Geometric Preferential Attachment Research Articles

Phase Transitions for Random Geometric Preferential Attachment Graphs

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.

Phase Transitions for Random Geometric Preferential Attachment Graphs

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.

Emergence of soft communities from geometric preferential attachment.

All real networks are different, but many have some structural properties in common. There seems to be no consensus on what the most common properties are, but scale-free degree distributions, strong clustering, and community structure are frequently mentioned without question. Surprisingly, there exists no simple generative mechanism explaining all the three properties at once in growing networks. Here we show how latent network geometry coupled with preferential attachment of nodes to this geometry fills this gap. We call this mechanism geometric preferential attachment (GPA), and validate it against the Internet. GPA gives rise to soft communities that provide a different perspective on the community structure in networks. The connections between GPA and cosmological models, including inflation, are also discussed.

Geometric preferential attachment in non-uniform metric spaces

We investigate the degree sequences of geometric preferential attachment graphs in general compact metric spaces. We show that, under certain conditions on the attractiveness function, the behaviour of the degree sequence is similar to that of the preferential attachment with multiplicative fitness models investigated by Borgs et al. When the metric space is finite, the degree distribution at each point of the space converges to a degree distribution which is an asymptotic power law whose index depends on the chosen point. For infinite metric spaces, we can show that for vertices in a Borel subset of S of positive measure the degree distribution converges to a distribution whose tail is close to that of a power law whose index again depends on the set.

Degree sequences of geometric preferential attachment graphs

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.

Degree sequences of geometric preferential attachment graphs

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.

A Geometric Preferential Attachment Model of Networks II

We study a random graph _G__<em>n</em> that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power law degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of _G__<em>n</em> are _n_ sequentially generated points, _x_₁, _x_₂, . . . , _x__<em>n</em>, chosen uniformly at random from the unit sphere in R³ After generating _x__<em>t</em>, we randomly connect it to _m_ points from those points _x_₁, _x_₂, . . . , _x__{<em>t</em>−1}.

A Geometric Preferential Attachment Model of Networks

We study a random graph _G_n_ that combines certain aspects of geometric random graphs and preferential attachment graphs. The vertices of _G_n_ are _n_ sequentially generated points _x_₁, _x_₂, . . . , _x__n chosen uniformly at random from the unit sphere in ℝ³. After generating _x_t_, we randomly connect that point to _m_ points from those points in _x_₁, _x_₂, . . . , _x_t_-1 that are within distance _r_ of _x_t_. Neighbors are chosen with probability proportional to their current degree, and a parameter a biases the choice towards self loops. We show that if _m_ is sufficiently large, if _r_ ≥ ln _n/n_^{1/2-<em>β</em>} for some constant _β_, and if α > 2, then with high probabilty (whp) at time n the number of vertices of degree _k_ follows a power law with exponent α + 1. Unlike the preferential attachment graph, this geometric preferential attachment graph has small separators, similar to experimental observations of [Blandford et al. 03]. We further show that if _m_ ≥ _K_ ln _n_, for _K_ sufficiently large, then _G_n_ is connected and has diameter _O_(ln _n/r_) whp.