Critical Branching Process Research Articles

Neuronal avalanches and time-frequency representations in stimulus-evoked activity

Neuronal avalanches are a hallmark feature of critical dynamics in the brain. While the theoretical framework of a critical branching processes is generally accepted for describing avalanches during ongoing brain activity, there is a current debate about the corresponding dynamical description during stimulus-evoked activity. As the brain activity evoked by external stimuli considerably varies in magnitude across time, it is not clear whether the parameters that govern the neuronal avalanche analysis (a threshold or a temporal scale) should be adaptively altered to accommodate these changes. Here, the relationship between neuronal avalanches and time-frequency representations of stimulus-evoked activity is explored. We show that neuronal avalanche metrics, calculated under a fixed threshold and temporal scale, reflect genuine changes in the underlying dynamics. In particular, event-related synchronization and de-synchronization are shown to align with variations in the power-law exponents of avalanche size distributions and the branching parameter (neural gain), as well as in the spatio-temporal spreading of avalanches. Nonetheless, the scale-invariant behavior associated with avalanches is shown to be a robust feature of healthy brain dynamics, preserved across various periods of stimulus-evoked activity and frequency bands. Taken together, the combined results suggest that throughout stimulus-evoked responses the operating point of the dynamics may drift within an extended-critical-like region.

Critical branching processes in digital memcomputing machines

Memcomputing is a novel computing paradigm that employs time non-locality (memory) to solve combinatorial optimization problems. It can be realized in practice by means of non-linear dynamical systems whose point attractors represent the solutions of the original problem. It has been previously shown that during the solution search digital memcomputing machines go through a transient phase of avalanches (instantons) that promote dynamical long-range order. By employing mean-field arguments we predict that the distribution of the avalanche sizes follows a Borel distribution typical of critical branching processes with exponent . We corroborate this analysis by solving various random 3-SAT instances of the Boolean satisfiability problem. The numerical results indicate a power-law distribution with exponent , in very good agreement with the mean-field analysis. This indicates that memcomputing machines self-tune to a critical state in which avalanches are characterized by a branching process, and that this state persists across the majority of their evolution.

Poisson random measures and critical Sevastyanov branching processes

We consider critical Sevastyanov branching processes with immigration at random time-points generated by a Poisson random measure with a local intensity for some slowly varying function The asymptotic behavior of the probability of non-extinction is studied and conditional limiting distributions of the processes with proper normalization are obtained.

Diffusion approximation of near critical branching processes in fixed and random environment

We consider Bienaymé-Galton-Watson and continuous-time Markov branching processes and prove diffusion approximation results in the near critical case, in fixed and random environment. In one hand, in the fixed environment case, we give new proofs and derive necessary and sufficient conditions for diffusion approximation to get hold of Feller-Jiřina and Jagers theorems. In the other hand, we propose a continuous-time Markov branching process with random environments and obtain diffusion approximation results. An averaging result is also presented. Proofs here are new, where weak convergence in the Skorohod space is proved via singular perturbation technique for convergence of generators and tightness of the distributions of the considered families of stochastic processes.

Asymptotic Behaviors for Critical Branching Processes with Immigration

In this paper, we investigate the asymptotic behaviors of the critical branching process with immigration $\{Z_n, n\ge 0\}$. First we get some estimation for the probability generating function of $Z_n$. Based on it, we get a large deviation for $Z_{n+1}/Z_n$. Lower and upper deviations for $Z_n$ are also studied. As a by-product, an upper deviation for $\max_{1\le i\le n} Z_i$ is obtained.

Оцiнка Беррi-Ессеєна для майже критичних гiллястих процесiв з iммiграцiєю

In this paper, we consider a nearly critical branching process with immigration. We obtain the rate of convergence in central limit theorem for nearly critical branching processes with immigration.

Branching processes in correlated random environment

We consider the critical branching processes in correlated random environment which is positively associated and study the probability of survival up to the $n$-th generation. Moreover, when the environment is given by fractional Brownian motion, we estimate also the tail of progeny as well as the tail of width.

Critical percolation and the incipient infinite cluster on Galton-Watson trees

We consider critical percolation on Galton-Watson trees and prove quenched analogues of classical theorems of critical branching processes. We show that the probability critical percolation reaches depth $n$ is asymptotic to a tree-dependent constant times $n^{-1}$. Similarly, conditioned on critical percolation reaching depth $n$, the number of vertices at depth $n$ in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution. The incipient infinite cluster (IIC) is constructed for a.e. Galton-Watson tree and we prove a limit law for the number of vertices in the IIC at depth $n$, again depending only on the offspring distribution. Provided the offspring distribution used to generate these Galton-Watson trees has all finite moments, each of these results holds almost-surely.

Reduced critical Bellman–Harris branching processes for small populations

Abstract A critical Bellman-Harris branching process {Z(t),t ≥ 0} with finite variance of the offspring number is considered. Assuming that 0 < Z(t) ≤ φ(t), where either φ(t) = o(t) as t → ∞ or φ(t) = at,a>0, we study the structure of the process where Z(s,t) is the number of particles in the initial process at moment s which either survive up to moment t or have a positive number of descendants at this moment.

The asymptotic behavior of the total number of particles in a critical branching process with immigration

The asymptotic behavior of the total number of particles in a critical branching process with immigration

Deterministic approximation for nearly critical branching processes with dependent immigration

Deterministic approximation for nearly critical branching processes with dependent immigration

Avalanche Behavior in Creep Failure of Disordered Materials

We present a mesoscale elastoplastic model of creep in disordered materials, which considers temperature-dependent stochastic activation of localized deformation events that are coupled by internal stresses, leading to collective avalanche dynamics. We generalize this stochastic plasticity model by introducing damage in terms of a local strength that decreases, on statistical average, with increasing local plastic strain. The model captures failure in terms of strain localization in a catastrophic shear band concomitant with a finite-time singularity of the creep rate. The statistics of avalanches in the run-up to failure is characterized by a decreasing avalanche exponent τ that, at failure, approaches the value τ=1.5 typical of a critical branching process. The average avalanche rate exhibits an inverse Omori law as a function of time to failure. The distribution of interavalanche times turns out to be consistent with the epidemic-type aftershock sequences (ETAS) model of earthquake statistics.

The survival probability of critical and subcritical branching processes in finite state space Markovian environment

Let (Zn)n≥0 be a branching process in a random environment defined by a Markov chain (Xn)n≥0 with values in a finite state space X. Let Pi be the probability law generated by the trajectories of Xnn≥0 starting at X0=i∈X. We study the asymptotic behaviour of the joint survival probability PiZn>0,Xn=j, j∈X as n→+∞ in the critical and strongly, intermediate and weakly subcritical cases.

REGENERATIVE MUTATION PROCESSES RELATED TO THE SELFDECOMPOSABILITY OF SIBUYA DISTRIBUTIONS

The Sibuya distribution is a discrete probability distribution on the positive integers which, while Poisson-compounding it, gives rise to the discrete-stable distribution of Steutel and van Harn. We first address the question of the discrete self-decomposability of Sibuya and Sibuya-related distributions. Discrete self-decomposable distributions arise as limit laws of pure-death branching processes with immigration, translating a balance between immigration events and systematic ageing and ultimate death of the immigrants at constant rate. Exploiting this fact, we design a new Luria–Delbrück-like model as an intertwining of a coexisting two-types (sensitive and mutant) population. In this model, a population of sensitive gently grows linearly with time. Mutants appear randomly at a rate proportional to the sensitive population size, very many at a time and with Sibuya-related distribution; each mutant is then immediately subject to random ageing and death upon appearance. The zero-set of the times free of mutants, when the sensitive population lacks immunity, is investigated using renewal theory. Finally, assuming each immigrant to die according to a critical binary branching processes, now with heavy-tailed extinction times, we observe that the local extinction events can become sparse, leading to a congestion of the mutants in the system.

Decomposable branching processes with two types of particles

Abstract A two-type critical decomposable branching process with discrete time is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. Assuming that the offspring distributions of particles of both types may have infinite variance, the asymptotic behavior of the tail distribution of the random variable Ξ 2, the total number of the second type particles ever born in the process is found. Limit theorems are proved describing (as N → ∞) the conditional distribution of the amount of the first type particles in different generations given that either Ξ 2 = N or Ξ 2 > N.

Tokunaga self-similarity arises naturally from time invariance.

The Tokunaga condition is an algebraic rule that provides a detailed description of the branching structure in a self-similar tree. Despite a solid empirical validation and practical convenience, the Tokunaga condition lacks a theoretical justification. Such a justification is suggested in this work. We define a geometric branching process G(s) that generates self-similar rooted trees. The main result establishes the equivalence between the invariance of G(s) with respect to a time shift and a one-parametric version of the Tokunaga condition. In the parameter region where the process satisfies the Tokunaga condition (and hence is time invariant), G(s) enjoys many of the symmetries observed in a critical binary Galton-Watson branching process and reproduces the latter for a particular parameter value.

Reduced multitype critical branching processes in random environment

Abstract We consider a multitype critical branching process Z n , n = 0, 1,…, in an i.i.d. random environment. Let Z m,n be the number of particles in this process at time m having descendants at time n. A limit theorem is proved for the logarithm of Z nt,n at moments nt,0 ≤ t ≤ 1, conditioned on the survival of the process Z n up to moment n when n → ∞.

The survival probability of a critical multi-type branching process in i.i.d. random environment

Conditioned on the generating functions of offspring distribution, we study the asymp-totic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments by using limits theorems for products of positive random matrices. Under some certain assumptions, the survival probability is proportional to 1/ √ n.

Branching processes seen from their extinction time via path decompositions of reflected Lévy processes

We consider a spectrally positive Levy process $X$ that does not drift to $+\infty $, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process [34, 29]. We denote by $I$ the past infimum process defined for each $t\geq 0$ by $I_t:= \inf _{[0,t]} X$ and we let $\gamma $ be the unique time at which the excursion of the reflected process $X-I$ away from 0 attains its supremum. We prove that the pre-$\gamma $ and the post-$\gamma $ subpaths of this excursion are invariant under space-time reversal, which has several other consequences in terms of duality for excursions of Levy processes. It implies in particular that the local time process of this excursion is also invariant when seen backward from its height. As a corollary, we obtain that some (sub)critical branching processes such as the binary, homogeneous (sub)critical Crump-Mode-Jagers (CMJ) processes and the excursion away from 0 of the critical Feller diffusion, which is the width process of the continuum random tree, are invariant under time reversal from their extinction time.

Conditional limit theorem for the near critical branching processes with final type of particles

Доказана предельная теорема о распределении числа частиц в близких к критическому ветвящихся процессах с финальным типом частиц при условии невырождения таких процессов к далекому моменту времени.