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We introduce and study the dynamics of an emph{immortal} critical branching process. In the classic, critical branching process, particles give birth to a single offspring or die at the same rates. Even though the average population is constant in time, the ultimate fate of the population is extinction. We augment this branching process with immortality by positing that either: (a) a single particle cannot die, or (b) there exists an immortal stem cell that gives birth to ordinary cells that can subsequently undergo critical branching. We discuss the new dynamical aspects of this immortal branching process.
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Spreading processes have been largely studied in the literature, both analytically and by means of large-scale numerical simulations. These processes mainly include the propagation of diseases, rumors and information on top of a given population. In the last two decades, with the advent of modern network science, we have witnessed significant advances in this field of research. Here we review the main theoretical and numerical methods developed for the study of spreading processes on complex networked systems. Specifically, we formally define epidemic processes on single and multilayer networks and discuss in detail the main methods used to perform numerical simulations. Throughout the review, we classify spreading processes (disease and rumor models) into two classes according to the nature of time: (i) continuous-time and (ii) cellular automata approach, where the second one can be further divided into synchronous and asynchronous updating schemes. Our revision includes the heterogeneous mean-field, the quenched-mean field, and the pair quenched mean field approaches, as well as their respective simulation techniques, emphasizing similarities and differences among the different techniques. The content presented here offers a whole suite of methods to study epidemic-like processes in complex networks, both for researchers without previous experience in the subject and for experts.
T. E. Harris was a pioneer par excellence in many fields of probability theory. In this paper, we give a brief survey of the many fundamental contributions of Harris to the theory of branching processes, starting with his doctoral work at Princeton in the late forties and culminating in his fundamental book The Theory of Branching Processes, published in 1963.
The decomposable branching processes are relatively less studied objects, particularly in the continuous time framework. In this paper, we consider various variants of decomposable continuous time branching processes. As usual practice in the theory of decomposable branching processes, we group various types into irreducible classes. These irreducible classes evolve according to the well-studied nondecomposable/ irreducible branching processes. And we investigate the time evolution of the population of various classes when the process is initiated by the other class particle(s). We obtained class-wise extinction probability and the time evolution of the population in the different classes. We then studied another peculiar type of decomposable branching process where any parent at the transition epoch either produces a random number of offspring, or its type gets changed (which may or may not be regarded as new offspring produced depending on the application). Such processes arise in modeling the content propagation of competing contents in online social networks. Here also, we obtain various performance measures. Additionally, we conjecture that the time evolution of the expected number of shares (different from the total progeny in irreducible branching processes) is given by the sum of two exponential curves corresponding to the two different classes.
The work continues the authors many-year research in theory of maximal branching processes, which are obtained from classical branching processes by replacing the summation of descendant numbers with taking the maximum. One can say that in each generation, descendants of only one particle survive, namely those of the particle that has the largest number of descendants. Earlier, the author generalized processes with integer values to processes with arbitrary nonnegative values, investigated their properties, and proved limit theorems. Then processes with several types of particles were introduced and studied. In the present paper we introduce the notion of maximal branching processes in random environment (with a single type of particles) and an important case of a power-law random environment. In the latter case, properties of maximal branching processes are studied and the ergodic theorem is proved. As applications, we consider gated infinite-server queues.
For any branching process, we demonstrate that the typical total number $r_{rm mp}( u tau)$ of events triggered over all generations within any sufficiently large time window $tau$ exhibits, at criticality, a super-linear dependence $r_{rm mp}( u tau) sim ( u tau)^gamma$ (with $gamma >1$) on the total number $ u tau$ of the immigrants arriving at the Poisson rate $ u$. In branching processes in which immigrants (or sources) are characterized by fertilities distributed according to an asymptotic power law tail with tail exponent $1 < gamma leqslant 2$, the exponent of the super-linear law for $r_{rm mp}( u tau)$ is identical to the exponent $gamma$ of the distribution of fertilities. For $gamma>2$ and for standard branching processes without power law distribution of fertilities, $r_{rm mp}( u tau) sim ( u tau)^2$. This novel scaling law replaces and tames the divergence $ u tau/(1-n)$ of the mean total number ${bar R}_t(tau)$ of events, as the branching ratio (defined as the average number of triggered events of first generation per source) tends to 1. The derivation uses the formalism of generating probability functions. The corresponding prediction is confirmed by numerical calculations and an heuristic derivation enlightens its underlying mechanism. We also show that ${bar R}_t(tau)$ is always linear in $ u tau$ even at criticality ($n=1$). Our results thus illustrate the fundamental difference between the mean total number, which is controlled by a few extremely rare realizations, and the typical behavior represented by $r_{rm mp}( u tau)$.
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