Robust Analysis of Preferential Attachment Models with Fitness

Dereich, Steffen; Ortgiese, Marcel

doi:10.1017/s0963548314000157

Cited by 34 publications

(44 citation statements)

References 25 publications

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“…we have (6) and clearly E( n+1 |  n ) = 0. ▪ Before continuing, we summarize the necessary results on one-dimensional stochastic approximations; recall the definitions of stable zeros, unstable zeros, and touchpoints from Section 2.2.…”

Section: Proofs Of Theorems 22 To 24 For Deterministic Choice Rulesmentioning

confidence: 97%

“…A key technique we use in our proofs is that of of stochastic approximation algorithms, originally developed by Robbins and Monro . Stochastic approximation methods appear naturally in preferential attachment models, and have been used, for example, by Malyshkin and Paquette and Dereich and Ortgiese . Stochastic approximation processes operate in discrete time with standard notation, based on Pemantle ,

X_{n + 1} - X_{n} = γ_{n} false(F false(X_{n} false) + ξ_{n + 1} + R_{n} false),

where { X n , n ≥ 1} is a sequence of random variables on

{double-struckR}^{d}

, γ n are step sizes satisfying

\sum_{n = 1}^{\infty} γ_{n} = \infty

and

\sum_{n = 1}^{\infty} γ_{n}^{2} < \infty

, F is a function from

{double-struckR}^{d}

to itself, R n are remainder terms which must tend to zero and satisfy

\sum_{n = 1}^{\infty} n^{- 1} false| R_{n} false| < \infty

, and ξ n + 1 are noise terms satisfying E( ξ n + 1 | n ) = 0.…”

Section: Proofsmentioning

confidence: 99%

“…A key technique we use in our proofs is that of of stochastic approximation algorithms, originally developed by Robbins and Monro [15]. Stochastic approximation methods appear naturally in preferential attachment models, and have been used, for example, by Malyshkin and Paquette [11] and Dereich and Ortgiese [6]. Stochastic approximation processes operate in discrete time with standard notation, based on Pemantle [14],…”

Section: Proofsmentioning

confidence: 99%

“…As a consequence, new, fitter, vertices can still compete against vertices which are well-established in the existing network. A particularly interesting feature of preferential attachment with fitness is the so-called condensation phenomena, where at time n a single vertex or a set of vertices of size o(n) (which vertices can depend on time) can have a total degree of order n. Condensation for preferential attachment with fitness is studied in detail by Borgs et al [4], Dereich and Ortgiese [6], and Dereich, Mailler, and Mörters [5].…”

mentioning

confidence: 99%

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Condensation in preferential attachment models with location‐based choice

Haslegrave

Jordan

Yarrow

2019

Random Struct Algorithms

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We introduce a model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex has an associated uniform random variable which we call its location. Our model evolves in discrete time by selecting r vertices from the graph with replacement, with probabilities proportional to their degrees plus a constant α. A new vertex joins the network and attaches to one of these vertices according to a given probability associated to the ranking of their locations. We give conditions for the occurrence of condensation, showing the existence of phase transitions in α below which condensation occurs. The condensation in our model differs from that in preferential attachment models with fitness in that the condensation can occur at a random location, that it can be due to a persistent hub, and that there can be more than one point of condensation.

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Section: Proofs Of Theorems 22 To 24 For Deterministic Choice Rulesmentioning

confidence: 97%

X_{n + 1} - X_{n} = γ_{n} false(F false(X_{n} false) + ξ_{n + 1} + R_{n} false),

where { X n , n ≥ 1} is a sequence of random variables on

{double-struckR}^{d}

, γ n are step sizes satisfying

\sum_{n = 1}^{\infty} γ_{n} = \infty

and

\sum_{n = 1}^{\infty} γ_{n}^{2} < \infty

, F is a function from

{double-struckR}^{d}

to itself, R n are remainder terms which must tend to zero and satisfy

\sum_{n = 1}^{\infty} n^{- 1} false| R_{n} false| < \infty

, and ξ n + 1 are noise terms satisfying E( ξ n + 1 | n ) = 0.…”

Section: Proofsmentioning

confidence: 99%

Section: Proofsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Condensation in preferential attachment models with location‐based choice

Haslegrave

Jordan

Yarrow

2019

Random Struct Algorithms

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show abstract

“…Borgs et al [6] and Dereich [9], [10] prove results on stationary CTBPs with fitness. In these works, the authors investigate models with affine dependence on the degree and bounded fitness distributions.…”

Section: 3mentioning

confidence: 99%

The Dynamics of Power laws: Fitness and Aging in Preferential Attachment Trees

2017

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Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment models in which the birth rates are linear functions. We are motivated by citation networks, where power-law citation counts are observed as well as aging in the citation patterns. To model this, we introduce fitness and age-dependence in these birth processes. The multiplicative fitness moderates the rate at which children are born, while the aging is integrable, so that individuals receives a finite number of children in their lifetime. We show the existence of a limiting degree distribution for such processes. In the preferential attachment case, where fitness and aging are absent, this limiting degree distribution is known to have power-law tails. We show that the limiting degree distribution has exponential tails for bounded fitnesses in the presence of integrable aging, while the power-law tail is restored when integrable aging is combined with fitness with unbounded support with at most exponential tails. In the absence of integrable aging, such processes are explosive.

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Diameter of P.A. random graphs with edge‐step functions

Alves

Ribeiro²,

Sanchis

2020

Random Struct Algorithms

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In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between existing vertices. Specifically, at each time step t, either a new vertex is added with probability f (t), or an edge is added between two existing vertices with probability 1 − f (t). We establish concentration inequalities for the global clustering and clique number of the resulting graphs under the assumption that f (t) is a regularly varying function at infinity with index of regular variation −γ, where γ ∈ [0, 1). We also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient.

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Robust Analysis of Preferential Attachment Models with Fitness

Cited by 34 publications

References 25 publications

Condensation in preferential attachment models with location‐based choice

Condensation in preferential attachment models with location‐based choice

The Dynamics of Power laws: Fitness and Aging in Preferential Attachment Trees

Diameter of P.A. random graphs with edge‐step functions

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