Poisson convergence and random graphs

1982 ◽  
Vol 92 (2) ◽  
pp. 349-359 ◽  
Author(s):  
A. D. Barbour
Keyword(s):  
Graph Theory ◽  
Poisson Distribution ◽  
Random Graphs ◽  
Factorial Moment ◽  
Expected Number ◽  
Factorial Moments ◽  
Random Graph Theory ◽  
The Mean ◽  
Fundamental Paper ◽  
Combinatorial Methods

Approximation by the Poisson distribution arises naturally in the theory of random graphs, as in many other fields, when counting the number of occurrences of individually rare and unrelated events within a large ensemble. For example, one may be concerned with the number of times that a particular small configuration is repeated in a large graph, such questions being considered, amongst others, in the fundamental paper of Erdös and Rényi (4). The technique normally used to obtain such approximations in random graph theory is based on showing that the factorial moments of the quantity concerned converge to those of a Poisson distribution as the size of the graph tends to infinity. Since the rth factorial moment is just the expected number of ordered r-tuples of events occurring, it is particularly well suited to evaluation by combinatorial methods. Unfortunately, such a technique becomes very difficult to manage if the mean of the approximating Poisson distribution is itself increasing with the size of the graph, and this limits the scope of the results obtainable.

Customer average and time average queue lengths and waiting times

1971 ◽  
Vol 8 (03) ◽  
pp. 535-542 ◽  
Author(s):  
Kneale T. Marshall ◽  
Ronald W. Wolff
Keyword(s):  
Upper Bound ◽  
Arrival Time ◽  
Waiting Times ◽  
Factorial Moment ◽  
Expected Number ◽  
Two Measures ◽  
Queue Lengths ◽  
The Mean ◽  
Time Average ◽  
The Difference

Bounds are obtained for the difference between the expected number in the queue found by an arrival and the time average expected number in the queue for the stationary GI/G/m queue. The lower bound is completely general but the upper bound requires that the class of inter-arrival distributions be restricted. When the upper bound applies, these quantities differ by at most one customer. Analogous results are obtained for the difference between the arrival average and time average number in the system for the GI/G/1 queue. An upper bound is also determined for the kth factorial moment of the number found in the queue by an arrival in terms of the kth. moment about the origin of the wait in the queue. Inequalities on the mean virtual wait are found in terms of the mean actual wait which show that under the same restrictions, these two measures of congestion differ by no more than half the mean inter-arrival time for the GI/G/1 queue.

Customer average and time average queue lengths and waiting times

1971 ◽  
Vol 8 (3) ◽  
pp. 535-542 ◽  
Author(s):  
Kneale T. Marshall ◽  
Ronald W. Wolff
Keyword(s):  
Upper Bound ◽  
Arrival Time ◽  
Waiting Times ◽  
Factorial Moment ◽  
Expected Number ◽  
Two Measures ◽  
Queue Lengths ◽  
The Mean ◽  
Time Average ◽  
The Difference

Bounds are obtained for the difference between the expected number in the queue found by an arrival and the time average expected number in the queue for the stationary GI/G/m queue. The lower bound is completely general but the upper bound requires that the class of inter-arrival distributions be restricted. When the upper bound applies, these quantities differ by at most one customer. Analogous results are obtained for the difference between the arrival average and time average number in the system for the GI/G/1 queue. An upper bound is also determined for the kth factorial moment of the number found in the queue by an arrival in terms of the kth. moment about the origin of the wait in the queue. Inequalities on the mean virtual wait are found in terms of the mean actual wait which show that under the same restrictions, these two measures of congestion differ by no more than half the mean inter-arrival time for the GI/G/1 queue.

SPATIAL PATTERN OF THE IMMATURE STAGES AND TENERAL ADULTS OF PHYLLOPHAGA SPP. (COLEOPTERA: SCARABAEIDAE) IN A PERMANENT MEADOW

1970 ◽  
Vol 102 (11) ◽  
pp. 1354-1359 ◽  
Author(s):  
J. C. Guppy ◽  
D. G. Harcourt
Keyword(s):  
Spatial Pattern ◽  
Poisson Distribution ◽  
Negative Binomial ◽  
Fractional Power ◽  
Immature Stages ◽  
Expected Number ◽  
White Grubs ◽  
Chi Square ◽  
First Instar ◽  
The Mean

Abstract Random counts of the white grubs, Phyllophaga fusca Froelich and P. anxia LeConte, in a permanent meadow did not conform to the Poisson distribution, there being an excess of uninfested and highly infested sample units over the expected number. But when the negative binomial series was fitted to the observed distribution, the discrepancies were not significant when tested by chi-square. Using a common k, the distribution of the various stages may be described by expansion of (q-p)−k, when values of k are as follows: egg 0.15, first instar 0.41, second instar 1.30, third instar 2.00, pupa 1.62, teneral adult 1.30. Aggregation resulted from the clumping of eggs at oviposition, and randomness increased with dispersal of the larvae. For all stages, the variance was proportional to a fractional power of the mean. Three transformations are offered for stabilizing the variance of field counts.

Distribution of the number of alleles in multigene families

1992 ◽  
Vol 29 (04) ◽  
pp. 759-769
Author(s):  
R. C. Griffiths
Keyword(s):  
Stochastic Model ◽  
Gene Conversion ◽  
Lower Bounds ◽  
Multigene Families ◽  
Expected Number ◽  
Allele Type ◽  
The Mean

The distribution of the number of alleles in samples from r chromosomes is studied. The stochastic model used includes gene conversion within chromosomes and mutation at loci on the chromosomes. A method is described for simulating the distribution of alleles and an algorithm given for computing lower bounds for the mean number of alleles. A formula is derived for the expected number of samples from r chromosomes which contain the allele type of a locus chosen at random.

scholarly journals Derivation of the mean highest reversal floor and expected number of stops in lift systems

1979 ◽  
Vol 3 (4) ◽  
pp. 275-279 ◽  
Author(s):  
N.A. Alexandris ◽  
G.C. Barney ◽  
C.J. Harris
Keyword(s):  
Expected Number ◽  
The Mean

SPATIAL PATTERN OF THE IMMATURE STAGES OF HYLEMYA BRASSICAE ON CABBAGE

1970 ◽  
Vol 102 (10) ◽  
pp. 1216-1222 ◽  
Author(s):  
M. K. Mukerji ◽  
D. G. Harcourt
Keyword(s):  
Spatial Pattern ◽  
Poisson Distribution ◽  
Negative Binomial ◽  
Immature Stages ◽  
Expected Number ◽  
Chi Square ◽  
Binomial Series ◽  
Binomial Parameter ◽  
Hylemya Brassicae ◽  
Cabbage Maggot

AbstractCounts of the cabbage maggot, Hylemya brassicae (Bouché), on cabbage did not conform to the Poisson distribution, there being an excess of uninfested and highly infested plants over the expected number. But when the negative binomial series was fitted to the observed distribution, the discrepancies were not significant when tested by chi-square. The negative binomial parameter k tended to increase with density. Using a common k, the distribution of the various stages may be described by expansion of (q − p)−k, when values of k are as follows: egg 0.78, larva 0.71, pupa 0.84. Three different transformations are offered for stabilizing the variance of field counts.

scholarly journals A FORMULA TO PREDICT THE TRANSMISSION FREQUENCY OF ACENTRIC FRAGMENTS

Genetics ◽  
1972 ◽  
Vol 72 (4) ◽  
pp. 777-782
Author(s):  
A V Carrano
Keyword(s):  
Cell Death ◽  
Poisson Distribution ◽  
Transmission Frequency ◽  
Chromosomal Deletions ◽  
Radiation Induced ◽  
Fragment Distribution ◽  
The Mean ◽  
Th 1

ABSTRACT A formula, based on the Poisson distribution of radiation-induced chromosomal deletions, was derived to predict the frequency of transmission of acentric fragments between subsequent mitoses. The frequency of deletions observed in the i  th + 1 division subsequent to fragment distribution at the i  th division anaphase is independent of the cell death resulting from fragment loss. Further, the transmission frequency of chromosome acentric fragments is mathematically equal to the fragment frequency observed in the i  th + 1 generation divided by the mean fragment frequency in the i  th generation. The formula was also extended to chromatid deletions.

scholarly journals Connectivity for Random Graphs from a Weighted Bridge-Addable Class

10.37236/2596 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Planar Graphs ◽  
Additional Assumption ◽  
General Distribution ◽  
Expected Number ◽  
Recent Interest ◽  
Main Conjecture

There has been much recent interest in random graphs sampled uniformly from the $n$-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general bridge-addable class $\cal A$ of graphs -- if a graph is in $\cal A$ and $u$ and $v$ are vertices in different components   then the graph obtained by adding an edge (bridge) between $u$ and $v$ must also be in $\cal A$. Various bounds are known concerning the probability of a random graph from such a   class being connected or having many components, sometimes under the additional assumption that bridges can be deleted as well as added. Here we improve or amplify or generalise these bounds (though we do not resolve the main conjecture). For example, we see that the expected number of vertices left when we remove a largest component is less than 2. The generalisation is to consider `weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.

Random Graphs from a Minor-Closed Class

2009 ◽  
Vol 18 (4) ◽  
pp. 583-599 ◽  
Author(s):  
COLIN McDIARMID
Keyword(s):  
Poisson Distribution ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Giant Component ◽  
Closed Class ◽  
Exponential Generating Function ◽  
Excluded Minor ◽  
A Minor ◽  
Labelled Graphs

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

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