Abstract
High dimensional embeddings of graph data into hyperbolic space have recently been shown to have great value in encoding hierarchical structures, especially in the area of natural language processing, named entity recognition, and machine generation of ontologies. Given the striking success of these approaches, we extend the famous hyperbolic geometric random graph models of Krioukov et al. to arbitrary dimension, providing a detailed analysis of the degree distribution behavior of the model in an expanded portion of the parameter space, considering several regimes which have yet to be considered. Our analysis includes a study of the asymptotic correlations of degree in the network, revealing a non-trivial dependence on the dimension and power law exponent. These results pave the way to using hyperbolic geometric random graph models in high dimensional contexts, which may provide a new window into the internal states of network nodes, manifested only by their external interconnectivity.
Highlights
Many complex networks, which arise from extremely diverse areas of study, surprisingly share a number of common properties
Perhaps the most well known is the Barabási–Albert model of preferential attachment [5]. This model is described in terms of a growing network, in which nodes of high degree are more likely to attract connections from newly born nodes. It is effective in capturing the power law degree sequence, but fails to generate graphs in which connected nodes tend to have a large fraction of their neighbors in common
It is reasonable to expect that, in order to capture the behavior of real world complex networks, it will be necessary to allow for similarity spaces of larger than one dimension
Summary
Many complex networks, which arise from extremely diverse areas of study, surprisingly share a number of common properties. We construct a model of random geometric graphs in a ball of Hd+1 , in direct analogy to the d = 1 construction of [7] In this model, the radial coordinate retains its effective meaning as a measure of the popularity of a node, while the similarity space becomes a full d-sphere, allowing for a much richer characterization of the interests of a person in a social network, or the role of an entity in some complex organizational structure. We generalize this result to arbitrary dimension, finding that the asymptotic independence of degree requires a steeper fall off for the degree distribution (governed by the parameter σ) at larger dimension d This dimensional dependence is reasonable, in the sense that, in higher dimensions, there are more directions in which nodes can interact. We do not know for certain whether the particular ranges of parameters that we explore in this paper generate δ-hyperbolic graphs for δ > 0, almost surely, but imagine it likely to be the case
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