Mercator: uncovering faithful hyperbolic embeddings of complex networks

García-Pérez, Guillermo; Allard, Antoine; Serrano, M. Ángeles; Boguñá, Marián

doi:10.1088/1367-2630/ab57d2

Cited by 74 publications

(123 citation statements)

References 35 publications

(122 reference statements)

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“…where constant K absorbs all terms independent of {x i }. Similar to other maximum-likelihood estimation (MLE) based embedders [34,35,43,48], node coordinatesx i are computed iteratively: starting with initial random coordinate configuration, the HYPERLINK embedder updates node coordinates at each iteration step to increase ln L({x i }|a i j ) and stops when we arrive to a stable configuration. One feature of the HYPERLINK embedder which is different from other MLE-based embedders is that at each iteration step the embedder adds synthetic noise of variable magnitude a( ) to angular node coordinates:…”

Section: Hyperlink Embedder In a Nutshellmentioning

confidence: 99%

Link prediction with hyperbolic geometry

Kitsak

Voitalov

Krioukov

2020

Phys. Rev. Research

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Link prediction is a paradigmatic problem in network science with a variety of applications. In latent space network models this problem boils down to ranking pairs of nodes in the order of increasing latent distances between them. The network model with hyperbolic latent spaces has a number of attractive properties suggesting it must be a powerful tool to predict links, but the past work in this direction reported mixed results. Here we perform a systematic investigation of the utility of latent hyperbolic geometry for link prediction in networks. We first show that some measures of link prediction accuracy are extremely sensitive with respect to inaccuracies in the inference of latent hyperbolic coordinates of nodes. This observation leads us to the development of a hyperbolic network embedding method, the HYPERLINK embedder, which we show maximizes the accuracy of such inference, compared to existing hyperbolic embedding methods. Applying this method to synthetic and real networks, we then find that when it comes to predicting obvious missing links hyperbolic link prediction-for short, HYPERLINK-is rarely the best but often competitive, compared to a multitude of other methods. However, HYPERLINK appears to be at its best, maximizing its competitive power, when the task is to predict less obvious missing links that are really hard to predict. These links include missing links in incomplete networks with large fractions of missing links, missing links between nodes that do not have any common neighbors, and missing links between dissimilar nodes at large latent distances. Overall these results suggest that the harder a specific link prediction task the more seriously one should consider using hyperbolic geometry.

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Section: Hyperlink Embedder In a Nutshellmentioning

confidence: 99%

Link prediction with hyperbolic geometry

Kitsak

Voitalov

Krioukov

2020

Phys. Rev. Research

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“…Notice that the distribution of angular positions of nodes in the S 1 model is assumed to be homogeneous, while typically real network embeddings show that nodes form geometric communities in similarity space [20,[34][35][36]. The geometric embeddings of real networks used in this work were computed using the mapping tool Mercator [21].…”

Section: Hierarchy Load Of Links and Nodesmentioning

confidence: 99%

“…All real complex networks used in this paper have been mapped into their hyperbolic latent geometry using the embedding method Mercator [21]. This method mixes machine learning and maximum likelihood approaches to infer the coordinates of the nodes in the underlying hyperbolic disk, while ensuring best congruency between the real network topology and the S 1 geometric model.…”

Section: A Empirical Datamentioning

confidence: 99%

“…Here, we integrate degree rank, or popularity, and similarity between nodes in an enriched interpretation of hierarchy, valid for real networks. For this purpose, we capture network architecture using geometric network maps [20,21] and models [19,22], which naturally encode popularity and similarity attributes of nodes as coordinates in an underlying metric space. Exploiting the geometric approach, we are able to characterize the individual contribution of each node and each link to the hierarchical structure of a network.…”

Section: Introductionmentioning

confidence: 99%

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Geometric detection of hierarchical backbones in real networks

Ortiz

García-Pérez

Serrano

2020

Phys. Rev. Research

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Hierarchies permeate the structure of real networks, whose nodes can be ranked according to different features. However, networks are far from treelike structures and the detection of hierarchical ordering remains a challenge, hindered by the small-world property and the presence of a large number of cycles, in particular clustering. Here, we use geometric representations of undirected networks to achieve an enriched interpretation of hierarchy that integrates features defining the popularity of nodes and similarity between them, such that the more similar a node is to a less popular neighbor the higher the hierarchical load of the relationship. The geometric approach allows us to measure the local contribution of nodes and links to the hierarchy within a unified framework. Additionally, we propose a link filtering method, the similarity filter, able to extract hierarchical backbones containing the links that represent statistically significant deviations with respect to the maximum entropy null model for geometric heterogeneous networks. We applied our geometric approach to the detection of similarity backbones of real networks in different domains and found that the backbones preserve local topological features at all scales. Interestingly, we also found that similarity backbones favor cooperation in evolutionary dynamics modeling social dilemmas.

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“…Given the ability of the S 1 /H 2 model to construct synthetic networks that resemble real networks, several methods have been developed to map real networks into the hyperbolic plane, i.e., to infer the nodes' latent coordinates r (or κ ) and θ , according to the model 23,[26][27][28][29][30] . The hyperbolic maps produced by these methods have been shown to be meaningful, and have been efficiently used in applications such as community detection, greedy routing and link prediction [26][27][28][29][30][31][32][33][34][35] .…”

mentioning

confidence: 99%

Hyperbolic mapping of human proximity networks

Rodríguez-Flores

Papadopoulos

2020

Sci Rep

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Human proximity networks are temporal networks representing the close-range proximity among humans in a physical space. They have been extensively studied in the past 15 years as they are critical for understanding the spreading of diseases and information among humans. Here we address the problem of mapping human proximity networks into hyperbolic spaces. Each snapshot of these networks is often very sparse, consisting of a small number of interacting (i.e., non-zero degree) nodes. Yet, we show that the time-aggregated representation of such systems over sufficiently large periods can be meaningfully embedded into the hyperbolic space, using methods developed for traditional (non-mobile) complex networks. We justify this compatibility theoretically and validate it experimentally. We produce hyperbolic maps of six different real systems, and show that the maps can be used to identify communities, facilitate efficient greedy routing on the temporal network, and predict future links with significant precision. Further, we show that epidemic arrival times are positively correlated with the hyperbolic distance from the infection sources in the maps. Thus, hyperbolic embedding could also provide a new perspective for understanding and predicting the behavior of epidemic spreading in human proximity systems.

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Mercator: uncovering faithful hyperbolic embeddings of complex networks

Cited by 74 publications

References 35 publications

Link prediction with hyperbolic geometry

Link prediction with hyperbolic geometry

Geometric detection of hierarchical backbones in real networks

Hyperbolic mapping of human proximity networks

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