On the asymptotics of degree distributions

Pal, Siddharth; Makowski, Armand M.

doi:10.1109/cdc.2015.7403082

Cited by 2 publications

(6 citation statements)

References 10 publications

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“…Corollary 3.2 was announced in the conference paper [15] when the fitness variables are exponentially distributed; in [16] the failure of the convergence (26) was shown in the exponential case with the help of asymptotic properties of order statistics. Here, a fuller picture is obtained: Corollary 3.2 is a by-product of the weak convergence (25) (which replaces the non-convergence (26) and requires only Assumption 1 to hold), and of the non-degenerate nature of the limiting rv Π(d).…”

Section: Resultsmentioning

confidence: 99%

“…and the desired conclusion ( 14)- (15) follows by standard arguments upon noting that the right-hand side is the probability generating function (pgf) of the pmf (15). Assumption 1 holds in a number of interesting cases; in what follows we use the standard notation x + = max(x, 0) for x in R: When ξ is exponentially distributed with parameter λ > 0, namely…”

Section: Existence Of a Limiting Degree Distributionmentioning

confidence: 99%

“…as we substitute (20) into the expression (15). Note that p Fuj (d) does not depend on λ since λξ is exponentially distributed with unit parameter if ξ is exponentially distributed with parameter λ.…”

Section: Existence Of a Limiting Degree Distributionmentioning

confidence: 99%

“…Thus, even in homogeneous graphs, the network-wide degree distribution and the nodal degree distribution may capture vastly different information. This issue was also investigated more broadly by the authors in the references [14,15,16]; see comments following Corollary 3.2.…”

Section: Introductionmentioning

confidence: 96%

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Asymptotic degree distributions in random threshold graphs

Makowski¹,

Pal²

2019

Preprint

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We discuss several limiting degree distributions for a class of random threshold graphs in the many node regime. This analysis is carried out under a weak assumption on the distribution of the underlying fitness variable. This assumption, which is satisfied by the exponential distribution, determines a natural scaling under which the following limiting results are shown: The nodal degree distribution, i.e., the distribution of any node, converges in distribution to a limiting pmf. However, for each d = 0, 1, . . ., the fraction of nodes with given degree d converges only in distribution to a non-degenerate random variable Π(d) (whose distribution depends on d), and not in probability to the aforementioned limiting nodal pmf as is customarily expected. The distribution of Π(d) is identified only through its characteristic function. Implications of this result include: (i) The empirical node distribution may not be used as a proxy for or as an estimate to the limiting nodal pmf; (ii) Even in homogeneous graphs, the network-wide degree distribution and the nodal degree distribution may capture vastly different information; and (iii) Random threshold graphs with exponential distributed fitness do not provide an alternative scale-free model to the Barabási-Albert model as was argued by some authors; the two models cannot be meaningfully compared in terms of their degree distributions! * This document does not contain technology or technical data controlled under either the U.S. International

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Section: Resultsmentioning

confidence: 99%

Section: Existence Of a Limiting Degree Distributionmentioning

confidence: 99%

Section: Existence Of a Limiting Degree Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

Asymptotic degree distributions in random threshold graphs

Makowski¹,

Pal²

2019

Preprint

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“…This highlights the fact that even in homogeneous graphs, the network-wide degree distribution and the nodal degree distribution may capture vastly different information. Some of the results discussed in this paper were announced in the conference paper [36], mostly without proofs. Different proofs to establish Proposition 8.2. were originally given in the Ph.D. thesis of the first author [35].…”

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confidence: 99%

Asymptotic Degree Distributions in Large (Homogeneous) Random Networks: A Little Theory and a Counterexample

Pal

Makowski

2020

IEEE Trans. Netw. Sci. Eng.

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In random graph models, the degree distribution of an individual node should be distinguished from the (empirical) degree distribution of the graph that records the fractions of nodes with given degree. We introduce a general framework to explore when these two degree distributions coincide asymptotically in a sequence of homogeneous random networks of increasingly large size. The discussion is carried under three basic statistical assumptions on the degree sequences: (i) distributional homogeneity; (ii) existence of an asymptotic (nodal) degree distribution; and (iii) asymptotic uncorrelatedness. It follows from the discussion that under (i)-(ii) the asymptotic equality of the two degree distributions occurs if and only if (iii) holds. We use this observation to show that the asymptotic equality may fail in some homogeneous random networks. The counterexample is found in the class of random threshold graphs for which (i) and (ii) hold but where (iii) does not. An implication of this finding is that these random threshold graphs cannot be used as a substitute to the Barabási-Albert model for scale-free network modeling, as was proposed by some authors. The results can also be formulated for non-homogeneous models by making use of a random sampling procedure over the nodes.

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On the asymptotics of degree distributions

Cited by 2 publications

References 10 publications

Asymptotic degree distributions in random threshold graphs

Asymptotic degree distributions in random threshold graphs

Asymptotic Degree Distributions in Large (Homogeneous) Random Networks: A Little Theory and a Counterexample

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