Asymptotic Enumeration by Degree Sequence of Graphs of High Degree

McKay, Brendan D.; Wormald, Nicholas C.

doi:10.1016/s0195-6698(13)80042-x

Cited by 115 publications

(170 citation statements)

References 10 publications

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“…Theorems 3.1 and 3.2 are proved by complex analysis, namely a multidimensional saddle-point calculation first demonstrated by McKay and Wormald [23] and McKay [25].…”

Section: Theorem 31 ([24]mentioning

confidence: 94%

Subgraphs of Random Graphs with Specified Degrees

McKay

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract. If a graph is chosen uniformly at random from all the graphs with a given degree sequence, what can be said about its subgraphs? The same can be asked of bipartite graphs, equivalently 0-1 matrices. These questions have been studied by many people. In this paper we provide a partial survey of the field, with emphasis on two general techniques: the method of switchings and the multidimensional saddle-point method.

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“…Theorems 3.1 and 3.2 are proved by complex analysis, namely a multidimensional saddle-point calculation first demonstrated by McKay and Wormald [23] and McKay [25].…”

Section: Theorem 31 ([24]mentioning

confidence: 94%

Subgraphs of Random Graphs with Specified Degrees

McKay

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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“…The most prominent example is switch randomization, where iteratively endpoints of randomly selected links are swapped (ensuring that bidirectional and unidirectional links are randomized independently and that no parallel links are produced in this way). The exact procedure of randomizing a graph while retaining the degree sequence of the graph has been debated both in fields of application [29] and in more mathematical communities [30][31][32][33]. Apart from pure convergence issues of the iterative randomization process, the individual randomization step is of interest.…”

Section: Terminology and Software Packagesmentioning

confidence: 99%

Artefacts in statistical analyses of network motifs: general framework and application to metabolic networks

Beber

Fretter

Jain

et al. 2012

J. R. Soc. Interface.

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Few-node subgraphs are the smallest collective units in a network that can be investigated. They are beyond the scale of individual nodes but more local than, for example, communities. When statistically over-or under-represented, they are called network motifs. Network motifs have been interpreted as building blocks that shape the dynamic behaviour of networks. It is this promise of potentially explaining emergent properties of complex systems with relatively simple structures that led to an interest in network motifs in an ever-growing number of studies and across disciplines. Here, we discuss artefacts in the analysis of network motifs arising from discrepancies between the network under investigation and the pool of random graphs serving as a null model. Our aim was to provide a clear and accessible catalogue of such incongruities and their effect on the motif signature. As a case study, we explore the metabolic network of Escherichia coli and show that only by excluding ever more artefacts from the motif signature a strong and plausible correlation with the essentiality profile of metabolic reactions emerges.

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“…Before quoting the McKay-Wormald formula, we wish to mention that the formulation below is similar to Proposition 3.1 of [26] which, in turn, is based on Theorem 3 of [33]. The later also contains an elegant and interesting probabilistic interpretation of this formula.…”

Section: Random Regular Graphsmentioning

confidence: 99%

“…Lemma 2.1 ( [33]). Assume thatd = (d j ) n j=1 is a dense and nearly regular degree sequence and let λ = ( n j=1 d j )/(n(n − 1)) be the normalized average ofd.…”

Section: Random Regular Graphsmentioning

confidence: 99%

The logic of random regular graphs

Haber

Krivelevich

2010

Journal of Combinatorics

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The first order language of graphs is a formal language in which one can express many properties of graphs -known as first order properties. The classic Zero-One law for random graphs states that if p is some constant probability then for every first order property the limiting probability of the binomial random graph G(n, p) having this property is either zero or one. The case of sparse random graphs has also been studied in detail for the binomial random graph model. We obtain results for random regular graphs that match the main results for G (n, p). In particular we prove that if the degree d is linear in the number of vertices n,obeys the Zero-One law. On the contrary, if d = n α for rational 0 < α < 1, then there is a (theoretically explicit) first order property with no limiting probability in G n,d .

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Asymptotic Enumeration by Degree Sequence of Graphs of High Degree

Cited by 115 publications

References 10 publications

Subgraphs of Random Graphs with Specified Degrees

Subgraphs of Random Graphs with Specified Degrees

Artefacts in statistical analyses of network motifs: general framework and application to metabolic networks

The logic of random regular graphs

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