Geometric preferential attachment in non-uniform metric spaces

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 e...

A Geometric Preferential Attachment Model of Networks

We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. The vertices of Gn are n sequentially generated points x1, x2, . . . , xn chosen uniformly at random from the unit sphere in R . After generating xt, we randomly connect it to m points from those points in x1, x2, . . . , xt−1 which are within distance r. Neighbors are chosen ...

Metric Spaces and Uniform Structures

The general notion of topology does not allow to compare neighborhoods of different points. Such a comparison is quite natural in various geometric contexts. The general setting for such a comparison is that of a uniform structure. The most common and natural way for a uniform structure to appear is via a metric, which was already mentioned on several occasions in Chapter 1, so we will postpone...

Geometric embeddings of metric spaces

Uniform Convergence and Uniform Continuity in Generalized Metric Spaces

In the paper [9] we introduced the class of generalized metric spaces. These spaces simultaneously generalize ‘standard’ metric spaces, probabilistic metric spaces and fuzzy metric spaces. We show that every generalized metric space is, naturally, a uniform space. Thus we can use standard topological techniques to study, for example, probabilistic metric spaces. We illustrate this by proving a ...