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Fig. 1 | Applied Network Science

Fig. 1

From: Manifold learning and maximum likelihood estimation for hyperbolic network embedding

Fig. 1

Issues with hyperbolic embeddings by LaBNE. a An artificial network produced by the PS model, with 500 nodes, 2m=10, γ=2.5 and T=0.3, is mapped to hyperbolic space with LaBNE and HyperMap. Even when LaBNE’s embedding is extremely fast and its inferred node positions, indicated with colours, coincide with those from the input network, panel b shows that HyperMap is more accurate, because the probability of finding connected nodes at small hyperbolic distances is higher than if LaBNE’s coordinates are used. We illustrate the reasons for this with the simple 5-node network of panel c and its embedding by LaBNE. Despite the fact that the inferred and real angular coordinates, together with the corresponding hyperbolic distances, are highly correlated (Pearson correlations of 0.97 and 0.98, respectively), it is clear that the angular positions of nodes 2, 3 and 4 are smaller than they should be. This is a consequence of LaBNE’s aim to map connected nodes as close as possible in the embedding space, disregarding that disconnected nodes should be far from each other. Although this does not have a big impact in Euclidean embeddings, it can be very problematic in hyperbolic space. Panel d shows how the hyperbolic distance between nodes 1 and 2 changes dramatically, even for small changes in the angular coordinate of the latter

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