Asymptotic enumeration by degree sequence of graphs with degreeso(n 1/2)

McKay, Brendan D.; Wormald, Nicholas C.

doi:10.1007/bf01275671

Cited by 159 publications

(166 citation statements)

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“…In fact McKay and Wormald's estimate [33] shows that this happens with very high probability except when d max = O(log 1/2 m). In order to fix this problem, Steger and Wormald [38] proposed that at any step one can only look at those pairs of mini-vertices that lead to simple graphs (denote these by suitable pairs) and pick one uniformly at random.…”

Section: Definitions and The Main Ideamentioning

confidence: 99%

“…Step (c) follows from equation (3.5) of [28] and writing McKay-Wormald's estimate [33] for |L(n, d)| as a multiple of the product…”

Section: Proof Of Inequality (48)mentioning

confidence: 99%

“…McKay and Wormald [31,33] give asymptotic estimates for the number of graphs with d max = O(m 1/3−τ ). However, the error terms in their estimates are larger than what is needed to apply Jerrum, Valiant and Vazirani's [20] reduction to achieve an asymptotically uniform sampling.…”

Section: Introductionmentioning

confidence: 99%

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A Sequential Algorithm for Generating Random Graphs

2009

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Section: Definitions and The Main Ideamentioning

confidence: 99%

“…Step (c) follows from equation (3.5) of [28] and writing McKay-Wormald's estimate [33] for |L(n, d)| as a multiple of the product…”

Section: Proof Of Inequality (48)mentioning

confidence: 99%

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A Sequential Algorithm for Generating Random Graphs

2009

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“…Another, more modern kind of switchings, was introduced by McKay and Wormald in [34] and [35], and this is the kind we apply here. While there are enumeration formulas parallel to Lemma 2.1 also for moderate degrees ( [34]), we prefer to apply the edge switching technique directly.…”

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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“…[3,14,32]. A theoretical example where the authors use a network model with hard truncation can be found in [26] or [40]. Perhaps surprisingly, the online social network of Facebook does not seem to follow a truncated power-law [60], even though the total number of friends of a person was limited to 5000 at the time of the measurement.…”

Section: Example 121 (Hard Truncation)mentioning

confidence: 99%

When is a Scale-Free Graph Ultra-Small?

Hofstad

2017

J Stat Phys

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In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent τ ∈ (2, 3), up to value n β n for some β n (log n) −γ and γ ∈ (0, 1). This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law empirical degree distributions where the (possibly exponential) truncation happens at n β n . These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around 2 log log(n β n )/| log(τ − 2)| + 1/(β n (3 − τ )), with tight fluctuations. Thus, the graph is an ultrasmall world whenever 1/β n = o(log log n). We determine the distribution of the fluctuations around this value, in particular we prove these form a sequence of tight random variables with distributions that show log log-periodicity, and as a result it is non-converging. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order n f n β n , where f n ∈ (0, 1) is a random variable that oscillates with n. We decompose shortest paths into three segments, two 'end-segments' starting at each of the two uniformly chosen vertices, and a middle segment. The two end-segments of any shortest path have length log log(n β n )/| log(τ − 2)|+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length 1/(β n (3 − τ ))+tight, and it contains only vertices with degree at least of order n (1− f n )β n , thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least n β n , and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees.

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Asymptotic enumeration by degree sequence of graphs with degreeso(n 1/2)

Cited by 159 publications

References 3 publications

A Sequential Algorithm for Generating Random Graphs

A Sequential Algorithm for Generating Random Graphs

The logic of random regular graphs

When is a Scale-Free Graph Ultra-Small?

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