High Dimensional Hyperbolic Geometry of Complex Networks

Yang, Weihua; Rideout, David

doi:10.3390/math8111861

Cited by 16 publications

(20 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Consider a set of N nodes, V = {x i } i=1,...,N ⊂ H D+1 , where each x i is a continuous random variable on H D+1 . A natural choice to study the effect of hyperbolic geometry on the graph is to sample uniformly in a subset of H D+1 [16,32]. The connection probability…”

Section: Hyperbolic Network Modelmentioning

confidence: 99%

“…II, increasing the dimension of hyperbolic random graphs boils down to considering more angular coordinates that maps to the sphere or higher dimensional D-spheres, but how does this impact the graph structure? Some properties of the graphs are almost unchanged, like the degree distribution [16], while some others, like the short cycle structure [17], are affected significantly.…”

Section: Effects Of Dimensionalitymentioning

confidence: 99%

“…The model has also been ex- * beatrice.desy.1@ulaval.ca tended to weighted [8], growing [5,9], bipartite [10], multilayer [11,12] or modular networks [13][14][15]. Now that hyperbolic networks of the lowest dimension have been shown to capture so many realistic properties, some attention has shifted to the study of higherdimensional models [16][17][18]. In these, there is still one radial coordinate for popularity, but there are D > 1 dimensions encoding similarity, or perhaps, similarities.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dimension matters when modeling network communities in hyperbolic spaces

Béatrice¹,

Desrosiers²,

Allard³

2022

Preprint

View full text Add to dashboard Cite

Over the last decade, random hyperbolic graphs have proved successful in providing geometric explanations to many key properties of real-world networks, including strong clustering, high navigability, and heterogeneous degree distributions. Although a few studies have shown that hyperbolic models can generate community structures, another salient feature observed in real networks, we argue that the current models are overlooking the choice of the latent space dimensionality that is required to adequately represent data with communities. We show that there is an important qualitative difference between the lowest-dimensional model and its higher-dimensional counterparts with respect to how similarity between nodes restricts connection probabilities. Since more dimensions also increases the number of nearest neighbors for angular clusters representing communities, considering only one more dimension allows us to generate more realistic and diverse community structures.

show abstract

Section: Hyperbolic Network Modelmentioning

confidence: 99%

Section: Effects Of Dimensionalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dimension matters when modeling network communities in hyperbolic spaces

Béatrice¹,

Desrosiers²,

Allard³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Along this line, the RHG model has recently been extended to d > 2 dimensions [41,42], placing the nodes in a d-dimensional hyperbolic ball. However, the extension of the PSO model to higher dimensions is still missing.…”

Section: Introductionmentioning

confidence: 99%

Growing hyperbolic networks beyond two dimensions: the generalised popularity-similarity optimisation model

Kovács,

Balogh,

Palla

2021

Preprint

View full text Add to dashboard Cite

Hyperbolic network models have gained considerable attention in recent years, mainly due to their capability of explaining many peculiar features of real-world networks. One of the most widely known models of this type is the popularitysimilarity optimisation (PSO) model, working in the native disk representation of the two-dimensional hyperbolic space and generating networks with small-world property, scale-free degree distribution, high clustering and strong community structure at the same time. With the motivation of better understanding hyperbolic random graphs, we hereby introduce the dPSO model, a generalisation of the PSO model to any arbitrary integer dimension d > 2. The analysis of the obtained networks shows that their major structural properties can be affected by the dimension of the underlying hyperbolic space in a non-trivial way. Our extended framework is not only interesting from a theoretical point of view but can also serve as a starting point for the generalisation of already existing two-dimensional hyperbolic embedding techniques.

show abstract

“…Grasping the above properties all at once with a simple network model is a challenging task for which hyperbolic approaches offer an intuitive framework. The basic idea of hyperbolic network models is to place the nodes in the hyperbolic space and connect them with a probability decaying as the function of the hyperbolic distance [14][15][16][17][18][19][20][21][22] . Remarkably, the networks generated in this way are usually small-world, highly clustered and scale-free 14,15 , and according to recent results they can easily display a strong community structure as well 18,20,[23][24][25][26][27] .…”

mentioning

confidence: 99%

Model-independent embedding of directed networks into Euclidean and hyperbolic spaces

Kovács

Palla

2023

Commun Phys

View full text Add to dashboard Cite

The arrangement of network nodes in hyperbolic spaces has become a widely studied problem, motivated by numerous results suggesting the existence of hidden metric spaces behind the structure of complex networks. Although several methods have already been developed for the hyperbolic embedding of undirected networks, approaches able to deal with directed networks are still in their infancy. Here, we present a framework based on the dimension reduction of proximity matrices reflecting the network topology, coupled with a general conversion method transforming Euclidean node coordinates into hyperbolic ones even for directed networks. While proposing a measure of proximity based on the shortest path length, we also incorporate an earlier Euclidean embedding method in our pipeline, demonstrating the widespread applicability of our Euclidean-hyperbolic conversion. Besides, we introduce a dimension reduction technique that maps the nodes directly into the hyperbolic space of any number of dimensions with the aim of reproducing a distance matrix measured on the given (un)directed network. According to various commonly used quality scores, our methods are capable of producing high-quality embeddings for several real networks.

show abstract

High Dimensional Hyperbolic Geometry of Complex Networks

Cited by 16 publications

References 20 publications

Dimension matters when modeling network communities in hyperbolic spaces

Dimension matters when modeling network communities in hyperbolic spaces

Growing hyperbolic networks beyond two dimensions: the generalised popularity-similarity optimisation model

Model-independent embedding of directed networks into Euclidean and hyperbolic spaces

Contact Info

Product

Resources

About