scholarly journals Election algorithms with random delays in trees

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jean-François Marckert ◽  
Nasser Saheb-Djahromi ◽  
Akka Zemmari
Keyword(s):  
Probability Distribution ◽  
Distributed Algorithm ◽  
Randomized Algorithms ◽  
Classical Problem ◽  
Random Delays ◽  
International Audience

International audience The election is a classical problem in distributed algorithmic. It aims to design and to analyze a distributed algorithm choosing a node in a graph, here, in a tree. In this paper, a class of randomized algorithms for the election is studied. The election amounts to removing leaves one by one until the tree is reduced to a unique node which is then elected. The algorithm assigns to each leaf a probability distribution (that may depends on the information transmitted by the eliminated nodes) used by the leaf to generate its remaining random lifetime. In the general case, the probability of each node to be elected is given. For two categories of algorithms, close formulas are provided.

THEORY OF RANDOM ADVECTION IN TWO DIMENSIONS

1996 ◽  
Vol 10 (18n19) ◽  
pp. 2273-2309 ◽  
Author(s):  
M. CHERTKOV ◽  
G. FALKOVICH ◽  
I. KOLOKOLOV ◽  
V. LEBEDEV
Keyword(s):  
Velocity Field ◽  
Probability Distribution ◽  
Probability Distributions ◽  
Classical Problem ◽  
Passive Scalar ◽  
Two Dimensions ◽  
Change Of Variables ◽  
The Matrix ◽  
Two Dimensional Flow ◽  
Non Gaussian

The steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described at scales less than the correlation length of the flow and larger than the diffusion scale. The probability distribution of the scalar is expressed via the probability distribution of the line stretching rate. The description of the line stretching can be reduced to the classical problem of studying the product of many matrices with a unit determinant. We found a change of variables which allows one to map the matrix problem into a scalar one and to prove thus a central limit theorem for the statistics of the stretching rate. The proof is valid for any finite correlation time of the velocity field. Whatever be the statistics of the velocity field, the statistics of the passive scalar in the inertial interval of scales is shown to approach Gaussianity as one increases the Peclet number Pe (the ratio of the pumping scale to the diffusion one). The first n < ln (Pe) simultaneous correlation functions are expressed via the flux of the squared scalar and only one unknown factor depending on the velocity field: the mean stretching rate. That factor can be calculated analytically for the limiting cases. The non-Gaussian tails of the probability distributions at finite Pe are found to be exponential.

scholarly journals On the spectral dimension of random trees

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Bergfinnur Durhuus ◽  
Thordur Jonsson ◽  
John Wheater
Keyword(s):  
Probability Distribution ◽  
Random Trees ◽  
Random Tree ◽  
Spectral Dimension ◽  
Fixed Size ◽  
Infinite Trees ◽  
Linear Chains ◽  
International Audience

International audience We determine the spectral dimensions of a variety of ensembles of infinite trees. Common to the ensembles considered is that sample trees have a distinguished infinite spine at whose vertices branches can be attached according to some probability distribution. In particular, we consider a family of ensembles of $\textit{combs}$, whose branches are linear chains, with spectral dimensions varying continuously between $1$ and $3/2$. We also introduce a class of ensembles of infinite trees, called $\textit{generic random trees}$, which are obtained as limits of ensembles of finite trees conditioned to have fixed size $N$, as $N \to \infty$. Among these ensembles is the so-called uniform random tree. We show that generic random trees have spectral dimension $d_s=4/3$.

scholarly journals Uncoupling Isaacs’equations in nonzero-sum two-player differential games : The example of conflict over parental care

2008 ◽  
Vol Volume 9, 2007 Conference in... ◽  
Author(s):  
Frédéric Hamelin ◽  
Pierre Bernhard
Keyword(s):  
Parental Care ◽  
Differential Games ◽  
Nash Equilibria ◽  
Closed Loop ◽  
Classical Problem ◽  
Complete Analysis ◽  
Behavioural Ecology ◽  
International Audience ◽  
Set Up

International audience We use a recently uncovered decoupling of Isaacs PDE’s of some mixed closed loop Nash equilibria to give a rather complete analysis of the classical problem of conflict over parental care in behavioural ecology, for a more general set up than had been considered heretofore. On utilise un découplage récemment mis en évidence des équations d’Isaacs d’un jeu différentiel pour des stratégies mixtes singulières particulières pour donner une analyse assez complète d’un problème classique en écologie comportementale concernant le conflit à propos des soins parentaux.

scholarly journals Structure of spanning trees on the two-dimensional Sierpinski gasket

2011 ◽  
Vol Vol. 12 no. 3 (Combinatorics) ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Probability Distribution ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Sierpinski Gasket ◽  
Rational Numbers ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Sierpiński Gasket ◽  
Average Probability ◽  
International Audience

Combinatorics International audience Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.

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 1-local 33/24-competitive Algorithm for Multicoloring Hexagonal Graphs

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Rafal Witkowski ◽  
Janez Žerovnik
Keyword(s):  
Graph Theory ◽  
Distributed Algorithm ◽  
Allocation Problem ◽  
Frequency Allocation ◽  
Cellular Telephone ◽  
Coverage Area ◽  
Competitive Algorithm ◽  
Phone Calls ◽  
International Audience ◽  
Hexagonal Graph

Graph Theory International audience In the frequency allocation problem, we are given a cellular telephone network whose geographical coverage area is divided into cells, where phone calls are serviced by assigned frequencies, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e. minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph, where each vertex knows its position in the graph. We present a 1-local 33/24-competitive distributed algorithm for multicoloring a hexagonal graph, thereby improving the previous 1-local 7/5-competitive algorithm.

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 Unifying description of the damping regimes of a stochastic particle in a periodic potential

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Antonio Piscitelli ◽  
Massimo Pica Ciamarra
Keyword(s):  
Probability Distribution ◽  
Periodic Potential ◽  
Stochastic Dynamics ◽  
Classical Problem ◽  
Normal Diffusion ◽  
Brownian Model ◽  
Gaussian Statistics ◽  
Asymptotic Dynamics ◽  
Non Gaussian ◽  
Typical Length

We analyze the classical problem of the stochastic dynamics of a particle confined in a periodic potential, through the so called Il’in and Khasminskii model, with a novel semi-analytical approach. Our approach gives access to the transient and the asymptotic dynamics in all damping regimes, which are difficult to investigate in the usual Brownian model. We show that the crossover from the overdamped to the underdamped regime is associated with the loss of a typical time scale and of a typical length scale, as signaled by the divergence of the probability distribution of a certain dynamical event. In the underdamped regime, normal diffusion coexists with a non-Gaussian displacement probability distribution for a long transient, as recently observed in a variety of different systems. We rationalize the microscopic physical processes leading to the non-Gaussian behavior, as well as the timescale to recover the Gaussian statistics. The theoretical results are supported by numerical calculations, and are compared to those obtained for the Brownian model.

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