Bridging ML and algorithms: comparison of hyperbolic embeddings
Authors: Dorota Celińska-Kopczyńska
Eryk Kopczyński
Venue: ICLR 2026 Poster (2026)
Keywords: hyperbolic embeddings
network theory
social networks
Primary area: learning on graphs and other geometries & topologies
TL;DR: A comparison of hyperbolic embeddings obtained in independent communities. It has nothing to do with GNNs despite naming!

Abstract: Hyperbolic embeddings are well-studied in the machine learning, network theory, and algorithm communities. However, as the research proceeds independently in those communities, comparisons and even awareness seem to be currently lacking. We compare the performance (computation time) and the quality of embeddings obtained by popular approaches as of 2025, both on real-life hierarchies and networks, and simulated networks. According to our results, the algorithm by Bläsius et al (ESA 2016) is about 100 times faster than the Poincar\'e embeddings (NIPS 2017) and Lorentz embeddings (ICML 2018) by Nickel and Kiela, while achieving results of similar (or, in some cases, even better) quality.

Published as a conference paper at ICLR 2026
BRIDGING ML AND ALGORITHMS: COMPARISON OF HYPERBOLIC EMBEDDINGS
Dorota Celinska-Kopezyyiske & Eryk Kopezyński Institute of Informatics
University of Warsaw
(dot, erykk) Smimuw.edu.e
ABSTRACT
Hyperbolic embeddings are well-studied in the machine learning, network theory, and algorithm communities. However, as the research proceeds independently in those communities, comparisons and even awareness seem to be currently lacking. We compare the performance (computation time) and the quality of embeddings obtained by popular approaches as of 2025, both on real-life hicrarchies and networks, and simulated networks. According our results, the algorithm by Bläsius et al (ESA 2016) is about 100 times faster than the Poincaré embeddings (NIPS 2017) and Lorentz embeddings (ICML 2018) by Nickel and Kiela, while achieving results of similar (or, in some cases, even better) quality.
INTRODUCTION
An embedding is an instance of some mathematical structure contained within another instance, such as group that a subgroup. In general topology, embedding homeomorphism onto its image. Homeomorphisms are the isomorphisms in the category of topological spaces they are the mappings that preserve all the topological properties a given space. Given network (V, E), where set of vertices and E is the set of edges, its embedding into some geometry G is map V - G.
the
In hyperbolic geometry, all the postulates Fuclid hold, except for the parallel axiom. While parallel lines stay a constant distance in Euclidean geometry, similar lines in hyperbolic geometry diverge exponentially. Recently, the area of hyperbolic embedders for networks -that is, algorithms for embedding networks into hyperbolic geometry- has gained popularity within the Machine Learning (ML) community. Those embedders exploit properties of hyperbolic geometry, such as exponential growth, making them perfect match for visualizing and modeling hierarchical structures.
Probably the most influential paper (Nickel and Kiela, 2017) (Poincaré embeddings) shows that hyperbolic embeddings achieve impressive results compared to Euclidean and translational ones. The results have been improved even further in the follow-up (Nickel and Kiela 2018) (Lorentz embeddings) by changing the used model of hyperbolic geometry. In the ML literature, those works are recognized as some of the first studies on hyperbolic embeddings (Gu et al.) 2019). However, it is worth noting that a rich history of hyperbolic embedding research precedes these papers. Hyperbolic embeddings were initially developed in the network theory (NT) community through the Hyperbolic Random Graph model (HRG) (Krioukov el al.) 2010). The algorithmic properties of this model, including embedding techniques. have been extensively studied in the algorithmic community. Surprisingly, there limited cross-referee cross-neferencing between these research communities. For example, machine learning papers have examined rarely cite algorithmic works, and vice versa. Also, lack comparative studies that bridge those communitics.
We believe the insights in the algorithmic/NT papers could significantly benefit the ML community. In this paper, we gather and experimentally compare 14 approaches from different communities using both worldworld (38 networks, including hierarchies, 21 connectomes, and 10 other networks) and simulated data (600 two-dimensional networks).
Against this background, contributions are follows:

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Figure 1: Tessellations of the hyperbolic plane. Bitruncated order-3 heptagonal tiling on the right.
We present the first experimental comparison of hyperbolic embedders from the ML, NT, and algorithmic communities, establishing crucial connections among these research areas.
We find that an (n)) algorithm for creating hyperbolic embeddings (BFKL) (Bläsius et al. 2016) that predates (Nickel and Kiela 2017) > orders of magnitude faster while achieving results of comparable quality, or in some cases, better. Mercator embeddings (García- al.) 2019) typically achieve cetemeddatt-qquliin results while also being slow; TreeRep Sonthalia and Gilbert (2020) achieves good embedding quality on hierarchies but bad quality on networks. The recent embedder CLOVE (Balogh el al. 2025) is also worth attention, both for its quality and time performance.
While higher dimensions yield better embeddings according standard quality measures (mAP, MeanRank, greedy routing success ratio, and efficiency), this is usually an artifact of optimization. Using information criteria principles, we introduce new measure (Informa- tion Control Value, ICV). Unlike the standard measures, ICV penalizes embeddings with large radius and/or dimension, thereby enhancing the robustness of our comparisons.
2 THEORETICAL BACKGROUND
2.1 PRELIMINARIES ON HYPERBOLIC GEOMETRY
We start with the basics of hyperbolic geometry. For simplicity, will focus the hyperbolic plane H?, although the same ideas work in higher dimensions. See, c.g., the book (Cannon et al.
1997) for more thorough formal exposition, the game HyperRogue (Kopezyński et al. 2017) to
gain intuitions. Recall the Euclidean space E" is R" with distance 8,(2, y) 9 + (x y. y).
where 9+ ((11...., 2n), (y1..... yn.) Σ 24344
In modern terms, the simplest non- -nn-Euciddean geometry spherical geometry. A two-dimensional sphere of radius is S² = {x E R 94 (x, =}. The distance is measured in terms of great circle ares; a point in distance r direction (angle) & from the central point C. = (0, 0, 1) has coordinates (sin(ø) sin (r), cos(Ø) sin(r), cos(r)). The spherical distance between and y can be computed as arccos( g+ (x. y)); ;; this is straightforward when Co. and also true in general, since S+ is invariant under the isometries (i.e., rotations) of the sphere.
Gaussian curvature is a measure of the difference of surface geometry from Euclidean geometry. sphere of radius R, RS² has constant Gaussian curvature K 1/ The hyperbolic plane is the opposite of spherical geometry, that is, it has constant negative Gaussian curvature. Hyperbolic surfaces less ubiquitous, because they do not embed symmetrically into E that would essentially require R to be imaginary. However, they appear in nature when maximizing surface area is needed (e.g.. lettuce leaves), and can be embedded symmetrically in the Minkowski spacetime. The hyperbolic plane H is thus {a: E IR3
> 0, g- (I, = -1}. where y- is the Minkowski inner product g- ((11, 22, ) (y1, /2, ys)) = #1 + T22//2 - T3y3 (the coordinate 23 works like time coordinate in special relativity). This is called the Minkowski hyperboloid model; many intuitions from spherical geometry work in this model, for example, a point in distance
direction (angle) 6 from the central point Co (0. 0, 1) has coordinates p(r, Φ) (sin()) sinh(r), cos( () sinh(r), cos))))) The hyperbolic distance between x and can be computed as arcosh (x,
While the formulas of the Minkowski hyperboloid model tend to be intuitively obtainable by analogy to the sphere model, this model not applicable to visualization, since it naturally lives in Minkowski spacetime rather than the usual three-dimensional space (we use Lorentz transformations rather than
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Euclidean rotations for isometries involving the time coordinate). The most common method of visualization of the hyperbolic plane is the Poincaré disk model, first devised by Eugenio Beltrami, obtained as the stereographic projection of the Minkowski hyperboloid: p(x, v. ( = This maps the (infinite) hyperbolic plane to a disk in the Euclidean plane. Figure shows some tessellations of the hyperbolic plane in the Poincaré disk model. Each shape of the same shade in each of these tessellations is of the same size; the Poincaré disk model distorts distances that the same hyperbolic distance appears smaller when closer to the boundary of the disk.
The Poincaré disk model is called a model (rather than projection) because it is often used directly, as an alternative representation of hyperbolic geometry. Many models are used; for us, the third important model is the native polar coordinates (r, (). The formulas for converting from native polar coordinates to the hyperboloid model are given above as p(r, Φ). All models describe the same (isometric) abstract metric space, so theoretically could be equivalently used in computations, although various models differ in how robust they are to numerical precision issues (as we will see later, hyperbolic geometry exhibits exponential growth, which makes such issues very significant (Celinska Kopczyńska and Kopczynski] (024b)). All can be generalized to higher dimensions and allow interpolation between possible values of curvature K. In our experience, people new to computational hyperbolic geometry use Poincaré model because introductory materials often focus on it; however, they have then difficulties computing distances and isometries, while such computations are straightforward in the hyperboloid model due to the full symmetry and spherical analogies. We see the difference between (Nickel and Kiela 2017) and (Nickel and Kiela 2018) as an example of this. The Minkowski hyperboloid is popular as the underlying model in the visualizations of hyperbolic geometry (Phillips and Gunn 1992 Kopezyńíski et al.) (017)) due to simplicity and being a generalization of the homogeneous coordinates commonly used in computer graphics. The choice of the model may affect numerical precision (Floyd el al.) 2002 Celinska-
Celinska-Kopezynskk Kopezyńska and Kopezyński
2024b). As we will see later, native polar coordinates are commonly used for hyperbolic embeddings of social networks (Friedrich et al. 2023).
2.2 HYPERBOLIC GEOMETRY IN VISUALIZATION, NT, AND ALGORITHMIC COMMUNITIES
While popular expositions of hyperbolic geometry usually focus on the sum of angles of a triangle being less than 180 degrees, what is actually important to us is exponential growth. As can be easily seen from the formula for p(r, x). a hyperbolic circle of radius r has circumference 2n sinh(r): sinh(r) grows exponentially with r. This exponential growth, as well as the tree-like nature of the hyperbolic space, can be seen in Figure Jand has found application in the visualization of hicrarchical data, such trees in the hyperbolic plane (Lamping et al. 1995) and three-dimensionee space (Munzner 1998). Drawing a binary tree of large depth h on Euclidean paper, while keeping all the edges to be of the same length, is difficult, because we eventually run out of space to fit all 2 leaves. The hyperbolic plane, with its exponential growth, solves this issue perfectly.
hyperbolic
This leads us another application of hyperbolic geometry: the modelling of scale-free networks. Scale-free networks are commonly found in nature, technology, and social structures. They are characterized by the power law distribution of degrees (the probability that a random vertex has degree 2 d is proportional to 9) as well as the high clustering coefficient (if node a is connected to b and c, the nodes b and c are also likely to be connected). Despite this ubiquity, it is not straightforward to find a mathematical model that exhibits both these properties. One such model is the Hyperbolic Random Graph model (HRG) (Krioukov el al. 2010)). characterized by parameters N. R, ax, T. In this model, N nodes {l,.... NJ are distributed randomly in hyperbolic disk of radius R. Their angular coordinates é are distributed uniformly, while their radial coordinates r are distributed according to the density function fr) sinh (ar) //Coosh(aR 1). Let denote with m(i) E H2 the position of node i. Every pair of nodes a and b is then connected with probability
(1) where 5(a, b) is the hyperbolic distance between the points in 2 representing the two nodes. The radial coordinates correspond to popularity (smaller r = more popular) while the angular coordinates correspond to similarity (closer á = more similar); the connections in network are based on popularity and similarity. It can be shown that a random graph thus obtained has a high clustering coefficient and degree distribution that follows a power law with 3 2a + 1. There is extensive literature on the HRG model, including its algorithmic properties. Hyperbolic random graphs can be generated naively in Onn2) (Aldecoa al. 2015). in subquadratic time (von Looz al. 2015).
p(a, b) (1 + exp((((m(()) m(b) - R))/2T))
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and in linear time (Bringmann al. 2 Earlier works include (Kleinberg 2007) and (Shavitt and Tankel 2008). Despite being relatively popular. HRG is not the only generative model based on similarity poppllary mechanism. Other approaches with similar properties include the earlier S! model (Serrano et al.) 2008) allowing arbitrary degree distributions, GIRG (Bringmann et al.) (2018) in which the similarity space is a torus of some dimension d, generalized PSO (Papadopoulos et al.)
2012) in which additional external edges are added as the network grows, E-PSO (Papadopoulos et all [0155b, which uses only the external edges, and nPSO (Muscoloni and Cannistraci 2018). which models realistic networks with communities using non-uniform angular distribution.
With the theoretical generative models came the embeddings of real scale-free networks into the hyperbolic plane. An embedding of a network (V, E') into geometry G is a mapping m. V + G. In (Boguñá et al. 2010), such an embedding of the Internet was obtained and found to be highly appropriate for greedy routing. In greedy routing, node wants to find a connection to another node b by finding one of its neighbors e which is the closest to b, then the neighbor of c which is closest to b, and Greedy routing is successful when we eventually reach b; in the original variant, it fails immediately when all the neighbors of e are further away from than e; in the modified variant, such hops are allowed, and the method fails when reach cycle.
However, the embedding method used by Boguñá et al. (2010) required substantial manual inter- vention and did not scale to large networks (Krioukov al.) 2010). Further research focused on finding unsupervised and efficient algorithms. An embedder is an algorithm that finds an embedding. While technically, any mapping is an embedding, we generally want the geometric structure of m to be consistent with the structure of the network. MLE embedders, based on the maximum likelihood estimation (MLE) method from statistics, work by finding an embedding that maximizes the loglikelihood (LL). LL is the logarithm of the probability that if, for every pair of nodes (a, b), we independently connect the nodes a and with the probability computed according to the formula [ (for some R and 7'.. Alternatively, spring embedders (Kobourov) (013) simulate forces acting on the graph: attractive forces pulling connected nodes together, and repulsive forces pushing unconnected nodes away. Spring embedders have been adapted to non- on-Fuulldden embeddings (Kobourov) 2013): however, the straightforward adaptation to hyperbolic geometry does not produce good embeddings of large radius (Bläsius et al. 2016).
Note that embedding is a difficult computational problem - even computing LL according to the for- mula requires time Onn?), which is significant for large networks. The first algorithm for embedding large networks, HyperMap. worked in time ((n²) (Papadopoulos et al. (2015b). later improved to O(n²) in HyperMapCN (Papadopoulos et al. 2015a) and Wang el af. (2016)). Bläsius et al. (016)) developed a quasilinear algorithm for finding hyperbolic embeddings. This a algorithm computes the HRG parameters based on the network's ' statistics. Then, it embeds the network in layers, starting from the nodes with the greatest degree, which form the center of the network. The algorithm, which we call the BFLL embedder, is evaluated on several scale-free networks from the SNAP database (Leskovec and Krevl (014) as well as randomly generated networks generated according to the HRG model. Eventually, Wang et al. (2016b) introduced a simple (nn) algorithm based on hierarchical community detection BBondel el al. (2008), ordering the communities based on the Community Intimacy between pairs of communities, and basing the angular coordinates on this order and the radial coordinates on the degree. HMCS (Wang el al. 2019) uses similar approach, but changing a few details to obtain higher-quality embeddings.
Time complexity is just one facet of the quality assessment. We need some measures of the goodness- of-fit of an embedder. Embedders specialized to solve specific tasks popularized different measures. greedy routing performance is now commonly assessed using the stretch factor (GSF), greedy success rate (GSR), and greedy routing efficiency (GRE). The stretch factor (GSF) is the average ratio of the number of steps to the minimum possible (for successful paths). Greedy success rate (GSR) yields the share of the successful routings. Boguñã el al. (2010) showed that using greedy routing with the distances from the hyperbolic embedding achieves (GSR) 90%, which is significantly higher than, e.g., greedy routing based on actual geographical distances between the network nodes; Blasius et al. (016) found that greedy routing based on the BFKL embeddings again achieves good GSR. In Bläsiis et al. 2018). the impact of numerical errors on the quality of hyperbolic embeddings and greedy routing is evaluated. Greedy Routing Efficiency (GRE) (Muscoloni et al. 2017) is the average of z/y over all pairs of nodes, where is the number of steps used by greedy routing and x the minimum possible; contrary to GSF, for failed routing we assume this ratio to be 0 (therefore, failed routings no longer can contribute positively to the measure). LL also became
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quality measure- good LL is achieved when connected nodes are placed close (distance less than R) and disconnected nodes are far away (distance greater than R). For tasks focused on angular positions (e.g.. problems related to the similarity space, such community detection), other measures, such as C-Score (Muscoloni et al. 2017). became a standard. Other methods include mapping accuracy /Zhan et al. 2021) and geometric congruence Cannistracci and Muscoloni (2022).
The official implementation of Bläsius et al. (2016) includes a spring embedder as a method of improving the result of the quasilinear algorithm; however, the running time of this step is ((n²), which is too slow for large graphs. In (Celinska-Kopezynskk and Kopezyńskil 2022), an alternative approach based on hyperbolic tilings, as shown in Figure I and previously used in HyperRogue (Kopozyński et al.) (017), was introduced. The nodes of our graph are mapped not to points of the hyperbolic plane, but rather to the tiles of such a tiling. Also, the distances are computed in a discrete way, as the number of tiles. This is called DHRG, the discrete HRG model. This works, because such tilings' distances are good approximation of hyperbolic distances (to a greater extent than similar approximations in Euclidean space (Celiñska-
Celinska-Kopczyyiskk Kopczyńska and Kopczyński 2022), and because the radii of HRG embeddings are large - the typical radii are on the order of = 30 tiles of the bitruncated order-3 heptagonal tiling (I). One benefit of such a discrete representation is avoiding numerical precision issues. The other benefit is algorithmic: given a tile t and a set of tiles T, we can compute array such that afi] is the number of tiles in T in distance from ( in time just ) The time of preprocessing (adding or removing a tile from 7) is (RR²) per tile. This gives us an efficient algorithm to compute the log-likelihood of DHRG embedding, and also to improve a DHRG embedding in terms of LL by local search. Muscoloni et al. (2017) (Coalescent embedding) and Garefa- et al. (2019) (Mercator) introduce ML algorithms obtain or improve embeddings.
Most research concentrates on 2D embeddings, including the recent state-of-the-art, CLOVE (Balogh et al.) (025). which arranges the communities using the algorithms for the Travelling Salesman Problem. Higher-dimensional embeddings have been studied recently (Bringmann et al. 2019| [Budll
el al. 2023 Kovács el al.) 2022 Jankowski et al. (023). A recent work (Celinska-
Beeliska-Kopeyynsse Kopezynska and Kopezynski 2024a) embeds into 3D Thurston geometries using tiles and simulated annealing.
2.3 HYPERBOLIC GEOMETRY IN ML COMMUNITY
Nickel and Kiela 2017) applied the Riemannian Stochastic Gradient Descent (RSGD) method to find hyperbolic embeddings. The algorithm is benchmarked on data that exhibit clear latent hierarchical structure (the WordNet noun hierarchy) and on social networks (scientific collaboration communities). The quality is evaluated using new measures: MeanRank (MR) and Mean Average Precision (MAP). MR is the average, over all edges u + c, of ru.,, which is the number of vertices w such that there is no edges from
w and w is closer MAP is the mcan of average precision scores (AP) for all vertices. The average precision score of vertex u is defined as Σ i/ru,,; where l is the number of vertices v such that u v, and v; is the i-th closest of these vertices. In the case of WordNet, u + v iff v is hypernym of u; this is a transitive relation. In (Nickel and Kiela 2018), the results are improved by using the hyperboloid model (referred to as the Lorentz model) instead of Poincaré model and evaluated using MR, MAP, and Spearman rank order multiple real- real-world taxonomies, including the WordNet noun and verb hierarchies, the Enron email corpus, and the historical linguistics data.
than v (including u, not including v, thus MR > 1).
While studies in NT hyperbolic geometry seem to be inspired by the theoretical and applicational premises (using geometry as the means to understand nature), ML researchers quickly recognized the potential of including hyperbolic geometry as a part of an analytic pipeline, even in classification tasks (see, e.g.. Chamberlain el al. (2017) application of hyperbolic embeddings to neural networks). That is why solutions to numerical precision issues have become a vibrant research area. [aaa er al. (2018) studied the tradeoff between the number of dimensions and the number of bits used for representing the angles. They also gave a combinatorial method of embedding tree-like graphs. Yu and De Sa 2019) suggested a blased model (LTiling) to combat the numerical precision issues. Their main idea is somewhat similar to DHRG, although while DHRG only uses the tiles, LTiling also includes the coordinates within the tiles. In TreeRep (Sonthalia and Gilbert| 2020), it is proposed that, instead of learning a hyperbolic embedding, we should instead learn a tree. Gu et al. (2019) embed networks not in H but in products of lower-dimensional spaces with hyperbolic, Euclidean, or spherical geometry, and in Guo et al. 2202)), a method for visualizing embeddings in H2 is proposed.
hyperbolic
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In Nickel and Kiela (2017). the early papers hyperbolic visualizations ( (Lamping et al., 1995), but not (Munzner] 1998)) and the HRG model are cited, although the authors and reviewers seem not to be aware of the extensive literature on hyperbolic embeddings, including the paper (Muscolomi et al. 2017) which uses MI. methods and has appeared on arXiv in Feb 2016. The Poincaré embeddings are thus compared only to Fuclidean and translational embeddings. This continues in the other papers mentioned in this section. As result, many papers even directly claim or suggest that Nickel and Kiela (017)) were the first to consider hyperbolic embeddings, e.g., "Initial works on hyperbolic embeddings include Nickel & Kiela (2017) [...]" (Gu et al.
2019
We have found citations to NT research in Ganea et al. (2018); in (Sothhaiaa and Gilbert (2020). Bläsius er al. (2016) is in the bibliography, but surprisingly. not referred to in text, despite the focus on speed; this paper also cites early work on hyperbolic embedding (Chepoi and Dragan 2000), hyperbolic multi--iimensionl scaling (Cvetkovski and Crovella (011). and embedding of d-hyperbolic graphs into trees (Chepoi and Dragan| 2000) Chepoi al. [2008] Abraham et al. 2007). Comparisons between the results of different communities seem lacking.
OUR RESULTS
3.1
COMPARISON ON REAL-WORLD TAXONOMIES AND SCALE-FREE NETWORKS
For every network, we use the following experimental setup.
Apply the following embedders to it: Poincaré embedding (PE) Nickel and Kiela 2017).
Lorentz embedding (LEy/Nickel and Kiela 2018). BF et al. (2016), 2-dimensional and 3-dimensional coalescent embedder Muscoloni et al. (2017). Hyperl.int iink embedder |iitskk et al. (2020), fast and full Mercator embedding García-Pérez et al. (2019).
3-dimensional Mercator embedding Jankowski et al. (2023). LTiling (Yu and De Sa 2019
TreeRep (Sonthalia and Gilbert, (020), Anneal (Celinska-Kopeznnen Kopezynska and Kopezynski
2024a). LPCS (Wang et al.]][]]10 2016b)) HMCS Wang et (2.19), CLOVE (Balogh 2025).
DHRG embedding improvement
Salniska-Kopzzyiskee and Kopezyński (2022) (applied to
BFKL, PE, LE, and CLOVE).
Evaluate the obtained embeddings according to quality measures from the literature: MAP, MR, GSF, GSR, GRE, and LL.
We also conduct analysis on hierarchies; in this case, we include the classic HypViewer tree embedder (Munzner 1998) (if the hierarchy is not a strict tree, the parent is picked randomly) and do not evaluate on measures meaningful only for networks (GSF, GRE, and GSR), For all hicrarchies, u - iff is superset (ancestor) of u; this is transitive relation. We the networks and hierarchies that have already been used as benchmarks in influential papers on hyperbolic embeddings. For the complete list of the networks and the hierarchies we used, see Appendix D
An implementation of MR and MAP is available with Nickel and Kiela (018)). However, on larger graphs, some embedders (such as BFKL) generate embeddings of large radius. This implementation fails to evaluate such embeddings due to a numerical precision error. Therefore, we use our own implementation which avoids this issue. See Appendix B
In the case of greedy routing measures, we prefer to use the original formulations, in which we immediately fail when there is no neighbor closer to the target. This is because some embedders use discrete tessellations, making it likely that some distances are equal. For original formulations, we can route randomly and use the expected route length (Celinska- Kopekkkakk and Kopezynskil (0024)). In the modified formulations, such an approach is ill-defined. Similar to (Celinska-Kopee) and Kopezyński 2022). to aid comparisons, we report the I.I. values for the and 7 values that maximize the log- likelihood (see Formula []. We restrict analysis to quality measures related to distance preservation; to our best knowledge, there are no measures that allow comparing the quality of angular positions in woaldwrldd embeddings.
The achievable quality of the embedding depends on the embedding dimension (achieving better results can be explained with higher dimensionality), therefore, in most cases, we compare 2D and 3D embeddings. (We include TreeRep because trees can be embedded into the hyperbolic plane.) For comparison, also evaluate 5DD PE, 50D, and 200D Fuclidean embeddings (EE) (Nickel and

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Figure 2: Quality assessment of embedders on real- world networks and hierarchies. Darker colors indicate that the given embedder occurred more frequently in the given percentile of ranks (higher percentiles are better) over all graphs benchmarked.
Product space embeddings (Gu et al. 2019) are an interesting approach, but they use higher-dimensinnal spaces, so they cannot be compared with 2D 3D methods. The hMDS method from (Sala et al. 2018) looks interesting, but it depends on the scaling factor, and it is not clear how learn this parameter; therefore, we do not include this method in our experiments, Most embedders are randomized, so we have repeated portion of experiments using different seeds; this does not usually change the rankings (Appendix H.). We use the official implementations and hyperparameters; Appendix
Kiela 2017).
A and the supplementary material.
Figure 2 shows the aggregate results, while details can be found in Appendices E (plots and tables) and [ (NOUN hierarchy). Surprisingly, while BFKL has been designed specifically for scale-free networks and greedy routing, and LE has been benchmarked on hierarchies and MAP and MR, our results show that BFKL DHRG achieves significantly better results on many hierarchies (BFKL: DHRG: mesh,,etrapodaa, while Lorentz embeddings tend to achieve better results on networks, especially for greedy routing (better GSR and GSF). Still, the quality of BFKL, BFKL+DHRG, and 2D LE is similar across scale-free networks in our experiments, as measured by MR and MAP. One counterexample in the YEAST network, where BFKL achieves significantly better results than Lorentz on MAP (0.756 vs 0.542). In all cases, BFKL (and even BFKL+DHRG) is orders of magnitude faster, making LE impractical for larger graphs. The new CLOVE embedder tends to achieve even better results on hierarchies, in even better time. In many cases, DHRG is able improve the results of fast embedders such as CLOVE while remaining reasonably quick.
Hyp Viewer (Munzner 1998) produces quite bad MR and MAP; however, it aims to put similar nodes close together, while due to how the transitive graphs are constructed for hypernymy hierarchies, high MR and MAP measures are achieved when similar categories (e.g., "lion" and "tiger") are closer to their hypernyms (feline, mammal, animal, entity) than to each other, which promotes longer edges on the outer levels of the hierarchy, and shorter edges in the center. The fast mode of Mercator usually produces worse embeddings than BFKL, while full Mercator usually achieves results between BFKL and 2D LE. Unfortunately, the full Mercator is slower than 2D LE for larger graphs. TreeRep is based on the idea of learning a tree instead of a hyperbolic embedding. We agree with this proposition for tree-like hierarchies, but for networks such as FACEBOOK and the connectomes, hyperbolic embeddings achieve significantly better results. (The hyperbolic plane is tree-like in large scale and Euclidean- like in small scale, and thus may potentially combine the advantages of both approaches). LTiling did not generally achieve better results than 2D LE in our experiments, while being significantly slower (contrary to DHRG, tiles are used only to improve numerical precision, not to make the process faster); however, this might be due to incorrectly set hyperparameters or testing on smaller, more shallow hierarchies, so the numerical precision issues did not yet become relevant. The results of LPCS are quite bad connectomes, but on hierarchies and other networks, its results are comparable to BFKL. The coalescent embedder also performed relatively poorly in our experiments. The KVK embedder often achieved excellent results, but unfortunately turned out to be very slow significantly slower than LE. Anneal works great for connectomes (which were its original area of application), but often turns out to be not that good for other data; this is probably
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because Anneal only produce embeddings of small radius, and connectomes, being physical networks, can have good embeddings of small radius.
Figure Top row: NOUN (Lorentz 2D). VERB (left right: Lorentz 2D, Lorentz 2D+DHRG, BFKL). Bottom row: DROSOPIILLIAI (Lorentz 2D, Lorentz 2D+DHRG, BFKL, BEKL+DHRG).
3.2 VISUALIZATION
One application of 2D embeddings is visualization. We rendered the embeddings using the tools from DHRG; see Figure 3 All pictures are in Poincaré model, centered on the center of the hyperbolic disk used for embedding. One observation that Lorentz embeddings tend put nodes close the center, while the center is generally avoided in BFKL, and DHRG improves the balance.
3.3 DIMENSIONALITY
According to all our experiments so far, ligher-dimensional embeddings achieve better results than lower-dimensional ones. This result is trivially an artifact of optimization. Reducing the number of dimensions could be seen as imposing a restriction on that dimension; usually, optimization without restrictions yields better results. To make comparisons fairer, we need use information criteria to control properly for this artifact. We introduce the information contro! value (ICV), based on the Minimum Description Length (MDL) principle (Rissanen, 1978). which takes into account both the
compations ofgredd SACCEOS, Todl
compatton of greeddyyy CuCCesS, T-E4
corpaston of grendy OACCROS, ToEL
Commaation of STENN ONECOSS,
Figure 4: Density plots of the differences between the values of quality measures (MAP and GSR) obtained by Lorentz 2D and BFKL. Negative values indicate that BFKL performed better.
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quality of edge prediction and the description length of the embedding; this description length is longer (worse) in more complex embeddings, such as those of higher dimension or radius. This is welcome, since more complex embeddings harder to visualize, and also embeddings of higher radius are more prone to numerical errors (Bläsius ef al. 2018 Sala et al. 2018 Celinska- Katisskaa-oezyssskk and Kopezyński 2024b). According our results, woodiiennioonal embeddings perform better for most real-world networks. The embedders we compare do not optimize the embedding radius, except Anneal, which enforces embeddings of small radii. To further improve ICV, we have also implemented a variant of DHRG that aims to reduce the embedding radius; the resulting improved BFKL called Penalty. See Appendix C for the description of ICV and the Penalty approach.
Powerd)
ARC
Figure 5: Quality assessment of embedders simulated networks. Darker colors indicate that the given embedder occured more frequently in the given percentile of ranks (higher percentiles are better) over all graphs benchmarked.
3.4 COMPARISON ON ARTIFICIAL SCALE-FREE NETWORKS
For more statistical analysis, we have also include comparison, especially BFKL and Lorentz 2D embeddings, on artificially generated scale-free networks. We use the generator from BFKL based on the HRG model, with default a = 0.75, network sizes E {500, 1000, 2000} and temperature Te (0.1. 0.4, 0.7, 0.9). This setup produces networks from the ultra-small world regime (Boguñá el al. 2020). For such networks, the average distance grows slower than any polynomial of the logarithm of the network size, which is typical to heterogeneous networks with hubs A significant share of real-world networks belongs to this regime, making it of greater interest to ML community.
Figure 5 depicts an aggregate ranking of all embedders. Regarding MAP and MR measures, we note apparent differences in the embedders' performance. LPCS tends to perform relatively poorly (usually in the bottom 10%), while 2D LE is significantly improved by discretization the case of MAP. In contrast to the analysis of the real-world networks, CLOVE's performance is mediocre here - occurs rarely within the top 10% of embedders. Regarding the greedy routing measures, see little difference. Interestingly, LPCS and Coalescent embeddings tend perform other embedders on simulated networks generated from the HRG model. An analysis of possible explanations for this finding could constitute future research line.
than
Fig depicts the densities of the differences between the values of quality measures obtained by 2D LE and BFKL, and Table Tableacontinns results of the logit regressions probability that BFKL, would perform better than 2D LE in terms of a given quality measure. No matter the quality measure, according to our results, the greater the graph, the higher the probability that BFKL. will perform better; however, with rising temperature, that probability decreases. Real- world networks are considered to have fairly large values of T, such as T 0.7 used for Internet mapping (Bläsius al.) 2016 Boguñá et al. 2010)), which is consistent with results on real-world scale-free networks. Although our models were aimed at interpretation instead of prediction, we included information the prediction quality, both from cross-validation and benchmarking. Our models are of satisfactory quality.
the determinants of the
Even if our results suggest that, in many cases, 2D LE outperforms BFKL, it still comes a high time cost. In Fig 6 we present the trade-off between the markup in time expenditure (how many times longer takes to compute) in comparison BFKL and the percentage gain in the quality of

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Published as a conference paper at ICLR 2026
MAP
GSR
GRE
Coeff.
Pr(> 1z])
Coeff.
Pr(> |2))
Coeff.
Pr(>> 12|
Intercept
-1.9701
7.09c-08
0.68510
0,007420
0.6402
0.01195
Temp=0.4
-0.8881
0.003800
-2.1156
3.12e-12
-2.1168
2.80c-12
Temp=0.7
44,4133
5,20e-14
-3,9175
<2e-16
-3,9746
<2e-16
Temp=0.9
44,413
5.20e-14
-4.3845
<2e-16
-4.3099
<2e-16
Size 1000
1.6316
7.70e-05
0.92710
0.001930
0.8945
0.00287
Size = 2000
3.7975
<6
2.53030
2.84c-15
2.6053
7.8e-16
N
600
600
600
ACC
0,8835
0.8231
0.8235
ACCbench
0.7867
0.6200
0.5356
0,6165
0.6133
0.6233
Table Results of logit regressions for the determinants of BFKL embedder outperforming Lorentz 2D embedder in terms of quality measures. ACC and x are average accuracy and Kappa from 10-fold vross-validation; ACCbench is the aceuracy of the naive model (always predict mode).
1000
632
600
510
000
Figure 6: Comparisons of percentage gains in quality (MAP) of the 2D Lorentz embedding against the markup in time expenditure in comparison BFKL embedder. X axis is the percentage gain in quality and the Y axis is how many times longer it takes.
the embedding (measured with MAP) resulting from using 2D LE. We conclude that there is no significant monotonic relationship between the time spent and the percentage gain in quality (p-values in Kendall-tau significance tests, we encounter ties in our data that may make Spearman's 's rho inappropriate to use, are: 0.9867, 0.2143, 0.0106, and 0.0364 if we control for temperature 0.1, 0.4, 0.7, and 0.9 respectively. The last two results are insignificant at 1% significance level).
4 CONCLUSION
We compared the popular hyperbolic embedders from three communities, paying special attention to BFKL against 2D Lorentz embeddings (LE). Our main motivation is the apparent lack of awareness of the algorithmic results hyperbolic embeddings in the ML community. Our results are of practical benefit: all the studied embedders generate static embeddings and thus, in ML pipelines, they can be easily replaced, to pick the best emhedder based on the relevant quality measures and resource usage. In all experiments, BFKL runs significantly (about 100 times) faster than LE, while achieving results generally of similar quality. Higher-dimension
LE generally gets better results than both kinds of 2D embeddings, even in 3D; however, this no longer holds when we take information criteria into account. more detailed study of our proposed criterion will be the subject of further research. In this study, we focus on ultra small world network regimes, generated with the HRG model. While we we do not perceive this limitation as serious threat to validity of our results, further research could involve other regimes and other generative models such as PSO and nPSO. There are new hyperbolic embedders emerging each year; our framework can be easily extended to include them.
We have also found discrepancies between our results and the results in (Nickel and Kiela 2017 2018). In particular, (Nickel and Kiela 2017) 200D SGD Euclidean embeddings are performing worse than even low-dimensional Poincaré embeddings, but in our experiments, they consistently achieve significantly higher results (this particular case of non ron-reprou has been previously observed and studied in (Bansal and Benton 202I)); ; in (Nickel and Kielal 2018) Lorentz embeddings achieve significantly better results than Poincaré, while in our experiments, their performance is similar, and Poincaré sometimes better. See Appendix G for details.
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ACKNOWLEDGMENTS
We are grateful our colleagues, the anonymous reviewers, and the participants of Network Geometry: Theory and Applications workshop at NetSci 2025 for their valuable feedback, comments, and suggestions on the various stages of the preparation of this paper. This work has been supported by the National Science Centre, Poland, grant UM0-2019//35/BSS660000
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A IMPLEMENTATION USED
The C++ implementation of our quality measures (MAP, MR, greedy routing measures, ICV) is published GitHub as part of the Rogue engine (nttps://githuu com/ zenorogue/ hyperroque) released under GPL v3 (subdirectory roquevia z embeadings). The exact version used in this paper, and the whole comparison framework (including auxiliary scripts), is available as the supplementary material.
We have downloaded the embedders from the following repositories and use the following settings:
Poincaré and Lorentz:
com/ http.///ccccccbbboreeeecc.... molncare-embeddingg embeddings (last commit on Sep 16, 2021), Attribution-NooCCommecoo
4.0 International
We use the hyperparameters -epochs 1500 -negs 50 -burnin 20
-dampening 75 -ndproc -eval_each 100 -fresh -sparse lbbunnnnmmltipplll
0.01 -neg_multipplier 0.1 -lr_type
constant 1r 1 -train 50 -gp from the example train-nouns. sh from the repository, except that we requested using the GPU (-train_threads 1 -gu 0). We also add the hyperparameters specifying method (-manifold poincare -dim 2). For Lorentz 2D, Poincaré 3D, Poincaré 5D, Lorentz 3D, we replace -Ir 1 with -ir 0.5 -no-maxnorm (this setting comes from the suggestion about Lorentz embeddings in train-noumd sh). The hyperparameters for SGD Fuclidean embeddings are not given in the current official repository; use the same parameters for Poincaré (learning rate l).
1 -dampning 1.0 -batchsize
BFKL:
overview (last commit on Sep 8, 2016), no license given
This method estimates the hyperparameters in est iaatmmyyeerrbbolllooooo a method. We do not modify the original settings. The temperature (T) parameter for embedding is set to a low value I which should work well for embeddings, the parameter is estimated based fitting the power law, and the radius (R) is computed using formula.
DHRG: rogueviz/ dhrg subdirectory in the Rigue engine
This method is parameterized by the tessellation used; we use the bitruncated order-3 heptagonal tiling. does not create embeddings from scratch, but rather improves them using local search; we allow up to 110 iterations of local search. Local search computes the log-likelihood using the logistic function. We test on the BFKL and Lorentz 2D embeddings. For the penalty variant, we set the parameter to 2 (see Appendix C and perform 100 iterations of local search.
TreeRep: httees//ggLaabb..c//roottt/////t 2023), G3L
(last commit on Jun 23,
This method has no settings hyperparameters (other than the number of threads, which we set to 8 as suggested in the repository).
LTiling:
ht attps: http::///ittule http://........yttyt///yooooolla Leariinn (last
commit Mar 19, 2020), Atrrbution-NonCCommerria 4.0 International
We use the hyperparameters from the -epochs 1000 -negs 50 -burnin 20 -dampening 7.75 -ndproc 4 -eval_each 100 -sparse -burnin_multiplls constant -trrain_threads 2 -dampening 1.0 -batchsize 50
01 -neg_multiplier 0.1 -lr_type
manifold Wriliingrrgd -dim 2 -com_n -Ir 0.3 mn-mmmmorrm from the train
example.
htft:s:/ggraphccs stanford. htw.mford.ett/mmmmneee////////// html (last modified 2003), license the COPYRIGHT file
Hyp
This method has no hyperparameters. For non-strict hierarchies we pick the parent randomly,
Mercator: http:///tt..t...oooo//o//ooo//ta
(last commit
Jun 21, 2022), GPI.
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Published as a conference paper at ICLR 2026
Mercator has setting for fast full embedding; the fast method skips the likelihood maximization step. We apply both variants. We do procpessn of the inferred values of the radial positions. The parameter 3 can be provided. but we use the default behavior, in which /3 is inferred to reproduce the average local clustering coefficient of the original edgelist.
d-Mercator,
(last com-
mit Nov 23, 2023), GPI. v3
The hyperparameters are similar to Mercator, except only full embedding is available for greater dimensions.
Simulated annealing: rogueviz / sag subdirectory in the Rogue Viz engine
This method is parameterized by the tessellation used; we use the bitruncated order-3 heptagonal tiling for 2D embeddings, and the subdivided(2) (4,3,5) honeycomb for 3D) embeddings (the g711 and g435b2 settings from the original paper). As in the original paper, we set the parameter controlling the number of tiles M 20000. The number of iterations of simulated annealing is Ng- 1/.00/V|- As in the original paper, we run the embedder twice; the first pass is to obtain good initial values of the and T parameters.
Coalescent:
par_lescentt_emeeddde https: : /gtphbttbb thub. com/ biomedica..... cayercareeeeeececcs.........
(last commit Jul 8, 2019) We use the hyperparameters and settings from
m. Specifically, 2D embed-
dings uses RAI-LE-EA. 3D embeddings use RA 1-ISO.. We the code in Octave (the free alternative of Matl Lb), which has no access to graphe 11
function; we
solve this issue by computing the table of shortest paths with our own C++ implementation.
KVK:
https://bi bttp:::/bitbbuckt.. tbucket....... http:///boccaooo//////aaaa////// ccwee_0ooo_____ http://000_ccce__yyeeeen/////r
master/ (last commit Jun 11, 2011)
This embedder has two parameters: the power-law exponent yy and the temperature T. For simulated networks use the actual temperature (providing more information the embedder), whereas for real-world networks we use 0.1, similar to BFKL. As explained in the paper, 7 2a + take a estimated by BFKL.
LPCS: the source code is included with the paper at httlttps://// www www.ssiieeccce www.eenceeeeioo 6000018
The source code is in MatLab. According to readme, before we call the Matl.ab func- tion main_ we need to use the Fast Modularity Optimization (FMO) algorithm proposed by Blondel et al. to detect the hierarchical community structure, using the multilevel. community function in R (the name of this function is currently depre- cated in favor of c cluster_ cluster_louval thus we had to write own, and also adapt the main LPCS function to the situation when cluster cttuuste__oovaiiiieeeeeeee returns a different number of community levels than 3. The code also has hyperparameter gama (the power law exponent); we the R function sit_power_lew aw (degree (g) ) to estimate this exponent. We also had to fix some bugs (the code did not work when only two communities were found) and Octave warnings. Since the official implementation ran very slow in Octave, we have also reimplemented the algorithm in C++ sscrip//pes-remake.cyy nectNextCom.m (which compares the intimacies of x(1) and x(2), while, according to the paper, intimacies of the first and last subcommunity in x should be compared here). In the following tables, embedding time is given for the original implementation, but not the reimplementation (the reimplementation is generally very fast).
louvaliiii in); the source code of this part has not been included,
Our reimplementation fixes a bug in Con-
HMCS: We were unable to find an official implementation of this embedder, so we have edited our C++ reimplementation of LPCSS to include the changes (obtaining the hierarchy by calling FMO algorithm repeatedly rather than just once; Community Closeness instead of Community Intimacy; angular size of a community based on the sum of degrees rather than the number of vertices). This embedder has one hyperparameter: the number of nested hierarchies to use. The authors suggest 2, or more for larger networks; we use levels.
CLOVE: https: hub. com/ samu wwwmmm2EEEEEhhycoo.. 2025)
(last commit Oct 16,
We use the default values of all settings and hyperparameters: y to fit the degree distribution, degree fitting sample size of 100, automatically detected dendrogram, Leiden community
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detection method, exponential coarsening, the number of anchor communities, Christofides algorithm for solving the Travelling Salesman Problem, degree eerrredyyyyyy node arrange- ment, community sector sizing based the number of nodes in the community, and PSO radial coordinates assigned. For some hierarchies, the official implementation CLOVE detects 7 2 and fails with parameter inference error. In these cases, 7 instead.
For replicability we also control the PRNG seed.
We have downloaded the connectome datasets brain maps data The scale-free networks are
from bttps://gittub..o com/
from the SNAP database ([eskovee and Krevl 2014). The tree-of-life and GitHub followers graph dataset have been included with DHRG.
We use the following hardware:
[1] Intel@ CoreTM 17-9700K CPU @ 3.60GHz, NVIDIA GeForce GTX 1060 6GB/PCIe/SSE2, 96
GB RAM (we used zram for the embedders which did not fit RAM)
[2] 11th Gen Intel TMre 17-118501 @ 2.50GHz, OpenGl. renderer string: NVIDIA RTX A3000
Laptop GPU/PCle/SSE2
Software: Arch Linux, g++ 12.2.1 to 15.2.1 (DHRG, BFKL, KVK, Anneal, Mercator), Julia 1.9.3 (TreeRep), Python 3.6 (Poincaré, Lorentz, Itiling, Mercator), Octave 10.3 (Coalescent, LPCS), 4.5.2 (LPCS, creation of graphs)
The times reported in the paper have been obtained on [1]. Some experiments have been run on [2].
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B COMPUTING DISTANCES, DISCRETE MAP AND MR
The distance between two points p(r1, Φ1) and p(r2, Φ2) in the hyperbolic plane can be computed as follows: (let á = Φ1 ②2)
(()( d2)) = ((p(r, (m,, (), p(r,, +))
arcosh g- ((sinh(r)) ): cosh(ri)).
(sinh(r2) cos Φ, sinh(r2) sim Φ, cosh(r2))
arcosh (sinh(ri) sinh(r2) ((inh(()))nh(r)) cos @ + cosh(r1) cosh(r2)) (
arcosh (cosh(r) - r2) + cos( sinh(rr) sinh(r2)) (inh(r))))(h(r()))
The last formula has better numerical properties (Bläsius et al.) 2016;
Kopezyáákzz and
Kopezyński, (024b)) The distance formula in the Poincaré disk model can be computed similarly, although converting from Poincaré hyperboloid needs solving a quadratic equation.
Still, the computation is somewhat slow: for each of (n)) nodes, O(n) distances from the other nodes need to be computed and sorted. It possible to apply the discretization method from DHRG to quickly compute discrete analogs of MAP and MR (which we call dMAP and dMR)) As mentioned in Section 2.2. discretization allows us compute, for every node t, an array a such that a[i] the number of tiles in T in distance i from f. in time O If t has e edges, we can compute a similar array bli] restricted connected tiles in time R.) Note that the formulas for MR and MAP given in Nickel et al. (2016) are for the case of continuous distances, and need to be adjusted for discrete values obtained from the DHRG model. In the case of MR, a non-edge with distance tie contributes 0.5 to Tu.e, and in the case of MAP. if there are bld] edges and a[d] total nodes in distance d, we assume 1--th of these edges to be ranked after a ald(k - /b5) nodes. We can compute such MR and MAP knowing a[i] and b[i] for every node in total time OnRR + mR), where m is the number of edges.
C CONTROL VALUE
This section describes the information control value (ICV), the embedding quality measure we propose. This value is based on the Minimum Description Length (MDL) principle (Rissanen 1978). According to this principle, the shortest description of the data is the best model. We need bit of information to describe an event happening with probability p.
loga (p)
In case of geometric embeddings, the description length consists of two parts: the description of the embedding itself, and the log-likelihood of obtaining the connections, given the embeddings. The second part is related to the log-likelihood used in the BFKL embedder. Recall that every pair of nodes a and b is then connected with probability p(a, b) p(ô(a, b)) (1 + exp((ō(a, 6) R)) / where 8(a, 6) is the hyperbolic distance between the points in H representing the two nodes. To compute the log-likelihood, we sum log(p(a, b)) for every connected pair of nodes, and log(1 p(a, b)) for every unconnected pair of nodes. To compute the description length in bits, use the same formula, except that we use - log, (p) instead of the natural logarithm log(p). The parameters R and T are chosen in order to maximize the log-likelihood (equivalently, minimize the description length).
In a d-dimensional embedding, every node i is described with coordinates (ri, 0t), where r; the distance from the center, and Φ; E gd-1 is the angular coordinate. We assume that r; has exponential distribution Exp(A) restricted to [0, Rmazl- We choose Rmma to be the maximum ry and which maximizes the likelihood. Let fn be the density of this distribution of the radial coordinates r,. For Øs we assume that is uniformly distributed in Sd-1
We assume that our coordinates are given with limited accuracy c. That is, instead of the precise (ra, (x) obtained in our embedding, we use (r/, of) such that p(r(; Φ() is close to p(r., Φi). To describe rf such that |r; - r/1 < e, we need - loga f [(r)dr bits. In case of angular coordinates, we need to divide the sphere of radius r;. whose volume is proportional to - by the area of (d - 1) region of diameter e, which is proportional to ed Therefore, to describe d'. we need d, (log2 log, sinh(r:)) bits (as long as r; > e). Since we know the positions of nodes a and b with error e, in the formula for p(a, b) we take not p(5(a, ()), but p'(a, b) (p(5(a,)) b) <) + p(ō(a, b) e)).
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For the given obtain the total description length L as of description lengths of r/,, of for all nodes i, and pl(a, b) for all pairs of nodes (a, b). We choose the e which minimalizes this description length L. Generally, smaller increases the description length for r and 0/+ but decreases the description length of p (a,b) b).
To normalize the total description length, compare L with the description length N of the naïve non-geometric representation, which simply assigns the same connection probability p to every pair of nodes. Theoretically, good geometric represesentation should obtain L < N; however, some of the algorithms we study obtain L > N. If there are n nodes and m edges, the minimum description length -m logz p (() m) loga (1 p) is obtained for p = m/ (2) Our control value is then oo) and from above by 1 (which would be obtained for I. = 0). Good geometric representation achieve the control value of at least 3. which corresponds to N.
N/ + L). This value is bounded from below by 0 (the worst
Note that, for di < d2, di-dimensional hyperbolic embedding e can be considered a d2- dimensional embedding, simply by considering the Hd that ei uses as subspace of db All quality measures found in the literature and studied in this paper (log-likelihood, MAP, MR, GSR and GSF) will give exactly the same result whether 61 considered d1 -d-imensiional or d2- dimensional; in other words, these measures will always give advantage to higher-dimensional embedding methods. In contrary, the control value will penalize higher-dimennionall embeddings, as the part of the description length which corresponds to (ø.) will be larger in higher dimension. Furthermore, control value will also penalize embeddings of larger radius. This is welcome, since
embeddings of large radius are harder to visualize, and also more prone to numerical errors (Bläsius) er al. 2018 Sala et al. 2018 Celinska Kooisssa-Kozzyskka and Kopezynski, 2024b).
In the Penalty variant of BFKL+DHRG, node placed in distance of steps from the center of the model costs K log((s). where ræ is the number of tiles in distance 6. Instead optimizing only the log-likelihood,
optimize the sum of log- logelikelihood and this cost. For K 1 this cost corresponds to the part of description used to describe the angular coordinate. our experiments we take K = make the embeddings even smaller.
D REAL-WORLD HIERARCHIES AND NETWORKS USED
Table 2 contains the list of real-world hierarchies and networks we benchmark all the embedders on. In the case of VERBF, we had to add an extra root node, since BFKL. requires the network to be connected. We have not included other networks used BFKL benchmarks because they are too large for slower algorithms such as Poincaré and Lorentz embeddings. In LE, the Enron email corpus and the historical linguistics data are analyzed using weighted edges, so we cannot compare them to BFKL DHRG.
E RESULTS ON REAL-WORLD HIERARCHIES AND NETWORKS
The detailed results of our evaluation on real-world hierarchies can be found in Table 3
We also include MAMMAL (the mammal subtree of Noun). The detailed results of our evaluation on real- world networks can be found in Tables 455 Figures 9 10 I1 and 12 contain visualizations of MAP, MR, GSR, GSF, and ICV on those hierarchies and networks. Figure 13 shows the aggregate information for the remaining measures.
Note that some algorithms are very slow, making them not feasible to run on large graphs. We do not provide the results in these cases.
F ARTIFICIAL NETWORKS
Table 6 and Figure 15 show the details our evaluation of BFKL versus Lorentz 2D on artificial networks.
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Published as a conference paper at ICLR 2026
lype
details
V
Source
ias
Internet
autonomous systems
23748
58414
IM (Boguñá et al. 2010
facebook
social
social circles
4039
88234
BDsG Leskovec and Krevl 2014
followers-2009
social
GitHub followers
74946
537952
D
openflights
transport
transport network
3397
38460
MC (OpenFlights website)
grqe
cilation
general relativity
4158
13422
PSTY Leskovec and Krevl
2014
astroph
citation
astrophysics
17903
196972
(Leskovec and Krevl 2014
condmat
cilation
condensed matter
21363
91286
P (Leskovec and Krevl
2014
hepph
citation
high-energe physics
11204
117619
P (eskovec and Krev
2014
yeast
biology
yeast metabolism
1458
1948
STY Ueong et al 2001
discasome
biology
disease relationships
516
1188
ST (Goh et al 2007
noun
hierarchy
WordNet
82115
743086
PLSTY Miller 1994
acm
hierarchy
ACM classification
2114
8121
I.
mammal
hierarchy
WordNet
1180
6540
PY Miilerr 1994
verbf
hierarchy
eardNet
13543
48621
LY Miillrr 1994
mesh
hierarchy
hierarchy
58737
300287
L (Rogers 1963
tetrapoda
hierarchy
hierarchy
11262
527580
Maddison et al.
2007
esphd
hierarchy
hierarchy
1025
3978
STG (De Nooy
2018
CElegans
cell
nervous system
279
2287
MAAT Varshney et al.
2011
Drosophilal
cell
optic medulla
350
2887
MA Shinomiva cf oz.
202
Drosophila2
cell
optic medulla
1770
8904
A
Shioomya er at.
Mouse2
cell
retina
916
77584
A Helmslaedter et al.
201
Mouse3
cell
retina
1076
90811
A Helmstaedier
2013
ZehraFinch2
cell
hasal-ganglia (Area X)
610
15342
Dorkenwald
2017
Macaque1
area
cortex
1515
Kaiser and Hilgetag
2006
Macaque2
area
cortex
71
438
Young 1993
Macaque3
area
cortex
242
3054
Harriger of
2012
Macaque4
area
curtex
29
322
A Markov et at.
2013
Catl
area
cortex
65
730
Scannell ef ol.
1995
Cat2
area
cortex and thalamus
95
1170
Scannell
1999
Cat3
cortex
52
515
Scannell
1999
Humanl
cortex
493
7773
Hagmann
20US
Human2
cortex
496
8037
Hagmann
2008
Human6
area
whole brain
116
1164
Gray Roncal el cl.
(2013
Human7
area
whole brain
110
965
A
Gray Roncal ot al. 2013
Human8
area
whole brain
246
11060
Gray Roncal of cl. A
2013
Ratl
area
nervous system
503
23029
Bota and Swanson A
2007
Rat2
area
nervous system
502
24655
A
Bota and Swanson
2007
Rat3
area
nervous system
493
25978
A
Bota and Swanson
2007
Table 2: Our benchmark graphs. 'Cell' and 'area' connectomes. The edges are directed in
hierarchies and undirected otherwise. Letters signify the embedders which used this benchmark: I
(Boguñá al.) 2010)) B (Bläsius et al., 2016). (Celiñska Kopeskkkkk and Kopczyński (022).
(Nickel and Kiela 2017), I. (Nickel and Kiela 2018), S (Sala el al.) (2018). G (Gu el al. 2019), Y
(Yu and De Sa 2019), M (García-Pérez el al.) 2019). T (Sonthalia and Gilbert 2020), (Allard and
Scrrano, 2020 Celinska- Kottaa-Kopyyykkaa and Kopczynski (024a)), C (Balogh al. 2025 Small letters
appear in the repository but are not discussed the paper.
G DISCREPANCIES
In Table our results are compared the results obtained in Nickel and Kiela (2017 2018). Note that VERB is different than VERBF used in our paper, which includes one extra node that is an ancestor of every other node. Furthermore, the ACM hierarchy Nickel and Kiela 2018) is given 2299 nodes and 6526 edges, while ours has 2114 nodes and 8121 edges, and the MESH hierarchy is given as 28470 nodes and 191849 edges, while ours has 58737 nodes and 300290 edges.
The discrepancy in the result of Euclidean higher-dimensional embeddings has been previously observed and studied in Bansal and Benton (202T): the reported values did arise as a result of using different setting where the Fuclidean embeddings were regularized!] Other differences in
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graph name nodes
Roun
miniitaa
verbf
06.11
mash
tetrap
espba
82115
1180
13543
2114
58737
11262
1025
edges
743088
6640)
48621
8121
300257
527580
3978
BPKL
42%
66
342
738
BFK1 + DHRG
1637
70
600
1191
1996
Poincare 2D
51791
408
2544
413
19137
26256
212
Poincare 2D DHRG
52104
410
2569
446
19339
26295
234
Pouncare
31347
369
20648
223
25)
38578
769
239
embedding time [4]
DDIRG
277
Larentz 3D
39890
279
22,90
15700
19892
coalescent2
3
23
67139
coalescenEs
12
56
7387366
KVK
1877
3561
1179
CLOVE
117
76
CLOVE DHRG LPPCS
3376
282
1162
377
HMUS
Mereater
37202
18
16617
Mercator full
4729
113
108459
1802
d-Merentor
151
4450
21.4
65067
24761
48
killng
63074
2013
2007
30.992
21:120
18,999
BPKL DHRGG
26.018
23.521
25. 25.159
17.634
26.163
24.706
15.915
Poinetre 2D)
12:207
12:305
12:208
12.205
12:207
13:205
12:300
Pouncare 3D
12.208
12.205
12.202
12.118
12.207
12.205
12.199
Lunentz 2D
14,509
14.509
14,509
13.290
14.500
14.509
14,509
DHRG
17.605
16.323
14.598
16,631
radiss (aboolute unis]
penalty
21,386
unneal2
7.751
7.739
7,740
anneals
3.712
3.712
conlesceer2 coalescent3
14.14b
15.111
565
14.146
15.313
86865
KVK
15.650
14.755
14.627
CLOVE
22.632
14.146
21.982
15,863
CLOVE DHRG
25.225
382
16.23
25, 35.302
964
HMCS
22.632
14.146
19
15.313
31.963
18.
Hyeee
22,525
102203
13.955
17.390
72.251
Mercator fest
$1,492
19,005
88.813
25. 654
49.317
38 38.5554
26.218
Mercator full
18.450
38, 88.899
20:254
38,002
23 055
26.055
d-Merenior
14.647
18:262
152
17.685
16.35 16.871
18.676
13:285
13.885
BFKL
0.281
0.19
0.345
0:423
0.321
0.276
0.667
BFKL DHIG
0.118
0.489
0,580
0.532
0.406
03707
0.615
Poincare 27)
0.105
0.92
0.350
0.657
0.195
0.530
0.565
Poinense
0.376
Loreniz 2D DIIRG
0.322
0.356
0.584
0.679
0.417
0.712
0.673
MAP: Mean Average Precisoon
Loientz 3D
0.500
0.951
0.528
0.376
01942
0.805
penally
0,330
0.209
0.294
0:410
0.293
0.689
0.691
anneal2
0.321
0.256
0.730
unneall
0.275
0.210
0.836
coaleszear2 coalescents
0.359
0.885
CLOVE
0.776
0.326
0.778
0.360
0.316
0.596
CLOVE DHKG 1PCC
0.291
03865
0.871
0.838
0.825
0.770
0.612
0.341
0.585
0.574
0.388
0.323
0.481
0.543
HIMCS
D.ISS
0.711
0.625
0.484
0.541
0.494
HypWVeewer fast
0.047
0.124
0.134
0.134
0.122
0.014
0.416
0.022
Mercator full
0.727
d-Mercator
0,009
0.014
Iillng
0.201
0.522
Eucurdea 30D
09921
01999
01925
01999
0.834
03997
1.000
Fachiceess 2007)
0.945
10000
0.931
0.871
0.998
100
102.1
DHRG
18
10.0
8.6
10.7
45.0
Poineare 21)
89.3
2.1
13.1
3.2
42.5
37.4
Poincare 3D
18.0
1.2
8.5
253
9.7
4.1
Poincase 5D
1.2
3.9
Lorentz 2D
42.3
25.7
4.0
54.5
DHHG
R0.7
14.1
20.4
25.1
penalty
48.2
16.2
39.7
MR MemnRnnk
unneal2
anneals
17.9
12.2
27.4
20.2
25.6
13.9
coalescent2
9.5
22.4
41.0
coalescent3
01.4
28.2
31.5
KVK
3.3
3.0
27.0
CLOVE
19.0
27.5
CLOVE DHRG
28.1
118.7
IMCSS
35.7
4.1
5.8
19.0
22.6
87.8
Viewer
4452.4
145.7
276.
77.0
522.2
5559.7
468.5
Mencator fost
177.8
8.8
16.2
20.7
60.3
106.1
20.8
Mercator full
2.8
7.7
4.2
29.6
19.9
21.7
MeMccaorr
250.8
804.2
237.2
51502
5522.4
40.3
Fuclicken 50D
Faclideus
Table 3: Our results real-world hierarchies. Darker cell color indicate better results.
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griph naiie
6stiop
ocaidmi
Sepph
Toccho
Nenst
discas
Follow
opend
165
CEllgs
Human]
Drose]
Monses
Rodes
17963
21363
41.58
11204
4039
1458
516
74946
3397
23748
279
493
350
1076
(undirected)
196972
91286
13422
117619
55234
1948
1188
537952
19230
58414
2287
7773
2887
90811
1FKL
179
9]
7
62
26
269
11
40
17
BFKL DHRG
230
579
79
168
43
26
20:34
29
440
45
Poincare 2D
20181
10025
1321
11920
3:50
235
155
65600
2020
1.96
254
807
325
8781
Poincare DHRG
20687
10265
1335
11993
8618
65878
2030
7476
255
815
324
8792
Poincary
20485
68445
2061
260
795
328
S716
Lovente 2D)
14741
7349
986
8761
49531
1501
5604
204
679
255
6413
eniboddingg time [s]
Locentz 2D + DIIG
14891
7475
1005
8807
6371
200
137
19814
1517
5691
208
612
256
6136
Lovenzz 3D
14612
7479
1155
0392
6319
184
134
51675
1485
5778
216
1554
248
0252
cualescemi2
13557
23340
162
3147
164
83
35
coslescen:3
173
15012
499
20
259
2
40
KVK
13910
15250
1278
20.5
7918
156
CLOVE
188
CLOVE DHRG LUPCS
279
1560
59
4055
HMCS
Merestoe fast
1565
2088
587
297077
56
2721
5
6
11
39
Menester Full
8906
13167
454
3707
408
47
195644
338
18055
7
9
13
65
G-Mareator
5110
7395
313
2291
403
44
27
250
9136
38
24
528
lilling
19721
3200
1392
25772
2795
7307
7237
BPKI.
15.400
17.677
21.792
21.881
12.576
16.267
12:800
20.0044
25.010
23.25
7.787
7:421
8.178
5.625
BFKL- - DEIRG
18.501
20.715
24.921
26.097
14. 1.506
18,80]
14.146
26.057
24.128
25. 25.991
9.250
8,747
9.796
10.060
Poincare 2D
12:199
12:1977
11:066
12:2009
12: 12:199
11,455
12:198
12.146
12 12:255
12.200
0:702
12. 196
8.205
10.753
Poincure
12:207
10.905
10.493
12.199
12:200
9.571
10.336
11.123
12.199
9.240
9.762
11.025
12.199
2D
069
12.335
11:555
9.563
10.608
13,250
11.
7.713
Loventz DIIRG
12.301
12.425
15.312
7.588
11.385
8.267
radis Jahookaze uniss
Lorentz 3D
12.415
12,649
12 12.930
9.707
10:67
12.323
12,589
9.541
9.349
10.523
12.521
penalty
14.229
13.977
14.613
15.825
10.267
15.077
8.4.18
16.588
23,586
24,909
8,038
7:713
8.015
10.060
annea12
7.794
7.782
7.775
7.774
7:761
7,600
7.602
7.680
7.751
anneal3
5.735
3.747
3.694
3.725
3.712
3,650
3.692
3.650
3,702
coalescemi2
19:585
19.939
16 666
18.6
16 608
14,570
12.492
261
11,262
12.401
11.716
13 13.962
coalescent3
16.666
18.645
16,008
14.570
12.492
16.261
11.262
12.401
11.716
13,962
KVK
18.332
17.367
12.97
10.063
11.699
CLOVE
19:585
66666
12.492
22/819
16.261
11,262
12:401
11.716
CLOVE DHRG LPCS
34.590
25.
700
23.591
738
16.208
27. 37.455
18.176
23, 23,944
13.283
13,835
13. 13,753
15.710
19:385
19.939
16.666
18.545
15,008
14,570
12.492
22.849
16.261
20.151
11:262
12:401
11:716
13.962
HMCS
19:585
19.939
16.666
18.648
16,608
12.492
22. .49
16.261
20.153
11. 11.262
12. 12.401
11.216
13.962
Mercator fast
70081
64.700
42. 144
33.310
31.315
22.056
26 306
173.262
33.330
63.496
16.438
20. 20.775
23. 148
28. 747
Merestre full
56.490
$2.331
40 500
$1,965
50.507
25.973
369
35,790
63.223
16 16.761
15 15.998
24 585
50 318
Mercator
22:544
19.129
17:288
10:118
17.850
13.956
14,604
16.498
16.730
hilling
8.N22
9.195
BEKL
0.208
0.278
0.130
0.320
0.531
0:756
0.827
0.128
0.459
0.547
0.454
0.575
0.381
0.358
BFKI- DERG
0.195
0.264
0.487
0.298
0.541
0.755
0.829
0.098
0.465
0.561
0.458
0.585
0.387
0.575
Poincare
0.392
0.003
0.685
0.389
0.348
0.438
0.492
0.630
0.384
0.576
Poincaro
0.597
0.772
0.135
Poincare
0.670
0,714
0.72
0.187
0.666
0.600
0.728
0.519
0.671
Lovenzz
0.375
0,4%
0.607
0:542
0.345
0.413
0.145
0.494
0.654
0.386
0.774
Locentz 2D DHKG
0.439
0.365
0.745
0.530
0.632
0,796
0.397
0,094
0.475
0.355
0.482
0.651
0.369
0.586
MAP: Mean Avenge Precision
Laventz 3D
0-464
0.597
0.785
0.576
0,68
0.758
0.020
0.129
0.509
0.471
0.577
0.722
0:495
0.651
penally
0,238
0,211
0.461
0.365
0.340
0.755
0.719
0,111
0.442
0,669
0.441
0.368
0.315
0.313
anneal2
0.61
0.603
0.653
0.841
0.393
0.535
0.675
0.479
0661
anncals
0.672
0.654
conlescent2
0.044
0.015
0.106
0.3K7
0.208
0.302
0.455
0.506
conlescent3
0.185
0.215
0.359
0,161
0,303
0.528
0.289
KVK
0.657
0.587
0.760
0.314
0.496
0.418
CLOVE
0.12
0.625
0.685
0.584
0.555
0,777
0.307
0:557
0.380
0.095
0.461
0.375
0.415
0.463
CLOVE DHRG LPCS
0.495
0.618
0.723
0.599
0.593
0.787
0.813
0.365
0,670
0.667
0.467
0.580
0.418
0.552
0.302
0.164
0.356
0.313
0.266
0,645
0.621
0.148
0.220
0.565
0.276
0.339
0.216
0.408
HMCS
0:469
0.567
0.613
0.524
0.519
0.673
0.372
0.517
0.581
0:443
0.567
0.384
0.539
Mercator fast Merentoe full
0.222
0.253
0.680
0.218
0.488
0.455
0.336
0.411
0.270
0.520
0.265
0.754
0.584
d-Mercator
0.078
0.310
0,216
0.113
0,045
0.370
0.339
orig TroeRap ree orig LeeeRop norec
0.437
0.355
0.817
0.572
0.205
0.241
0.243
0.333
0.392
0.668
0.360
0.816
0.852
0.561
0.204
0.259
0.250
0.227
hillnn
0.599
0.319
0.577
0.402
01488
D3637
Euchideam 30D
B996
0.968
1000
0.990
1.00
1.000
1.000
0.452
1100
0.831
1:000
1.000
1:000
019133
Euclidean 200D
0.994
0.975
1000
0.984
1.00
1,000
1000
0.836
1.000
0.856
1.000
1.000
1.000
1.000
BPKL.
1890.0
1717.0
169:2
558.2
X4.0
50.4
9.2
7660.
56.1
709.7
38.0
50.9
52.0
104.0
BFKL- - DHRG
1763.9
1719.2
156.5
558.6
62.1
45.9
8.7
6897.2
53.8
680.1
37.6
48.6
49.7
99.7
Poincare 2D
1153.7
887.3
76.0
298.5
45.3
32.7
6.3
6567.3
67.8
537.2
21.6
31.4
46.6
96.2
Poincare 3D
924.3
647.8
40.8
242.3
43.0
21.2
3.8
5522.7
359.9
28.1
25:1
39.3
84.1
Poincare
4913
363
4894.0
80.0
2D
81.4
205.3
55.2
31.5
44.5
Lorentz -DHRG
116088
1069.2
89.8
320.5
57.3
6688.5
78.4
677.2
22.8
47.6
96.5
Lorentz 3D
$76.1
617.1
51.5
202.6
11.9
22.2
$585.8
58.0
337.7
27/2
25.0
38.7
$4.2
penalty
1571.0
1496.5
144.6
481.6
62.9
52.0
3.5
6989.7
65.3
911.0
37.0
46.2
49.1
97.9
MR: Mcanlankk
anneaf2
161.1
71.8
69.9
16.1
70.3
13.4
40.4
46.3 3
94.0
anneal3
116.5
25.6
63.9
7.8
$2.11
27.3
233.3
27.8
29.7
cualescen12
2086.1
3404.7
437.4
623.6
96.3
63.2
14.1
115.6
38.6
41.11
39.1
107.8
conlescent5
512.6
213.8
11.6
164.9
2807
49.8
113.8
KVK
47.6
CLOVE
1821.7
750.5
12.2
111.1
685,0
52.7
166.0
CLOVE+ DIIRG LICS
1244.6
1124.8
106.1
375.0
$1.1
42.2
11.2
3082
68.0
704.5 8
11.8
51.2
52.6
120.5
1853.9
1359.6
110.5
398.2
140.6
46.3
11.8
7656.8
158.9
686.1
30.3
64.6
73.8
149.9
HMCS
1827.8
1355.5
120.6
420.5
91.7
13.1
7563.
90.8
670
41.9
54.6
66.9
127.1
Mencator East
2081.3
20-b..1
209.5
686.9
101.4
$1.0
7.7
8073.4
69.0
951_3
27.7
41.5
54.1
103.5
Mercator full
1513.3
1539.9
153.9
476.8
77.8
45.7
6.7
61.5
770.3
34.2
41.2
47.5
99.7
MeMrrctor
3350.7
1659.3
172.5
27.3
430.9
45660.
24.3
92.8
hilling
43.0
Buchideam
1:0
1.0
110
1.0
Euchidean 200D
1.0
1.0
1.0
1.0
1.0
1.0
1.1
1.0
1.0
1.0
1.0
1.0
Table 4: Our results on real-world networks. Darker cell color indicate better results.
23

---

Published as a conference paper at ICLR 2026
Bragh noase
astrop
condma
0rge
hegph
faoebo
yesst
discos
follow
opcn8
185
CHlegs
Human]
Diasal
Mouses
BFKL
0,080
0.026
0.052
0.072
0.163
0,061
0.155
0.047
0.515
0.191
0.775
0.778
0.519
0.905
BPKI DHRG
0:077
0.029
0.070
0.451
0.064
0.186
0.051
0.475
0.77%
0.300
0.652
0.919
Poincarr
0.080
0:515
0.149
0.235
0.036
0.409
0.897
0.541
0.727
Poincare
0.250
0.146
0.0
0.542
0.16
0.261
0.08
0.607
0.935
0.915
0.847
0.964
Lorentz 2D
0.053
0.099
0.169
0.460
0.141
0.220
0.035
0.395
0.300
0.899
0.880
0.747
0.963
GSSR Greoddy Success Ratio
Locenzz 2D DHRG
0.227
0.103
0.115
0.173
0.487
0.135
0.175
0U637
0.400
0.506
0.840
0.925
0.686
0.925
Losentz 3D
0,272
0.144
01176
0.208
0.474
0.229
12.770
0.996
0.455
02712
03929
0.921
0.863
0,982
penilty
0.075
0.017
01044
0.085
0.184
0.063
0.250
0.026
0.517
0.47
0.775
0.757
0.6%
0.915
anneal2
0.072
0.396
0.086
0.152
0.358
0.929
0.835
0.965
anneals
0,390
0.89
0.847
coalescent2
0.034
0.006
0.019
0,271
0.045
0.128
0.249
0.405
0.5.9
0.427
coalescent3
0.055
0.072
3.00
0.051
0:160
0.226
0.625
0.616
0.569
KVK
0.109
0.113
0.098
0.164
0.892
0.779
CLOVE
11242
0.140
0.306
0.471
0.111
0.166
0.506
11/02
0.764
0.866
0.862
0.844
1L972
CLOVE + DHRG L,PCS
0,384
0.211
0.124
0.250
0.457
0.101
0,100
0.404
0.024
0.720
0,825
0.800
0,780
0,934
0.152
140
0.084
0. 119
0.327
0,084
0.128
0.107
0.189
0.61
0.356
0.471
0.453
0.780
IMCS
0.367
0.191
0.105
193
011
0.156
0.083
0.680
0.820
0.828
0.793
Mercatoe fast
0.014
0.048
0.155
0.062
0.430
0.524
0.457
Mercator
0.013
01068
0.482
0.068
0.568
0.536
0.868
0.784
0.783
d-Mereator
0.014
0.005
0.013
0.029
0.127
0.023
0.044
0.127
0.066
0.587
0.78%
0.562
lilling
001002
0.126
0:205
0.369
0.897
0.855
BFRI.
0:035
0:035
0.00M
0.00
0.688
0.574
BFKL- DEIRG
0.027
0.066
0.063
0.119
0.018
0018
0,166
0.696
0.582
Poincare
0.120
0.054
0.094
137
0.228
0.032
0 0,300
0.778
0.733
0.630
0.870
Poincure 313
0:221
0.127
01138
0.227
0.534
0.201
0.253
00087
0.000
0.830
0.815
0.758
0.553
GREE: Greoddy Routing E euuy
Lacentz 2D
0.047
01192
0. 150
0.442
0.1 14
0.215
0,030
0.372
1.344
0.778
0.769
0.644
0.867
Lovente 2D + DIIRG
0.191
0.050
0.106
0.154
0.428
0.130
0.172
0.033
0.378
0.487
0,735
0.799
0.598
0.853
Lorentz 3D
0:231
0.126
183
0.468
0.26
0.085
0.435
00681
0.852
0.822
0.756
0.951
penalty
0,066
0,010
00078
0
0,025
0.403
0.359
0.382
0
0:812
0.814
0.745
anneal3
0.488
0,362
0.865
0.812
0:751
cualescent2
0.032
0.006
0018
0.052
0.370
0.044
0.125
0.235
0.452
0.4K]
0.305
0.776
conlescent3
0.053
0.069
0.298
0,050
0:157
0.216
0.584
0.553
0.529
0,805
KVK
01099
0.406
0.094
0.161
0.782
0.684
CLOVE
B.300
19194
0.122
01240
0:454
0.306
0.162
0:419
0.747
0.749
0.727
0,740
0,922
CLOVE DHRG LPCS
0:311
0.175
0.113
0.11
0.448
0.097
0.157
0.345
0.501
0.609
0.724
0.683
0.608
0.900
0.125
0.117
0.078
0.107
0.319
0.081
0.126
0.095
0.007
0.488
0,416
0.344
0.152
HMCS
0:161
0.152
0.649
Mercator Merestoe d: &MMercator hillig
0.014
0.037
0,363
0.44
0.150
0.058
0,421
0.424
034800
0.476
0,4077
0.755
D:121
0.040
0.061
0.434
0.067
0.54
0.514
0.761
0.683
0.698
0.915
0.013
0.005
0.013
0.027
0.113
0,023
0.0.9
0.117
0.004
0.358
0.702
0.520
0,839
01094
10120
0.348
0:275
0.743
BEKL
0.271
0:452
0.554
0.586
0.230
0.457
0.799
0.283
0.371
0.259
BFKL - DIIRG
0.321
0.298
0.534
0.496
0.605
8.608
0.83
0.308
0.514
0.616
0.302
0.397
0.293
0.901
Poincare 213
0:415
01456
0.675
0.649
0.714
0.614
0100
02722
0.621
0.510
0.348
0.43%
0.301
0,409
Poincaur 3D)
0:548
0.994
0.770
0.7
0.748
0.655
88556
0.631
0:415
0.552
0.368
0.461
LI L.:.oomallee and logi kothhoo
Lorente 2D+ DIRR
Locentz 2D
0:501
0.542
0.445
0.0622
0.727
0
0,638
0070
0.720
0.708
0,343
0.650
0,799
0.794
0.314
0.265
0.6 8365
0.806
0.179
0.590
0.352
0,349
0.463
0,450
0,318
0,311
0.417
0.007
3D
0.746
0.381
0.634
penalry
0.541
0.675
0.388
0.701
0.385
0.555
0.311
0.413
0.303
anneal2
0,703
0.557
0.718
0.367
0.875
0.344
anneal3
0.72
0.565
0:812
0.619
0:415
0.597
0.385
0.475
conlescent2
0:210
0.108
0.251
0.470
0,188
0,415
0.381
0.501
0.222
0.329
0.181
0.348
coalescent3
0 1939
0.467
0.174
0.18)
0.425
0.399
0.236
0.386
0.242
0.352
KVK
01669
D0662
1.:42
0.744
0.345
0.327
CLOVE
B1419
0.532
0.606
0:444
0.567
0.640
0.694
0.385
0.507
0.284
0.358
0.240
0.180
CLOVE DHRG
0.570
0.639
0.719
0.435
0.085
0.303
0,380
0.364
LPPSS
0.500
0.586
0.594
0.575
0.46
0.641
0.168
0.201
0.123
0.250
HMCS
0.375
0.519
0.488
0.587
0.347
0.57%
0.629
0.266
0.354
0.188
0.322
Merestoe faet
0,253
0.246
0.425
0.47
0.545
0.535
250
0.567
0.483
0.262
0.308
0.251
0.368
Mercatoe full
0,384
0.361
0.554
0.574
0.625
0.805
0.030
0.003
0,360
0.39]
0.327
0
d-Mescator
0.022
0.033
0.254
0.183
0.425
0.174
0.354
0.122
0103
0.290
0,490
0.313
0.414
BFKL
0.466
0.550
0.644
0.353
0.456
0.487
0,467
0.428
0.513
0.567
0.510
BFKL- - DEIRG
0,536
0.462
0.447
0.558
0.662
0,335
0.450
0.494
0,488
0.428
0.507
0.571
0,510
0.609
Poincare 2D)
B:587
013532
0.575
0.066
0.725
0.433
0.515
0.538
115995
01513
0.530
0.575
0.535
0.613
Poincare 3D
0.595
0.537
0.531
0.656
0.696
0.373
0,440
0.535
0.45
0. 0.462
0.572
0.4k]
0.612
ICV. Informaiion Control Value
Losentz 2D
0.553
0.378
0.662
0.725
0,450
0.329
0.538
0.597
0.501
0,500
0.588
0,533
0.613
Lorentz 2D+ DIRR
8.807
0.565
0.584
0.681
0.725
0,436
0,520
0.513
0.593
0.199
0.534
0.588
0,531
0.615
3D
0.598
0.537
0.573
0.439
0.536
0.544
0,450
0.484
0.579
0.488
penalty
0,359
0.519
0.368
0.459
0.523
0.586
anneal2
0.730
0.558
0.535
0,500
0.534
0.620
anneal3
0.733
0.555
0.614
0.511
0.635
0,537
0.634
conlescent2
0.425
0.427
0.567
0.616
0.553
0.44]
0,552
0,475
0.3.5
0,467
0,584
coalescent
0. 1666
0. 506
0.325
0.285
3.34
0,456
0.408
0.4k7
0,412
0.563
KVK
0.193
0.683
0,368
03:16
0.319
0,512
CLOVE CLOVE DHHIG LIBSS
0.553
0.510
0.507
0.561
0.634
0,375
0.457
0.517
0.525
0.169
0.185
0.54]
0.478
0,530
0.567
0.458
0.619
0.658
0.333
0.438
0.525
0.456
0.478
0.5.5
0.482
0.530
0.501
0.606
0.448
0.400
0.520
0:464
0.465
0.504
0.464
0.351
HMCS
0.541
8.506
0.496
0.612
0.605
0.448
0.508
0.551
0465
0.482
0,540
0.476
0.573
Mercator fast
0.435
0.544
0.349
0.482
0.601
0.290
0.348
0.299
0.667
0.250
0.459
0.506
0.454
0.579
Mercatoe full
0.479
0.580
0.370
0.520
0.632
0.274
0.302
0.47]
0.300
0,475
0.54]
0.442
0.589
d-Mercator
0.480
01452
0. 0.386
0.485
0.575
0.353
D:414
0.474
0412
0.419
0.525
0.412
0.589
Table 5: Our results on real-world networks. Darker cell color indicate better results. logeliicl (LL) is normalized by dividing by the value of N from Section Cland subtracting from 1.
24

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Published as a conference paper at ICLR 2026
Figure Quality assessment of embedders on real-world hierarchies and networks: MAP.
Figure 8: Quality assessment of embedders on real-world hierarchies and networks: -log(MR).
experimental results might be caused by differences in hyperparameters; the repository only gives the values of hyperparameters used to reproduce the 10-dimensional embeddings.
H REPEATED EXPERIMENTS
In Tables 8 and 9 we list the results of repeated experiments the NOUN hierarchy and the GRQC, YEAST, MOUSE3, HUMANI, DROSOPHILAI and CELEGANS networks. most cases, the differences minor and do not affect the rankings.
25

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Published as a conference paper at ICLR 2026
Figure 9: Quality assessment of embedders real-world networks: greedy success rate (GSR).
Figure 10: Quality assessment of embedders on real-world networks: greedy routing efficiency (GRE).
26

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Published as a conference paper at ICLR 2026
Figure 11: Quality assessment of embedders real-world networks: -log(GSFF)
3,5
Figure 12: Quality assessment of embedders on real-world networks: information control value (ICV).
27

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Published as a conference paper at ICLR 2026
MMS
MMC
HO
LPC
Pryttons dee3
Caloe
PARC
LOWOLE COR2
BC C002
AMMOO
RIEL oven
CHIE
Figure 13: Aggregates for MR, ICV, GSR, and GSF measures real-world networks and hierarchies.
Misaao
Lewe 220-000
Paoo sireno
Nerrate Bat
IMCS
DEAL
Figure 14: Aggregates for MR, ICV, GSR, and GSF measures on simulated networks.
28

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Published as a conference paper at ICLR 2026
comparison
comparison
comparison
rottas +0000
cot3 +400
--b0 +450
+400
4031
10
400
400
comparison -log(MR), T=0.1
comparison -lg(MRI, T=0.4
comparison -lggMRR, T=0.7
comparison HoggMRL, T-0.9
1-000
comparison of greedy success, T-0.1
comparison of greedl success, T-0.4
comparison of greedy success, T-0.7
comparison of greedy success, T-0.9
641000
comparison of -lggsttetcch,, T-01
compartson of -logistretch,, T-04
compartson of -logistrezch,, T-077
comparison of -logistretchi T-0.9
144000
141000
541000
2-
comparison of GREE, T:0.1
comparison GRE, T=0.4
comparison SRE.RR..ee T:0.7
comparison G T=0.9
1-1000
1-1000
+1100
comparison 14000
comparison
comparison
14000
14000 1000
14200
Figure 15: Density plots of the differences between the values of quality measures obtained by Lorentz 2D and BFKL embedders. Top to bottom: MAP, -log(MR), GSR, -log(GSF)) GRE, ICV. Left to right: 0.1, = 0.4, T 0.7, 0.9. Negative values indicate that BFKL embedder performed better.
29

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Published as a conference paper at ICLR 2026
MAP
GSR
GSF
ICV
Coett.
Pit FD)
Coeti.
Pr(> EE)
Coeff.
Pr( FD
Coeft.
Prc 1ED
Coeft.
Intercept
1,9701
7.09--08
-1,4818
2.
0.685I0
0.007420
-1.2612
1,28e-07
43.3744
9.77e-14 14
Temp=0.4
-0.8881
0,003800
-0,3620
0.159600
-2.1166
3.12e-12
-1.1232
5.73e-05
-1.4332
0.000874
Temp=0.7
4,4173
5.20c-14
-0.5747
0.028400
-3.9175
<22-16
-0.6677
0.010650
-41914
8.69e-10
Temp=0.9
4,4173
5.20c-14
-0,2939
0.251100
-4,3455
<2c-16
0.15180
0.538150
-4,6607
1.99c-07
1.6316
0.70c-05
1.0513
3.07e-05
0.92710
0.001930
0.76950
0.001960
2.37400
7.85e-05
Size 2000
3.7975
<6
1,8908
1.95e 14
2.53030
2.84c-15
1,58290
6.58e-11
6,30360
1.13e-08
RaFKi//pp
-53.9572
5.63c-14
600
600
600
ACC.v
0.8835
0.6917
0.8231
0.7266
0,9266
ACCaench
0.7867
0.6683
0.6200
0.6833
0.8967
0.6165
0.2486
0.6133
0.2717
0,5011
Table 6: Results of logit regressions for the determinants of BFKL embedder outperforming Lorentz 2D embedder in terms of quality measures. ACC and are average accuracy and Kappa from 20-fold cross vooos-vvlii,tion; ACCbench is the accuracy of the naive model (always predict mode). Results for GRE in Table
graph name
NOUN
VERB
ASTRO
COND
GRQC
HEPPH
Poincaré 2D MR (ours)
88.0
15.7
1127.0
889.4
68.8
302.2
Poincaré 2D MR
90.7
10.7
Poincaré 2D MAP (ours)
0.105
0.314
0.324
0.391
0.660
0.472
Poincaré 2D MAP
0.118
0.365
Lorentz 2D MR (ours)
43.0
42.1
1104.8
949.2
81.4
293.2
Lorentz 2D MR
22.8
3.64
Lorentz 2D MAP (ours)
0.168
0.184
0.306
0.345
0,599
0,417
Lorentz 2D MAP
0.305
0.579
Euclidean 50D MR (ours)
1.5
1.2
1.0
1.0
1.0
1.0
Euclidean 50D MR
1281.7
Euclidean 50D MAP (ours)
0.921
0.908
0.988 0.376
0.968 0.356
1.000 0.522
0.980
Euclidean 50D MAP
0.140
0.434
Table 7: Our results compared with the results from [Nickel and Kiela, 2017; Nickel and Kiela, 2018].
I TRIVIA ABOUT THE NOUN DATASET
This Appendix gives details about our experiments with the NOUN dataset, i.e., the WordNet hyper- nymy structure. This the first hierarchy that PE/LE have been benchmarked on, common in ML studies.
We get MAP of 0.284 using BFKL which is significantly better than the result of Poincaré 2D of 0.118, but not the result of Lorentz 2D of 0.305, according to NNickel and Kiela (2018). However, our results are different: 0.105 for 2D PE and 0.168 for 2D LE. Furthermore, while the PE/LE papers mention the good performance of their embedding methods, on our machine, BFKL is almost 100 times faster than LE, which is especially impressive given that BFKL runs on single CPU. Furthermore, the DHRG improvement improves the BFKL embedding from dMAP 0,050 to dMAP 0.411, while LE is improved from dMAP 0.192 to dMAP 0.320. This suggests that the layered approach of BFKL. produces a better structure of the embedding. Furthermore, the combination of BFKL+DDRG still more than 10 times faster than 2D LE. (The dMAP result of 0.050 very low compared to the continuous result of 0.284; this seems to be an outlier, in other experiments the results of MAP and dMAP are very similar.)
This is very large hierarchy, so it is not feasible to run slower embedders on it. We have also run the new CLOVE embedder, which achieves MAP of 0.769, which is significantly better than the earlier two-dimensional embedders. Furthermore, it runs over 3 times faster than BFKL. It is possible to apply the DHRG imrpovement to this embedding, obtaining an even better value of MAP (0.791).
These results consistent across multiple runs (Table 10
30

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Published as a conference paper at ICLR 2026
graph name
grqe
yeast
CElegs
Human
Drosol
Mouse3
nodes
4158
1458
279
493
350
1076
edges (undirected)
13422
1948
2287
7773
2887
90811
BFKI.
0.059
0.062
0.781
0.779
0.629
0.905
(0. 035.0 0.063)
(0.060.0.003)
(0.770.0.7922 792)
(0. 7.2742...0 9033
(0.618,0. (..41)
(0.902.0.900)
BFKL DHRG
0.057
0.062
0.777
0.638
0.921
053,0
(0.629.0.646)
(0.918,0.925)
Poincare 2D
0.146
0,894
0.864
0.733
0.942
(0. 10.13200.154 1044
(0.879.0 900)
(0.845.0 880;
(0.721.0.2
(0 935,00 (36)))))))))))))))))))
Poincare 3D
01.332
0.939
0.917
0.830
0.968
(0.913:0.922)
(6.19900.11))
Lorentz 2D
0.107
0.137
0,889
0.882
0.743
0.938
GSR
(0. 102,0.114
(0.131 .0: 144)
(0.876,00 501)
(0 456. 9077
[0. 0515
(0.9 10-935,00 03310.041)
Lorentz 3D
0.24.0
0.921
0.922
0.834
0,966
(0.284.0.2455)
928)
penalty
0.058
0.062
0,782
0.785
U. 0.622
0.915
(0. 00000,.000 1.4.0.00r 0.06G)
(0. (0.058 (0.058,0.064)
(0.771.0.790)
(0. 10.762.00 (62.0.807)
(0.602 0..33)
(0.908.0.919)
anneal2
0.076
0,087
0.994
0.918
0,841
0.968
10.073.0.00)
(0.063.0.091)
(0.913,0.
808.08.000
(0.530,0.851)
11.966.0.37
anneal3
0.091
0.119
0.931
1.948
0.849
0.950
(0. 085, O 000)
00.113.0.122 125)
(0 (0.9225,00 1.500.000 905)
B6S)
83G 864)
10.947.0.800 0522)
KVK
0.119
0.099
0.871
0.79.
(0. 105,0
(0.098.0.100
5844
(0.784.0.812)
CLOVE
0. 158
0.105
0.844
0.83
0.829
0.968
1:6,0 100)
(0 0..064..00 1099
(0 8511 957)
(0-81100880 85
85000 2599
(7D)
LPCS
0.085
0.090
0.518
0.526
0.432
0.760
(0:081,
(0. 0.0066.00 06050.00))
(0.494,,0
(0.280,0.5085
(0.417.0.443)
(0.7 (0.745.0.74)
HMCS
0.106
0.8 18
0.804
0.784
0.939
105,0
(0.806.00
(0. (0.770.0.830)
0.796)
(..99.0..41)
Mercator fast
0.048
0.524
0.534
0.440
0.830
(0. 0..048.0.049)
(0.525,0 525)
(0. 0..532,0.3 535;
(0. 00.436,0.449
828,0.0.000 (2.)0.0.332)
Mercator full
0.008
0,845
0.797
0.755
0.960
(0.067.0.0m) 0.0700
00.83.0
(0. (0.787.0.811))
00.744.0.772
1..99.0..00
d-Mercator
0.015
0.021
0,582
0,788
0.563
0,881
(0.014.0.016)
(0.019.0.023)
(0.379.0. 588))
(0. (0.787.0.789)
(0.562 1.62.0.200 (.32)
(0.880.0.883
BFKL
0.056
0.061
0.699
0.674
0.561
0.850
(0.053.0.00
(0.059.0 0.0.)
(0.685.0.50 6009)
(0.643.0.00 695)
(0.563300.500)
(0.849.0.88
BFKI, DHRG
0.054
0.061
0.689
0.698
0.571
0.857
(0.0 (0.051.0.061)
(0. (0.067.0. (8.63)
(0.081.0.00 694)
(0.672.0. 7.350
(0.564.0.077)
(0.853.0.80) (602)
Poincure 2D
0.139
0.776
0.750
0.633
0.869
(0.126.0.
(0.765,00
(0-138.0.761)
(0. (0.624,0.64) 0.2400.644
863,663,997
Poincare 3D
0.217
0.835
0.817
0.745
0.937
(0.829.0)
[0. 11.756.0.75
Lorentz 2D
0.099
0.131
0.771
0.768
0.644
0.865
GRB
(0. 00..09...11 105)
(0.120 02.126,0. 136)
(0.762,0 (70))
(0. 0..74700786)
(0.637 (.650)
801 D. 0.800)
Lorentz 3D
0.225
0,824
0.820
0,750
0.935
(0.219.00
(0.818.0
818.0.8223
0.777
penalty
0.055
0.061
0,686
0.683
0.546
0,842
10.051.0.00)
(0.067.0 (6.66.0022)
(0.677.0 0L677,007011 701)
(0.661.0.60)
(0.928, (0.5260..588)
(0.857.0.845)
anneal2
0.070
0.083
0,807
0.798
1.733
0.894
(0.080,00 0.087
(0.799.0.
10.788.0.802)
00.728.0.741
(0.890.0.80)
anneal3
0.084
0.113
0.825
0.860
0.752
0.891
(0.078.0.08)) 088)
(0.168.0.115)
(0.816,0. SUT)
(0 223.0.8789
(0. (0.741.1.765)
(0.889.0.900 0999)
KYK
0.765
(0.094.0.00 (..000
(0.756.0
(0.682.0.77)
CLOVE
0.136
0.100
0.738
0.701
0.727
0.918
(0. 127.0 142)
00.090.0.110
(0.727.0 745)
(0. 00.892.0.717
(0.2 7L.788. (1350.735)
(0. 119.1.00 (20)
LPCS
0.079
0.087
0.458
0.734
(0.429:0.d85;
(0.719.0.547)
HMCS
0.098
0.720
0.682
0.688
0.901
(0. 100)
(0.710.0 738)
(0.653.00 (0.659.0.702)
(0.654.8.cm .01.)
(0.595.0.mm 906)
Mercator fast
0.048
0,481
0.475
0.410
0.786
(0.047
(0.174:0.476)
00.40600.444)
Mercator full
0.066
0.744
0.693
0.679
0.916
(0.065 (066.00068)
(a. 00.751.0..3 7061
(0.686. 1.96.00.00 7.70)
(0. 6.668 6 (.00
(8.9 (1.915.0.916
d-Mereator
0.014 (6.014.0.016)
0.021
0.533
00.004
0.522
0.840
(0.019.0.025)
00708.00705
(0.521 (0.54)
Table Repeated experiments: greedy routing measures. Mean values from 5 runs. Bootstrapped confidence intervals in brackets.
31

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Published as a conference paper at ICLR 2026
graph
grge
yeast
CElegs 279
Human
Drosol
Mouse3
nodes
4158
1458
493
350
1076
edges (undirected)
13422
1948
2287
7773
2887
90811
BFKL
0.735
0.462
0.568
0.384
505)
409)
(0,300,0
0.387
1.50900.564
BFKI. + DHRG
0.497
0.735
0.464
0.575
0.391
0.527
(0.489 (15)))
(0. T23.0.747
(0.460,0.467)
(0.560,0.50) 587)
(0. (0.384.0. 11644,..977
(0.674.0.575)
Poincare
0.671
0.493
0.640
0.387
0.575
(0,003,0
Poincare 3D
0.765
01576
0.719
0.482
01.632
60 T20.0.771)
(0.5 (0.573,0.550)
[0.716,0 T22)
(0 476.4.660000 0.466)
(0, (0.651.0) 653)
Lorentz
0.612
0.535
0.645
0.390
0.573
5.5.0.540)
(0.57100.544)
MAP
Lorentz 3D
0.764
0.574
0.715
0.489
0.633
60 758 0.700.720
00.521003770
[0:711,0:7110
(0. del 181.0.00 0.44533
(O (1.52200.00
penalty
0.470
0.733
0.451
0.577
0.370
0.574
(0.464.0.477)
720.0.740
00.444.0.457)
(0.561.0.55
(0. 5777)
anneal2
0.613
0.644
0.537
0.670
0,479
0.610
[0.611.0. 614)
638 o (.649)
(0 00.5366.0 (.539)
(0. [0.008,0 673)
(0. 47..0.4.554)
(0. 00.07. (.111)
anneal3
0.6448
0.675
0.585
0,800
0.515
0.652
(0.644.0.60
671.6.6675
(0.769.0
00.112.00 (11200.020)
(0. 6.63).333
KVK
633
0.750
0.489
0,432
10.374.0 0.44..000 683)
742.0.700 7.6))
(0. 063.863...0
(0. 422.0..266
CLOVE
0.688
0.775
(1.460
0.561
0.415
0.466
10.683.0 685)
4G8.0.463)
(0.546.0.35 571)
(0. 10..000..222
(0.46..04773)
LPCS
0.558
0.652
0.277
0.360
0.212
0.397
10.587 0 581)
664,0 081
(0.255.4) (...0.2599)
[0.346,0 375)
(0. (0.200: 0.944
(0.367.0 0.47.04407
HMCS
0.616
0.451
0.566
0.388
0.533
10.614.0.00 619)
(0.444.0.43)
[0.507,0 572)
(0. 10.378.0.20
1..31.0.331
Mercator fast
0.682
0.337
0.41
0.270
0.520
(0.336.0 00.336.0.337)
(0.411.0.413) 41..)
(0.270.0.250)
(0.520.0) 5200)
Mereator full
0.753
0.486
0.549
02.423
0.583
952 05760-755
(0. 06..6.9..495
(0.548,0 (.51,0.551)
(0 (0-418.0.1k) 0.300
(0. (0.584,0 585)
d-Mercator
0,317
0.215
0.370
0.636
0.346
0.583
(0.314.0.319)
236.0.233)
00.367..0 0..70)
(0.362.0.300)
orig TreeRep rec
0.199
0,264
0,233
0,236
(0.191.0.200)
(0.249.0.211) 217)
(0. 00.220,.000
(0.227.0.244
orig TreeRep norec
0,266
0.233
0.244
(0.207
(0.254,0 278)
0.246)
(0. (0.232,0.22
BFKL.
171.1
53.4
37.5
50.1
53.0
102.
1.811.1181.5)
(81.6.56.5)
(38.7.38.7
(48. 6.5.6.
(52 (52.4.33.8)
(101.5.163
BFKL DIRG
153.8
49.0
36.9
48
49.9
98.9
(180 1,0.5,163. 163.
(46 (63.5,350)
(36.0.37
(46.3, 49.8)
(49.6,50.4)
(0e 4,09
Poincare
32.7
31.4
42.8
47.
96.5
(30. 7.33.5)
(91.1.31.7]
(39.9. 10.1)
(46.6.47.9)
(96 26.3
Poincare 3D
22.1
27.3
25.7
39.6
84.3
(21 5.22
(20.9 27.87
(26 3, 3,96
(30.2.30.8)
(64.1,6) 5)
Lorentz
75.3
38,3
31.6
40.5
47.2
MR
4,79
(37 38.98.9
(31.4.915
(47.0.47.4)
(96
Lorentz 3D
21.6
27.2
26.5
39.3
$4.0
(20 000.8.22
(27.0.
(26.0,27
(38 $..99)
13.9,844
penalty
143.0
56.3
36.2
45.6
49.8
97
(53.0.64.2)
(35.3.37.2
(44.0.46.3)
(49.3,50.3)
(96.7.97.
anneal2
156.8
74.2
33.2
42.2
46.4
93.8
(.107.1
(71.2.76.7)
(32.5,88.8
(41. (41.0.43.3)
(46.3.46.))
(93 63.6.94.00
anneal3
59.2
27.3
19.3
38
4,127.0)
KVK
130.1
49.6
36.1
47.4
(125 (125.4.141
(48 2230.77
(35.2 36
(47.2.47.6)
CLOVE
42.9
55.6
63.0
164.9
(6)
(48
[42.0.33
(61.7.86.0)
(161.0,107:0
LPCS
119.8
47.8
50.7
63.4
73.2
157.7
(116.9.122.0
(46.
(48.9.52.1) 1)
(61-4,65.0)
(72.6.78.7)
(152:2116..7)
HMCS
119.6
43.6
54.6
68.2
133.1
111822,1209)
(42.6,15.1
(53-5.55.4)
(67.3,09.4)
Mercator fast
50.8
37.8
41.5
54..3
103.5
(30.7 (0.7750.5)
(37.7.87.5)
(41.5.41.6)
(54.3.54.4)
(108.5.168
Mercator full
45.7
34.2
41.2
47.8
99.5
(45.
(34.0.34.3)
(41.1.41.5)
(47.6.47.9)
d-Mercator
500.6 (478.4. 478.44500.2
187.8 201.77
34.7
24
41.0
82.8
(34.3.35.0)
(24 (2221)
(40.9:41.1)
(92 8.02
Table 9: Repeated experiments: MAP and MR measures. Mean values from 5 runs. Bootstrapped confidence intervals in brackets.
32

---

Published as a conference paper at ICLR 2026
embedder
MAP
dMAP
MR
dMR
BFKL.
0.280
0.049
64.7
807.5
(0.276,0.cm 286)
(0.045,0.049)
(63.3,66.0)
(784.3,824.3)
BFKL + DHRG
0.452
0.442
34.8
34.9
(0. (0.4300 0040))
40)))
(33. (.3.7.37.3
(33.9,37.3)
Poincare 2D
0.105
0.018
89.5
2536.2
(0.104,0. 101))
(0.018.0.018)
(88.0.90.7)
(2300.6, 25,5665.1
Poincare 2D DHRG
0.062
370.3
(0.062.0.063)
(351.7,388.9)
Poincare 3D
0.490
16.1
(0.487 .0.492))
(15.7 (15.7.16.3)
Lorentz 2D
0.196
0.193
41.8
42.0
(0.195.0.1m) 1966)
(U.191,0.193)
(41.2, 42.3)
(41.4,42.4)
Lorentz 2D DHRG
0..26
0.324
30.1
28.4
(0.323,0.320)
(0.321.0.3)
(29.2,30.9)
(27.6,29.0)
Lorentz 3D
0.506
14.8
(0. (0.506,0 505)
(14.7.15.0)
CLOVE
0.769
0.630
18.8
23.1
(0. (0.763 0.777)
(U (0.6244.0000 636)
(18.3.19.7)
(22.6 (22.6.24.1)
CLOVE + DHRG
0.791 (0.788.0.70)
0.783
21.7
20.9
(0.779.0.787)
(21.1,22.6)
(20.5,21.6)
Table 10: Repeated experiments on the NOUN hierarchy. Mean values from runs. Bootstrapped confidence intervals in brackets.
33


Reviews:
[{"review_id": "Za1JlePeSd", "reviewer": "Reviewer_1A68", "summary": "The paper aims to provide a systematic, cross-community comparison of hyperbolic embedding algorithms developed in machine learning (ML), network theory (NT), and algorithmic graph research.\nThe authors aim to close the gap between these communities by experimentally evaluating 14 embedding methods on 30 real-world and 450 simulated networks, comparing both embedding quality and computational efficiency.\nThis work aims to bridge ML and algorithmic perspectives on hyperbolic embeddings, attempting to show that older algorithmic methods are far more computationally efficient without sacrificing accuracy, and proposes ICV which is intended to be a fairer, theory-grounded quality metric for comparing embeddings.", "questions": "I kindly ask to the authors to address all the concerns I raised in the section Weaknesses above.\nHowever, in my opinion this study needs to be totally restructured and re-written and my recommendation  is to withdraw and resubmit in a next conference a majorly improved version of this study.", "limitations": "", "rating": 2, "confidence": 5, "soundness": 1, "presentation": 1, "contribution": 1, "strengths": "The main strength of the paper is the mission to offer a wide comparison of algorithms for hyperbolic embedding of networks across domains.", "weaknesses": "The comparison on artificial networks is misleading. The authors should use benchmarks that create ground-truth networks generated in the hyperbolic space using the nonuniform popularity similarity model (nPSO) that allows also the creation of mesoscale community structure in artificial networks as many real networks have. Then they should use measures for the order of nodes on angular coordinates, the hyperbolic distance correlation and the angular separability index of the community. In particular, the presence of community in the benchmark is fundamental because they are a property that justifies why methods such as Hypermap and HypermapCN did not work well, because being based on the simple PSO model they did not have a parameter to model the community organization of real networks. \n\nThe test on real networks are not convincing because as a matter of fact the authors do not offer any quantitative evidence that these networks are hyperbolic and the measures they use are all empirical and ill posed. For instance, assuming that a network is embedded better because their greedy stretch factor is higher is misleading. In reality, every network has a complex system on the back with an intrinsic navigability and over-estimating it is wrong. \n\nThe measure of Greedy stretch factor for navigability is old and surpassed. Measures such as the greedy routing efficiency and the geometrical congruency are more advanced tools to quantify the navigability.\n\nThe authors do not consider in their comparison important studies:\n+ one of the recent states of the art proposed in this article: \nCLOVE, a Travelling Salesman’s approach to hyperbolic embeddings of complex networks with communities. SG Balogh, B Sulyok, T Vicsek, G Palla. Communications Physics 8 (1), 397\n+ a fast method based on network automata: \nMinimum curvilinear automata with similarity attachment for network embedding and link prediction in the hyperbolic space.A Muscoloni, CV Cannistraci. arXiv preprint arXiv:1802.01183\n+ the studies of Filippo Radicchi on this topic seems to be neglected. \n+ It does not seem that the Authors mentions explicitly the algorithms Hypermap and HypermapCN in their initial review. These are inefficient algorithm with low performance but they are between the first methods offered. Papadopoulos, F., Psomas, C. & Krioukov, D. Network mapping by replaying hyperbolic growth. IEEE/ACM Trans. Netw 23, 198–211 (2015). \nPapadopoulos, F., Aldecoa, R. & Krioukov, D. Network geometry inference using common neighbors. Phys. Rev. E 92, 22807 (2015).\n+  It does not seem that the Authors mentions explicitly the algorithms of LPCS of Wang, Z., Wu, Y., Li, Q., Jin, F. & Xiong, W. Link prediction based on hyperbolic mapping with community structure for complex networks. Phys. A 450, 609–623 (2016).\nThis is one of the fastest ever algorithm proposed, and it should be considered together with the other evolutions proposed by the same authors in subsequent articles that the Authors can find themselves by reviewing the literature of these relevant scientists. \n7. This study seems to be neglected together with the following of the same authors: Martin Keller‑Ressel and Stephanie Nargang authored “HYDRA: a method for strain-minimizing hyperbolic embedding of network- and distance-based data”.\n\nCoalescent embedding, which is a machine learning method, was published on arXiv on [Submitted on 21 Feb 2016, Machine learning meets network science: dimensionality reduction for fast and efficient embedding of networks in the hyperbolic space], hence before the other machine learning studies listed. \n\nThe study does not report enough information to replicate the experiments. For many algorithms, it is not reported the way their versions or hyperparameters are selected. \n\nThe organization of the article needs a strong effort to improve  the quality of the information reported.\nFor instance the authors could propose a first figure that organizes the different algorithms in a genealogical tree in order of date of publication and relationship of methodology used. \nThe main article could report the selected results for the best methods and the appendix the full results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4eLAUduqBc", "reviewer": "Reviewer_61kf", "summary": "Hyperbolic embedding is a well-studied technique for graph representation learning. This paper presents a comprehensive comparison of algorithms from the ML/DL community and the traditional algorithm community. It presents a convincing survey that shows: the most popular methods derived by the ML/DL community (e.g. by Nickel and Kiela) are inferior to classic algorithms (e.g. by Blasius et al) in terms of both computational efficiency and embedding quality. This is a problem that many ML/DL researchers vaguely noticed but failed to explicitly point out. I highly appreciate the contributions of this work.", "questions": "1. Line 288-290, the authors claim that \"The hMDA method from (Sala et al., 2018) looks interesting, but it depends on the scaling factor, and it is not clear how to learn this parameter; therefore we do not include this method in our experiments\". Do you mean the hMDS method? You need to explain more clearly why this method does apply to your experimental setting.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. This paper presents a comprehensive and clear survey of both ML/DL approaches to hyperbolic embedding as well as traditional algorithmic approaches. The computational complexity of each approach is clearly discussed.\n\n2. Very comprehensive experimental results to support the authors' claims.\n\n3. Good writing styles -- the authors wrote a very clear preliminary section on hyperbolic geometry for the readers' information, followed by separate sections on algorithmic approaches and ML/DL approaches.", "weaknesses": "1. Some citations should be in brackets, e.g. on Line 42.\n\n2. Figure 2, 3, and 5 are difficult to read -- the markers are too small and too similar. The same problem applies to the figures in the Appendix. If these figures are scaled to fit the page limit, I'd suggest re-design Table 1 to save some space instead of sacrificing the readability of your most illustrative figures.\n\n3. Different hyperbolic embedding strategies perform differently on different structures, be it a tree, a deep but sparse graph, a shallow but dense graph, or a nearly fully-connected graph. You can refer to this ICLR workshop paper: https://arxiv.org/pdf/2407.16641?. The concept of local capacity explains why some methods work better on certain datasets. You may want to include more varied network datasets in your experiments, or generate datasets with distinct structures to test the methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JfLaBN3ayP", "reviewer": "Reviewer_VCXG", "summary": "The paper presents more like a survey with experiments bridging hyperbolic embedding methods in machine learning and classical network embedding algorithms. It reviews learning-based hyperbolic embeddings algorithms commonly used in ML with embeddings algorithms from network theory. The authors argue that the ML community often focuses on gradient-based methods, overlooking algorithmic approaches from network theory, that could show better efficiency with comparable performances.", "questions": "- How do the authors envision static embeddings from network theory adapting to downstream ML tasks that require trainable representations?\n\n- what causes the non-reproducibility of Poincaré/Lorentz experiment results?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "The paper highlights the diversity of hyperbolic embedding methods and draws attention to classical embedding algorithms from network theory. Conducted experiments for both sides algorithms and draw comparison. \n\nIt makes the point that representations from network theory is more efficient and could even performs better than ML learning-based algorithms tracing from Nickel and Kiela (2017, 2018) and thus should be favored, cited and adopted. \n\nThe survey-style overview may help readers unfamiliar with the historical algorithmic side of hyperbolic embeddings algorithms.", "weaknesses": "The criticism that ML work “fails to cite/use” algorithmic hyperbolic embedding literature is not very compelling. \n\n- The reason the ML community focuses more on works tracing from Nickel and Kiela (2017, 2018) is that, they allow gradient flow to the embedding layer, and trainable, end-to-end representations that adapt to downstream tasks, not just for embedding data (rather the reported performances serves as a demo of the learnt representation). Static embeddings from algorithmic approaches (network theory) cannot serve this role. Ultimately in my opinion, the hyperbolic learning community targets at an end-to-end hyperbolic networks that can replace existing Euclidean-based networks on some tasks. And it's already well-established how important learnable embedding layers are.\n\n- The focus on metrics is also somewhat misplaced, since for downstream tasks (e.g., node or graph classification), the rank and precision of distances is very relevant to downstream performance, that truly matters, especially considering current e.g. classification algorithms are largely dependent on the hyperbolic distance. \n\n- Some reported results appear unreliable—for example, the 2D Poincaré and Lorentz embeddings from Nickel and Kiela (2017, 2018) were not reproduced, even though they were already shown in many following ML works, can be reproduced. \n\n- The framing of the paper is also misleading: it reads more like a survey rather than a genuine attempt to bridge ML and algorithms due to the reasons stated above.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zPIhTbIdsw", "reviewer": "Reviewer_XXeR", "summary": "The paper systematically compares hyperbolic embedding methods spanning three communities. The headline result is that the BFKL algorithm is typically ~100× faster than Poincaré (NIPS’17) and Lorentz (ICML’18) embeddings while achieving similar quality under MAP/MR and greedy routing metrics. The authors also introduce an Information Control Value (ICV) criterion to penalize excessive radius/dimension, arguing that apparent gains from higher dimension are often optimization artifacts.", "questions": "No questions", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The work fills a real gap by bridging communities that have studied hyperbolic embeddings in isolation, delivering a broad, apples-to-apples experimental comparison across many methods, datasets, and metrics.\n2. The paper thoughtfully treats numerical stability and contributes the ICV criterion to counter dimension/radius inflation, leading to more balanced method selection. The inclusion of both real hierarchies and scale-free networks, plus synthetic HRG controls with regression analyses on temperature/size, strengthens the empirical narrative.", "weaknesses": "1. There is no theoretical support in the paper. \n2. Figures 2 and 5 are difficult to read.\n3. Despite the breadth, some baseline coverage choices limit conclusions, which may bias the landscape.\n4. Some evaluations rely on discrete variants (dMAP/dMR) to avoid precision issues, which can shift scores relative to standard definitions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)