Geometric preferential attachment in non-uniform metric spaces

Jordan, Jonathan

doi:10.1214/ejp.v18-2271

Cited by 18 publications

(28 citation statements)

References 15 publications

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“…We speak of spatially induced clustering. Spatial preferential attachment models were studied by Manna and Sen [22], Flaxman, Frieze and Vera [11,12], Aiello, Bonato, Cooper, Janssen and Pra lat [1], Jordan [17,18], Janssen, Pra lat, Wilson [16], Jordan and Wade [19], and Jacob and Mörters [14,15].…”

Section: Potential Features Of Network We Are Interested In Includementioning

confidence: 99%

The age-dependent random connection model

et al. 2019

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We investigate a class of growing graphs embedded into the d-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative ages. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.MSc classification (2010): Primary 05C80 Secondary 60K35.

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Section: Potential Features Of Network We Are Interested In Includementioning

confidence: 99%

The age-dependent random connection model

et al. 2019

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“…The following is proved in [6], by appealing to the theory of stochastic approximation. For convenience we give essentially the same proof, using our notation, in Appendix A.…”

Section: B Empirical Degree Distribution For Large Tmentioning

confidence: 99%

“…Still, the convergence of the empirical distribution of the induced labels of half edges (see Proposition 2 below) makes the analysis tractable without the exact formulas. A sequence of models evolved from preferential attachment with fitness [7], towards the case examined in [6], such that the attachment probability is weighted by a factor depending on the labels of both the new vertex and a potential target vertex. The model of [16] is a special case, for which attachment is possible if the labels are sufficiently close.…”

Section: Introductionmentioning

confidence: 99%

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Community Recovery in a Preferential Attachment Graph

Hajek

Sankagiri

2019

IEEE Trans. Inform. Theory

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A message passing algorithm is derived for recovering communities within a graph generated by a variation of the Barabási-Albert preferential attachment model. The estimator is assumed to know the arrival times, or order of attachment, of the vertices. The derivation of the algorithm is based on belief propagation under an independence assumption. Two precursors to the message passing algorithm are analyzed: the first is a degree thresholding (DT) algorithm and the second is an algorithm based on the arrival times of the children (C) of a given vertex, where the children of a given vertex are the vertices that attached to it. Comparison of the performance of the algorithms shows it is beneficial to know the arrival times, not just the number, of the children. The probability of correct classification of a vertex is asymptotically determined by the fraction of vertices arriving before it. Two extensions of Algorithm C are given: the first is based on joint likelihood of the children of a fixed set of vertices; it can sometimes be used to seed the message passing algorithm. The second is the message passing algorithm. Simulation results are given. 1

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“…Thus, nearby nodes are likely to be connected, thereby inducing clustering [31]. In this sense, spatial preferential attachment models (S-PAMs) combine the virtues of PA models and classical geometric random graphs [38], which exhibit strong local clustering but are not scale-free.The literature offers a variety of definitions and results for spatial PA models [1,11,12,23,24,[29][30][31][32][33][34][35]. In the present paper, we investigate typical distances and large deviation principles (LDPs) in the model introduced in [29], as illustrated in Figure 1.…”

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Distances and large deviations in the spatial preferential attachment model

Hirsch¹,

Mönch²

2020

Bernoulli

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We investigate two asymptotic properties of a spatial preferential-attachment model introduced by E. Jacob and P. Mörters [29]. First, in a regime of strong linear reinforcement, we show that typical distances are at most of doubly-logarithmic order. Second, we derive a large deviation principle for the empirical neighbourhood structure and express the rate function as solution to an entropy minimisation problem in the space of stationary marked point processes.An elegant approach to incorporate clustering into PA models is to use geometry. We embed the network nodes into Euclidean or hyperbolic space and make the PA mechanism aware of spatial distances. Thus, nearby nodes are likely to be connected, thereby inducing clustering [31]. In this sense, spatial preferential attachment models (S-PAMs) combine the virtues of PA models and classical geometric random graphs [38], which exhibit strong local clustering but are not scale-free.The literature offers a variety of definitions and results for spatial PA models [1,11,12,23,24,[29][30][31][32][33][34][35]. In the present paper, we investigate typical distances and large deviation principles (LDPs) in the model introduced in [29], as illustrated in Figure 1. Our methods are fairly robust and we believe that in essence our analysis could be transferred to many of the other architectures mentioned above.Firstly, we verify an intriguing conjecture in [32, Remark 3] about typical distances in the largest connected component of the S-PAM. As made precise in Theorem 2.1 below, the asymptotic behaviour depends on both the strength of the preferential attachment as well as on the influence of vertex distances on the connection probability. This is in contrast to the situation for the asymptotic degree distribution. Indeed, here only the strength of the preferential attachment but not the geometry affects the power-law exponent [31, Remark 1].There are a number of small-world results for complex networks endowed with a geometric structure already available [13,24,28,36,39]. Our work complements these earlier investigations naturally by providing a distance result in a spatial model with preferential attachment in the ultra-small regime of doubly-logarithmic typical distances. Our proof shows that, in this regime, the geometry of the underlying space has an influence on the length of shortest paths in the graph. However, the geometry does not change their fundamental architecture as it can already be observed in non-geometric PA models [7,15,19]. Secondly, we establish an LDP for the in-degree evolution and the evolution of the neighbourhood structure of a typical vertex. The most fundamental building blocks for this result are the process-level LDP of a marked Poisson point process [26] and the contraction principle. The rate function is expressed via a constraint minimisation problem of the specific relative entropy. Loosely speaking, large deviations from the neighbourhood structure are induced by stationary modifications of the original Poisson point process of vertices. ...

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Geometric preferential attachment in non-uniform metric spaces

Cited by 18 publications

References 15 publications

The age-dependent random connection model

The age-dependent random connection model

Community Recovery in a Preferential Attachment Graph

Distances and large deviations in the spatial preferential attachment model

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