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.

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.

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.

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 The Degree Analysis of an Inhomogeneous Growing Network with Two Types of Vertices

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Huilin Huang
Keyword(s):  
Probability Distribution ◽  
Law Of Large Numbers ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Degree Sequences ◽  
Strong Law ◽  
Large Numbers ◽  
Different Types ◽  
Azuma's Inequality ◽  
Growing Network

We consider an inhomogeneous growing network with two types of vertices. The degree sequences of two different types of vertices are investigated, respectively. We not only prove that the asymptotical degree distribution of typesfor this process is power law with exponent2+1+δqs+β1-qs/αqs, but also give the strong law of large numbers for degree sequences of two different types of vertices by using a different method instead of Azuma’s inequality. Then we determine asymptotically the joint probability distribution of degree for pairs of adjacent vertices with the same type and with different types, respectively.

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