scholarly journals Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

2006 ◽  
Vol 38 (2) ◽  
pp. 287-298 ◽  
Author(s):  
Maria Deijfen ◽  
Ronald Meester
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Graphs ◽  
Random Number ◽  
Random Variables ◽  
Nearest Neighbors ◽  
Expected Length ◽  
Vertex Set ◽  
Finitary Isomorphisms ◽  
Independent Identically Distributed

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

scholarly journals Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

2006 ◽  
Vol 38 (02) ◽  
pp. 287-298 ◽  
Author(s):  
Maria Deijfen ◽  
Ronald Meester
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Graphs ◽  
Random Number ◽  
Random Variables ◽  
Nearest Neighbors ◽  
Expected Length ◽  
Vertex Set ◽  
Finitary Isomorphisms ◽  
Independent Identically Distributed

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

On random mappings with a single attracting centre

1987 ◽  
Vol 24 (1) ◽  
pp. 258-264 ◽  
Author(s):  
Ljuben R. Mutafchiev
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Vector ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Random Mappings ◽  
Vertex Set ◽  
Independent Identically Distributed ◽  
Independent And Identically Distributed

We consider the random vector T = (T(0), ···, T(n)) with independent identically distributed coordinates such that Pr{T(i) = j} = Pj, j = 0, 1, ···, n, Σ . A realization of T can be viewed as a random graph GT with vertices {0, ···, n} and arcs {(0, T(0)), ···, (n, T(n))}. For each T we partition the vertex-set of GT into three disjoint groups and study the joint probability distribution of their cardinalities. Assuming that we observe the asymptotics of this distribution, as n → ∞, for all possible values of P0. It turns out that in some cases these cardinalities are asymptotically independent and identically distributed.

Sequential selection of an increasing subsequence from a sample of random size

1999 ◽  
Vol 36 (04) ◽  
pp. 1074-1085 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Random Number ◽  
Random Variables ◽  
Random Size ◽  
Sequential Selection ◽  
Expected Length ◽  
Decision Policies ◽  
Independent Identically Distributed ◽  
Selection Of

A random number of independent identically distributed random variables is inspected in strict succession. As a variable is inspected, it can either be selected or rejected and this decision becomes final at once. The selected sequence must increase. The problem is to maximize the expected length of the selected sequence. We demonstrate decision policies which approach optimality when the number of observations becomes in a sense large and show that the maximum expected length is close to an easily computable value.

On random mappings with a single attracting centre

1987 ◽  
Vol 24 (01) ◽  
pp. 258-264 ◽  
Author(s):  
Ljuben R. Mutafchiev
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Vector ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Random Mappings ◽  
Vertex Set ◽  
Independent Identically Distributed ◽  
Independent And Identically Distributed

We consider the random vector T = (T(0), ···, T(n)) with independent identically distributed coordinates such that Pr{T(i) = j} = P j , j = 0, 1, ···, n, Σ . A realization of T can be viewed as a random graph G T with vertices {0, ···, n} and arcs {(0, T(0)), ···, (n, T(n))}. For each T we partition the vertex-set of G T into three disjoint groups and study the joint probability distribution of their cardinalities. Assuming that we observe the asymptotics of this distribution, as n → ∞, for all possible values of P 0 . It turns out that in some cases these cardinalities are asymptotically independent and identically distributed.

Sequential selection of an increasing subsequence from a sample of random size

1999 ◽  
Vol 36 (4) ◽  
pp. 1074-1085 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Random Number ◽  
Random Variables ◽  
Random Size ◽  
Sequential Selection ◽  
Expected Length ◽  
Decision Policies ◽  
Independent Identically Distributed ◽  
Selection Of

A random number of independent identically distributed random variables is inspected in strict succession. As a variable is inspected, it can either be selected or rejected and this decision becomes final at once. The selected sequence must increase. The problem is to maximize the expected length of the selected sequence.We demonstrate decision policies which approach optimality when the number of observations becomes in a sense large and show that the maximum expected length is close to an easily computable value.

On colouring random graphs

1975 ◽  
Vol 77 (2) ◽  
pp. 313-324 ◽  
Author(s):  
G. R. Grimmett ◽  
C. J. H. McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Subgraph ◽  
Vertex Set ◽  
Image Position

AbstractLet ωn denote a random graph with vertex set {1, 2, …, n}, such that each edge is present with a prescribed probability p, independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ωn is, with probability one,

scholarly journals Metric Dimension for Random Graphs

10.37236/2639 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Béla Bollobás ◽  
Dieter Mitsche ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Metric Dimension ◽  
Wide Range ◽  
Minimum Number ◽  
Vertex Set

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$.

scholarly journals Theorems on large deviations for the sum of random number of summands

2010 ◽  
Vol 51 ◽  
Author(s):  
Aurelija Kasparavičiūtė ◽  
Leonas Saulis
Keyword(s):  
Large Deviations ◽  
Rate Of Convergence ◽  
Random Number ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Compound Process ◽  
Independent Identically Distributed

In this paper, we present the rate of convergence of normal approximation and the theorem on large deviations for a compound process Zt = \sumNt i=1 t aiXi, where Z0 = 0 and ai > 0, of weighted independent identically distributed random variables Xi, i = 1, 2, . . . with  mean EXi = µ and variance DXi = σ2 > 0. It is assumed that Nt is a non-negative integervalued random variable, which depends on t > 0 and is independent of Xi, i = 1, 2, . . . .

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