Search Results (11)

Diffusion is a commonly used technique for spreading information from point to point on a graph. The rationale behind diffusion is not clear. The multi-type Galton–Watson forest is a random model of population growth without space or any other resource constraints. In this paper, we use the degenerated multi-type Galton–Watson forest (MGWF) to interpret the diffusion process, corresponding vertices to types and establishing an equivalence relationship between them. With the two-phase setting of the MGWF, one can interpret the diffusion process and the Google PageRank system explicitly. It also improves the convergence behavior of the iterative diffusion process and Google PageRank system. We validate the proposal by experiment while providing new research directions. Full article

The nonparametric bootstrap has been used in cluster analysis for various purposes. One of those purposes is to account for sampling variability. This can be achieved by obtaining a bootstrap approximation of the sampling distribution function of the estimator of interest and then clustering those distribution functions. Although the consistency of the nonparametric bootstrap in estimating transformations of the sample mean has been known for decades, little is known about how it carries over to clustering. Here, we investigated this problem with a simulation study. We considered single-linkage agglomerative hierarchical clustering and a three-type branching process for parametrized transformations of random vectors of relative frequencies of possible types of the index case of each process. In total, there were nine factors and 216 simulation scenarios in a fully-factorial design. The ability of the bootstrap-based clustering to recover the ground truth clusterings was quantified by the adjusted transfer distance between partitions. The results showed that in the best 18 scenarios, the average value of the distance was less than 20 percent of the maximum possible distance value. We noticed that the results most notably depended on the number of retained clusters, the distribution for sampling the prevalence of types, and the sample size appearing in the denominators of relative frequency types. The comparison of the bootstrap-based clustering results with so-called uninformed random partitioning results showed that in the vast majority of scenarios considered, the bootstrap-based approach led, on average, to remarkably lower classification errors than the random partitioning. Full article

Consider a supercritical Galton–Watson process with immigration $({\mathrm{X}}_n;n\ge 0)$. The Lotka–Nagaev estimator $\frac{{\mathrm{X}}_{n+1}}{{\mathrm{X}}_n}$ is a common estimator for the offspring mean. In this work, we used the Martingale method to establish several types of Cramér moderate deviation results for the Lotka–Nagaev estimator. To satisfy our needs, we employed the well-known Cramér approach for our proofs, which establishes the moderate deviation of the sum of the independent variables. Simultaneously, we provided a concrete example of its applicability in constructing confidence intervals. Full article

Let $\partial \mathsf{T}$ be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of $\partial \mathsf{T}$ along which the sequence $S_n\mathsf{X}\left(t\right)/S_n\tilde{\mathsf{X}}\left(t\right)$ has a given set of limit points, where $S_n\mathsf{X}\left(t\right)$ and $S_n\tilde{\mathsf{X}}\left(t\right)$ are two branching random walks defined on $\partial \mathsf{T}$. In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence $s=\left(s_n\right)$, we consider $E_{\alpha ,s}=\left\{t\in \partial \mathsf{T}:S_n\mathsf{X}\left(t\right)-\alpha S_n\tilde{\mathsf{X}}\left(t\right)\sim s_n\mathrm{as}n\to +\infty \right\}.$ We will give a sufficient condition on $\left(s_n\right)$ so that $E_{\alpha ,s}$ has a maximal Hausdorff and packing dimension. Full article

We consider a discrete-time single server queueing system, where arrivals stem from a multi-type Galton–Watson branching process with migration. This branching-type arrival process exhibits intricate correlation, and the performance of the corresponding queueing process can be assessed analytically. We find closed-form expressions for various moments of both the queue content and packet delay. Close inspection of the arrival process at hand, however, reveals that sample paths consist of large independent bursts of arrivals followed by geometrically distributed periods without arrivals. Allowing for non-geometric periods without arrivals, and correlated bursts, we apply $\pi$-thinning on the arrival process. As no closed-form expressions can be obtained for the performance of the corresponding queueing system, we focus on approximations of the main performance measures in the light and heavy traffic regimes. Full article

As the classic branching process, the Galton-Watson process has obtained intensive attentions in the past decades. However, this model has two idealized assumptions–discrete states and time-homogeneity. In the present paper, we consider a branching process with continuous states, and for any given $n\in \mathbb{N}$, the branching law of every particle in generation n is determined by the population size of generation $n.$ We consider the case that the process is extinct with Probability 1 since in this case the process will be substantially different from the size-dependent branching process with discrete states. We give the extinction rate in the sense of $L^2$ and almost surely by the form of harmonic moments, that is to say, we show how fast $\left\{Z_n^{-1}\right\}$ grows under a group of sufficient conditions. From the result of the present paper, we observe that the extinction rate will be determined by an asymptotic behavior of the mean of the branching law. The results obtained in this paper have the more superiority than the counterpart from the existing literature. Full article

We propose and analyze a model for optimizing the prefetching of documents, in the situation where the connection between documents is discovered progressively. A random surfer moves along the edges of a random tree representing possible sequences of documents, which is known to a controller only up to depth d. A quantity k of documents can be prefetched between two movements. The question is to determine which nodes of the known tree should be prefetched so as to minimize the probability of the surfer moving to a node not prefetched. We analyzed the model with the tools of Markov decision process theory. We formally identified the optimal policy in several situations, and we identified it numerically in others. Full article

Few studies have examined the transmission dynamics of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in rural areas and clarified rural–urban differences. Moreover, the effectiveness of non-pharmaceutical interventions (NPIs) relative to vaccination in rural areas is uncertain. We addressed this knowledge gap through using an improved statistical stochastic method based on the Galton–Watson branching process, considering both symptomatic and asymptomatic cases. Data included 1136 SARS-2-CoV infections of the rural outbreak in Hebei, China, and 135 infections of the urban outbreak in Tianjin, China. We reconstructed SARS-CoV-2 transmission chains and analyzed the effectiveness of vaccination and NPIs by simulation studies. The transmission of SARS-CoV-2 showed strong heterogeneity in urban and rural areas, with the dispersion parameters k = 0.14 and 0.35, respectively (k < 1 indicating strong heterogeneity). Although age group and contact-type distributions significantly differed between urban and rural areas, the average reproductive number (R) and k did not. Further, simulation results based on pre-control parameters (R = 0.81, k = 0.27) showed that in the vaccination scenario (80% efficacy and 55% coverage), the cumulative secondary infections will be reduced by more than half; however, NPIs are more effective than vaccinating 65% of the population. These findings could inform government policies regarding vaccination and NPIs in rural and urban areas. Full article

A two-type two-sex branching process is considered to model the evolution of the number of carriers of an allele and its mutations of a Y-linked gene. The limiting growth rates of the different types of couples and males (depending on the allele, mutated or not, that they carry on) on the set of coexistence of both alleles and on the fixation set of the mutant allele are obtained. In addition, the limiting genotype of the Y-linked gene and the limiting sex frequencies on those sets are established. Finally, the main results have been illustrated with simulated studies contextualized in problems of population genetics. Full article

We compute exact values respectively bounds of dissimilarity/distinguishability measures–in the sense of the Kullback-Leibler information distance (relative entropy) and some transforms of more general power divergences and Renyi divergences–between two competing discrete-time Galton-Watson branching processes with immigration GWI for which the offspring as well as the immigration (importation) is arbitrarily Poisson-distributed; especially, we allow for arbitrary type of extinction-concerning criticality and thus for non-stationarity. We apply this to optimal decision making in the context of the spread of potentially pandemic infectious diseases (such as e.g., the current COVID-19 pandemic), e.g., covering different levels of dangerousness and different kinds of intervention/mitigation strategies. Asymptotic distinguishability behaviour and diffusion limits are investigated, too. Full article

Branching processes are stochastic individual-based processes leading consequently to a bottom-up approach. In addition, since the state variables are random integer variables (representing population sizes), the extinction occurs at random finite time on the extinction set, thus leading to fine and realistic predictions. Starting from the simplest and well-known single-type Bienaymé-Galton-Watson branching process that was used by several authors for approximating the beginning of an epidemic, we then present a general branching model with age and population dependent individual transitions. However contrary to the classical Bienaymé-Galton-Watson or asymptotically Bienaymé-Galton-Watson setting, where the asymptotic behavior of the process, as time tends to infinity, is well understood, the asymptotic behavior of this general process is a new question. Here we give some solutions for dealing with this problem depending on whether the initial population size is large or small, and whether the disease is rare or non-rare when the initial population size is large. Full article