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.

Getting a Directed Hamilton Cycle Two Times Faster

2012 ◽  
Vol 21 (5) ◽  
pp. 773-801 ◽  
Author(s):  
CHOONGBUM LEE ◽  
BENNY SUDAKOV ◽  
DAN VILENCHIK
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Directed Graph ◽  
Classical Result ◽  
Hamilton Cycle ◽  
Empty Graph ◽  
Underlying Graph ◽  
Random Direction ◽  
Random Graph Theory ◽  
On Line

Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if, at time t, instead of a random direction one is allowed to choose the orientation of et, then whether or not it is possible to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p. the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.

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.

A Weighted Network Topology Model of WSN Based on Random Geometric Graph Method

2014 ◽  
Vol 962-965 ◽  
pp. 2898-2902 ◽  
Author(s):  
Yue Qing Ren ◽  
Zhi Qiang Zhang
Keyword(s):  
Graph Theory ◽  
Energy Consumption ◽  
Network Topology ◽  
Real World ◽  
Geometric Theory ◽  
Geometric Graph ◽  
Random Geometric Graph ◽  
Communication Signal ◽  
Random Graph Theory ◽  
Topology Model

Focus on the weakness of modeling WSN topology by the means of random graph theory, a new weighted topology model of WSN based on random geometric theory is proposed in this paper. For the proposed new network model, the weighting scheme of network edge take into account the property of distance decreasing effect for communication signal. It also defines the differential weight which can embody the energy consumption for network communication. Based on simulations and calculations of the measures for the network topology as well as comparison with other forms of network, the results indicate that the proposed network topology model can not only describe the interrelated connected relationship between different nodes but also verify the degree of node is subject to Poisson distribution. It also has prominent clustering effects. These properties are consist with real world networks.

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.

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