently of the period of high or low volatility analyzed. The evidence shows that during these periods, slight changes in the second-order interactions are enough to induce large changes in assets correlations but the proportion of positive and negative interactions remains virtually unchanged. Although some interactions change signs, the proportion of these changes are the same period to period, which keeps the system in a ferromagnetic state. These results are similar even when analyzing triadic structures in the signed network of couplings.

Nonlinearity in Global Crude Oil Benchmarks: Disentangling the Effect of Time Aggregation

We model the first and second moments of global crude oil benchmarks, using iterative pre-whitened generalized autoregressive conditional heteroskedasticity (GARCH) models and, in doing so, validate the efficacy of such models in assimilating the neglected nonlinearities in the underlying data-generating processes. The benchmarks considered for this study are Brent, Dubai/Oman, and West Texas Intermediate (WTI) crude oil. While nonlinear serial dependence happens to be a stylized fact across different asset classes, it is our view that prior scholarly contributions have not adequately untangled the effect of data aggregation (in time) in the examination of nonlinear dependencies. In this context, the present study strives to untangle the critical role that time aggregation plays in the examination of nonlinearity in global crude oil benchmarks using data at daily, weekly as well as monthly time frequencies. Our findings are as follows: the optimum GARCH models perform well in capturing all of the neglected nonlinearity in monthly returns of the crude benchmarks. When it comes to daily and weekly returns, our study reveals traces of neglected nonlinearities that are not completely captured by GARCH models. Moreover, such residual traces of neglected nonlinear dependencies are relatively more pronounced at the granular levels and become more and more elusory as the data get aggregated in time. JEL codes: C22, C53, C58, G1, Q47

ГАУССОВ ПУЧОК КАК МОДЕЛЬ ИСТОЧНИКА ИОНОВ

Для расчета статистического аксептанса КФМ использовался траекторный метод. Функция плотности вероятности захваченных фазовых точек предназначена для определения матриц вторых моментов. Элементы этих матриц описывают эллипсы захвата на X и Y фазовых плоскостях. Мерой согласования Гауссова пучка и аксептанса квадруполя служат площади эллипсов. При постоянных параметрах Гауссова пучка ионов эффективность согласования слабо уменьшается с увеличением разрешающей способности. Полученные данные будут полезны при проектировании современных источников ионов. To calculate the statistical QMF acceptance, an ion trajectory method has been used. The probability density functions of accepted points allow fitting the matrix of the second moments. The elements of these matrices describe the acceptance ellipses on phase X and Y planes. The measure of the coupling Gaussian beam and quadrupole acceptance is ellipse area. Colored distributions of the input and output coordinates and velocities are presented, in which the initial phases are marked with different colors. It was found that with increasing resolution, the statistical acceptance ellipses are nested into each other. At constant parameters of the input Gaussian beam, the matching efficiency weakly decreases with resolution. The obtained data will be useful for creation a new modern ion sources.

Risk Analysis of the Copula Dependent Aggregate Discounted Claims with Weibull Inter-Arrival Time

We model the recursive moments of aggregate discounted claims, assuming the inter-claim arrival time follows a Weibull distribution to accommodate overdispersed and underdispersed data set. We use a copula to represent the dependence structure between the inter-claim arrival time and its subsequent claim amount. We then use the Laplace inversion via the Gaver-Stehfest algorithm to solve numerically the first and second moments, which takes the form of a Volterra integral equation (VIE). We compute the average and variance of the aggregate discounted claims under the Farlie-Gumbel-Morgenstern (FGM) copula and conduct a sensitivity analysis under various Weibull inter-claim parameters and claim-size parameters. The comparison between the equidispersed, overdispersed and underdispersed counting processes shows that when claims arrive at times that vary more than is expected, insured lives can expect to pay higher premium, and vice versa for the case of claims arriving at times that vary less than expected. Upon comparing the Weibull risk process with an equivalent Poisson process, we also found that copulas with a wider range of dependency parameter such as the Frank and Heavy Right Tail (HRT), have a greater impact on the value of moments as opposed to modeling under FGM copula with weak dependence structure.

The Phase Transition Analysis for the Random Regular Exact 2-(d,k)-SAT Problem

In a regular (d,k)-CNF formula, each clause has length k and each variable appears d times. A regular structure such as this is symmetric, and the satisfiability problem of this symmetric structure is called the (d,k)-SAT problem for short. The regular exact 2-(d,k)-SAT problem is that for a (d,k)-CNF formula F, if there is a truth assignment T, then exactly two literals of each clause in F are true. If the formula F contains only positive or negative literals, then there is a satisfiable assignment T with a size of 2n/k such that F is 2-exactly satisfiable. This paper introduces the (d,k)-SAT instance generation model, constructs the solution space, and employs the method of the first and second moments to present the phase transition point d* of the 2-(d,k)-SAT instance with only positive literals. When d<d*, the 2-(d,k)-SAT instance can be satisfied with high probability. When d>d*, the 2-(d,k)-SAT instance can not be satisfied with high probability. Finally, the verification results demonstrate that the theoretical results are consistent with the experimental results.