scholarly journals Trees with product-form random weights

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Konstantin Borovkov ◽  
Vladimir Vatutin
Keyword(s):  
Asymptotic Behavior ◽  
Probability Distribution ◽  
Random Environment ◽  
Product Form ◽  
Random Weights ◽  
International Audience ◽  
The Mean ◽  
Recursive Trees ◽  
Independent Identically Distributed

International audience We consider growing random recursive trees in random environment, in which at each step a new vertex is attached according to a probability distribution that assigns the tree vertices masses proportional to their random weights.The main aim of the paper is to study the asymptotic behavior of the mean numbers of outgoing vertices as the number of steps tends to infinity, under the assumption that the random weights have a product form with independent identically distributed factors.

scholarly journals On the asymptotic behaviour of random recursive trees in random environments

2006 ◽  
Vol 38 (4) ◽  
pp. 1047-1070
Author(s):  
K. A. Borovkov ◽  
V. A. Vatutin
Keyword(s):  
Probability Distribution ◽  
Asymptotic Behaviour ◽  
Product Form ◽  
Random Environments ◽  
Random Weights ◽  
The Mean ◽  
Recursive Trees ◽  
Tree Vertex ◽  
Independent Identically Distributed

We consider growing random recursive trees in random environments, in which at each step a new vertex is attached (by an edge of random length) to an existing tree vertex according to a probability distribution that assigns the tree vertices masses proportional to their random weights. The main aim of the paper is to study the asymptotic behaviour of the distance from the newly inserted vertex to the tree's root and that of the mean numbers of outgoing vertices as the number of steps tends to ∞. Most of the results are obtained under the assumption that the random weights have a product form with independent, identically distributed factors.

scholarly journals Epidemic Size in the SIS Model of Endemic Infections

2008 ◽  
Vol 45 (3) ◽  
pp. 757-778 ◽  
Author(s):  
David A. Kessler
Keyword(s):  
Asymptotic Behavior ◽  
Probability Distribution ◽  
Population Size ◽  
Critical Region ◽  
Exact Expression ◽  
Sis Model ◽  
Analytical Expression ◽  
Epidemic Size ◽  
The Mean ◽  
Endemic Infections

We study the susceptible-infected-susceptible model of the spread of an endemic infection. We calculate an exact expression for the mean number of transmissions for all values of the population size (N) and the infectivity. We derive the large-N asymptotic behavior for the infectivitiy below, above, and in the critical region. We obtain an analytical expression for the probability distribution of the number of transmissions, n, in the critical region. We show that this distribution has an n-3/2 singularity for small n and decays exponentially for large n. The exponent decreases with the distance from the threshold, diverging to ∞ far below and approaching 0 far above.

scholarly journals Epidemic Size in the SIS Model of Endemic Infections

2008 ◽  
Vol 45 (03) ◽  
pp. 757-778 ◽  
Author(s):  
David A. Kessler
Keyword(s):  
Asymptotic Behavior ◽  
Probability Distribution ◽  
Population Size ◽  
Critical Region ◽  
Exact Expression ◽  
Sis Model ◽  
Analytical Expression ◽  
Epidemic Size ◽  
The Mean ◽  
Endemic Infections

We study the susceptible-infected-susceptible model of the spread of an endemic infection. We calculate an exact expression for the mean number of transmissions for all values of the population size (N) and the infectivity. We derive the large-N asymptotic behavior for the infectivitiy below, above, and in the critical region. We obtain an analytical expression for the probability distribution of the number of transmissions, n, in the critical region. We show that this distribution has an n -3/2 singularity for small n and decays exponentially for large n. The exponent decreases with the distance from the threshold, diverging to ∞ far below and approaching 0 far above.

Asymptotic Behavior of a Stochastic E-rumor Model with Lévy Jump

2020 ◽  
pp. 149-169
Author(s):  
Alain Pietrus ◽  
Séverine Bernard ◽  
Kendy Valmont
Keyword(s):  
Asymptotic Behavior ◽  
Social Network ◽  
Random Environment ◽  
Persistence In The Mean ◽  
The Mean ◽  
The Difference ◽  
White Noises

In this paper, we investigate the effects of a Lévy jump on the dynamic of propagation of a rumor on a social network. The random environment is characterized by white noises and Lévy jump and we establish suffcient conditions for extinction and persistence in the mean of an e-rumor. At the end, we compare our study with our previous one[7] to see the difference with only white noises.

scholarly journals Label-based parameters in increasing trees

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Probability Distribution ◽  
Random Variable ◽  
Growth Behavior ◽  
Limiting Distribution ◽  
Factorial Moments ◽  
International Audience ◽  
Recursive Trees ◽  
Insertion Process

International audience Grown simple families of increasing trees are a subclass of increasing trees, which can be constructed by an insertion process. Three such tree families contained in the grown simple families of increasing trees are of particular interest: $\textit{recursive trees}$, $\textit{plane-oriented recursive trees}$ and $\textit{binary increasing trees}$. Here we present a general approach for the analysis of a number of label-based parameters in a random grown simple increasing tree of size $n$ as, e.g., $\textit{the degree of the node labeled j}$, $\textit{the subtree-size of the node labeled j}$, etc. Further we apply the approach to the random variable $X_{n,j,a}$, which counts the number of size-$a$ branches attached to the node labeled $j$ (= subtrees of size $a$ rooted at the children of the node labeled $j$) in a random grown simple increasing tree of size $n$. We can give closed formulæ for the probability distribution and the factorial moments. Furthermore limiting distribution results for $X_{n,j,a}$ are given dependent on the growth behavior of $j=j(n)$ compared to $n$.

scholarly journals Limit distribution of the size of the giant component in a web random graph

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Yuri Pavlov
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Limit Distribution ◽  
Maximum Degree ◽  
Arbitrary Order ◽  
Graph Models ◽  
Random Graph Models ◽  
The Past ◽  
International Audience ◽  
Independent Identically Distributed

International audience Consider random graph with $N+ 1$ vertices as follows. The degrees of vertices $1,2,\ldots, N$ are the independent identically distributed random variables $\xi_1, \xi_2, \ldots , \xi_N$ with distribution $\mathbf{P}\{\xi_1 \geq k\}=k^{− \tau},$ $k= 1,2,\ldots,$ $\tau \in (1,2)$,(1) and the vertex $N+1$ has degree $0$, if the sum $\zeta_N=\xi_1+ \ldots +\xi_N$ is even, else degree is $1$. From (1) we get that $p_k=\mathbf{P}\{\xi_1=k\}=k^{−\tau}−(k+ 1)^{−\tau}$, $k= 1,2,\ldots$ Let $G(k_1, \ldots , k_N)$ be a set of graphs with $\xi_1=k_1,\ldots, \xi_N=k_N$. If $g$ is a realization of random graph then $\mathbf{P}\{g \in G(k_1, \ldots , k_N)\}=p_{k_1} \cdot \ldots \cdot p_{k_N}$. The probability distribution on the set of graph is defined such that for a vector $(k_1, \ldots, k_N)$ all graphs, lying in $G(k_1, \ldots , k_N)$, are equiprobable. Studies of the past few years show that such graphs are good random graph models for Internet and other networks topology description (see, for example, H. Reittu and I. Norros (2004)).To build the graph, we have $N$ numbered vertices and incident to vertex $i \xi_i$ stubs, $i= 1, \ldots , N$.All stubs need to be connected to another stub to construct the graph. The stubs are numbered in an arbitrary order from $1$ to $\zeta_N$. Let $\eta_{(N)}$ be the maximum degree of the vertices.

scholarly journals One-sided Variations on Tries: Path Imbalance, Climbing, and Key Sampling

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Costas A. Christophi ◽  
Hosam M. Mahmoud
Keyword(s):  
Probability Distribution ◽  
Generating Function ◽  
Path Length ◽  
Mellin Transform ◽  
Moment Generating Function ◽  
Exact Probability ◽  
Digital Trees ◽  
Exact Probability Distribution ◽  
International Audience ◽  
The Mean

International audience One-sided variations on path length in a trie (a sort of digital trees) are investigated: They include imbalance factors, climbing under different strategies, and key sampling. For the imbalance factor accurate asymptotics for the mean are derived for a randomly chosen key in the trie via poissonization and the Mellin transform, and the inverse of the two operations. It is also shown from an analysis of the moving poles of the Mellin transform of the poissonized moment generating function that the imbalance factor (under appropriate centering and scaling) follows a Gaussian limit law. The method extends to several variations of sampling keys from a trie and we sketch results of climbing under different strategies. The exact probability distribution is computed in one case, to demonstrate that such calculations can be done, at least in principle.

Multitype weakly subcritical branching processes in random environment

2021 ◽  
Vol 31 (3) ◽  
pp. 207-222
Author(s):  
Vladimir A. Vatutin ◽  
Elena E. Dyakonova
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
Branching Processes ◽  
Left Eigenvector ◽  
Mean Values ◽  
Long Time ◽  
The Mean ◽  
Independent Identically Distributed

Abstract A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvector and the increment X of the associated random walk generated by the logarithms of the Perron roots of these matrices satisfies conditions E X < 0 and E XeX > 0.

scholarly journals On subcritical multi-type branching process in random environment

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Elena Dyakonova
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Survival Probability ◽  
Branching Process ◽  
Random Variables ◽  
International Audience ◽  
Watson Process ◽  
Independent Identically Distributed

International audience We investigate a multi-type Galton-Watson process in a random environment generated by a sequence of independent identically distributed random variables. Suppose that the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices has negative mean and assuming some additional conditions, we find the asymptotics of the survival probability at time $n$ as $n \to \infty$.

Determination of the Probability Distribution of the Mean Sound Level

2010 ◽  
Vol 35 (4) ◽  
pp. 543-550 ◽  
Author(s):  
Wojciech Batko ◽  
Bartosz Przysucha
Keyword(s):  
Probability Distribution ◽  
Probability Distribution Function ◽  
Mean Value ◽  
Sound Level ◽  
Function Form ◽  
Noise Levels ◽  
Logarithmic Mean ◽  
The Mean ◽  
Estimation Of Uncertainty

AbstractAssessment of several noise indicators are determined by the logarithmic mean <img src="/fulltext-image.asp?format=htmlnonpaginated&src=P42524002G141TV8_html\05_paper.gif" alt=""/>, from the sum of independent random resultsL1;L2; : : : ;Lnof the sound level, being under testing. The estimation of uncertainty of such averaging requires knowledge of probability distribution of the function form of their calculations. The developed solution, leading to the recurrent determination of the probability distribution function for the estimation of the mean value of noise levels and its variance, is shown in this paper.

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