Growth of preferential attachment random graphs via continuous-time branching processes

2008 ◽  
Vol 118 (3) ◽  
pp. 473-494 ◽  
Author(s):  
Krishna B. Athreya ◽  
Arka P. Ghosh ◽  
Sunder Sethuraman
Keyword(s):  
Random Graphs ◽  
Continuous Time ◽  
Preferential Attachment ◽  
Branching Processes ◽  
Continuous Time Branching Processes

scholarly journals From trees to graphs: collapsing continuous-time branching processes

2018 ◽  
Vol 55 (3) ◽  
pp. 900-919
Author(s):  
A. Garavaglia ◽  
R. van der Hofstad
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Real World ◽  
Continuous Time ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Branching Processes ◽  
Random Graph Theory ◽  
First Time ◽  
Continuous Time Branching Processes

Abstract Continuous-time branching processes (CTBPs) are powerful tools in random graph theory, but are not appropriate to describe real-world networks since they produce trees rather than (multi)graphs. In this paper we analyze collapsed branching processes (CBPs), obtained by a collapsing procedure on CTBPs, in order to define multigraphs where vertices have fixed out-degree m≥2. A key example consists of preferential attachment models (PAMs), as well as generalized PAMs where vertices are chosen according to their degree and age. We identify the degree distribution of CBPs, showing that it is closely related to the limiting distribution of the CTBP before collapsing. In particular, this is the first time that CTBPs are used to investigate the degree distribution of PAMs beyond the tree setting.

Exponential ergodicity for single-birth processes

2004 ◽  
Vol 41 (4) ◽  
pp. 1022-1032 ◽  
Author(s):  
Yong-Hua Mao ◽  
Yu-Hui Zhang
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
New Method ◽  
Sufficient Condition ◽  
Exponential Ergodicity ◽  
Comparison Methods ◽  
Birth Processes ◽  
Single Birth ◽  
Continuous Time Branching Processes

An explicit, computable, and sufficient condition for exponential ergodicity of single-birth processes is presented. The corresponding criterion for birth–death processes is proved using a new method. As an application, some sufficient conditions are obtained for exponential ergodicity of an extended class of continuous-time branching processes and of multidimensional Q-processes, by comparison methods.

Continuous-time branching processes with decreasing state-dependent immigration

1984 ◽  
Vol 16 (4) ◽  
pp. 697-714 ◽  
Author(s):  
K. V. Mitov ◽  
V. A. Vatutin ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Continuous Time ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Continuous Time Branching Processes

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

scholarly journals Asymptotic behavior of critical, irreducible multi-type continuous state and continuous time branching processes with immigration

2016 ◽  
Vol 16 (04) ◽  
pp. 1650008 ◽  
Author(s):  
Mátyás Barczy ◽  
Gyula Pap
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Bessel Process ◽  
Step Functions ◽  
Squared Bessel Process ◽  
Continuous State ◽  
Random Step ◽  
Branching Process With Immigration ◽  
Continuous Time Branching Processes

Under natural assumptions, a Feller type diffusion approximation is derived for critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Namely, it is proved that a sequence of appropriately scaled random step functions formed from a critical, irreducible multi-type continuous state and continuous time branching process with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of a matrix related to the branching mechanism of the branching process in question.

Continuous-time branching processes with decreasing state-dependent immigration

1984 ◽  
Vol 16 (04) ◽  
pp. 697-714 ◽  
Author(s):  
K. V. Mitov ◽  
V. A. Vatutin ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Continuous Time ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Continuous Time Branching Processes

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

Strong ergodicity for single-birth processes

2001 ◽  
Vol 38 (01) ◽  
pp. 270-277 ◽  
Author(s):  
Yu-Hui Zhang
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Strong Ergodicity ◽  
Finite Dimensional ◽  
Comparison Methods ◽  
Schlögl Model ◽  
Birth Processes ◽  
Single Birth ◽  
Continuous Time Branching Processes

An explicit and computable criterion for strong ergodicity of single-birth processes is presented. As an application, some sufficient conditions are obtained for strong ergodicity of an extended class of continuous-time branching processes and multi-dimensional Q-processes by comparison methods respectively. Consequently strong ergodicity of the Q-process corresponding to the finite-dimensional Schlögl model is proven.

scholarly journals Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models

2016 ◽  
Vol 162 (6) ◽  
pp. 1608-1638 ◽  
Author(s):  
Angelica Pachon ◽  
Federico Polito ◽  
Laura Sacerdote
Keyword(s):  
Random Graphs ◽  
Continuous Time ◽  
Preferential Attachment

scholarly journals Asymptotic behavior of projections of supercritical multi-type continuous-state and continuous-time branching processes with immigration

2021 ◽  
Vol 53 (4) ◽  
pp. 1023-1060
Author(s):  
Mátyás Barczy ◽  
Sandra Palau ◽  
Gyula Pap
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Order Moment ◽  
Moment Condition ◽  
First Order ◽  
Continuous State ◽  
Branching Process With Immigration ◽  
Continuous Time Branching Processes

AbstractUnder a fourth-order moment condition on the branching and a second-order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous-state and continuous-time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In the case of a non-trivial process, under a first-order moment condition on the immigration mechanism, we also prove the convergence of the relative frequencies of distinct types of individuals on a suitable event; for instance, if the immigration mechanism does not vanish, then this convergence holds almost surely.

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