The diameter of random regular graphs

Bollobás, Béla; Véga, W. Fernandez de la

doi:10.1007/bf02579310

Cited by 152 publications

(129 citation statements)

References 4 publications

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“…There are two examples where the law of W is known. The first is when all degrees in the graph are equal to some r > 2, and we obtain the r-regular graph (see also [15], where the diameter of this graph is studied). In this case, we have that µ = r, ν = r − 1, and W = 1 a.s.…”

Section: Resultsmentioning

confidence: 99%

Distances in random graphs with finite variance degrees

Hofstad

Hooghiemstra

Mieghem

2005

Random Struct Algorithms

151

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In this paper we study a random graph with N nodes, where node j has degree D j and. We assume that 1 − F (x) ≤ cx −τ +1 for some τ > 3 and some constant c > 0. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance.The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N → ∞. We prove that the graph distance grows like log ν N , when the base of the logarithm equalsThis confirms the heuristic argument of Newman, Strogatz and Watts [35]. In addition, the random fluctuations around this asymptotic mean log ν N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.

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Section: Resultsmentioning

confidence: 99%

Distances in random graphs with finite variance degrees

Hofstad

Hooghiemstra

Mieghem

2005

Random Struct Algorithms

151

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“…Each such interval is contained in one of the intervals J t = [tδ, (t + 2)δ], 0 ≤ t ≤ n 3/5 . The probability that a particular r i lies in one such J t is exactly (4t +4)δ 2 , so the probability that at least two r i lie in J t is at most mn(mn − 1)(4t + 4) 2 δ 4 /2 < (log n) 9 (t + 1) 2 /n 2 . Thus…”

Section: Proof Of Lemma 7 Let Us Write R(x)mentioning

confidence: 99%

“…Attention has been focused particularly on the 'smallworld' phenomenon, that graphs with a very large number n of vertices often have diameter around log n. In the context of various standard random graph models, this phenomenon has of course been known for a long time; see for example [13,14,6]. That such a small diameter can be expected even when the vertex degrees are constant was shown in [9]. Related results in different contexts include [19,18,10]; see also chapter X of [7].…”

Section: Introductionmentioning

confidence: 99%

The Diameter of a Scale-Free Random Graph

Bollobas¹,

2004

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We consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree. This process was introduced by Barabási and Albert [3], as a simple model of the growth of real-world graphs such as the world-wide web. Computer experiments presented by Barabási, Albert and Jeong [1,5] and heuristic arguments given by Newman, Strogatz and Watts [23] suggest that after n steps the resulting graph should have diameter approximately log n. We show that while this holds for m = 1, for m ≥ 2 the diameter is asymptotically log n/ log log n.

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“…Earlier we suggested that interesting empirical findings about networks often motivate the development of new random graph models, but we have to be careful in framing the issue here: a simple abundance of short paths is in fact something that most basic models of random graphs already "get right." As a paradigmatic example of such a result, consider the following theorem of Bollobás and de la Vega [14], [17].…”

Section: Basic Models Of Small-world Networkmentioning

confidence: 99%

“…We say that a function is O(f (n)) if there is a constant c so that for all sufficiently large n, the function is bounded by cf (n).) In fact, [17] states a much more detailed result concerning the dependence on n, but this will not be crucial for our purposes here.…”

Section: Basic Models Of Small-world Networkmentioning

confidence: 99%

Complex networks and decentralized search algorithms

Kleinberg¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

151

194

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Abstract. The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that capture some of the qualitative properties observed in large-scale network data; such models have the potential to help us reason, at a general level, about the ways in which real-world networks are organized.We survey one particular line of network research, concerned with small-world phenomena and decentralized search algorithms, that illustrates this style of analysis. We begin by describing a well-known experiment that provided the first empirical basis for the "six degrees of separation" phenomenon in social networks; we then discuss some probabilistic network models motivated by this work, illustrating how these models lead to novel algorithmic and graph-theoretic questions, and how they are supported by recent empirical studies of large social networks. Mathematics Subject Classification (2000). Primary 68R10; Secondary 05C80, 91D30.

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The diameter of random regular graphs

Cited by 152 publications

References 4 publications

Distances in random graphs with finite variance degrees

Distances in random graphs with finite variance degrees

The Diameter of a Scale-Free Random Graph

Complex networks and decentralized search algorithms

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