In this paper, we study a new discrete tree and the resulting branching process, which we call the \textbf{E}rlang \textbf{W}eighted \textbf{T}ree(\textbf{EWT}). The EWT appears as the local weak limit of a random graph model proposed in~\cite{La2015}. In contrast to the local weak limit of well-known random graph models, the EWT has an interdependent structure. In particular, its vertices encode a multi-type branching process with uncountably many types.
We derive the main properties of the EWT, such as the probability of extinction, growth rate, etc. We show that the probability of extinction is the smallest fixed point of an operator. We then take a point process perspective and analyze the growth rate operator. We derive the Krein--Rutman eigenvalue $β_0$ and the corresponding eigenfunctions of the growth operator, and show that the probability of extinction equals one if and only if $β_0 \leq 1$.
Certain Markov processes, or deterministic evolution equations, have the property that they are dual to a stochastic process that exhibits extinction versus unbounded growth, i.e., the total mass in such a process either becomes zero, or grows without bounds as time tends to infinity. If this is the case, then this phenomenon can often be used to determine the invariant measures, or fixed points, of the process originally under consideration, and to study convergence to equilibrium. This principle, which has been known since early work on multitype branching processes, is here demonstrated on …
This paper considers linear functions constructed on two different weighted branching processes and provides explicit bounds for their Kantorovich-Rubinstein distance in terms of couplings of their corresponding generic branching vectors. Motivated by applications to the analysis of random graphs, we also consider a variation of the weighted branching process where the generic branching vector has a different dependence structure from the usual one. By applying the bounds to sequences of weighted branching processes, we derive sufficient conditions for the convergence in the Kantorovich-Rubins…
We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.
We consider the down/up crossing property of weighted Markov branching processes. The joint probability distribution of multi crossing numbers of such processes are obtained. In particular, for Markov branching processes, the probability distribution of death number is given and for bulk-arrival queueing model, the joint probability distributions of service number and arrival number is also given.
Let $(Z_n)$ be a supercritical branching process in a random environment $% ζ$, and $W$ be the limit of the normalized population size $Z_n/\mathbb{E%}(Z_n|ζ)$. We show necessary and sufficient conditions for the existence of weighted moments of $W$ of the form $\E W^α\ell(W)$, where $α\geq 1$, $\ell$ is a positive function slowly varying at $\infty$. In the Galton-Watson case, the results improve those of Bingham and Doney (1974).
Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space, $r\in\mathcal{T}$ is a distinguished root, and $p$ is a probability measure on this space. The underlying branching structure is carried implicitly in the metric $d$. We explore various ways of describing the interaction between branching structure and mass in $(\mathcal{T},d,r,p)$ in a way that depends on $d$ only by way of this branching structure. We introduce a notion of ma…
In this paper we study first-passage percolation in the configuration model with empirical degree distribution that follows a power-law with exponent $τ\in (2,3)$. We assign independent and identically distributed (i.i.d.)\ weights to the edges of the graph. We investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical distances. When the underlying age-dependent branching process approximating the local neighborhoods of vertices is found to produce infinitely many individuals in finite time -- called explosive branching p…
We study a model for the spread of an infectious disease which incorporates spatial and temporal effects. The model is a delayed multi-type branching process in which types represent geographic regions while infected individuals reproduce offspring during a finite time interval and have convalescence times and random death/recovery outcomes. We give simple expressions for the limit of the geometrically weighted mean evolution of the process.
Continuous time branching models are used to create random fractals in a Euclidean space, whose Hausdorff dimension is controlled by an input parameter. Finite realizations are applied in modelling the set of sites visited in models of human and animal mobility.
In O'Connell-Pei(2013) a q-weighted version of the Robinson-Schensted algorithm was introduced. In this paper we show that this algorithm has a symmetry property analogous to the well known symmetry property of the normal Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin(1979,1986,1994,1995). This approach, which uses "growth graphs", can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other q-weighted versions of the Robinson-Schensted algorithm which have recently b…
We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R))$, as $N\to\infty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-value…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of weights, the first order for the height of such trees has been recently established in [Electron. J. Probab. 26 (2021), Paper No. 80] by one of the authors. In this paper, we obtain the second and third orders in the asymptotic expansion of the height of weighted recursive trees, under similar assumptions. Our methods are inspired from those used to prove similar r…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total weight of the tree at step $n$ grows polynomially in $n$, we obtained in (Pain-Sénizergues 2022) an asymptotic expansion for the height of the tree, which falls into the university class of the maximum of branching random walks. In this paper, we consider the case of a total weight growing sub-polynomially in $n$ and obtain asymptotics for the height of the tree …
Extinction and extinguishment properties for a nonlinear predator-prey branching model
Shenzhen MSU-BIT University（深圳莫斯科国立鲍曼技术大学）；Southern University of Science and Technology（南方科技大学）；Peking University（北京大学）；North Minzu University（北方民族大学）
We study the minimal/endogenous solution $R$ to the maximum recursion on weighted branching trees given by $$R\stackrel{\mathcal{D}}{=}\left(\bigvee_{i=1}^NC_iR_i \right)\vee Q,$$ where $(Q,N,C_1,C_2,\dots)$ is a random vector with $N\in \mathbb{N}\cup\{\infty\}$, $P(|Q|>0)>0$ and nonnegative weights $\{C_i\}$, and $\{R_i\}_{i\in\mathbb{N}}$ is a sequence of i.i.d. copies of $R$ independent of $(Q,N,C_1,C_2,\dots)$; $\stackrel{\mathcal{D}}{=}$ denotes equality in distribution. Furthermore, when $Q>0$ this recursion can be transformed into its additive equivalent, which corresponds to the maxim…
We study the situations when the solution to a weighted stochastic recursion has a power law tail. To this end, we develop two complementary approaches, the first one extends Goldie's (1991) implicit renewal theorem to cover recursions on trees; and the second one is based on a direct sample path large deviations analysis of weighted recursive random sums. We believe that these methods may be of independent interest in the analysis of more general weighted branching processes as well as in the analysis of algorithms.
We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling-Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space $(C_0[0,1])^{\infty}$, with the product topol…
We introduce an algorithm for generating a random sequence of fragmentation trees, which we call the ancestral branching algorithm. This algorithm builds on the recursive partitioning structure of a tree and gives rise to an associated family of Markovian transition kernels whose finite-dimensional transition probabilities can be written in closed-form as the product over partition-valued Markov kernels. The associated tree-valued Markov process is infinitely exchangeable provided its associated partition-valued kernel is infinitely exchangeable. We also identify a transition procedure on part…
Let $(p_k)_{k\in\mathbb{N}}$ be a discrete probability distribution for which the counting function $x\mapsto \#\{k\in\mathbb{N}: p_k\geq 1/x\}$ belongs to the de Haan class $Π$. Consider a deterministic weighted branching process generated by $(p_k)_{k\in\mathbb{N}}$. A nested Karlin's occupancy scheme is the sequence of Karlin balls-in-boxes schemes in which boxes of the $j$th level, $j=1,2,\ldots$ are identified with the $j$th generation individuals and the hitting probabilities of boxes are identified with the corresponding weights. The collection of balls is the same for all generations, …
In this paper we prove a functional limit theorem for the weighted profile of a $b$-ary tree. For the proof we use classical martingales connected to branching Markov processes and a generalized version of the profile-polynomial martingale. By embedding, choosing weights and a branch factor in a right way, we finally rediscover the profiles of some well-known discrete time trees.
To implement the Wigner branching random walk, the particle carrying a signed weight, either $-1$ or $+1$, is more friendly to data storage and arithmetic manipulations than that taking a real-valued weight continuously from $-1$ to $+1$. The former is called a signed particle and the latter a weighted particle. In this paper, we propose two efficient strategies to realize the signed-particle implementation. One is to interpret the multiplicative functional as the probability to generate pairs of particles instead of the incremental weight, and the other is to utilize a bootstrap filter to adj…
