scholarly journals Phase Transitions for Random Geometric Preferential Attachment Graphs

2015 ◽  
Vol 47 (2) ◽  
pp. 565-588 ◽  
Author(s):  
Jonathan Jordan ◽  
Andrew R. Wade
Keyword(s):  
Phase Transitions ◽  
Preferential Attachment ◽  
Geometric Model ◽  
Degree Sequence ◽  
Spatial Proximity ◽  
Nearest Neighbour ◽  
Intermediate Regime ◽  
On Line ◽  
Attachment Model ◽  
Preferential Rule

Vertices arrive sequentially in space and are joined to existing vertices at random according to a preferential rule combining degree and spatial proximity. We investigate phase transitions in the resulting graph as the relative strengths of these two components of the attachment rule are varied.Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence mimics the standard Barabási-Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence mimics a purely geometric model, the on-line nearest-neighbour graph, for which we prove some extensions of known results. We also show the presence of an intermediate regime, with behaviour distinct from both the on-line nearest-neighbour graph and the Barabási-Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence.

scholarly journals Phase Transitions for Random Geometric Preferential Attachment Graphs

2015 ◽  
Vol 47 (02) ◽  
pp. 565-588
Author(s):  
Jonathan Jordan ◽  
Andrew R. Wade
Keyword(s):  
Phase Transitions ◽  
Preferential Attachment ◽  
Geometric Model ◽  
Degree Sequence ◽  
Spatial Proximity ◽  
Nearest Neighbour ◽  
Intermediate Regime ◽  
On Line ◽  
Attachment Model ◽  
Preferential Rule

Vertices arrive sequentially in space and are joined to existing vertices at random according to a preferential rule combining degree and spatial proximity. We investigate phase transitions in the resulting graph as the relative strengths of these two components of the attachment rule are varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence mimics the standard Barabási-Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence mimics a purely geometric model, the on-line nearest-neighbour graph, for which we prove some extensions of known results. We also show the presence of an intermediate regime, with behaviour distinct from both the on-line nearest-neighbour graph and the Barabási-Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence.

scholarly journals Asymptotic Degree Distribution of a Kind of Asymmetric Evolving Network

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Zhimin Li ◽  
Zhaolin He ◽  
Chunhua Hu
Keyword(s):  
Exponential Distribution ◽  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Degree Sequence ◽  
Tree Structure ◽  
Uniform Model ◽  
Attachment Model

We propose a kind of evolving network which shows tree structure. The model is a combination of preferential attachment model and uniform model. We show that the proportional degree sequencepkk>1obeys power law, exponential distribution, and other forms according to the relation ofkand parameterm.

scholarly journals Degree sequences of geometric preferential attachment graphs

2010 ◽  
Vol 42 (02) ◽  
pp. 319-330 ◽  
Author(s):  
Jonathan Jordan
Keyword(s):  
Preferential Attachment ◽  
Degree Sequence ◽  
The Self ◽  
Extended Model ◽  
Degree Sequences ◽  
Attachment Model

We investigate the degree sequence of the geometric preferential attachment model of Flaxman, Frieze and Vera (2006), (2007) in the case where the self-loop parameter α is set to 0. We show that, given certain conditions on the attractiveness function F, the degree sequence converges to the same sequence as found for standard preferential attachment in Bollobás et al. (2001). We also apply our method to the extended model introduced in van der Esker (2008) which allows for an initial attractiveness term, proving similar results.

scholarly journals Degree sequences of geometric preferential attachment graphs

2010 ◽  
Vol 42 (2) ◽  
pp. 319-330 ◽  
Author(s):  
Jonathan Jordan
Keyword(s):  
Preferential Attachment ◽  
Degree Sequence ◽  
The Self ◽  
Extended Model ◽  
Degree Sequences ◽  
Attachment Model

We investigate the degree sequence of the geometric preferential attachment model of Flaxman, Frieze and Vera (2006), (2007) in the case where the self-loop parameter α is set to 0. We show that, given certain conditions on the attractiveness function F, the degree sequence converges to the same sequence as found for standard preferential attachment in Bollobás et al. (2001). We also apply our method to the extended model introduced in van der Esker (2008) which allows for an initial attractiveness term, proving similar results.

Models of network formation

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Distribution ◽  
Network Formation ◽  
Preferential Attachment ◽  
Network Size ◽  
Large Network ◽  
Citation Networks ◽  
Airline Networks ◽  
Attachment Model ◽  
Delivery Networks

This chapter describes models of the growth or formation of networks, with a particular focus on preferential attachment models. It starts with a discussion of the classic preferential attachment model for citation networks introduced by Price, including a complete derivation of the degree distribution in the limit of large network size. Subsequent sections introduce the Barabasi-Albert model and various generalized preferential attachment models, including models with addition or removal of extra nodes or edges and models with nonlinear preferential attachment. Also discussed are node copying models and models in which networks are formed by optimization processes, such as delivery networks or airline networks.

Clustering Coefficient of a Spatial Preferential Attachment Model

2018 ◽  
Vol 98 (1) ◽  
pp. 304-307 ◽  
Author(s):  
L. N. Iskhakov ◽  
M. S. Mironov ◽  
L. A. Prokhorenkova ◽  
B. Kamiński ◽  
P. Prałat
Keyword(s):  
Preferential Attachment ◽  
Clustering Coefficient ◽  
Attachment Model

scholarly journals Nonuniform Distribution of Nodes in the Spatial Preferential Attachment Model

2015 ◽  
Vol 12 (1-2) ◽  
pp. 121-144 ◽  
Author(s):  
Jeannette Janssen ◽  
Paweł Prałat ◽  
Rory Wilson
Keyword(s):  
Preferential Attachment ◽  
Nonuniform Distribution ◽  
Attachment Model

scholarly journals Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

2013 ◽  
Vol 50 (3) ◽  
pp. 721-740 ◽  
Author(s):  
Jennie C. Hansen ◽  
Jerzy Jaworski
Keyword(s):  
Asymptotic Behaviour ◽  
Preferential Attachment ◽  
Degree Sequence ◽  
Discrete Distributions ◽  
Threshold Behaviour ◽  
Mapping Model ◽  
Random Mapping ◽  
Exact Distributions ◽  
Random Mappings ◽  
Expected Values

In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, TnD̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂1,D̂2,…,D̂n). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂1,D̂2,…,D̂n. As an application of these results, we consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

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