scholarly journals Nilpotent adjacency matrices, random graphs and quantum random variables

2008 ◽  
Vol 41 (15) ◽  
pp. 155205 ◽  
Author(s):  
René Schott ◽  
George Stacey Staples
Keyword(s):  
Random Graphs ◽  
Random Variables ◽  
Adjacency Matrices

scholarly journals CLT-related large deviation bounds based on Stein's method

2007 ◽  
Vol 39 (3) ◽  
pp. 731-752 ◽  
Author(s):  
Martin Raič
Keyword(s):  
Random Graphs ◽  
Finite Population ◽  
Large Deviation ◽  
Random Variables ◽  
Stein’S Method ◽  
Stein's Method ◽  
Population Statistics ◽  
Sums Of Random Variables ◽  
Dependence Structures

Large deviation estimates are derived for sums of random variables with certain dependence structures, including finite population statistics and random graphs. The argument is based on Stein's method, but with a novel modification of Stein's equation inspired by the Cramér transform.

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 the asymptotics of degree structure of configuration graphs with bounded number of edges

2019 ◽  
Vol 29 (4) ◽  
pp. 219-232 ◽  
Author(s):  
Yurii L. Pavlov ◽  
Irina A. Cheplyukova
Keyword(s):  
Random Graphs ◽  
Power Law ◽  
Random Variables ◽  
Random Variable ◽  
Positive Parameter ◽  
Power Law Distribution ◽  
Bounded Number ◽  
Degree Structure ◽  
Vertex Degrees ◽  
Independent Identically Distributed

Abstract We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with positive parameter $\tau .$We study properties of random graphs such that the sum of vertex degrees does not exceed n and the parameter is a random variable uniformly distributed on the interval $\left[ a,\,\,b \right],0<a<b<\infty .$We find limit distributions of the number ${{\mu }_{r}}$of vertices with degree r for various types of variation of N, n and r.

Cliques in random graphs

1976 ◽  
Vol 80 (3) ◽  
pp. 419-427 ◽  
Author(s):  
B. Bollobas ◽  
P. Erdös
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Random Variables ◽  
Natural Numbers ◽  
Complete Subgraph ◽  
Maximal Size ◽  
Point Set ◽  
Image Position

Let 0 < p < 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i < j, defined byare independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variablesthe number of Kr's in Gn, andthe maximal size of a clique in Gn.

scholarly journals CLT-related large deviation bounds based on Stein's method

2007 ◽  
Vol 39 (03) ◽  
pp. 731-752 ◽  
Author(s):  
Martin Raič
Keyword(s):  
Random Graphs ◽  
Finite Population ◽  
Large Deviation ◽  
Random Variables ◽  
Stein’S Method ◽  
Stein's Method ◽  
Population Statistics ◽  
Sums Of Random Variables ◽  
Dependence Structures

Large deviation estimates are derived for sums of random variables with certain dependence structures, including finite population statistics and random graphs. The argument is based on Stein's method, but with a novel modification of Stein's equation inspired by the Cramér transform.

scholarly journals Markov Processes Involving q-Stirling Numbers

1997 ◽  
Vol 6 (2) ◽  
pp. 165-178 ◽  
Author(s):  
D. CRIPPA ◽  
K. SIMON ◽  
P. TRUNZ
Keyword(s):  
Markov Process ◽  
Normal Distribution ◽  
Markov Processes ◽  
Random Graphs ◽  
Transition Probabilities ◽  
Analysis Of Algorithms ◽  
Random Variables ◽  
Stirling Numbers ◽  
Time Dependent

In this paper we consider the Markov process defined byP1,1=1, Pn,[lscr ]=(1−λn,[lscr ]) ·Pn−1,[lscr ] +λn,[lscr ]−1 ·Pn−1,[lscr ]−1for transition probabilities λn,[lscr ]=q[lscr ] and λn,[lscr ]=qn−1. We give closed forms for the distributions and the moments of the underlying random variables. Thereby we observe that the distributions can be easily described in terms of q-Stirling numbers of the second kind. Their occurrence in a purely time dependent Markov process allows a natural approximation for these numbers through the normal distribution. We also show that these Markov processes describe some parameters related to the study of random graphs as well as to the analysis of algorithms.

General percolation and random graphs

1981 ◽  
Vol 13 (1) ◽  
pp. 40-60 ◽  
Author(s):  
Colin McDiarmid
Keyword(s):  
Random Graphs ◽  
Random Variables ◽  
Binary Random Variables

I introduce some useful general results concerning clutter percolation and families of binary random variables arranged in independent subfamilies, and give several applications to the study of random graphs and digraphs.

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