Branching Process Research Articles

Population Dependent Two-Sex Branching Process with Random Mating and Overlapping Generations

Population Dependent Two-Sex Branching Process with Random Mating and Overlapping Generations

Branching processes in nearly degenerate varying environment

Abstract We investigate branching processes in varying environment, for which $\overline{f}_n \to 1$ and $\sum_{n=1}^\infty (1-\overline{f}_n)_+ = \infty$ , $\sum_{n=1}^\infty (\overline{f}_n - 1)_+ < \infty$ , where $\overline{f}_n$ stands for the offspring mean in generation n. Since subcritical regimes dominate, such processes die out almost surely, therefore to obtain a nontrivial limit we consider two scenarios: conditioning on nonextinction, and adding immigration. In both cases we show that the process converges in distribution without normalization to a nondegenerate compound-Poisson limit law. The proofs rely on the shape function technique, worked out by Kersting (2020).

Collinear fragmentation at NNLL: generating functionals, groomed correlators and angularities

Jet calculus offers a unique mathematical technique to bridge the area of QCD resummation with Monte Carlo parton showers. With the ultimate goal of constructing next-to-next-to-leading logarithmic (NNLL) parton showers we study, using the language of generating functionals, the collinear fragmentation of final-state partons. In particular, we focus on the definition and calculation of the Sudakov form factor, which physically describes the no-emission probability in an ordered branching process. We review recent results for quark jets and compute the Sudakov form factor for the collinear fragmentation of gluon jets at NNLL. The NNLL corrections are encoded in a z dependent two-loop anomalous dimension B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document}2(z), with z being a suitably defined longitudinal momentum fraction. This is obtained from the integration of the relevant 1 → 3 collinear splitting kernels combined with the one-loop corrections to the 1 → 2 counterpart. This work provides the missing ingredients to extend the methods of jet calculus in the collinear limit to NNLL and gives an important element of the next generation of NNLL parton shower algorithms. As an application we derive new NNLL results for both the fractional moments of energy-energy correlation FCx and the angularities λx measured on mMDT/Soft-Drop (β = 0) groomed jets.

Critical Multitype Branching Processes with Random Migration

Abstract The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton–Watson process with migration introduced in [N. M. Yanev and K. V. Mitov, Controlled branching processes: The case of random migration, C. R. Acad. Bulgare Sci. 33 1980, 4, 473–475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.

Dynamic analysis of HIV infection model with CTL immune response and cell-to-cell transmission

HIV infection is still a serious worldwide public health problem. During the early stage of HIV infection, the number of infected cells and viruses are extremely low and the process of infection is stochastic. In this paper, we use a stochastic model consisting of continuous-time Markov chain to investigate CTL immune response and two modes of infection (viral transmission and cellular transmission). A multitype branching process is used to approximate the probability of clearance and outbreak of HIV infection near the infection-free equilibrium. In addition, we compare the dynamics of the deterministic model with the results of the stochastic model, and the differences are as follows: (1) when R0>1, the deterministic model shows HIV infection outbreak, while the stochastic model indicates a positive probability of HIV infection elimination; (2) the initial number of infected cells and viruses have no effect on the dynamics of the deterministic model, whereas the dynamics of the stochastic model are highly dependent on the initial sizes and cannot be ignored.

Diffusion approximation of critical controlled multi-type branching processes

Branching processes form an important family of stochastic processes that have been successfully applied in many fields. In this paper, we focus our attention on controlled multi-type branching processes (CMBPs). A Feller-type diffusion approximation is derived for some critical CMBPs. Namely, we consider a sequence of appropriately scaled random step functions formed from a critical CMBP with control distributions having expectations that satisfy a kind of linearity assumption. It is proved that such a sequence converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of a matrix related to the offspring mean matrix and the control distributions of the branching process in question. As applications, among others, we derive Feller-type diffusion approximations of critical, primitive multi-type branching processes with immigration and some two-sex branching processes. We also describe the asymptotic behaviour of the relative frequencies of distinct types of individuals for critical CMBPs.

Occupation times for age-structured branching processes

We consider occupation time processes for the age-structured branching processes with or without immigration introduced by Ji and Li (2022). Two large number laws and a central limit theorem for occupation times of the model with immigration are obtained.

Uniform Cramér moderate deviations and Berry–Esseen bounds for superadditive bisexual branching processes in random environments

Uniform Cramér moderate deviations and Berry–Esseen bounds for superadditive bisexual branching processes in random environments

Study on Gesture Recognition Method with Two-Stream Residual Network Fusing sEMG Signals and Acceleration Signals.

Currently, surface EMG signals have a wide range of applications in human-computer interaction systems. However, selecting features for gesture recognition models based on traditional machine learning can be challenging and may not yield satisfactory results. Considering the strong nonlinear generalization ability of neural networks, this paper proposes a two-stream residual network model with an attention mechanism for gesture recognition. One branch processes surface EMG signals, while the other processes hand acceleration signals. Segmented networks are utilized to fully extract the physiological and kinematic features of the hand. To enhance the model's capacity to learn crucial information, we introduce an attention mechanism after global average pooling. This mechanism strengthens relevant features and weakens irrelevant ones. Finally, the deep features obtained from the two branches of learning are fused to further improve the accuracy of multi-gesture recognition. The experiments conducted on the NinaPro DB2 public dataset resulted in a recognition accuracy of 88.25% for 49 gestures. This demonstrates that our network model can effectively capture gesture features, enhancing accuracy and robustness across various gestures. This approach to multi-source information fusion is expected to provide more accurate and real-time commands for exoskeleton robots and myoelectric prosthetic control systems, thereby enhancing the user experience and the naturalness of robot operation.

Limits to selection on standing variation in an asexual population

We consider how a population of N haploid individuals responds to directional selection on standing variation, with no new variation from recombination or mutation. Individuals have trait values z1,…,zN, which are drawn from a distribution ψ; the fitness of individual i is proportional to ezi. For illustration, we consider the Laplace and Gaussian distributions, which are parametrised only by the variance V0, and show that for large N, there is a scaling limit which depends on a single parameter NV0. When selection is weak relative to drift (NV0≪1), the variance decreases exponentially at rate 1/N, and the expected ultimate gain in log fitness (scaled by V0), is just NV0, which is the same as Robertson’s (1960) prediction for a sexual population. In contrast, when selection is strong relative to drift (NV0≫1), the ultimate gain can be found by approximating the establishment of alleles by a branching process in which each allele competes independently with the population mean and the fittest allele to establish is certain to fix. Then, if the probability of survival to time t∼1/V0 of an allele with value z is P(z), with mean P¯, the winning allele is the fittest of NP¯ survivors drawn from a distribution ψP/P¯. The expected ultimate change is ∼2log(1.15NV0) for a Gaussian distribution, and ∼−12log0.36NV0−log−log0.36NV0 for a Laplace distribution. This approach also predicts the variability of the process, and its dynamics; we show that in the strong selection regime, the expected genetic variance decreases as ∼t−3 at large times. We discuss how these results may be related to selection on standing variation that is spread along a linear chromosome.

Identification of tomato C2 gene family and functional analysis of the inflorescence branching regulatory gene SlC2–25

The C2 domain plays an important role in plant growth and development. We performed genome-wide identification of SlC2 gene family members by bioinformatics, and predictive analysis of chromosome distribution, evolutionary relationships, gene structure and colinearity of their family members. The results showed that a total of 103 SlC2 genes were screened, according to the order of their physical coordinates, 103 SlC2 genes were named from SlC2–1 to SlC2–103. The amino acid lengths ranged from 107 to 2133aa, the molecular weights ranged from 12,416.96 to 229,427.7 Da, and the isoelectric points ranged from 4.18 to 9.94. The cluster analysis was divided into six subgroups, and the gene sequences were highly conserved. Based on the pre-laboratory transcriptome data, we screened the SlC2–25 gene, analyzed the expression pattern of SlC2–25 in two kinds of tomato inflorescences by RT-qPCR, and preliminarily explored the function of SlC2–25 in the regulation of tomato inflorescence meristem by using gene editing technology, and found that SlC2–25 was a negative regulator of tomato inflorescence branching. In addition, the endogenous hormone content of tomato inflorescences was determined, and it was hypothesized that the SlC2s genes might be involved in the regulatory process of tomato inflorescence branching by altering the content of endogenous hormones in the plant or by interacting with a variety of proteins.

Fitting Distances by Tree Metrics Minimizing the Total Error within a Constant Factor

We consider the numerical taxonomy problem of fitting a positive distance function \({\mathcal {D}:{S\choose 2}\rightarrow \mathbb {R}_{\gt 0}}\) by a tree metric. We want a tree T with positive edge weights and including S among the vertices so that their distances in T match those in \(\mathcal {D}\) . A nice application is in evolutionary biology where the tree T aims to approximate thebranching process leading to the observed distances in \(\mathcal {D}\) [Cavalli-Sforza and Edwards 1967]. We consider the total error, that is, the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees and for the special case of ultrametrics with a root having the same distance to all vertices in S . The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was O ((log n )(log log n )) by Ailon and Charikar [2005], who wrote “determining whether an O (1) approximation can be obtained is a fascinating question.”

Synthesis of Soluble High Molar Mass Poly(Phenylene Methylene)-Based Polymers.

Poly(phenylene methylene) (PPM) is a multifunctional polymer that is also active as an anticorrosion fluorescent coating material. Although this polymer was synthesized already more than 100 years ago, a versatile synthetic route to obtain soluble high molar mass polymers based on PPM has yet to be achieved. In this article, the influence of bifunctional bis-chloromethyl durene (BCMD) as a branching agent in the synthesis of PPM is reported. The progress of the reaction was followed by gel permeation chromatography (GPC) and NMR analysis. PPM-based copolymers with the highest molar mass reported so far for this class of materials (up to Mn of 205,300 g mol-1) were isolated. The versatile approach of using BCMD was confirmed by employing different catalysts. Interestingly, thermal and optical characterization established that the branching process does not affect the thermoplastic behavior and the fluorescence of the material, thus opening up PPM-based compounds with high molar mass for applications.

Redesigning the US Army’s Branching Process: A Case Study in Minimalist Market Design

We present a proof-of-concept for minimalist market design (Sönmez 2023) as an effective methodology to enhance an institution based on stakeholders’ desiderata with minimal interference. Four objectives— respecting merit, increasing retention, aligning talent, and enhancing trust—guided reforms to the US Army’s centralized branching process of cadets to military specialties since 2006. USMA’s mechanism for the class of 2020 exacerbated challenges in implementing these objectives. Formulating the Army’s desiderata as rigorous axioms, we analyze their implications. Under our minimalist approach to institution redesign, the Army’s objectives uniquely identify a branching mechanism. Our design is now adopted at USMA and ROTC. (JEL D47, H56, J45)

Correcting for cutoff dependence in backward evolution of QCD parton showers

Monte Carlo event generators for hard hadronic collisions depend on the evolution of parton showers backwards from a high-scale subprocess to the hadronization scale. The evolution is treated as a branching process with a sequence of resolvable parton emissions. The criterion of resolvability involves cutoffs that determine the no-emission probability (NEP) for a given range of the evolution scale. Existing event generators neglect cutoff-dependent terms in the NEP that, although formally power-suppressed, can have significant phenomenological effects. We compute such terms and study their consequences. One important result is that it is not possible for the backward shower to faithfully reproduce the cutoff-independent parton distribution functions (PDFs) used to generate it. We show that the computed NEP corrections mitigate but do not eliminate this problem. An alternative approach is to use cutoff-dependent PDFs that are consistent with the uncorrected NEP. Then one must apply cutoff-dependent corrections to hard subprocess matrix elements. We compute those corrections to the first nontrivial order for the Drell-Yan process and for Higgs production by gluon fusion.

The impact of effective participation in stopping misinformation: an approach based on branching processes

The emergence of research that focuses on understanding the spreading and impact of disinformation is increasing year after year. Most of the time, the purpose of those who start the spreading of intentionally false information that is designed to cause harm is to catalyze its fast transformation into misinformation, which is the false content shared by people who do not realize it is false or misleading. Our interest is in discussing the role of people who decide to adopt an active role in stopping the propagation of information when they realize that it is false. For this, we formulate two simple probabilistic models to compare misinformation spreading in possible scenarios for which there is a passive or an active environment of aware individuals. With aware individuals, we mean those individuals who realize that a piece of given information is false or misleading. In the passive environment, we assume that if one of an aware individual is exposed to the misinformation then he/she will not spread it. In the active environment, we assume that if one of an aware individual is exposed to the misinformation then he/she will not spread it, but also he/she will stop the propagation to other individuals from the individual who contacted him/her. We appeal to the theory of branching processes to analyze propagation in both scenarios, and we discuss the role and the impact of effective participation in stopping misinformation. We show that the propagation reduces drastically, provided we assume an active environment. We also obtain theoretical and computational results to measure such a reduction, which in turn depends on the proportion of aware individuals and the number of potential contacts of each individual, which is assumed to be random.

On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law

Abstract The probability generating function (pgf) B ⁢ ( s ) {B(s)} of the limiting conditional law (LCL) of a subcritical Markov branching process ( Z t : t ≥ 0 ) {(Z_{t}:t\geq 0)} (MBP) has a certain integral representation and it satisfies B ⁢ ( 0 ) = 0 {B(0)=0} and B ′ ⁢ ( 0 ) > 0 {B^{\prime}(0)>0} . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 ( Z t = j ) {P_{1}(Z_{t}=j)} have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL.

Behaviours of honeybees can reduce the probability of deformed wing virus outbreaks in Varroa destructor-infested colonies

AbstractHoneybees are important plant pollinators. Unfortunately, there is a growing increase in the loss of honeybee colonies, and this is having a serious economic impact on crop farmers. A major cause of these losses is the parasitic mite Varroadestructor, which is a vector of deformed wing virus (DWV). Some bee species have resistant mechanisms, such as grooming and hygienic behaviours, against Varroa mites. A clear understanding of the effects of these control behaviours on the mites and the viruses they transmit can be important in reducing colony losses. Here, a stochastic model is formulated and analysed to consider the extent to which these control behaviours reduce the probability of an outbreak of DWV in honeybee colonies. Vector and bee-to-bee transmission routes are considered. Using branching process theory, it is shown that without any hygienic or grooming behaviour, a large probability of a DWV outbreak is possible. Also, if bees apply grooming or hygienic behaviour, this can reduce the probability of a virus outbreak, especially in the case of vector transmission, where it can be reduced to zero. Hygienic behaviour is the most significant factor in reducing a DWV outbreak. Thus, bee selection for hygienic behaviour may be important to reduce honeybee colony losses caused by DWV.

Abstract 2286: Clonal sweep dynamics: A marker of tumor fitness and predictor of clinical outcomes

Abstract Cancer evolution is a complex process driven by cellular adaptations to environmental pressures. Understanding a tumor's evolutionary path is crucial for tailoring therapeutic strategies based on its genomic profile. We developed a method that estimates tumor 'fitness' (w) by analyzing the time of every clonal sweep. This is quantified by driver copy number or point mutations through analytical solutions derived from the branching process of tumor evolution. The method incorporates tumor size and growth rate, assessed through in-vivo BrdU staining, to determine the tumor's age, diversity, and aggressiveness. Using our method in the Yates et al. 2015 multi-region study, we analyzed 303 breast cancer tumors from 45 patients using whole-genome and exome sequencing (WES), focusing on ER+ (N=26) and ER- (N=19) subtypes. A negative correlation was observed between clonal cytoband alterations and fitness (ER+, R=-0.41; P=0.04; ER-, R=-0.72; P<0.001). Expanding the analysis with the METABRIC study (Curtis et al. 2012), clonal cytoband alterations (CA) were linked to survival outcomes for both ER+ and ER- subtypes. Additionally, in the TRACERx NSCLC study (Jamal-Hanjani et al. 2017) of 327 tumors from 100 patients who underwent WES, the median tumor fitness was associated with a higher risk of recurrence or death. Survival analyses are detailed in Table 1 below. Multivariate analysis was adjusted for tumor size, nodal status, and CD8α expression in the breast cancer study, and according to the authors' methodology in the TRACERx study, respectively. The observed tumor growth rates in both studies align with known doubling times, demonstrating our method's accuracy. This approach successfully identifies tumor profiles linked to worse outcomes in breast and NSCLC cancers. Our findings suggest that analyzing clonal composition in tumors can improve patient stratification, particularly in infiltrated tumors, potentially enhancing immunotherapy response due to increased neoantigen exposure. Table 1 Survival Analysis Study and Endpoint Parameter Univariate HR (95% CI) Multivariate HR (95% CI) METABRIC: ER- (N=128) on RFS CA > 3 (Ref: CA ≤ 3) 0.61 (0.40, 0.93), P=0.02 0.49 (0.32, 0.75), P<0.001 METABRIC: ER- (N=128) on OS CA > 3 (Ref: CA ≤ 3) 0.68 (0.47, 1), P=0.05 0.60 (0.4, 0.88), P=0.01 METABRIC: ER+ (N=458) on RFS CA > 3 (Ref: CA ≤ 3) 1.61 (1.33, 1.94), P<0.001 1.51 (1.25, 1.82), P<0.001 METABRIC: ER+ (N=458) on OS CA > 3 (Ref: CA ≤3) 1.44 (1.23, 1.68), P<0.001 1.38 (1.18, 1.61), P<0.001 TRACERx NSCLC (N=43) on RFS Med. w > 1.031 (Ref: w ≤ 1.031) 3.24 (1.38, 7.62),P=0.007 4.11 (1.12, 15.1), P=0.033 Citation Format: Luis E. Lara Gonzalez, Sherene Loi, Anthony T. Papenfuss, David L. Goode. Clonal sweep dynamics: A marker of tumor fitness and predictor of clinical outcomes [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 2286.

An approximation of populations on a habitat with large carrying capacity

We consider stochastic dynamics of a population which starts from a small colony on a habitat with large but limited carrying capacity. A common heuristics suggests that such population grows initially as a Galton–Watson branching process and then its size follows an almost deterministic path until reaching its maximum, sustainable by the habitat. In this paper we put forward an alternative and, in fact, more accurate approximation which suggests that the population size behaves as a special nonlinear transformation of the Galton–Watson process from the very beginning.