Random networks with sublinear preferential attachment: Degree evolutions

Dereich, Steffen; Mörters, Peter

doi:10.1214/ejp.v14-647

Cited by 83 publications

(131 citation statements)

References 11 publications

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“…Results for other regimes are at present still on a conjectural level and will be discussed in the forthcoming work [DMM11]. We conjecture the existence of two further regimes:…”

Section: Small Worldsmentioning

confidence: 75%

“…Theorem 2 (Asymptotic outdegree distribution, see Theorem 1.1(b) in [DM09]). The conditional distribution of the outdegree of the (n + 1)st incoming node given the graph at time n converges almost surely in the total variation norm to the Poisson distribution with parameter…”

Section: Empirical Degree Distributionsmentioning

confidence: 99%

“…Theorem 3 (Existence of a perpetual hub, see Theorem 1.5 in [DM09]). A perpetual hub exists almost surely, if and only if…”

Section: Existence Of a Perpetual Hubmentioning

confidence: 99%

“…An analysis of the small world phenomenon for the given class of preferential attachment models is currently undertaken by Dereich, Mönch and Mörters in [DMM10] and [DMM11]. Recall that the distance of two vertices v, w ∈ V in a network with edge set E ⊂ V × V is…”

Section: Small Worldsmentioning

confidence: 99%

“…The investigation in [DMM10,DMM11] is naturally concentrated on the case where a giant component exists, and focuses on the typical distance between vertices in the giant component. We expect the occurrence of three different regimes, depending on the strength of the preference in the attachment rule.…”

Section: Small Worldsmentioning

confidence: 99%

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Random Networks with Concave Preferential Attachment Rule

Dereich

Mörters

2010

Jahresber. Dtsch. Math. Ver.

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Many of the phenomena in the complex world in which we live have a rough description as a large network of interacting components. Random network theory tries to describe the global structure of such networks from basic local principles. One such principle is the preferential attachment paradigm which suggests that networks are built by adding nodes and links successively, in such a way that new nodes prefer to be connected to existing nodes if they have a high degree. Our research gives the first comprehensive and mathematically rigorous treatment of the case when this preference follows a nonlinear, or more precisely concave, rule. We survey results obtained so far and some ongoing developments. NetworksAlthough networks are in principle simple objects, they can show immense complexity when their size is very large. They can therefore often offer a meaningful rough description of complex systems in nature, society or technology. Examples of the kind of networks we have in mind are the following:• a cell may be described by its metabolism as a network of chemicals with edges representing chemical reactions transforming one substance into another one;• a social network has human individuals as nodes and edges representing either friendship, acts of communication or other social interactions;• in a collaboration graph the nodes represent scientists which are connected by an edge if they have collaborated or if they have authored a joint paper;• the world-wide web consists of web pages connected by hyperlinks;• the internet has routers and computers as nodes which have physical or wireless links.The mathematical notion of networks is easy to derive. Definition 1.A network is defined as a finite set V of nodes, or vertices, together with a set E ⊂ V × V of links, or edges. The degree of a vertex v ∈ V is the number of vertices w ∈ V with (w, v) ∈ E (called the outdegree) plus the number of vertices w ∈ V with (v, w) ∈ E (called the indegree). Two vertices v, w ∈ V are connected if there exist finitely many vertices v = v 0 , v 1 , . . . , v n = w such that for every i ∈ {1, . . . , n} we have (v i−1 , v i ) ∈ E or (v i , v i−1 ) ∈ E. This defines an equivalence relation on V and therefore a partition of the set V of vertices into connected components.

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“…Results for other regimes are at present still on a conjectural level and will be discussed in the forthcoming work [DMM11]. We conjecture the existence of two further regimes:…”

Section: Small Worldsmentioning

confidence: 75%

Section: Empirical Degree Distributionsmentioning

confidence: 99%

“…Theorem 3 (Existence of a perpetual hub, see Theorem 1.5 in [DM09]). A perpetual hub exists almost surely, if and only if…”

Section: Existence Of a Perpetual Hubmentioning

confidence: 99%

Section: Small Worldsmentioning

confidence: 99%

Section: Small Worldsmentioning

confidence: 99%

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Random Networks with Concave Preferential Attachment Rule

Dereich

Mörters

2010

Jahresber. Dtsch. Math. Ver.

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A scaling limit for the degree distribution in sublinear preferential attachment schemes

Choi

Sethuraman

Venkataramani

2015

Random Struct. Alg.

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ABSTRACT:We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures.In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to 'non-tree' evolutions where cycles may develop in the network.A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C 0 -semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003).

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Fluctuations in a general preferential attachment model via Stein's method

Betken

Döring

Ortgiese³

2019

Random Struct Algorithms

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We consider a class of dynamic random graphs known as preferential attachment models, where the probability that a new vertex connects to an older vertex is proportional to a sublinear function of the indegree of the older vertex at that time. It is well known that the distribution of a uniformly chosen vertex converges to a limiting distribution. Depending on the parameters, the tail of the limiting distribution may behave like a power law or a stretched exponential. Using Stein's method we provide rates of convergence to zero of the total variation distance between the finite distribution and its limit. Our proof uses the fact that the limiting distribution is the stationary distribution of a Markov chain together with the generator method of Barbour.

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Random networks with sublinear preferential attachment: Degree evolutions

Cited by 83 publications

References 11 publications

Random Networks with Concave Preferential Attachment Rule

Random Networks with Concave Preferential Attachment Rule

A scaling limit for the degree distribution in sublinear preferential attachment schemes

Fluctuations in a general preferential attachment model via Stein's method

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