scholarly journals A Limit Theorem for the Six-length of Random Functional Graphs with a Fixed Degree Sequence

10.37236/7710 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Kevin Leckey ◽  
Nick Wormald
Keyword(s):  
Limit Theorem ◽  
Degree Sequence ◽  
Limiting Distribution ◽  
Fixed Degree ◽  
Random Mapping ◽  
Functional Graph ◽  
Functional Digraph

We obtain results on the limiting distribution of the six-length of a random functional graph, also called a   functional digraph or random mapping, with given in-degree sequence. The six-length  of a vertex $v\in V$  is defined from the associated mapping, $f:V\to V$, to be the maximum $i\in V$ such that the elements $v, f(v), \ldots, f^{i-1}(v)$ are all distinct. This has relevance to the study of algorithms for integer factorisation.

scholarly journals How to Determine if a Random Graph with a Fixed Degree Sequence Has a Giant Component

2016 ◽  
Author(s):  
Felix Joos ◽  
Guillem Perarnau ◽  
Dieter Rautenbach ◽  
Bruce Reed
Keyword(s):  
Random Graph ◽  
Degree Sequence ◽  
Giant Component ◽  
Fixed Degree

A Limit Theorem for Brownian Motion in a Random Scenery

1991 ◽  
Vol 34 (3) ◽  
pp. 385-391 ◽  
Author(s):  
Bruno Remillard ◽  
Donald A. Dawson
Keyword(s):  
Brownian Motion ◽  
Limit Theorem ◽  
Random Potential ◽  
Limiting Distribution ◽  
Standard Brownian Motion ◽  
Random Scenery

AbstractWe find the limiting distribution of , where {Bu}u≧0 is the standard Brownian motion on ℝd, V is a particular random potential and {an}n≧1 is a normalizing sequence.

scholarly journals Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

2013 ◽  
Vol 50 (3) ◽  
pp. 721-740 ◽  
Author(s):  
Jennie C. Hansen ◽  
Jerzy Jaworski
Keyword(s):  
Asymptotic Behaviour ◽  
Preferential Attachment ◽  
Degree Sequence ◽  
Discrete Distributions ◽  
Threshold Behaviour ◽  
Mapping Model ◽  
Random Mapping ◽  
Exact Distributions ◽  
Random Mappings ◽  
Expected Values

In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, TnD̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂1,D̂2,…,D̂n). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂1,D̂2,…,D̂n. As an application of these results, we consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

scholarly journals On the Random Satisfiable Process

2009 ◽  
Vol 18 (5) ◽  
pp. 775-801 ◽  
Author(s):  
MICHAEL KRIVELEVICH ◽  
BENNY SUDAKOV ◽  
DAN VILENCHIK
Keyword(s):  
Degree Sequence ◽  
Natural Extension ◽  
Solution Space ◽  
Polynomial Time Algorithm ◽  
Random Processes ◽  
Time Algorithm ◽  
Satisfying Assignment ◽  
Fixed Degree ◽  
Graph Properties ◽  
Cnf Formulas

In this work we suggest a new model for generating random satisfiable k-CNF formulas. To generate such formulas. randomly permute all $2^k\binom{n}{k}$ possible clauses over the variables x1,. . .,xn, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if, after its addition, the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first m clauses (in the random permutation's order).Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruciński and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erdős, Suen and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties have been studied, such as planarity and H-freeness. Thus our model is a natural extension of this approach to the satisfiability setting.Our main contribution is as follows. For m ≥ cn, c = c(k) a sufficiently large constant, we are able to characterize the structure of the solution space of a typical formula in this distribution. Specifically, we show that typically all satisfying assignments are essentially clustered in one cluster, and all but e−Ω(m/n)n of the variables take the same value in all satisfying assignments. We also describe a polynomial-time algorithm that finds w.h.p. a satisfying assignment for such formulas.

Regenerative stochastic processes

1955 ◽  
Vol 232 (1188) ◽  
pp. 6-31 ◽  
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Distribution ◽  
Stochastic Processes ◽  
Wide Class ◽  
Initial Conditions ◽  
Limiting Distribution ◽  
Regenerative Processes ◽  
Exponential Distributions ◽  
Negative Exponential

A wide class of stochastic processes, called regenerative, is defined, and it is shown that under general conditions the instantaneous probability distribution of such a process tends with time to a unique limiting distribution, whatever the initial conditions. The general results are then applied to 'S.M.-processes’, a generalization of Markov chains, and it is shown that the limiting distribution of the process may always be obtained by assuming negative exponential distributions for the ‘waits’ in the different ‘states’. Lastly, the behaviour of integrals of regenerative processes is considered and, amongst other results, an ergodic and a multi-dimensional central limit theorem are proved.

scholarly journals Extremal trees with fixed degree sequence for atom-bond connectivity index

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 683-688 ◽  
Author(s):  
Rundan Xing ◽  
Bo Zhou
Keyword(s):  
Degree Sequence ◽  
Connectivity Index ◽  
Fixed Degree ◽  
Degree Of Vertex ◽  
Abc Index ◽  
Atom Bond

The atom-bond connectivity (ABC) index of a graph G is the sum of ?d(u)+d(v)?2/d(u)d(v) over all edges uv of G, where d(u) is the degree of vertex u in G. We characterize the extremal trees with fixed degree sequence that maximize and minimize the ABC index, respectively. We also provide algorithms to construct such trees.

scholarly journals Karp–Sipser on Random Graphs with a Fixed Degree Sequence

2011 ◽  
Vol 20 (5) ◽  
pp. 721-741 ◽  
Author(s):  
TOM BOHMAN ◽  
ALAN FRIEZE
Keyword(s):  
Random Graphs ◽  
High Probability ◽  
Degree Sequence ◽  
Fixed Degree

Let Δ ≥ 3 be an integer. Given a fixed z ∈ +Δ such that zΔ > 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1,. . .,Δ. We study the performance of the Karp–Sipser algorithm when applied to Gz. If there is an index δ > 1 such that z1 = . . . = zδ−1 = 0 and δzδ,. . .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp–Sipser algorithm succeeds in finding a matching with n ∥ z ∥ 1/2 − o(n1−ε) edges in Gz, where ε = ε (Δ, z) is a constant.

A limit theorem for the replication time of a DNA molecule

1995 ◽  
Vol 32 (02) ◽  
pp. 296-303 ◽  
Author(s):  
Richard Cowan ◽  
S. N. Chiu ◽  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Completion Time ◽  
Limiting Distribution ◽  
Dna Molecule ◽  
Replication Time ◽  
Coverage Problems

The DNA of higher animals replicates by an interesting mechanism. Enzymes recognise specific sites randomly scattered on the molecule and establish a bidirectional process of unwinding and replication from these sites. We investigate the limiting distribution of the completion time for this process by considering related coverage problems investigated by Janson (1983) and Hall (1988).

scholarly journals Hamilton Cycles in Random Graphs with a Fixed Degree Sequence

2010 ◽  
Vol 24 (2) ◽  
pp. 558-569 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Michael Krivelevich
Keyword(s):  
Random Graphs ◽  
Degree Sequence ◽  
Hamilton Cycles ◽  
Fixed Degree
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