Optimisation of the coalescent hyperbolic embedding of complex networks

Kovács, Béla; Palla, Gergely

doi:10.1038/s41598-021-87333-5

Cited by 11 publications

(15 citation statements)

References 45 publications

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“…Hyperbolic networks are also closely related to network models based on simplicial complexes 32 , 33 , where the emergent geometry of the generated random graphs was shown to be hyperbolic. In addition, significant achievements were obtained related to the problem of hyperbolic embedding as well 13 , 19 , 23 , 34 – 36 , where the task is to find the most suitable node coordinates in a hyperbolic space given an input network topology.…”

Section: Introductionmentioning

confidence: 99%

The inherent community structure of hyperbolic networks

Kovács

Palla

2021

Sci Rep

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A remarkable approach for grasping the relevant statistical features of real networks with the help of random graphs is offered by hyperbolic models, centred around the idea of placing nodes in a low-dimensional hyperbolic space, and connecting node pairs with a probability depending on the hyperbolic distance. It is widely appreciated that these models can generate random graphs that are small-world, highly clustered and scale-free at the same time; thus, reproducing the most fundamental common features of real networks. In the present work, we focus on a less well-known property of the popularity-similarity optimisation model and the $${\mathbb {S}}^1/{\mathbb {H}}^2$$ S 1 / H 2 model from this model family, namely that the networks generated by these approaches also contain communities for a wide range of the parameters, which was certainly not an intention at the design of the models. We extracted the communities from the studied networks using well-established community finding methods such as Louvain, Infomap and label propagation. The observed high modularity values indicate that the community structure can become very pronounced under certain conditions. In addition, the modules found by the different algorithms show good consistency, implying that these are indeed relevant and apparent structural units. Since the appearance of communities is rather common in networks representing real systems as well, this feature of hyperbolic models makes them even more suitable for describing real networks than thought before.

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Section: Introductionmentioning

confidence: 99%

The inherent community structure of hyperbolic networks

Kovács

Palla

2021

Sci Rep

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“…Examples include the E-PSO model that inherently accounts for the creation of internal links, i.e. connections emerging between old nodes in the network 22 , or alternatively, the deletion of already existing links 28 . Or, the nonuniform popularity-similarity optimisation (nPSO) model 24 , 29 that allows the generation of networks with an adjustable community structure by assuming a heterogeneous angular node distribution with multiple peaks is also worth mentioning.…”

Section: Introductionmentioning

confidence: 99%

“…In parallel with the developments of hyperbolic network models, another closely related field given by hyperbolic embedding techniques has also received great attention 22 , 27 , 28 , 35 – 38 . Briefly, this tackles the problem of inferring the most plausible coordinates for the network nodes based on the topology of a given network.…”

Section: Introductionmentioning

confidence: 99%

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Generalised popularity-similarity optimisation model for growing hyperbolic networks beyond two dimensions

Kovács

Balogh

Palla

2022

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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 popularity-similarity 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 $$d>2$$ 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.

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“…In parallel with the developments of hyperbolic network models, another closely related field given by hyperbolic embedding techniques has also received great attention [22,27,28,[35][36][37][38]. Briefly, this tackles the problem of inferring the most plausible coordinates for the network nodes based on the topology of a given network.…”

Section: Introductionmentioning

confidence: 99%

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

Kovács,

Balogh,

Palla

2021

Preprint

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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

Optimisation of the coalescent hyperbolic embedding of complex networks

Cited by 11 publications

References 45 publications

The inherent community structure of hyperbolic networks

The inherent community structure of hyperbolic networks

Generalised popularity-similarity optimisation model for growing hyperbolic networks beyond two dimensions

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

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