We consider a model of binary opinion dynamics where one opinion is inherently “superior” than the other, and social agents exhibit a “bias” toward the superior alternative. Specifically, it is assumed that an agent updates its choice to the superior alternative with probability α > 0 irrespective of its current opinion and opinions of other agents. With probability $1-\alpha$, it adopts majority opinion among two randomly sampled neighbors and itself. We are interested in the time it takes for the network to converge to a consensus on the superior alternative. In a complete graph of size n, we show that irrespective of the initial configuration of the network, the average time to reach consensus scales as $\Theta(n\,\log n)$ when the bias parameter α is sufficiently high, that is, $\alpha \gt \alpha_c$ where αc is a threshold parameter that is uniquely characterized. When the bias is low, that is, when $\alpha \in (0,\alpha_c]$, we show that the same rate of convergence can only be achieved if the initial proportion of agents with the superior opinion is above certain threshold $p_c(\alpha)$. If this is not the case, then we show that the network takes $\Omega(\exp(\Theta(n)))$ time on average to reach consensus.

In this paper, we establish some stochastic comparison results for largest claim amounts of two sets of independent and also for interdependent portfolios under the setup of the proportional odds model. We also establish stochastic comparison results for aggregate claim amounts of two sets of independent portfolios. Further, stochastic comparisons for largest claim amounts from two sets of independent multiple-outlier claims have also been studied. The results we obtained apply to the whole family of extended distributions, also known as the Marshall–Olkin family of distributions. We have given many numerical examples to illustrate the results obtained.

We study site and bond percolation in simple directed random graphs with a given degree distribution. We derive the percolation threshold for the giant strongly connected component and the fraction of vertices in this component as a function of the percolation probability. The results are obtained for degree sequences in which the maximum degree may depend on the total number of nodes n, being asymptotically bounded by $n^{\frac{1}{9}}$.

In this paper, we consider the friendship paradox in the context of random walks and paths. Among our results, we give an equality connecting long-range degree correlation, degree variability, and the degree-wise effect of additional steps for a random walk on a graph. Random paths are also considered, as well as applications to acquaintance sampling in the context of core-periphery structure.

Finite state Markov processes and their aggregated Markov processes have been extensively studied, especially in ion channel modeling and reliability modeling. In reliability field, the asymptotic behaviors of repairable systems modeled by both processes have been paid much attention to. For a Markov process, it is well-known that limiting measures such as availability and transition probability do not depend on the initial state of the process. However, for an aggregated Markov process, it is difficult to directly know whether this conclusion holds true or not from the limiting measure formulas expressed by the Laplace transforms. In this paper, four limiting measures expressed by Laplace transforms are proved to be independent of the initial state through Tauber’s theorem. The proof is presented under the assumption that the rank of transition rate matrix is one less than the dimension of state space for the Markov process, which includes the case that all states communicate with each other. Some numerical examples and discussions based on these are presented to illustrate the results directly and to show future related research topics. Finally, the conclusion of the paper is given.

Omega ratio, a risk-return performance measure, is defined as the ratio of the expected upside deviation of return to the expected downside deviation of return from a predetermined threshold described by an investor. Motivated by finding a solution protected against sampling errors, in this paper, we focus on the worst-case Omega ratio under distributional uncertainty and its application to robust portfolio selection. The main idea is to deal with optimization problems with all uncertain parameters within an uncertainty set. The uncertainty set of the distribution of returns given characteristic information, including the first two orders of moments and the Wasserstein distance, can handle data problems with uncertainty while making the calculation feasible.

In this paper, a new point process is introduced. It combines the nonhomogeneous Poisson process with the generalized Polya process (GPP) studied in recent literature. In reliability interpretation, each event (failure) from this process is minimally repaired with a given probability and GPP-repaired with the complementary probability. Characterization of the new process via the corresponding bivariate point process is presented. The mean numbers of events for marginal processes are obtained via the corresponding rates, which are used for considering an optimal replacement problem as an application.

We consider the problem of optimally maintaining an offshore wind farm in which major components progressively degrade over time due to normal usage and exposure to a randomly varying environment. The turbines exhibit both economic and stochastic dependence due to shared maintenance setup costs and their common environment. Our aim is to identify optimal replacement policies that minimize the expected total discounted setup, replacement, and lost power production costs over an infinite horizon. The problem is formulated using a Markov decision process (MDP) model from which we establish monotonicity of the cost function jointly in the degradation level and environment state and characterize the structure of the optimal replacement policy. For the special case of a two-turbine farm, we prove that the replacement threshold of one turbine depends not only on its own state of degradation but also on the state of degradation of the other turbine in the farm. This result yields a complete characterization of the replacement policy of both turbines by a monotone curve. The policies characterized herein can be used to optimally prescribe timely replacements of major components and suggest when it is most beneficial to share costly maintenance resources.

Let $(X_{1},\ldots,X_{n})$ be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions. Let for $1\leq i\leq n$, $X_{i:n}$ denote the corresponding ith-order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is in particular proved that for $m=2,\ldots,n$, the dependence of $X_{2:m}$ on $X_{1:m}$ is more than that of X2 on X1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that $(X_{1:m},X_{2:m})$ is more concordant than $(X_{1},X_{2})$. It will be interesting to examine whether these results can be extended to other exchangeable models.

The purpose of this paper is twofold. The first part is to introduce relative-$\chi_{\alpha}^{2}$, Jensen-$\chi_{\alpha}^{2}$ and (p, w)-Jensen-$\chi_{\alpha}^2$ divergence measures and then examine their properties. In addition, we also explore possible connections between these divergence measures and Jensen–Shannon entropy measure. In the second part, we introduce $(p,\eta)$-mixture model and then show it to be an optimal solution to three different optimization problems based on $\chi_{\alpha}^{2}$ divergence measure. We further study the relative-$\chi_{\alpha}^{2}$ divergence measure for escort and arithmetic mixture densities. We also provide some results associated with relative-$\chi_{\alpha}^{2}$ divergence measure of mixed reliability systems. Finally, to demonstrate the usefulness of the Jensen-$\chi_{\alpha}^{2}$ divergence measure, we apply it to a real example in image processing and present some numerical results. Our findings in this regard show that the Jensen-$\chi_{\alpha}^{2}$ is an effective criteria for quantifying the similarity between two images.

This paper studies an M/M/1 retrial queue with negative customers, passive breakdown, and delayed repairs. Assume that the breakdown behavior of the server during idle periods is different from that during busy periods. Passive breakdowns may occur when the server is idle, due to the lack of monitoring of the server during idle periods. When the passive breakdown occurs, the server does not get repaired immediately and enters a delayed repair phase. Negative customers arrive during the busy period, which will cause the server to break down and remove the serving customers. Under steady-state conditions, we obtain explicit expressions of the probability generating functions for the steady-state distribution, together with some important performance measures for the system. In addition, we present some numerical examples to illustrate the effects of some system parameters on important performance measures and the cost function. Finally, based on the reward-cost structure, we discuss Nash equilibrium and socially optimal strategy and numerically analyze the influence of system parameters on optimal strategies and optimal social benefits.

In this paper, we consider a nonstandard multidimensional risk model, in which the claim sizes $\{\vec{X}_k, k\ge 1\}$ form an independent and identically distributed random vector sequence with dependent components. By assuming that there exists the regression dependence structure between inter-arrival time and the claim-size vectors, we extend the regression dependence to a more practical multidimensional risk model. For the univariate marginal distributions of claim vectors with consistently varying tails, we obtain the precise large deviation formulas for the multidimensional risk model with the regression size-dependent structure.