Original: G LOBAL GRAPH CURVATURE
Revised: G LOBAL GRAPH CURVATURE

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Recently, non-Euclidean spaces became popular for embedding structured data.
Revised: Recently, non-Euclidean spaces became popular for embedding structured data.

Original: However, determining suitable geometry and, in particular, curvature for a given dataset is still an open problem.
Revised: However, determining suitable geometry and, in particular, curvature for a given dataset is still an open problem.

Original: In this paper, we deﬁne a notion of global graph curvature, speciﬁcally catered to the problem of embedding graphs, and analyze the problem of estimating this curvature using only graph-based characteristics (without actual embedding the graph).
Revised: In this paper, we deﬁne a notion of global graph curvature, speciﬁcally catered to the problem of embedding graphs, and analyze the problem of estimating this curvature using only graph-based characteristics (without actual embedding the graph).

Original: We show that optimal curvature essentially depends on the dimensionality of the embedding space and loss function one aims to minimize via embedding.
Revised: We show that optimal curvature essentially depends on the dimensionality of the embedding space and loss function one aims to minimize via embedding.

Original: We review existing notions of local curvature (e.g., Ollivier-Ricci curvature) and analyze their properties theoretically and empirically.
Revised: We review existing notions of local curvature (e.g., Ollivier-Ricci curvature) and analyze their properties theoretically and empirically.

Original: In particular, we show that such curvatures are often unable to properly estimate the global one.
Revised: In particular, we show that such curvatures are often unable to properly estimate the global one.

Original: We also make an important observation: larger dimensionality usually leads to less negative optimal curvature.
Revised: We also make an important observation: larger dimensionality usually leads to less negative optimal curvature.

Original: We propose a new estimator of global graph curvature which, in contrast to the existing ones, reﬂects this property as it is dimension-speciﬁc.
Revised: We propose a new estimator of global graph curvature which, in contrast to the existing ones, reﬂects this property as it is dimension-speciﬁc.

Original: We demonstrate that for a given dimension the new estimator is able to distinguish between “hyperbolic” and “Euclidean” networks.
Revised: We demonstrate that for a given dimension the new estimator is able to distinguish between “hyperbolic” and “Euclidean” networks.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Representation learning is an important tool for learning from structured data such as graphs or texts (Grover & Leskovec, 2016; Perozzi et al., 2014; Mikolov et al., 2013).
Revised: Representation learning is an important tool for learning from structured data such as graphs or texts (Grover & Leskovec, 2016; Perozzi et al., 2014; Mikolov et al., 2013).

Original: State-of-the-art algorithms typically use Euclidean space for embedding.
Revised: State-of-the-art algorithms typically use Euclidean space for embedding.

Original: Recently, however, it was found that hyperbolic spaces demonstrate superior performance for various tasks (Nickel & Kiela, 2018; Sala et al., 2018), while in some cases spherical spaces can be useful (Liu et al., 2017).
Revised: Recently, however, it was found that hyperbolic spaces demonstrate superior performance for various tasks (Nickel & Kiela, 2018; Sala et al., 2018), while in some cases spherical spaces can be useful (Liu et al., 2017).

Original: A key characteristic classifying the above-mentioned spaces is curvature, which is negative for hyperbolic spaces, zero for Euclidean and positive for spherical spaces.
Revised: A key characteristic classifying the above-mentioned spaces is curvature, which is negative for hyperbolic spaces, zero for Euclidean and positive for spherical spaces.

Original: These ﬁndings, therefore, show that certain graphs are better represented in spaces with non-zero curvature.
Revised: These ﬁndings, therefore, show that certain graphs are better represented in spaces with non-zero curvature.

Original: While some methods simply ﬁx curvature (e.g., -1 for hyperbolic space) and then ﬁnd the optimal embedding of the graph in the corresponding space (Nickel & Kiela, 2018), others try to learn the right curvature and embedding simultaneously (Gu et al., 2019).
Revised: While some methods simply ﬁx curvature (e.g., -1 for hyperbolic space) and then ﬁnd the optimal embedding of the graph in the corresponding space (Nickel & Kiela, 2018), others try to learn the right curvature and embedding simultaneously (Gu et al., 2019).

Original: In this paper, we analyze the problem of determining a graph curvature suitable for embedding without actual embedding the graph itself.
Revised: In this paper, we analyze the problem of determining a graph curvature suitable for embedding without actual embedding the graph itself.

Original: The aim is to use some simple graph characteristics to understand whether it is reasonable to embed a graph into some non-Euclidean space and if yes, then which curvature to choose.
Revised: The aim is to use some simple graph characteristics to understand whether it is reasonable to embed a graph into some non-Euclidean space and if yes, then which curvature to choose.

Original: Having such an estimator would save computational resources, since learning an embedding is computationally expensive and should be done for each space separately.
Revised: Having such an estimator would save computational resources, since learning an embedding is computationally expensive and should be done for each space separately.

Original: More importantly, it can also save human resources by showing whether it is worth investing resources in the implementation of a more complex embedding algorithm.
Revised: More importantly, it can also save human resources by showing whether it is worth investing resources in the implementation of a more complex embedding algorithm.

Original: We make an important ﬁrst step in this direction by introducing a concept of global graph curvature, which depends on both the dimension and loss function used for the embedding.
Revised: We make an important ﬁrst step in this direction by introducing a concept of global graph curvature, which depends on both the dimension and loss function used for the embedding.

Original: We consider two loss functions: distortion , which is widely used in embedding literature, and threshold-based loss (see Section 2), which is more suitable for some practical applications.
Revised: We consider two loss functions: distortion , which is widely used in embedding literature, and threshold-based loss (see Section 2), which is more suitable for some practical applications.

Original: We demonstrate, both theoretically and empirically, that these loss functions lead to fundamentally different graph curvatures.
Revised: We demonstrate, both theoretically and empirically, that these loss functions lead to fundamentally different graph curvatures.

Original: We also compare several estimators of the global curvature, in particular, the ones based on well-known Ollivier-Ricci and Forman-Ricci local graph curvatures.
Revised: We also compare several estimators of the global curvature, in particular, the ones based on well-known Ollivier-Ricci and Forman-Ricci local graph curvatures.

Original: Our analysis shows that all these notions give curvatures that are far from the optimal curvature for embedding the graph (in particular, because they are dimension-independent).
Revised: Our analysis shows that all these notions give curvatures that are far from the optimal curvature for embedding the graph (in particular, because they are dimension-independent).

Original: This raises the important question of how to properly estimate the best curvature for embedding graphs.
Revised: This raises the important question of how to properly estimate the best curvature for embedding graphs.

Original: We approach this problem by introduc- ing a new curvature estimator which, in contrast to all existing ones, is dimension-speciﬁc.
Revised: We approach this problem by introduc- ing a new curvature estimator which, in contrast to all existing ones, is dimension-speciﬁc.

Original: This estimator agrees with our theoretical results and is also able to distinguish between “hyperbolic” and “Euclidean” networks.
Revised: This estimator agrees with our theoretical results and is also able to distinguish between “hyperbolic” and “Euclidean” networks.

Original: One important insight, which we gained based on both theoretical and empirical analysis, is the fact that the optimal curvature is usually smaller for smaller dimensions.
Revised: One important insight, which we gained based on both theoretical and empirical analysis, is the fact that the optimal curvature is usually smaller for smaller dimensions.

Original: In other words, a network which seems to be negatively curved (for some small dimension) usually becomes more neutral as dimensionality grows.
Revised: In other words, a network which seems to be negatively curved (for some small dimension) usually becomes more neutral as dimensionality grows.

Original: Finally, while in the current research we assume that the global graph curvature is constant, this concept can be extended in the future to cover, e.g., the product spaces from (Gu et al., 2019).
Revised: Finally, while in the current research we assume that the global graph curvature is constant, this concept can be extended in the future to cover, e.g., the product spaces from (Gu et al., 2019).

Original: We focused on the constant curvature since, ﬁrst, such spaces are widely used in various applications (Tifrea et al., 2018) and, second, even for constant curvature we observed many non-trivial theoretical and empirical results.
Revised: We focused on the constant curvature since, ﬁrst, such spaces are widely used in various applications (Tifrea et al., 2018) and, second, even for constant curvature we observed many non-trivial theoretical and empirical results.

Original: These results and insights can eventually be extrapolated to analyze curvature in other, more complicated spaces.
Revised: These results and insights can eventually be extrapolated to analyze curvature in other, more complicated spaces.

Original: To the best of our knowledge, the problem of analyzing different notions of curvature for embedding graphs (even in simple spaces) was not considered before, so we hope that more results will follow.
Revised: To the best of our knowledge, the problem of analyzing different notions of curvature for embedding graphs (even in simple spaces) was not considered before, so we hope that more results will follow.

Original:  B ACKGROUND AND RELATED WORK
Revised:  B ACKGROUND AND RELATED WORK

Original: Graph embeddings For an unweighted connected graph G =
Revised: Graph embeddings For an unweighted connected graph G =

Original: ( V, E ) equipped with the shortest path distance metric d G , a graph embedding f is a map f : V → U , where U is a metric space.
Revised: ( V, E ) equipped with the shortest path distance metric d G , a graph embedding f is a map f : V → U , where U is a metric space.

Original: We refer to Goyal & Ferrara (2018) for a survey of several graph embedding methods and their applications.
Revised: We refer to Goyal & Ferrara (2018) for a survey of several graph embedding methods and their applications.

Original: The goal of an embedding is to preserve some structural properties of a graph.
Revised: The goal of an embedding is to preserve some structural properties of a graph.

Original: Depending on an application, different loss/quality functions are used to measure the quality of a graph embedding.
Revised: Depending on an application, different loss/quality functions are used to measure the quality of a graph embedding.

Original: The most well-known is distortion :
Revised: The most well-known is distortion :

Original: [Equation]
Revised: [Equation]

Original: Distortion is a global metric, it takes into account all graph distances.
Revised: Distortion is a global metric, it takes into account all graph distances.

Original: However, in some practical applications, it may not be the best choice.
Revised: However, in some practical applications, it may not be the best choice.

Original: For example, in recommendation tasks, we usually deal with a partially observed graph, so a huge graph distance between nodes in the observed part does not necessarily mean that the nodes are not connected in the full graph.
Revised: For example, in recommendation tasks, we usually deal with a partially observed graph, so a huge graph distance between nodes in the observed part does not necessarily mean that the nodes are not connected in the full graph.

Original: Additionally, as discussed in Section 4.3.1, graph distances are hard to preserve: there are simple graphs on just 4 nodes that can be perfectly embedded only in a space of curvature −∞ for any dimension.
Revised: Additionally, as discussed in Section 4.3.1, graph distances are hard to preserve: there are simple graphs on just 4 nodes that can be perfectly embedded only in a space of curvature −∞ for any dimension.

Original: Another measure, often used for embeddings, is Mean Average Precision (MAP), which, for a given node v , compares the distance-based ranking of other embedded nodes with the graphneighborhood-based ranking.
Revised: Another measure, often used for embeddings, is Mean Average Precision (MAP), which, for a given node v , compares the distance-based ranking of other embedded nodes with the graphneighborhood-based ranking.

Original: In contrast to distortion, MAP is scale-invariant, as it cares only about the order.
Revised: In contrast to distortion, MAP is scale-invariant, as it cares only about the order.

Original: Since changing curvature is equivalent to changing scale, MAP is insensitive to curvature.
Revised: Since changing curvature is equivalent to changing scale, MAP is insensitive to curvature.

Original: However, in practical applications like recommendation systems, ranking scores are usually complemented by minimum score thresholds to have personalized sizes of top recommended items.
Revised: However, in practical applications like recommendation systems, ranking scores are usually complemented by minimum score thresholds to have personalized sizes of top recommended items.

Original: This motivated us to consider another loss function, which is at the same time local and scale-dependent.
Revised: This motivated us to consider another loss function, which is at the same time local and scale-dependent.

Original: Let us deﬁne the following class of threshold-based loss functions.
Revised: Let us deﬁne the following class of threshold-based loss functions.

Original: Given an embedding f of a graph
Revised: Given an embedding f of a graph

Original: G , we (re)construct a graph G (cid:48) in the following way: v and u are connected in G (cid:48)
Revised: G , we (re)construct a graph G (cid:48) in the following way: v and u are connected in G (cid:48)

Original: iff d ( f ( v ) , f ( u )) ≤ 1 .
Revised: iff d ( f ( v ) , f ( u )) ≤ 1 .

Original: Then, any loss function which is based on the comparison of G and G (cid:48) is called thresholdbased.
Revised: Then, any loss function which is based on the comparison of G and G (cid:48) is called thresholdbased.

Original: While all our theoretical results hold for any threshold-based loss function (see Section 4), in our experiments we measure and compare several functions of this type including, e.g., zeroone loss for the edge classiﬁcation problem (whether a given node pair is connected or not), see
Revised: While all our theoretical results hold for any threshold-based loss function (see Section 4), in our experiments we measure and compare several functions of this type including, e.g., zeroone loss for the edge classiﬁcation problem (whether a given node pair is connected or not), see

Original: Appendix E.2 for deﬁnitions and discussion.
Revised: Appendix E.2 for deﬁnitions and discussion.

Original: Such loss functions are natural in many applications (graph reconstruction, link prediction, recommendations).
Revised: Such loss functions are natural in many applications (graph reconstruction, link prediction, recommendations).

Original: -1, 0, 1, corresponding to hyperbolic, Eu- clidean and spherical spaces.
Revised: -1, 0, 1, corresponding to hyperbolic, Eu- clidean and spherical spaces.

Original: Moreover, by considering a small enough region in hyperbolic or spherical space we get geometry similar to the Euclidean one, so for MAP it is important to distinguish only between -1 and 1.
Revised: Moreover, by considering a small enough region in hyperbolic or spherical space we get geometry similar to the Euclidean one, so for MAP it is important to distinguish only between -1 and 1.

Original: Hyperbolic and Spherical Spaces For many years, Euclidean space was the primary choice for data embeddings (Goyal & Ferrara, 2018).
Revised: Hyperbolic and Spherical Spaces For many years, Euclidean space was the primary choice for data embeddings (Goyal & Ferrara, 2018).

Original: However, it turned out that many observed datasets are well ﬁtted into hyperbolic space (Krioukov et al., 2010).
Revised: However, it turned out that many observed datasets are well ﬁtted into hyperbolic space (Krioukov et al., 2010).

Original: Nickel & Kiela (2017) have shown that such hyperbolic embeddings can improve state-of-the-art quality in several practical tasks, e.g., lexical entailment and link prediction.
Revised: Nickel & Kiela (2017) have shown that such hyperbolic embeddings can improve state-of-the-art quality in several practical tasks, e.g., lexical entailment and link prediction.

Original: On the other hand, spherical spaces are also used for embeddings (Liu et al., 2017).
Revised: On the other hand, spherical spaces are also used for embeddings (Liu et al., 2017).

Original: Gu et al.
Revised: Gu et al.

Original: (2019) goes even further by suggesting mixed spaces: product manifolds combining multiple copies of spherical, hyperbolic, and Euclidean spaces.
Revised: (2019) goes even further by suggesting mixed spaces: product manifolds combining multiple copies of spherical, hyperbolic, and Euclidean spaces.

Original: The main advantage of hyperbolic space is that it is “larger”: the volume of a ball grows exponentially with its radius.
Revised: The main advantage of hyperbolic space is that it is “larger”: the volume of a ball grows exponentially with its radius.

Original: Hence, such spaces are well suited for tree-like structures.
Revised: Hence, such spaces are well suited for tree-like structures.

Original: On the other hand, spherical spaces are suitable for embedding cycles (Gu et al., 2019).
Revised: On the other hand, spherical spaces are suitable for embedding cycles (Gu et al., 2019).

Original: Spherical and hyperbolic spaces are parametrized by curvature c , which is positive for spherical space and negative for hyperbolic space.
Revised: Spherical and hyperbolic spaces are parametrized by curvature c , which is positive for spherical space and negative for hyperbolic space.

Original: As c → 0 , geometry of both these spaces becomes similar to the Euclidean one.
Revised: As c → 0 , geometry of both these spaces becomes similar to the Euclidean one.

Original: We discuss some geometrical properties of these different spaces in Appendix A.
Revised: We discuss some geometrical properties of these different spaces in Appendix A.

Original:  L OCAL GRAPH CURVATURES
Revised:  L OCAL GRAPH CURVATURES

Original: While in this paper we analyze the global graph curvature, there are several local ones proposed in the literature.
Revised: While in this paper we analyze the global graph curvature, there are several local ones proposed in the literature.

Original: Many of them are based on the notion of sectional curvature and Ricci curvature deﬁned for Riemannian manifolds.
Revised: Many of them are based on the notion of sectional curvature and Ricci curvature deﬁned for Riemannian manifolds.

Original: Intuitively, Ricci curvature controls whether the distance between small spheres is smaller or larger than the distance between the centers of the spheres.
Revised: Intuitively, Ricci curvature controls whether the distance between small spheres is smaller or larger than the distance between the centers of the spheres.

Original: Ricci curvature is positive if small spheres are closer than their centers are.
Revised: Ricci curvature is positive if small spheres are closer than their centers are.

Original: We refer to (Jost, 2009; O’neill, 1983) for more details on Ricci curvature.
Revised: We refer to (Jost, 2009; O’neill, 1983) for more details on Ricci curvature.

Original: Ollivier curvature Ollivier curvature translates the deﬁnition of Ricci curvature to graphs.
Revised: Ollivier curvature Ollivier curvature translates the deﬁnition of Ricci curvature to graphs.

Original: Again, the idea is to compare the distance between two small balls with the distance between their centers.
Revised: Again, the idea is to compare the distance between two small balls with the distance between their centers.

Original: The distance between balls is deﬁned by the well-known optimal transport distance (a.k.a.
Revised: The distance between balls is deﬁned by the well-known optimal transport distance (a.k.a.

Original: Wasserstein distance or earth-mover distance).
Revised: Wasserstein distance or earth-mover distance).

Original: Formally, for a graph G we consider the shortest path metric on G , denoted by d G , and let W G 1 denote the Wasserstein metric with respect to the metric space ( G, d G ) .
Revised: Formally, for a graph G we consider the shortest path metric on G , denoted by d G , and let W G 1 denote the Wasserstein metric with respect to the metric space ( G, d G ) .

Original: Furthermore, for each node v we let m v denote the uniform probability measure on the neighbors of v , i.e., m v ( u ) = 1 u ∼ v deg( v ) , where deg( v ) denotes the degree of v .
Revised: Furthermore, for each node v we let m v denote the uniform probability measure on the neighbors of v , i.e., m v ( u ) = 1 u ∼ v deg( v ) , where deg( v ) denotes the degree of v .

Original: Then, the classic deﬁnition 3 of Ollivier curvature between two neighboring nodes v ∼ u in G is deﬁned as
Revised: Then, the classic deﬁnition 3 of Ollivier curvature between two neighboring nodes v ∼ u in G is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: We note that Ollivier curvature always belongs to the interval [ − 2 , 1] (Jost & Liu, 2014).
Revised: We note that Ollivier curvature always belongs to the interval [ − 2 , 1] (Jost & Liu, 2014).

Original: Forman curvature Forman curvature (Sreejith et al., 2016) is based on the discretization of Ricci curvature proposed by Forman (2003).
Revised: Forman curvature Forman curvature (Sreejith et al., 2016) is based on the discretization of Ricci curvature proposed by Forman (2003).

Original: It is deﬁned for a general weighted graph G , with both node and edge weights.
Revised: It is deﬁned for a general weighted graph G , with both node and edge weights.

Original: When the graph G is not weighted, the deﬁnition becomes:
Revised: When the graph G is not weighted, the deﬁnition becomes:

Original: [Equation]
Revised: [Equation]

Original: Forman was interested in a general discretization of curvature for Riemannian manifolds and his formula includes faces of any dimension.
Revised: Forman was interested in a general discretization of curvature for Riemannian manifolds and his formula includes faces of any dimension.

Original: Although this can be translated to graphs (Weber et al., 2017), the formula becomes quite cumbersome.
Revised: Although this can be translated to graphs (Weber et al., 2017), the formula becomes quite cumbersome.

Original: Therefore, in Equation 3 only 1 -dimensional faces (edges) are included.
Revised: Therefore, in Equation 3 only 1 -dimensional faces (edges) are included.

Original: One can extend this expression by including higher dimensional faces.
Revised: One can extend this expression by including higher dimensional faces.

Original: This was considered in (Samal et al., 2018), where 2 -dimensional faces on three nodes (triangles) were included.
Revised: This was considered in (Samal et al., 2018), where 2 -dimensional faces on three nodes (triangles) were included.

Original: In the case of an unweighted graph, we then obtain
Revised: In the case of an unweighted graph, we then obtain

Original: [Equation]
Revised: [Equation]

Original: where ∆ uv is the number of triangles that contain the edge ( u, v ) .
Revised: where ∆ uv is the number of triangles that contain the edge ( u, v ) .

Original: Based on the deﬁnitions, both Forman curvatures, especially F G ( u, v ) , are expected to often be highly negative.
Revised: Based on the deﬁnitions, both Forman curvatures, especially F G ( u, v ) , are expected to often be highly negative.

Original: Indeed, this is what we observe in Sections 4.3 and 5.
Revised: Indeed, this is what we observe in Sections 4.3 and 5.

Original: Heuristic sectional curvature
Revised: Heuristic sectional curvature

Original: A different notion of curvature used by Gu et al.
Revised: A different notion of curvature used by Gu et al.

Original: (2019) is based on the following geometric fact.
Revised: (2019) is based on the following geometric fact.

Original: Let abc be a geodesic triangle and let m be the (geodesic) midpoint of bc .
Revised: Let abc be a geodesic triangle and let m be the (geodesic) midpoint of bc .

Original: Then the value d ( a, m ) 2 + d ( b,c ) 2− d ( a,b ) 2 + d ( a,c ) 2is equal to zero in euclidean space, is positive in spherical space and negative in hyperbolic space.
Revised: Then the value d ( a, m ) 2 + d ( b,c ) 2− d ( a,b ) 2 + d ( a,c ) 2is equal to zero in euclidean space, is positive in spherical space and negative in hyperbolic space.

Original: For graphs, let v be a node in G , b, c neighbors of v and a any other node.
Revised: For graphs, let v be a node in G , b, c neighbors of v and a any other node.

Original: Then, we deﬁne
Revised: Then, we deﬁne

Original: [Equation]
Revised: [Equation]

Original: This resembles the formula above with m = v and the normalization constant 2 d G ( v, a ) is included to yield the right scalings for trees and cycles.
Revised: This resembles the formula above with m = v and the normalization constant 2 d G ( v, a ) is included to yield the right scalings for trees and cycles.

Original: To deﬁne the graph sectional curvature of a node v and its neighbors b, c , we average ξ G ( v ; b, c ; a ) over all possible a : ξ G ( v ; b, c ) =
Revised: To deﬁne the graph sectional curvature of a node v and its neighbors b, c , we average ξ G ( v ; b, c ; a ) over all possible a : ξ G ( v ; b, c ) =

Original: [Equation]
Revised: [Equation]

Original: |
Revised: |

Original: [Equation]
Revised: [Equation]

Original: |
Revised: |

Original: [Equation]
Revised: [Equation]

Original: v,b,c }
Revised: v,b,c }

Original: [Equation]
Revised: [Equation]

Original: G ( v ; b, c ;
Revised: G ( v ; b, c ;

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original:  G LOBAL GRAPH CURVATURE
Revised:  G LOBAL GRAPH CURVATURE

Original: The problem with these different notions of local graph curvature is that they cannot be easily used in practical applications, where data is usually embedded in a space of constant curvature.
Revised: The problem with these different notions of local graph curvature is that they cannot be easily used in practical applications, where data is usually embedded in a space of constant curvature.

Original: Hence, a global notion of curvature is needed.
Revised: Hence, a global notion of curvature is needed.

Original: In this section, we propose a practice-driven deﬁnition of global graph curvature, discuss how to estimate this curvature based on local notions and compare all curvatures theoretically and empirically for several simple graphs.
Revised: In this section, we propose a practice-driven deﬁnition of global graph curvature, discuss how to estimate this curvature based on local notions and compare all curvatures theoretically and empirically for several simple graphs.

Original: 4. D EFINITION
Revised: 4. D EFINITION

Original: For a graph G , let f ( G ) be an embedding of this graph into a d -dimensional space of constant curvature c (spherical, Euclidean or hyperbolic).
Revised: For a graph G , let f ( G ) be an embedding of this graph into a d -dimensional space of constant curvature c (spherical, Euclidean or hyperbolic).

Original: Assume that we are given a loss function L ( f ) for the embedding task (see Section 2).
Revised: Assume that we are given a loss function L ( f ) for the embedding task (see Section 2).

Original: Now, let L opt ( G, d, c ) be the optimal loss for given d and c :
Revised: Now, let L opt ( G, d, c ) be the optimal loss for given d and c :

Original: L opt ( G, d, c ) =
Revised: L opt ( G, d, c ) =

Original: min f L ( f ) .
Revised: min f L ( f ) .

Original: Then, we deﬁne d -dimensional curvature of G in the following way:
Revised: Then, we deﬁne d -dimensional curvature of G in the following way:

Original: [Equation]
Revised: [Equation]

Original: Note that there may be several values of curvature c delivering the minimum of L opt ( G, d, c ) , in this case we say that C Ld ( G ) consists of all such points.
Revised: Note that there may be several values of curvature c delivering the minimum of L opt ( G, d, c ) , in this case we say that C Ld ( G ) consists of all such points.

Original: Below we analyze global curvatures based on distortion ( C distd ( G ) ) and threshold-based ( C thrd ( G ) ) loss functions.
Revised: Below we analyze global curvatures based on distortion ( C distd ( G ) ) and threshold-based ( C thrd ( G ) ) loss functions.

Original: In the latter case, in experiments, we measure zero-one loss, but our theoretical results apply to any threshold-based loss, since L opt ( G, d, c ) reaches its minimum on “perfect” embeddings, where we precisely reconstruct the graph G .
Revised: In the latter case, in experiments, we measure zero-one loss, but our theoretical results apply to any threshold-based loss, since L opt ( G, d, c ) reaches its minimum on “perfect” embeddings, where we precisely reconstruct the graph G .

Original: 4. A PPROXIMATIONS
Revised: 4. A PPROXIMATIONS

Original: Let us discuss how local graph curvatures can be used to estimate the global one.
Revised: Let us discuss how local graph curvatures can be used to estimate the global one.

Original: In all cases, the standard practice is to average edge or sectional curvature over the graph.
Revised: In all cases, the standard practice is to average edge or sectional curvature over the graph.

Original: Ollivier
Revised: Ollivier

Original: [Equation]
Revised: [Equation]

Original: Average sectional curvature Let P 3 denote the number of paths of length 3 in G , then ξ ( G ) =
Revised: Average sectional curvature Let P 3 denote the number of paths of length 3 in G , then ξ ( G ) =

Original: [Equation]
Revised: [Equation]

Original: Further we slightly abuse notation by writing that C Ld ( G ) is a real value if such c is unique and a set of
Revised: Further we slightly abuse notation by writing that C Ld ( G ) is a real value if such c is unique and a set of

Original: d i s t o r t
Revised: d i s t o r t

Original: i o n curvature n = 5, d = 2 n = 5, d = 10 n = 100, d =n = 100, d = 0.
Revised: i o n curvature n = 5, d = 2 n = 5, d = 10 n = 100, d =n = 100, d = 0.

Original: z e r o - o n e l o ss curvature
Revised: z e r o - o n e l o ss curvature

Original: n = 5, d = 2 n = 6 d = 2 n = 7, d =n = 5, d = 10 n = 100, d = 2 n = 100, d =
Revised: n = 5, d = 2 n = 6 d = 2 n = 7, d =n = 5, d = 10 n = 100, d = 2 n = 100, d =

Original: It is important to note that all curvatures discussed above do not depend on neither dimension d nor loss function L .
Revised: It is important to note that all curvatures discussed above do not depend on neither dimension d nor loss function L .

Original: However, as we show below, global curvature deﬁned in Section 4.1 signiﬁcantly depends on them.
Revised: However, as we show below, global curvature deﬁned in Section 4.1 signiﬁcantly depends on them.

Original: Let us also mention that there is a concept of Gromov’s hyperbolicity (Gromov, 1987), which is sometimes used to decide whether it is reasonable to embed a graph to a hyperbolic space.
Revised: Let us also mention that there is a concept of Gromov’s hyperbolicity (Gromov, 1987), which is sometimes used to decide whether it is reasonable to embed a graph to a hyperbolic space.

Original: However, ﬁrst, this estimator cannot be easily converted to curvature, and, second, it does not say anything about the sphericity of data.
Revised: However, ﬁrst, this estimator cannot be easily converted to curvature, and, second, it does not say anything about the sphericity of data.

Original: 4. T HEORETICAL ANALYSIS OF GLOBAL CURVATURE AND ITS APPROXIMATIONS
Revised: 4. T HEORETICAL ANALYSIS OF GLOBAL CURVATURE AND ITS APPROXIMATIONS

Original: To better understand the performance of the proposed approximations of global graph curvature, we shall consider several basic graphs and analyze their global curvature and approximations both theoretically and empirically.
Revised: To better understand the performance of the proposed approximations of global graph curvature, we shall consider several basic graphs and analyze their global curvature and approximations both theoretically and empirically.

Original: By studying these graphs we also hope to gain insights into how classic graph topologies inﬂuence the curvature of the space in which they can be properly embedded.
Revised: By studying these graphs we also hope to gain insights into how classic graph topologies inﬂuence the curvature of the space in which they can be properly embedded.

Original: For more complex graphs, e.g., for combinations of simple graphs studied in this section, it becomes extremely hard to prove rigorous theoretical results.
Revised: For more complex graphs, e.g., for combinations of simple graphs studied in this section, it becomes extremely hard to prove rigorous theoretical results.

Original: However, the obtained intuitions work in such scenarios, as we illustrate and discuss in Appendix C.
Revised: However, the obtained intuitions work in such scenarios, as we illustrate and discuss in Appendix C.

Original: The experimental setup for our empirical illustrations is described in Appendix E.3.
Revised: The experimental setup for our empirical illustrations is described in Appendix E.3.

Original: 4.3. S TAR S n
Revised: 4.3. S TAR S n

Original: It is pointed out in numerous papers that trees are negatively curved.
Revised: It is pointed out in numerous papers that trees are negatively curved.

Original: We analyze this theoretically and start with the simplest tree: one central node and n leaves.
Revised: We analyze this theoretically and start with the simplest tree: one central node and n leaves.

Original: We denote this graph by S n and assume that n ≥ 3 .
Revised: We denote this graph by S n and assume that n ≥ 3 .

Original: Ollivier curvature Consider any tree graph T , let v, u be two neighbors.
Revised: Ollivier curvature Consider any tree graph T , let v, u be two neighbors.

Original: Then Propositionin (Jost & Liu, 2014) states that
Revised: Then Propositionin (Jost & Liu, 2014) states that

Original: [Equation]
Revised: [Equation]

Original: where t + = max { 0 , t } .
Revised: where t + = max { 0 , t } .

Original: In particular, if either deg( v ) = 1 or deg( u ) = 1 , then κ T ( u, v ) = 0 .
Revised: In particular, if either deg( v ) = 1 or deg( u ) = 1 , then κ T ( u, v ) = 0 .

Original: As a result, for a star we have κ S n ( u, v ) = 0 , so κ ( S n ) = 0 and stars are not negatively curved according to Ollivier curvature.
Revised: As a result, for a star we have κ S n ( u, v ) = 0 , so κ ( S n ) = 0 and stars are not negatively curved according to Ollivier curvature.

Original: Forman curvature If follows from Equation 2 and Equation 

Revised: Forman curvature If follows from Equation 2 and Equation

Original: that F ( S n ) =
Revised: that F ( S n ) =

Original: ˆ F ( S n ) = 3 − n , so stars are highly negatively curved for large n according to Forman curvature.
Revised: ˆ F ( S n ) = 3 − n , so stars are highly negatively curved for large n according to Forman curvature.

Original: Average sectional curvature Heuristic sectional curvature is deﬁned for a node and its two neighbors.
Revised: Average sectional curvature Heuristic sectional curvature is deﬁned for a node and its two neighbors.

Original: In case of a star we can only take a central node v and two neighboring ones b and c .
Revised: In case of a star we can only take a central node v and two neighboring ones b and c .

Original: For any other node a we obtain ξ S n ( v ; b, c ; a ) = − 1 .
Revised: For any other node a we obtain ξ S n ( v ; b, c ; a ) = − 1 .

Original: Therefore, by averaging we obtain ξ ( S n ) = − 1 .
Revised: Therefore, by averaging we obtain ξ ( S n ) = − 1 .

Original: Distortion-based curvature We prove the following theorem (the proof is in Appendix B.3).
Revised: Distortion-based curvature We prove the following theorem (the proof is in Appendix B.3).

Original: Theorem 4.1.
Revised: Theorem 4.1.

Original: Assume that d and n are ﬁxed.
Revised: Assume that d and n are ﬁxed.

Original: If c is bounded below by a constant, then D ( S n ) is also bounded below by a constant.
Revised: If c is bounded below by a constant, then D ( S n ) is also bounded below by a constant.

Original: If c → −∞ , then the optimal distortion D opt ( S n ) =
Revised: If c → −∞ , then the optimal distortion D opt ( S n ) =

Original: Θ (cid:16) 1 √− c (cid:17) .
Revised: Θ (cid:16) 1 √− c (cid:17) .

Original: Therefore, for any S n we have C distd ( S n ) =
Revised: Therefore, for any S n we have C distd ( S n ) =

Original: −∞ .
Revised: −∞ .

Original: This result is illustrated in Figure 6 (left): distortion decreases as curvature becomes smaller.
Revised: This result is illustrated in Figure 6 (left): distortion decreases as curvature becomes smaller.

Original: The intuition behind this result is the following: we cannot embed a star S 3 with zero distortion into a space of any constant curvature and any dimension, because in case of zero distortion the central node v has to lie on the geodesics between all pairs of leaves, so all 4 nodes have to belong to one geodesics, which is impossible.
Revised: The intuition behind this result is the following: we cannot embed a star S 3 with zero distortion into a space of any constant curvature and any dimension, because in case of zero distortion the central node v has to lie on the geodesics between all pairs of leaves, so all 4 nodes have to belong to one geodesics, which is impossible.

Original: Moreover, the same problem occurs if any graph G contains S 3 as an induced subgraph.
Revised: Moreover, the same problem occurs if any graph G contains S 3 as an induced subgraph.

Original: On the other hand, if c → −∞ , we can spread all leaves of S n uniformly on a circle of radius 1 around the central node and distortion of such construction will tend to zero since the distance between the pairs of leaves will tend to 2 (triangles become thinner).
Revised: On the other hand, if c → −∞ , we can spread all leaves of S n uniformly on a circle of radius 1 around the central node and distortion of such construction will tend to zero since the distance between the pairs of leaves will tend to 2 (triangles become thinner).

Original: Further we will see that if we minimize a threshold-based loss, then any tree can be perfectly embedded with d = 2 .
Revised: Further we will see that if we minimize a threshold-based loss, then any tree can be perfectly embedded with d = 2 .

Original: Threshold-based curvature
Revised: Threshold-based curvature

Original: Here we have the following theorem (the proof is in Appendix B.5).
Revised: Here we have the following theorem (the proof is in Appendix B.5).

Original: Theorem 4.2.
Revised: Theorem 4.2.

Original: C thrd ( S n ) =
Revised: C thrd ( S n ) =

Original: ( −∞ , C ) for some C = C ( n, d ) , which increases with d and decreases with n .
Revised: ( −∞ , C ) for some C = C ( n, d ) , which increases with d and decreases with n .

Original: In particular, for d = 2 , if n < 6 , then C = (cid:16) arccos
Revised: In particular, for d = 2 , if n < 6 , then C = (cid:16) arccos

Original: [Equation]
Revised: [Equation]

Original: (cid:17) 2 ; if n = 6 , then C = 0 ; if n > 6 , then C = − (cid:16) 2 arccosh
Revised: (cid:17) 2 ; if n = 6 , then C = 0 ; if n > 6 , then C = − (cid:16) 2 arccosh

Original: [Equation]
Revised: [Equation]

Original: The result is illustrated in Figure 6 (right).
Revised: The result is illustrated in Figure 6 (right).

Original: Note that zero-one loss can be noisy, especially on small graphs, since it is threshold-based, i.e., discrete.
Revised: Note that zero-one loss can be noisy, especially on small graphs, since it is threshold-based, i.e., discrete.

Original: 4.3. T REE T b WITH BRANCHING FACTOR b
Revised: 4.3. T REE T b WITH BRANCHING FACTOR b

Original: We consider a tree T b , b ≥ 2 .
Revised: We consider a tree T b , b ≥ 2 .

Original: For symmetry, assume that the ﬁrst node has b + 1 children, while all other nodes have a parent and b children.
Revised: For symmetry, assume that the ﬁrst node has b + 1 children, while all other nodes have a parent and b children.

Original: Also, for Ollivier, Forman and threshold-based curvatures, we assume that the depth of T b is inﬁnite.
Revised: Also, for Ollivier, Forman and threshold-based curvatures, we assume that the depth of T b is inﬁnite.

Original: In other cases, our statements hold for any ﬁnite tree T .
Revised: In other cases, our statements hold for any ﬁnite tree T .

Original: Ollivier curvature We can use Equation 6 and get, for any edge, κ T b ( u, v ) =
Revised: Ollivier curvature We can use Equation 6 and get, for any edge, κ T b ( u, v ) =

Original: − 2 (cid:16) 1 − 1 b +1 − 1 b +1 (cid:17)
Revised: − 2 (cid:16) 1 − 1 b +1 − 1 b +1 (cid:17)

Original: = − 2 + 4 b
Revised: = − 2 + 4 b

Original: +1 .
Revised: +1 .

Original: So, κ ( T b ) = − 2 + 4 b +1 , which is negative.
Revised: So, κ ( T b ) = − 2 + 4 b +1 , which is negative.

Original: Forman curvature It is easy to see that we have F ( T b )
Revised: Forman curvature It is easy to see that we have F ( T b )

Original: =
Revised: =

Original: ˆ F ( T b ) = 4 − 2( b + 1) =
Revised: ˆ F ( T b ) = 4 − 2( b + 1) =

Original: 2(1 − b ) .
Revised: 2(1 − b ) .

Original: Average sectional curvature In contrast to Ollivier and Forman curvatures, heuristic sectional curvature is global, i.e., it depends on the whole graph, which has to be ﬁnite.
Revised: Average sectional curvature In contrast to Ollivier and Forman curvatures, heuristic sectional curvature is global, i.e., it depends on the whole graph, which has to be ﬁnite.

Original: Note that for any tree, to compute sectional curvature, we average 0 and -1.
Revised: Note that for any tree, to compute sectional curvature, we average 0 and -1.

Original: As a result, for any tree T we have ξ ( T ) ∈ [ − 1 , 0] (Gu et al., 2019).
Revised: As a result, for any tree T we have ξ ( T ) ∈ [ − 1 , 0] (Gu et al., 2019).

Original: Distortion-based curvature On the one hand
Revised: Distortion-based curvature On the one hand

Original: , our result for S n implies that if a graph contains
Revised: , our result for S n implies that if a graph contains

Original: S 3 , then it cannot be embedded with zero distortion in any space.
Revised: S 3 , then it cannot be embedded with zero distortion in any space.

Original: One the other hand, Sarkar (2011) proves that if we scale all edges by a sufﬁciently large factor τ , then the obtained tree can be embedded to the hyperbolic plane with distortion at most 1 + ε with arbitrary small ε .
Revised: One the other hand, Sarkar (2011) proves that if we scale all edges by a sufﬁciently large factor τ , then the obtained tree can be embedded to the hyperbolic plane with distortion at most 1 + ε with arbitrary small ε .

Original: Note that multiplying graph edges by τ is equivalent to changing curvature from 1 to τ 2 .
Revised: Note that multiplying graph edges by τ is equivalent to changing curvature from 1 to τ 2 .

Original: As a result, Sarkar (2011) proves that we can achieve an arbitrary small distortion if c → −∞ .
Revised: As a result, Sarkar (2011) proves that we can achieve an arbitrary small distortion if c → −∞ .

Original: Hence, C distd ( T ) =
Revised: Hence, C distd ( T ) =

Original: −∞ for any T .
Revised: −∞ for any T .

Original: Threshold-based curvature We prove the following theorem (see Appendix B.6).
Revised: Threshold-based curvature We prove the following theorem (see Appendix B.6).

Original: Theorem 4.3.
Revised: Theorem 4.3.

Original: C thr d ( T b ) =
Revised: C thr d ( T b ) =

Original: ( −∞ , C ) for
Revised: ( −∞ , C ) for

Original: some C = C ( b, d ) , which increases with d and decreases with b .
Revised: some C = C ( b, d ) , which increases with d and decreases with b .

Original: The following lower bound holds:
Revised: The following lower bound holds:

Original: [Equation]
Revised: [Equation]

Original: Actually, the bound above holds for any tree whose branching is bounded by b .
Revised: Actually, the bound above holds for any tree whose branching is bounded by b .

Original: Interestingly, while it is often claimed that trees are intrinsically hyperbolic, to the best of our knowledge, we are the ﬁrst to formally prove that trees can be perfectly embedded in a hyperbolic plane of some curvature.
Revised: Interestingly, while it is often claimed that trees are intrinsically hyperbolic, to the best of our knowledge, we are the ﬁrst to formally prove that trees can be perfectly embedded in a hyperbolic plane of some curvature.

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: 4.3. C OMPLETE GRAPH K n
Revised: 4.3. C OMPLETE GRAPH K n

Original: Ollivier curvature For any two nodes u and v , it follows from Example 1 in (Jost & Liu, 2014) that κ K n ( v, u ) = n − 2 n − 1 .
Revised: Ollivier curvature For any two nodes u and v , it follows from Example 1 in (Jost & Liu, 2014) that κ K n ( v, u ) = n − 2 n − 1 .

Original: Thus, κ ( K n ) = n − 2 n − 1 and it tends to 1 as n → ∞ .
Revised: Thus, κ ( K n ) = n − 2 n − 1 and it tends to 1 as n → ∞ .

Original: Forman curvature Simple computations yield: F ( K n ) = 6 − 2 n , ˆ F ( K n ) = n , i.e., we get either highly positive or highly negative value.
Revised: Forman curvature Simple computations yield: F ( K n ) = 6 − 2 n , ˆ F ( K n ) = n , i.e., we get either highly positive or highly negative value.

Original: Average sectional curvature It is easy to compute that ξ ( K n ) = 18 .
Revised: Average sectional curvature It is easy to compute that ξ ( K n ) = 18 .

Original: Distortion-based curvature
Revised: Distortion-based curvature

Original: The following theorem analyzes C dist d ( K n ) if d = n − 2 (see Appendix B.4 for the proof).
Revised: The following theorem analyzes C dist d ( K n ) if d = n − 2 (see Appendix B.4 for the proof).

Original: Theorem 4.4.
Revised: Theorem 4.4.

Original: [Equation]
Revised: [Equation]

Original: Note that the optimal curvature −∞ appears here since embedding of a complete graph can be reduced to embedding of a weighted star.
Revised: Note that the optimal curvature −∞ appears here since embedding of a complete graph can be reduced to embedding of a weighted star.

Original: This theorem is illustrated in Figure 2.
Revised: This theorem is illustrated in Figure 2.

Original: For n = 4 , d =we expect to see the minimum at about 3 . 65 , for n = 12 , d = 

Revised: For n = 4 , d =we expect to see the minimum at about 3 . 65 , for n = 12 , d =

Original: we expect 2 .
Revised: we expect 2 .

Original: 76 , which is indeed the case.
Revised: 76 , which is indeed the case.

Original: For other parameters, we do not have theoretical results, however we see sudden drops in positive curvature.
Revised: For other parameters, we do not have theoretical results, however we see sudden drops in positive curvature.

Original: Finally, as expected, distortion decreases as curvature becomes small.
Revised: Finally, as expected, distortion decreases as curvature becomes small.

Original: Threshold-based curvature C thrd ( K n ) =
Revised: Threshold-based curvature C thrd ( K n ) =

Original: R , since we can embed any complete graph perfectly by mapping all nodes to one point.
Revised: R , since we can embed any complete graph perfectly by mapping all nodes to one point.

Original: 4.3. C YCLE GRAPH C n
Revised: 4.3. C YCLE GRAPH C n

Original: We consider a cycle C n with n ≥ 4 .
Revised: We consider a cycle C n with n ≥ 4 .

Original: [Figure]
Revised: [Figure]

Original: d i s t o r t
Revised: d i s t o r t

Original: i o n
Revised: i o n

Original: [Figure]
Revised: [Figure]

Original: Ollivier curvature Let v ∼ u be two neighbors.
Revised: Ollivier curvature Let v ∼ u be two neighbors.

Original: Then it is easy to see that W G 1 ( m u , m v )
Revised: Then it is easy to see that W G 1 ( m u , m v )

Original: =
Revised: =

Original: and hence κ G ( u, v ) = 0 .
Revised: and hence κ G ( u, v ) = 0 .

Original: Thus, κ ( C n ) = 0 .
Revised: Thus, κ ( C n ) = 0 .

Original: Forman curvature Similarly, it is easy to see that F ( C n )
Revised: Forman curvature Similarly, it is easy to see that F ( C n )

Original: =
Revised: =

Original: ˆ F ( C n ) = 0 .
Revised: ˆ F ( C n ) = 0 .

Original: Average sectional curvature If n is even, then ξ C n ( v ; b, c ; a ) = 

Revised: Average sectional curvature If n is even, then ξ C n ( v ; b, c ; a ) =

Original: for all points except the one diametrically opposite to v for which we have ξ C n ( v ; b, c ; a ) = 1 .
Revised: for all points except the one diametrically opposite to v for which we have ξ C n ( v ; b, c ; a ) = 1 .

Original: If n is odd, then for two points we have ξ C n ( v ; b, c ; a ) = n 2( n − 1) .
Revised: If n is odd, then for two points we have ξ C n ( v ; b, c ; a ) = n 2( n − 1) .

Original: As a result, ξ ( C n ) = 1 n − 3 for even n and ξ ( C n ) = n ( n − 1)( n − 3) for odd n .
Revised: As a result, ξ ( C n ) = 1 n − 3 for even n and ξ ( C n ) = n ( n − 1)( n − 3) for odd n .

Original: Distortion-based curvature
Revised: Distortion-based curvature

Original: Here we have that C distd ( C n ) =
Revised: Here we have that C distd ( C n ) =

Original: (cid:0) 2 πn (cid:1) 2 .
Revised: (cid:0) 2 πn (cid:1) 2 .

Original: Indeed, if we consider any three consequent nodes, then the middle one should lie on the geodetic between the other two.
Revised: Indeed, if we consider any three consequent nodes, then the middle one should lie on the geodetic between the other two.

Original: So, they all lie on a great circle (of length n ) from which the result follows.
Revised: So, they all lie on a great circle (of length n ) from which the result follows.

Original: In particular, C distd ( C 5 ) ≈ 1 .
Revised: In particular, C distd ( C 5 ) ≈ 1 .

Original: 58 , C distd ( C 100 ) ≈ 0 .
Revised: 58 , C distd ( C 100 ) ≈ 0 .

Original: 004 , which is illustrated in Figure 3 (left).
Revised: 004 , which is illustrated in Figure 3 (left).

Original: Threshold-based curvature It is easy to see that C thrd ( C n ) =
Revised: Threshold-based curvature It is easy to see that C thrd ( C n ) =

Original: ( −∞ , C ) with some C > 0 , which decreases with n and increases with d (see Figure 3, right).
Revised: ( −∞ , C ) with some C > 0 , which decreases with n and increases with d (see Figure 3, right).

Original: A simple lower bound for C is
Revised: A simple lower bound for C is

Original: C ≥ (cid:0) 4 πn (cid:1) 2 , since for such curvature we can embed all nodes into a great circle with distances 1/ between the closest ones.
Revised: C ≥ (cid:0) 4 πn (cid:1) 2 , since for such curvature we can embed all nodes into a great circle with distances 1/ between the closest ones.

Original: 4.3. C OMPLETE BIPARTITE GRAPH K l,m
Revised: 4.3. C OMPLETE BIPARTITE GRAPH K l,m

Original: W.l.o.g.
Revised: W.l.o.g.

Original: we assume that l ≥ m ≥ 2 (the remaining cases are stars and are already considered).
Revised: we assume that l ≥ m ≥ 2 (the remaining cases are stars and are already considered).

Original: Ollivier curvature We prove the following lemma (the proof is in Appendix B.1).
Revised: Ollivier curvature We prove the following lemma (the proof is in Appendix B.1).

Original: Lemma 4.5.
Revised: Lemma 4.5.

Original: κ ( K l,m ) = 0 .
Revised: κ ( K l,m ) = 0 .

Original: Forman curvature F ( K l,m )
Revised: Forman curvature F ( K l,m )

Original: =
Revised: =

Original: ˆ F ( K l,m ) = 4 − l − m .
Revised: ˆ F ( K l,m ) = 4 − l − m .

Original: Average sectional curvature The following lemma holds (see Appendix B.2).
Revised: Average sectional curvature The following lemma holds (see Appendix B.2).

Original: Lemma 4.6.
Revised: Lemma 4.6.

Original: ξ ( K l,m ) =
Revised: ξ ( K l,m ) =

Original: − ( l − m ) 2 + m + l − 2 ( m + l − 2)( l + m − 3) .
Revised: − ( l − m ) 2 + m + l − 2 ( m + l − 2)( l + m − 3) .

Original: In particular, if m = l we get ξ ( K l,m ) = 1 2 m − 3 .
Revised: In particular, if m = l we get ξ ( K l,m ) = 1 2 m − 3 .

Original: This lemma implies that for balanced complete bipartite graphs ξ ( K l,m ) is positive, but tends to zero as the graph grows.
Revised: This lemma implies that for balanced complete bipartite graphs ξ ( K l,m ) is positive, but tends to zero as the graph grows.

Original: Distortion-based curvature Figure 4 (left) shows the empirical behavior of distortion depending on curvature.
Revised: Distortion-based curvature Figure 4 (left) shows the empirical behavior of distortion depending on curvature.

Original: We observe that for all graphs optimal curvature (for the considered range) is between 2 and 2.5, where distortion has a drop.
Revised: We observe that for all graphs optimal curvature (for the considered range) is between 2 and 2.5, where distortion has a drop.

Original: The following proposition gives an intuition about why one could expect this.
Revised: The following proposition gives an intuition about why one could expect this.

Original: Proposition 4.7.
Revised: Proposition 4.7.

Original: For any d , C distd ( K 2 , 2 ) = (cid:0) π 2 (cid:1) 2 ≈ 2 . 47 and K 2 , 2 is the only complete bipartite graph (with at least two nodes in each part) for which zero distortion is achievable.
Revised: For any d , C distd ( K 2 , 2 ) = (cid:0) π 2 (cid:1) 2 ≈ 2 . 47 and K 2 , 2 is the only complete bipartite graph (with at least two nodes in each part) for which zero distortion is achievable.

Original: [Table]
Revised: [Table]

Original: Power /
Revised: Power /

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: Indeed, the result for K 2 , 2 follows from the corresponding result on cycle C 4 .
Revised: Indeed, the result for K 2 , 2 follows from the corresponding result on cycle C 4 .

Original: Moreover, if for
Revised: Moreover, if for

Original: K l,m we have l ≥ 3 and m ≥ 2 , then for any two nodes in the part of size l there are at leastdifferent geodesics of length 2 between them.
Revised: K l,m we have l ≥ 3 and m ≥ 2 , then for any two nodes in the part of size l there are at leastdifferent geodesics of length 2 between them.

Original: Therefore, all such pairs lie at opposite poles of the hypersphere, which is impossible since l ≥ 3 .
Revised: Therefore, all such pairs lie at opposite poles of the hypersphere, which is impossible since l ≥ 3 .

Original: Threshold-based curvature Threshold-based embeddings to Euclidean spaces are well studied.
Revised: Threshold-based curvature Threshold-based embeddings to Euclidean spaces are well studied.

Original: In particular, it is know that for any graph G ( V, E ) a perfect threshold embedding exists for some d ≤ | V | (Maehara, 1984).
Revised: In particular, it is know that for any graph G ( V, E ) a perfect threshold embedding exists for some d ≤ | V | (Maehara, 1984).

Original: However, undirected bipartite graphs are hard to embed in Euclidean space: the bound d = O ( n ) is known (Maehara, 1984).
Revised: However, undirected bipartite graphs are hard to embed in Euclidean space: the bound d = O ( n ) is known (Maehara, 1984).

Original: We empirically observe that they are similarly hard to embed to both spherical and hyperbolic spaces, as shown in Figure 4 (right).
Revised: We empirically observe that they are similarly hard to embed to both spherical and hyperbolic spaces, as shown in Figure 4 (right).

Original: Empirically, these graphs are insensitive to curvature (given that it is not too large).
Revised: Empirically, these graphs are insensitive to curvature (given that it is not too large).

Original: 4. V OLUME - BASED APPROXIMATION
Revised: 4. V OLUME - BASED APPROXIMATION

Original: Let us propose a new heuristic curvature estimator which is motivated by the intuition obtained in the previous section as well as by extensive literature arguing that the most important feature of a hyperbolic space is the fact that it has “more volume” in the neighborhood.
Revised: Let us propose a new heuristic curvature estimator which is motivated by the intuition obtained in the previous section as well as by extensive literature arguing that the most important feature of a hyperbolic space is the fact that it has “more volume” in the neighborhood.

Original: Therefore, the natural idea is to estimate the volume we need for embedding nodes that are at distance at most k from a given one and then compute the curvature needed to get such volume.
Revised: Therefore, the natural idea is to estimate the volume we need for embedding nodes that are at distance at most k from a given one and then compute the curvature needed to get such volume.

Original: We compare the number of nodes at distance exactly k from a node (discounted using the number of edges between them) with the area of a hypersphere of radius k in a space of curvature c .
Revised: We compare the number of nodes at distance exactly k from a node (discounted using the number of edges between them) with the area of a hypersphere of radius k in a space of curvature c .

Original: The detailed description of the procedure is given in Appendix D. A nice feature of the proposed estimator is that it depends on dimension.
Revised: The detailed description of the procedure is given in Appendix D. A nice feature of the proposed estimator is that it depends on dimension.

Original: Moreover, this dependence is natural: when d becomes larger, curvature becomes smaller in absolute value.
Revised: Moreover, this dependence is natural: when d becomes larger, curvature becomes smaller in absolute value.

Original: As we show in the experiments on various datasets, a similar property is observed for the optimal curvature.
Revised: As we show in the experiments on various datasets, a similar property is observed for the optimal curvature.

Original: Also, it turns out that for a given dimension the new estimator is able to distinguish well between negatively curved and neutral networks.
Revised: Also, it turns out that for a given dimension the new estimator is able to distinguish well between negatively curved and neutral networks.

Original: However, despite the potential of this new estimator, the problem of proper global curvature estimation is still wide open.
Revised: However, despite the potential of this new estimator, the problem of proper global curvature estimation is still wide open.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: In this section, we compare all discussed curvature estimators empirically.
Revised: In this section, we compare all discussed curvature estimators empirically.

Original: Our experiments are based on a publicly available implementation of graph embedding by Gu et al.
Revised: Our experiments are based on a publicly available implementation of graph embedding by Gu et al.

Original: The main advantage of this algorithm is that it works for any curvature: negative, positive or zero.
Revised: The main advantage of this algorithm is that it works for any curvature: negative, positive or zero.

Original: We modiﬁed the implementation for our task: in particular, we added a regime responsible for minimizing
Revised: We modiﬁed the implementation for our task: in particular, we added a regime responsible for minimizing

Original: [Table]
Revised: [Table]

Original: threshold-based loss functions.
Revised: threshold-based loss functions.

Original: The detailed description of our experimental setup is given in Appendix E.3, the datasets and their properties are listed in Appendix E.1.
Revised: The detailed description of our experimental setup is given in Appendix E.3, the datasets and their properties are listed in Appendix E.1.

Original: The results of the comparison for distortion and zero-one loss are presented in Tables 1 and 2, respectively.
Revised: The results of the comparison for distortion and zero-one loss are presented in Tables 1 and 2, respectively.

Original: 7 We also added a method based on the gradient descent, proposed by Gu et al.
Revised: 7 We also added a method based on the gradient descent, proposed by Gu et al.

Original: This method is used here only as an illustration, since our aim is to compare estimators which do not do actual embedding.
Revised: This method is used here only as an illustration, since our aim is to compare estimators which do not do actual embedding.

Original: Also, the gradient descent algorithm uses the best space based on the value of the loss function, while all other curvature estimators do not use such information.
Revised: Also, the gradient descent algorithm uses the best space based on the value of the loss function, while all other curvature estimators do not use such information.

Original: We marked in bold both the global winner and the winner among all curvatures except the gradient descent one.
Revised: We marked in bold both the global winner and the winner among all curvatures except the gradient descent one.

Original: The main goal of this experiment is to show that all known curvatures (Ollivier, Forman and average sectional) have unstable performance and even gradient descent method is not always the best (in both tables).
Revised: The main goal of this experiment is to show that all known curvatures (Ollivier, Forman and average sectional) have unstable performance and even gradient descent method is not always the best (in both tables).

Original: Another important conclusion is that the optimal curvature depends on both dimension and loss function.
Revised: Another important conclusion is that the optimal curvature depends on both dimension and loss function.

Original: We show more evidence for that in our extensive analysis in Appendix E.5, where we consider more datasets and more threshold-based loss functions.
Revised: We show more evidence for that in our extensive analysis in Appendix E.5, where we consider more datasets and more threshold-based loss functions.

Original: The experiments also conﬁrm that networks which seems to be negatively curved (for some small dimension) become more neutral as dimensionality grows
Revised: The experiments also conﬁrm that networks which seems to be negatively curved (for some small dimension) become more neutral as dimensionality grows

Original: In our experiments with the proposed volume-based estimator (see Appendix E.6), we noticed that it is able to predict well whether the hyperbolic space is needed for an embedding or Euclidean space is enough.
Revised: In our experiments with the proposed volume-based estimator (see Appendix E.6), we noticed that it is able to predict well whether the hyperbolic space is needed for an embedding or Euclidean space is enough.

Original: However, the magnitude for the predicted curvature can be not optimal.
Revised: However, the magnitude for the predicted curvature can be not optimal.

Original: Volume-based estimator also predicts that for large dimensions Euclidean space is often a good choice.
Revised: Volume-based estimator also predicts that for large dimensions Euclidean space is often a good choice.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we introduced a concept of global graph curvature motivated by the practical task of embedding graphs.
Revised: In this paper, we introduced a concept of global graph curvature motivated by the practical task of embedding graphs.

Original: This curvature depends on the loss function and dimension of a space.
Revised: This curvature depends on the loss function and dimension of a space.

Original: To get an intuition about how global graph curvature behaves, we theoretically and empirically analyzed it for several simple graphs.
Revised: To get an intuition about how global graph curvature behaves, we theoretically and empirically analyzed it for several simple graphs.

Original: We compared (theoretically and empirically) the global graph curvature and several approximations based on well-known local graph curvatures and showed that they essentially differ.
Revised: We compared (theoretically and empirically) the global graph curvature and several approximations based on well-known local graph curvatures and showed that they essentially differ.

Original: We demonstrated that dimensionality and the choice of a loss function fundamentally affect the global curvature and, in particular, when dimension is larger the optimal curvature usually becomes less negative, which agrees with our theoretical analysis.
Revised: We demonstrated that dimensionality and the choice of a loss function fundamentally affect the global curvature and, in particular, when dimension is larger the optimal curvature usually becomes less negative, which agrees with our theoretical analysis.

Original: We also proposed a simple dimension-speciﬁc estimator which decides whether it is reasonable to embed into a negatively curved space.
Revised: We also proposed a simple dimension-speciﬁc estimator which decides whether it is reasonable to embed into a negatively curved space.

Original: Our work shows that the problem of ﬁnding the rights space for graph embedding is interesting and non-trivial and we hope our results will encourage further research on global graph curvature.
Revised: Our work shows that the problem of ﬁnding the rights space for graph embedding is interesting and non-trivial and we hope our results will encourage further research on global graph curvature.

Original: A G EOMETRICAL PROPERTIES OF SPACES OF CONSTANT CURVATURE
Revised: A G EOMETRICAL PROPERTIES OF SPACES OF CONSTANT CURVATURE

Original: In this section we recall some useful equalities which will be used throughout the proofs.
Revised: In this section we recall some useful equalities which will be used throughout the proofs.

Original: In all proofs, we use notation R , where R = 1 √ c in spherical space (corresponds to the radius of a sphere) and in the hyperbolic case ( c < 0) R = 1 √− c can be considered as a scaling factor compared to the space of curvature 1.
Revised: In all proofs, we use notation R , where R = 1 √ c in spherical space (corresponds to the radius of a sphere) and in the hyperbolic case ( c < 0) R = 1 √− c can be considered as a scaling factor compared to the space of curvature 1.

Original: Law of cosines Let us consider a triangle with angles A, B, C and the lengths of opposite sides a, b, c , respectively.
Revised: Law of cosines Let us consider a triangle with angles A, B, C and the lengths of opposite sides a, b, c , respectively.

Original: In Euclidean, space we have:
Revised: In Euclidean, space we have:

Original: [Equation]
Revised: [Equation]

Original: In spherical space, the ﬁrst law of cosines is:
Revised: In spherical space, the ﬁrst law of cosines is:

Original: [Equation]
Revised: [Equation]

Original: and the second law of cosines is:
Revised: and the second law of cosines is:

Original: [Equation]
Revised: [Equation]

Original: In hyperbolic space, we have
Revised: In hyperbolic space, we have

Original: [Equation]
Revised: [Equation]

Original: Equilateral triangle
Revised: Equilateral triangle

Original: The following equalities follow from the corresponding laws of cosines, assuming that all sides (and angles) are equal.
Revised: The following equalities follow from the corresponding laws of cosines, assuming that all sides (and angles) are equal.

Original: For hyperbolic space:
Revised: For hyperbolic space:

Original: [Equation]
Revised: [Equation]

Original: For spherical space:
Revised: For spherical space:

Original: [Equation]
Revised: [Equation]

Original: Area and volume of hypersphere Let S d ( r ) and V d ( r ) denote area of a hypersphere and volume of a ball of radius r in d -dimensional space.
Revised: Area and volume of hypersphere Let S d ( r ) and V d ( r ) denote area of a hypersphere and volume of a ball of radius r in d -dimensional space.

Original: In euclidean space,
Revised: In euclidean space,

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: In spherical space, sphere of radius r is isometric to Euclidean sphere of radius R sin rR .
Revised: In spherical space, sphere of radius r is isometric to Euclidean sphere of radius R sin rR .

Original: Therefore, the area is
Revised: Therefore, the area is

Original: [Equation]
Revised: [Equation]

Original: Similarly, in hyperbolic space,
Revised: Similarly, in hyperbolic space,

Original: [Equation]
Revised: [Equation]

Original: B C URVATURE OF SIMPLE GRAPHS B. P ROOF OF L EMMA 4.5 (O LLIVIER CURVATURE FOR BIPARTITE GRAPHS )
Revised: B C URVATURE OF SIMPLE GRAPHS B. P ROOF OF L EMMA 4.5 (O LLIVIER CURVATURE FOR BIPARTITE GRAPHS )

Original: Let use denote the node sets in K l,m by U := { u 1 , . .
Revised: Let use denote the node sets in K l,m by U := { u 1 , . .

Original: . , u l } and V := { v 1 , . .
Revised: . , u l } and V := { v 1 , . .

Original: . , v m } .
Revised: . , v m } .

Original: We will prove that for any edge ( u, v ) , W K l,m( m u , m v ) = 1 , which then implies that κ ( K l,m ) = 0 .
Revised: We will prove that for any edge ( u, v ) , W K l,m( m u , m v ) = 1 , which then implies that κ ( K l,m ) = 0 .

Original: For this we use the dual representation for the Wasserstein distance, see Ollivier (2009) for more details.
Revised: For this we use the dual representation for the Wasserstein distance, see Ollivier (2009) for more details.

Original: This states that on the one hand
Revised: This states that on the one hand

Original: [Equation]
Revised: [Equation]

Original: where the inﬁmum is taken over all joint probability measures on the product of the neighborhoods of v and u , while on the other hand
Revised: where the inﬁmum is taken over all joint probability measures on the product of the neighborhoods of v and u , while on the other hand

Original: [Equation]
Revised: [Equation]

Original: where the supremum is taken over all 1 -Lipschitz functions, i.e | f ( u ) − f ( v ) | ≤ d G ( u, v ) .
Revised: where the supremum is taken over all 1 -Lipschitz functions, i.e | f ( u ) − f ( v ) | ≤ d G ( u, v ) .

Original: Note that for any u ∈ U, v ∈ V the joint neighborhood is V × U .
Revised: Note that for any u ∈ U, v ∈ V the joint neighborhood is V × U .

Original: First we establish an upper bound by considering the product joint probability density on V × U ρ
Revised: First we establish an upper bound by considering the product joint probability density on V × U ρ

Original: [Equation]
Revised: [Equation]

Original: x, y
Revised: x, y

Original: [Equation]
Revised: [Equation]

Original: = 1 ml.
Revised: = 1 ml.

Original: It then follows that
Revised: It then follows that

Original: [Equation]
Revised: [Equation]

Original: , u j )
Revised: , u j )

Original: [Equation]
Revised: [Equation]

Original: u
Revised: u

Original: [Equation]
Revised: [Equation]

Original: = 1 .
Revised: = 1 .

Original: For the lower bound we deﬁne the function
Revised: For the lower bound we deﬁne the function

Original: [Equation]
Revised: [Equation]

Original: Observe that if u ∈ U and v ∈ V then | f ( u ) − f
Revised: Observe that if u ∈ U and v ∈ V then | f ( u ) − f

Original: ( z ) |
Revised: ( z ) |

Original: =
Revised: =

Original: =
Revised: =

Original: d G ( u, v ) .
Revised: d G ( u, v ) .

Original: On the other hand, if u, u (cid:48) ∈ U then | f ( u ) − f ( u (cid:48) )
Revised: On the other hand, if u, u (cid:48) ∈ U then | f ( u ) − f ( u (cid:48) )

Original: | = 0 ≤ 2 = d G ( u, u (cid:48) ) and similar for v, v (cid:48) ∈ V .
Revised: | = 0 ≤ 2 = d G ( u, u (cid:48) ) and similar for v, v (cid:48) ∈ V .

Original: Thus we conclude that f is 1-Lipschitz.
Revised: Thus we conclude that f is 1-Lipschitz.

Original: It now follows that
Revised: It now follows that

Original: [Equation]
Revised: [Equation]

Original: which completes the proof.
Revised: which completes the proof.

Original: B. P ROOF OF L EMMA 4.6 ( AVERAGE SECTIONAL CURVATURE FOR BIPARTITE GRAPHS )
Revised: B. P ROOF OF L EMMA 4.6 ( AVERAGE SECTIONAL CURVATURE FOR BIPARTITE GRAPHS )

Original: If v and a are in the same part of the bipartite graph, then ξ K l,m ( v ; b, c ; a ) = 1 , otherwise ξ K l,m ( v ; b, c ; a ) = − 1 .
Revised: If v and a are in the same part of the bipartite graph, then ξ K l,m ( v ; b, c ; a ) = 1 , otherwise ξ K l,m ( v ; b, c ; a ) = − 1 .

Original: Therefore, if a belongs to the part of size l , sectional curvature is ξ K l,m ( v ; b, c ) =
Revised: Therefore, if a belongs to the part of size l , sectional curvature is ξ K l,m ( v ; b, c ) =

Original: l − m +1 l + m − 3 , otherwise it is ξ K l,m ( v ; b, c )
Revised: l − m +1 l + m − 3 , otherwise it is ξ K l,m ( v ; b, c )

Original: = m − l +1 l + m − 3 .
Revised: = m − l +1 l + m − 3 .

Original: As a result, by averaging over all triplets, we get
Revised: As a result, by averaging over all triplets, we get

Original: [Equation]
Revised: [Equation]

Original: K l
Revised: K l

Original: [Equation]
Revised: [Equation]

Original: B. P ROOF OF T HEOREM 4.1 ( DISTORTION - BASED CURVATURE FOR STARS )
Revised: B. P ROOF OF T HEOREM 4.1 ( DISTORTION - BASED CURVATURE FOR STARS )

Original: The general idea is the following: we take any curvature c and prove a lower bound on distortion (for any given dimension d ).
Revised: The general idea is the following: we take any curvature c and prove a lower bound on distortion (for any given dimension d ).

Original: Then, we obtain an upper bound on optimal distortion which tends to zero as c → −∞ , which gives the claimed result C dist d ( S n ) =
Revised: Then, we obtain an upper bound on optimal distortion which tends to zero as c → −∞ , which gives the claimed result C dist d ( S n ) =

Original: −∞ .
Revised: −∞ .

Original: First, let us analyze the lower bound on distortion.
Revised: First, let us analyze the lower bound on distortion.

Original: Recall that distortion of a graph is the average distortion over all pairs of nodes.
Revised: Recall that distortion of a graph is the average distortion over all pairs of nodes.

Original: Let v be the central node and v 1 , . .
Revised: Let v be the central node and v 1 , . .

Original: . , v n ( n ≥ 3 ) be its neighbors.
Revised: . , v n ( n ≥ 3 ) be its neighbors.

Original: Then, for any embedding f , we have
Revised: Then, for any embedding f , we have

Original: [Equation]
Revised: [Equation]

Original: Let D min be the minimum value of the following weighted distortion of a star with 3 leaves:
Revised: Let D min be the minimum value of the following weighted distortion of a star with 3 leaves:

Original: D min = min f
Revised: D min = min f

Original: [Equation]
Revised: [Equation]

Original: then
Revised: then

Original: [Equation]
Revised: [Equation]

Original: Hence, it remains to ﬁnd a lower bound on D min , i.e., a lower bound for a weighted distortion of
Revised: Hence, it remains to ﬁnd a lower bound on D min , i.e., a lower bound for a weighted distortion of

Original: S 3 with central node v and three leaves v 1 , v 2 , v 3 .
Revised: S 3 with central node v and three leaves v 1 , v 2 , v 3 .

Original: If we consider three angles at the node v , then at least one of them is α ≤ 2 π/ 3 , so we can get a lower bound by only considering this triangle, which is, w.l.o.g., formed by v, v 1 , v 2 .
Revised: If we consider three angles at the node v , then at least one of them is α ≤ 2 π/ 3 , so we can get a lower bound by only considering this triangle, which is, w.l.o.g., formed by v, v 1 , v 2 .

Original: [Equation]
Revised: [Equation]

Original: Denote d ( f ( v 1 ) , f ( v ))
Revised: Denote d ( f ( v 1 ) , f ( v ))

Original: = x = 1 + ε , d ( f ( v 2 ) , f ( v )) =
Revised: = x = 1 + ε , d ( f ( v 2 ) , f ( v )) =

Original: y
Revised: y

Original: = 1 + δ , d
Revised: = 1 + δ , d

Original: ( f ( v 1 ) , f ( v 2 ))
Revised: ( f ( v 1 ) , f ( v 2 ))

Original: =
Revised: =

Original: z = 2 + ε + δ − ϕ
Revised: z = 2 + ε + δ − ϕ

Original: with some ε, δ and some ϕ > 0 (from triangle inequality).
Revised: with some ε, δ and some ϕ > 0 (from triangle inequality).

Original: Assume that | ε | < 1 / 2 and | δ | < 1 / 2 (otherwise the lower bound is trivial).
Revised: Assume that | ε | < 1 / 2 and | δ | < 1 / 2 (otherwise the lower bound is trivial).

Original: Now we use the law of cosines to get a lower bound on ϕ .
Revised: Now we use the law of cosines to get a lower bound on ϕ .

Original: We consider Euclidean and hyperbolic spaces separately and note that the bound obtained in
Revised: We consider Euclidean and hyperbolic spaces separately and note that the bound obtained in

Original: Euclidean space also holds in spherical spaces (with any c ).
Revised: Euclidean space also holds in spherical spaces (with any c ).

Original: In Euclidean space, using triangle inequality, we get ϕ
Revised: In Euclidean space, using triangle inequality, we get ϕ

Original: = x
Revised: = x

Original: + y − z > 0 .
Revised: + y − z > 0 .

Original: So, in Euclidean and spherical spaces ϕ is bounded below by a constant.
Revised: So, in Euclidean and spherical spaces ϕ is bounded below by a constant.

Original: In hyperbolic space the law of cosines gives (recall that R =
Revised: In hyperbolic space the law of cosines gives (recall that R =

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: from which, using cosh( x + y ) =
Revised: from which, using cosh( x + y ) =

Original: cosh x cosh y + sinh x sinh y , we get
Revised: cosh x cosh y + sinh x sinh y , we get

Original: [Equation]
Revised: [Equation]

Original: If R → ∞ , then, similarly to Euclidean case, we get ϕ
Revised: If R → ∞ , then, similarly to Euclidean case, we get ϕ

Original: = x
Revised: = x

Original: + y − z = Ω(1) .
Revised: + y − z = Ω(1) .

Original: On the other hand, if R → 0 , we get ϕ
Revised: On the other hand, if R → 0 , we get ϕ

Original: = Ω( R ) .
Revised: = Ω( R ) .

Original: Note that D min ≥ | z − 2 | + | x − 1 | + | y − 1 | ( n − 1) /=
Revised: Note that D min ≥ | z − 2 | + | x − 1 | + | y − 1 | ( n − 1) /=

Original: | ε + δ − ϕ
Revised: | ε + δ − ϕ

Original: | + | ε | + | δ | ( n − 1) / 2 .
Revised: | + | ε | + | δ | ( n − 1) / 2 .

Original: This gives us a lower bound D min = Ω(1 /n ) in spherical and Euclidean spaces and D min = Ω(min
Revised: This gives us a lower bound D min = Ω(1 /n ) in spherical and Euclidean spaces and D min = Ω(min

Original: ( R, 1) /
Revised: ( R, 1) /

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: Ω (cid:18) 1 n max ( √− c, 1 ) (cid:19) in hyperbolic space.
Revised: Ω (cid:18) 1 n max ( √− c, 1 ) (cid:19) in hyperbolic space.

Original: From this and
Revised: From this and

Original: Equation 9 the bound on D ( S n ) follows.
Revised: Equation 9 the bound on D ( S n ) follows.

Original: Now, let us get an upper bound on optimal distortion D opt ( S n ) .
Revised: Now, let us get an upper bound on optimal distortion D opt ( S n ) .

Original: To do this, we explicitly construct an embedding with sufﬁciently low distortion D ( S n ) .
Revised: To do this, we explicitly construct an embedding with sufﬁciently low distortion D ( S n ) .

Original: Let v be the central node, then we spread all other nodes uniformly on a 2-dimensional circle of radius 1 centred at v .
Revised: Let v be the central node, then we spread all other nodes uniformly on a 2-dimensional circle of radius 1 centred at v .

Original: The smallest angle between two points is 2 π/n .
Revised: The smallest angle between two points is 2 π/n .

Original: Therefore, from the law of cosines, the distance between leaves is at least k with
Revised: Therefore, from the law of cosines, the distance between leaves is at least k with

Original: [Equation]
Revised: [Equation]

Original: Note that for any two leaves v i and v j we have that d ( f ( v i ) , f ( v j )) ≤ 2 .
Revised: Note that for any two leaves v i and v j we have that d ( f ( v i ) , f ( v j )) ≤ 2 .

Original: In particular, the closer two leaves are, the greater the difference 2 − d ( f ( v i ) , f ( v j )) is.
Revised: In particular, the closer two leaves are, the greater the difference 2 − d ( f ( v i ) , f ( v j )) is.

Original: Hence, the distance between adjacent leaves is the worst case and thus D opt ( S n ) can be upper bounded as
Revised: Hence, the distance between adjacent leaves is the worst case and thus D opt ( S n ) can be upper bounded as

Original: D
Revised: D

Original: [Equation]
Revised: [Equation]

Original: Note that 1 − cos 2 πn = Θ (cid:0) 1 n (cid:1) and sinh 2 (cid:0) 1 R (cid:1) =
Revised: Note that 1 − cos 2 πn = Θ (cid:0) 1 n (cid:1) and sinh 2 (cid:0) 1 R (cid:1) =

Original: Θ (cid:0) e 2 /R (cid:1) .
Revised: Θ (cid:0) e 2 /R (cid:1) .

Original: Then, arccosh (cid:0) Θ (cid:0) 1 n e 2 /R (cid:1) + 1 (cid:1) behaves as (cid:112) 2 e 

Revised: Then, arccosh (cid:0) Θ (cid:0) 1 n e 2 /R (cid:1) + 1 (cid:1) behaves as (cid:112) 2 e

Original: /R /n
Revised: /R /n

Original: if 2 /R (cid:28) log n and as 2 R − log n
Revised: if 2 /R (cid:28) log n and as 2 R − log n

Original: if 2 /R
Revised: if 2 /R

Original: (cid:29) log n .
Revised: (cid:29) log n .

Original: Therefore, we get
Revised: Therefore, we get

Original: [Equation]
Revised: [Equation]

Original: B. P ROOF OF T HEOREM 4.4 ( DISTORTION - BASED CURVATURE FOR COMPLETE GRAPHS )
Revised: B. P ROOF OF T HEOREM 4.4 ( DISTORTION - BASED CURVATURE FOR COMPLETE GRAPHS )

Original: If d = n − 2 , then we are given a ( n − 1) -simplex, which can be embedded into n − 2 -dimensional spherical space.
Revised: If d = n − 2 , then we are given a ( n − 1) -simplex, which can be embedded into n − 2 -dimensional spherical space.

Original: Indeed, the radius of circumscribed hypersphere for the ( n − 1) -simplex with side length a is known to be R = a (cid:113) n − 1 2 n .
Revised: Indeed, the radius of circumscribed hypersphere for the ( n − 1) -simplex with side length a is known to be R = a (cid:113) n − 1 2 n .

Original: Since we want the spherical distance between all points to be equal to one, we need to choose a accordingly:
Revised: Since we want the spherical distance between all points to be equal to one, we need to choose a accordingly:

Original: [Equation]
Revised: [Equation]

Original: Here α corresponds to the angle giving the arc length 1, while the condition on sin α 2 relates α and a since α is the angle in a triangle with side lengths α, R, R .
Revised: Here α corresponds to the angle giving the arc length 1, while the condition on sin α 2 relates α and a since α is the angle in a triangle with side lengths α, R, R .

Original: This implies that
Revised: This implies that

Original: [Equation]
Revised: [Equation]

Original: and solving this equation for a yields
Revised: and solving this equation for a yields

Original: [Equation]
Revised: [Equation]

Original: Plugging this back into the formula for the radius R we obtain
Revised: Plugging this back into the formula for the radius R we obtain

Original: [Equation]
Revised: [Equation]

Original: so that we have
Revised: so that we have

Original: [Equation]
Revised: [Equation]

Original: Finally, let us show that if c → −∞ , then optimal distortion D opt ( K n ) → 0 .
Revised: Finally, let us show that if c → −∞ , then optimal distortion D opt ( K n ) → 0 .

Original: This result follows from the fact that D opt ( S n ) → 0 , because to embed a clique, it is sufﬁcient to embed a star on n +node with edge lengths 1/2 and then remove the central node.
Revised: This result follows from the fact that D opt ( S n ) → 0 , because to embed a clique, it is sufﬁcient to embed a star on n +node with edge lengths 1/2 and then remove the central node.

Original: B. P ROOF OF T HEOREM 4.2 ( THRESHOLD - BASED CURVATURE FOR STAR )
Revised: B. P ROOF OF T HEOREM 4.2 ( THRESHOLD - BASED CURVATURE FOR STAR )

Original: We show below that for any n , any dimension d and some curvature c there exists a perfect embedding.
Revised: We show below that for any n , any dimension d and some curvature c there exists a perfect embedding.

Original: Therefore C dist d ( S n ) consists of curvatures for which such perfect embedding exists.
Revised: Therefore C dist d ( S n ) consists of curvatures for which such perfect embedding exists.

Original: First, let us note that if there exists a perfect embedding f for some curvature c , then there exists a perfect embedding for any curvature c (cid:48) < c .
Revised: First, let us note that if there exists a perfect embedding f for some curvature c , then there exists a perfect embedding for any curvature c (cid:48) < c .

Original: Indeed, w.l.o.g., we assume that the central node v is mapped to the origin of a hyperspherical coordinate system and other points v 1 , . .
Revised: Indeed, w.l.o.g., we assume that the central node v is mapped to the origin of a hyperspherical coordinate system and other points v 1 , . .

Original: . , v n can be described by their radii and angles.
Revised: . , v n can be described by their radii and angles.

Original: We know that the distance between v and any v i is at mostand the distance between any pair v i , v j is larger than one.
Revised: We know that the distance between v and any v i is at mostand the distance between any pair v i , v j is larger than one.

Original: Now we change curvature to c (cid:48)
Revised: Now we change curvature to c (cid:48)

Original: <
Revised: <

Original: c and keep hyperspherical coordinates the same.
Revised: c and keep hyperspherical coordinates the same.

Original: Then the distance between v and any v i does not change, while the distance between nodes v i , v j increases.
Revised: Then the distance between v and any v i does not change, while the distance between nodes v i , v j increases.

Original: Now, it is easy to see that C increases with d : if there exists an embedding to some dimension d , then, obviously, the same embedding works for d (cid:48) >
Revised: Now, it is easy to see that C increases with d : if there exists an embedding to some dimension d , then, obviously, the same embedding works for d (cid:48) >

Original: d .
Revised: d .

Original: Further, C decreases with n since if there exists embedding of S n , then we can easily construct an embedding of S n (cid:48) for n (cid:48) < n by removing some nodes.
Revised: Further, C decreases with n since if there exists embedding of S n , then we can easily construct an embedding of S n (cid:48) for n (cid:48) < n by removing some nodes.

Original: Now, let us construct a perfect embedding of S n and estimate the required curvature.
Revised: Now, let us construct a perfect embedding of S n and estimate the required curvature.

Original: Recall that in a perfect embedding all leaves have to be inside the ball of radius 1 around the central node v and also the distance between any two leaves has to be larger than one.
Revised: Recall that in a perfect embedding all leaves have to be inside the ball of radius 1 around the central node v and also the distance between any two leaves has to be larger than one.

Original: It is easy to see that if we managed to spread n points inside the ball of radius 1 with distances more than 1 between them, then we can move each point along the radius up to distance 1 from v preserving this property.
Revised: It is easy to see that if we managed to spread n points inside the ball of radius 1 with distances more than 1 between them, then we can move each point along the radius up to distance 1 from v preserving this property.

Original: Therefore, it is sufﬁcient to spread all points on a hypersphere.
Revised: Therefore, it is sufﬁcient to spread all points on a hypersphere.

Original: Let us ﬁrst consider d = 2 .
Revised: Let us ﬁrst consider d = 2 .

Original: In this case we can ﬁnd an explicit analytic expression for C .
Revised: In this case we can ﬁnd an explicit analytic expression for C .

Original: This will be an upper bound for any d > 2 .
Revised: This will be an upper bound for any d > 2 .

Original: First, assume that n > 6 .
Revised: First, assume that n > 6 .

Original: In this case we have to consider only hyperbolic space, since n neighbors would not ﬁt to a circle of radius 1 in neither spherical or Euclidean spaces.
Revised: In this case we have to consider only hyperbolic space, since n neighbors would not ﬁt to a circle of radius 1 in neither spherical or Euclidean spaces.

Original: We will ﬁnd such largest curvature C which allows to have distance exactly 1 between the closest leaves.
Revised: We will ﬁnd such largest curvature C which allows to have distance exactly 1 between the closest leaves.

Original: In this case we cannot embed S n in a space of curvature C , but can embed in a space of any smaller curvature.
Revised: In this case we cannot embed S n in a space of curvature C , but can embed in a space of any smaller curvature.

Original: We use Equation 7 and let α = 2 πn :
Revised: We use Equation 7 and let α = 2 πn :

Original: [Equation]
Revised: [Equation]

Original: Note that if n is large, then sin α 2 = sin πn − 1 ∼ πn − 1 .
Revised: Note that if n is large, then sin α 2 = sin πn − 1 ∼ πn − 1 .

Original: Then, arccosh
Revised: Then, arccosh

Original: [Equation]
Revised: [Equation]

Original: so we get C ∼ − 4 log 2 n .
Revised: so we get C ∼ − 4 log 2 n .

Original: Now, let us consider n ≤ 6 .
Revised: Now, let us consider n ≤ 6 .

Original: Obviously, for n = 6 we have C = 0 .
Revised: Obviously, for n = 6 we have C = 0 .

Original: If n < 6 , then C > 0 .
Revised: If n < 6 , then C > 0 .

Original: We use Equation 8:
Revised: We use Equation 8:

Original: [Equation]
Revised: [Equation]

Original: B. P ROOF OF T HEOREM 4.3 ( THRESHOLD - BASED CURVATURE FOR TREES )
Revised: B. P ROOF OF T HEOREM 4.3 ( THRESHOLD - BASED CURVATURE FOR TREES )

Original: As for S n , we prove that for T b a perfect embedding exists for d = 2 and some curvature c .
Revised: As for S n , we prove that for T b a perfect embedding exists for d = 2 and some curvature c .

Original: Then, similarly to the previous section, it is clear that C increases with b and decreases with n .
Revised: Then, similarly to the previous section, it is clear that C increases with b and decreases with n .

Original: For the lower bound on C , we have to guarantee that an embedding exists.
Revised: For the lower bound on C , we have to guarantee that an embedding exists.

Original: For this, we provide the following construction (see Figure 5 for an illustration).
Revised: For this, we provide the following construction (see Figure 5 for an illustration).

Original: (Below we assume that the curvature is large enough for our construction to work, then we estimate the required curvature.)
Revised: (Below we assume that the curvature is large enough for our construction to work, then we estimate the required curvature.)

Original: First, we take the node v and consider a circle of radius 1 around this node.
Revised: First, we take the node v and consider a circle of radius 1 around this node.

Original: We spread b + 1 neighbors uniformly around this node.
Revised: We spread b + 1 neighbors uniformly around this node.

Original: For our construction to work, we need all distances between these nodes to be larger than 1.
Revised: For our construction to work, we need all distances between these nodes to be larger than 1.

Original: Now, at some step of the algorithm, assume that we have all nodes at level l placed at some circle centered at v and all distances between the nodes at level l are larger than 1.
Revised: Now, at some step of the algorithm, assume that we have all nodes at level l placed at some circle centered at v and all distances between the nodes at level l are larger than 1.

Original: Our aim is then to ﬁnd positions for all nodes at level l + 1 .
Revised: Our aim is then to ﬁnd positions for all nodes at level l + 1 .

Original: Let us take any node at l -th level.
Revised: Let us take any node at l -th level.

Original: Consider two points u l and u r on the same circle at distancefrom the node u to the left and to the right, respectively.
Revised: Consider two points u l and u r on the same circle at distancefrom the node u to the left and to the right, respectively.

Original: Let u, u l , z l and u, u r , z r form equilateral м 𝑣 𝑢 𝑢 l 𝑢 r 𝑧 l 𝑧 r 𝑢 1 𝑢 2 𝑢      triangles (with sides equal to 1).
Revised: Let u, u l , z l and u, u r , z r form equilateral м 𝑣 𝑢 𝑢 l 𝑢 r 𝑧 l 𝑧 r 𝑢 1 𝑢 2 𝑢      triangles (with sides equal to 1).

Original: Then we let the points at l + 1 -th level to be spread on the circle centred at v and passing through z l and z r .
Revised: Then we let the points at l + 1 -th level to be spread on the circle centred at v and passing through z l and z r .

Original: The children of u ( u 1 , . . .
Revised: The children of u ( u 1 , . . .

Original: , u b ) will be placed on the circular arc between z l and z r .
Revised: , u b ) will be placed on the circular arc between z l and z r .

Original: As usual, we want u i and u i +1 to be at distance at least 1 from each other.
Revised: As usual, we want u i and u i +1 to be at distance at least 1 from each other.

Original: Moreover, they have to be at distance at least 1 from children of other nodes.
Revised: Moreover, they have to be at distance at least 1 from children of other nodes.

Original: Also, note that placing u 1 , . . .
Revised: Also, note that placing u 1 , . . .

Original: , u b between z l and z r guarantees that these nodes are closer than 1 to their parent node u but at the same time at a distance larger than 1 from all other nodes at l -th level.
Revised: , u b between z l and z r guarantees that these nodes are closer than 1 to their parent node u but at the same time at a distance larger than 1 from all other nodes at l -th level.

Original: Also, all points at l -th level are far enough from points at l + 2 -th level.
Revised: Also, all points at l -th level are far enough from points at l + 2 -th level.

Original: It remains to ﬁnd a maximum curvature such that the required conditions are satisﬁed.
Revised: It remains to ﬁnd a maximum curvature such that the required conditions are satisﬁed.

Original: Let r and r (cid:48) be radii of circles at l -th and l + 1 -th levels and let 2 α = ∠ u l vu .
Revised: Let r and r (cid:48) be radii of circles at l -th and l + 1 -th levels and let 2 α = ∠ u l vu .

Original: We know (the law of cosines and cos 2 α = 1 − 2 sin 2 α )
Revised: We know (the law of cosines and cos 2 α = 1 − 2 sin 2 α )

Original: that
Revised: that

Original: [Equation]
Revised: [Equation]

Original: And the only condition we need for the whole procedure to work is that we have enough space on the circular arc for placing b nodes there:
Revised: And the only condition we need for the whole procedure to work is that we have enough space on the circular arc for placing b nodes there:

Original: [Equation]
Revised: [Equation]

Original: Now we note that sin 2 αb
Revised: Now we note that sin 2 αb

Original: [Equation]
Revised: [Equation]

Original: bfor all b ≥ 1 .
Revised: bfor all b ≥ 1 .

Original: Therefore, it is sufﬁcient to have
Revised: Therefore, it is sufﬁcient to have

Original: [Equation]
Revised: [Equation]

Original: Combining Equation 10 and Equation 11, we obtain:
Revised: Combining Equation 10 and Equation 11, we obtain:

Original: [Equation]
Revised: [Equation]

Original: To achieve this, it is sufﬁcient to have:
Revised: To achieve this, it is sufﬁcient to have:

Original: [Equation]
Revised: [Equation]

Original: It remains to ﬁnd the lower bound for r (cid:48)
Revised: It remains to ﬁnd the lower bound for r (cid:48)

Original: − r
Revised: − r

Original: and it is easy to see that r (cid:48)
Revised: and it is easy to see that r (cid:48)

Original: − r decreases with r .
Revised: − r decreases with r .

Original: Therefore, it is sufﬁcient to consider only the second step of the construction procedure, when we move from the circle of radius 1 to the next one.
Revised: Therefore, it is sufﬁcient to consider only the second step of the construction procedure, when we move from the circle of radius 1 to the next one.

Original: In this case, r = 1 and
Revised: In this case, r = 1 and

Original: [Equation]
Revised: [Equation]

Original: So, we have
Revised: So, we have

Original: [Equation]
Revised: [Equation]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Original: i s t
Revised: i s t

Original: [Figure]
Revised: [Figure]

Original: r t i
Revised: r t i

Original: [Figure]
Revised: [Figure]

Original: n
Revised: n

Original: [Figure]
Revised: [Figure]

Original: k
Revised: k

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: C C OMBINATIONS OF SIMPLE GRAPHS
Revised: C C OMBINATIONS OF SIMPLE GRAPHS

Original: In this section, in order to get additional intuition, we illustrate how combining several simple subgraphs may affect the optimal curvature.
Revised: In this section, in order to get additional intuition, we illustrate how combining several simple subgraphs may affect the optimal curvature.

Original: We choose distortion for this illustrative example, since for this loss we theoretically obtained non-trivial results: for some simple graphs the optimal curvature is negative, while for other it is positive.
Revised: We choose distortion for this illustrative example, since for this loss we theoretically obtained non-trivial results: for some simple graphs the optimal curvature is negative, while for other it is positive.

Original: Indeed, for trees (and, in particular, stars) the optimal curvature is minus inﬁnity and for cycles the optimal curvature is positive.
Revised: Indeed, for trees (and, in particular, stars) the optimal curvature is minus inﬁnity and for cycles the optimal curvature is positive.

Original: Our intuition is that combining stars and cycles in one graph would give optimal curvature located between the curvatures of these simple graph.
Revised: Our intuition is that combining stars and cycles in one graph would give optimal curvature located between the curvatures of these simple graph.

Original: Indeed, we observe such expected behavior for two illustrative examples shown in Figures 6 and 7.
Revised: Indeed, we observe such expected behavior for two illustrative examples shown in Figures 6 and 7.

Original: In Figure 6 we take a cycle of length 10 and iteratively add edges to one of its nodes, i.e., we create a star on one node.
Revised: In Figure 6 we take a cycle of length 10 and iteratively add edges to one of its nodes, i.e., we create a star on one node.

Original: In Figure 7 we take the same cycle and add small stars sequentially to several of its nodes: in the left ﬁgure we just add one dangling edge for each node (i.e., create a star subgraph on 4 nodes), in the right ﬁgure we add two edges.
Revised: In Figure 7 we take the same cycle and add small stars sequentially to several of its nodes: in the left ﬁgure we just add one dangling edge for each node (i.e., create a star subgraph on 4 nodes), in the right ﬁgure we add two edges.

Original: In all cases, the optimal curvature drifts from a positive one for a cycle to negative values.
Revised: In all cases, the optimal curvature drifts from a positive one for a cycle to negative values.

Original: This drift is more pronounced In Figure 6, where we create just one (relatively) large star.
Revised: This drift is more pronounced In Figure 6, where we create just one (relatively) large star.

Original: D V OLUME - BASED CURVATURE APPROXIMATION
Revised: D V OLUME - BASED CURVATURE APPROXIMATION

Original: In this section, we describe the proposed curvature approximation.
Revised: In this section, we describe the proposed curvature approximation.

Original: An input parameter of this estimator is the maximum neighborhood size k max (we ﬁx k max = 3 in the experiments).
Revised: An input parameter of this estimator is the maximum neighborhood size k max (we ﬁx k max = 3 in the experiments).

Original: We take a node v and for any k ≤ k max we compute the number of nodes
Revised: We take a node v and for any k ≤ k max we compute the number of nodes

Original: N k ( v ) at distance exactly k from it.
Revised: N k ( v ) at distance exactly k from it.

Original: Then, we discount this value using the number of edges between the nodes.
Revised: Then, we discount this value using the number of edges between the nodes.

Original: This captures our intuition that we need enough space to place the neighbors, but the connected neighbors are not required to be far away from each other.
Revised: This captures our intuition that we need enough space to place the neighbors, but the connected neighbors are not required to be far away from each other.

Original: Formally, let E k ( v ) be the number of non-edges between the pairs of nodes at distance k .
Revised: Formally, let E k ( v ) be the number of non-edges between the pairs of nodes at distance k .

Original: Then, we say that N (cid:48) k
Revised: Then, we say that N (cid:48) k

Original: ( v ) is such that (cid:0) N (cid:48)
Revised: ( v ) is such that (cid:0) N (cid:48)

Original: k ( v ) 2 (cid:1) =
Revised: k ( v ) 2 (cid:1) =

Original: E k ( v ) .
Revised: E k ( v ) .

Original: Finally, we compute the curvature
Revised: Finally, we compute the curvature

Original: c k ( v ) such that the area of a hypersphere or radius k is equal to N (cid:48) k
Revised: c k ( v ) such that the area of a hypersphere or radius k is equal to N (cid:48) k

Original: ( v ) .
Revised: ( v ) .

Original: [Table]
Revised: [Table]

Original: And our estimate for v is c ( v ) =
Revised: And our estimate for v is c ( v ) =

Original: min k ≤ k max c k ( v ) (the worst-case curvature).
Revised: min k ≤ k max c k ( v ) (the worst-case curvature).

Original: To get a curvature of the whole graph, we take a median of all c k ( v ) .
Revised: To get a curvature of the whole graph, we take a median of all c k ( v ) .

Original: (Similarly to other curvature estimators, one can consider a subsample of nodes instead of the whole node set to speed up the computation).
Revised: (Similarly to other curvature estimators, one can consider a subsample of nodes instead of the whole node set to speed up the computation).

Original: E E XPERIMENTS E. D ATASETS
Revised: E E XPERIMENTS E. D ATASETS

Original: Datasets used in our experiments are listed in Table 3.
Revised: Datasets used in our experiments are listed in Table 3.

Original: In all cases, we take only the giant connected component (numbers of nodes and edges in Table 3 correspond to already ﬁltered dataset).
Revised: In all cases, we take only the giant connected component (numbers of nodes and edges in Table 3 correspond to already ﬁltered dataset).

Original: Synthetic datasets are described in Section 4.3.
Revised: Synthetic datasets are described in Section 4.3.

Original: The dataset T 3 , 6 is a tree with branching factor 3 and depth 6.
Revised: The dataset T 3 , 6 is a tree with branching factor 3 and depth 6.

Original: E. T HRESHOLD - BASED LOSS FUNCTIONS
Revised: E. T HRESHOLD - BASED LOSS FUNCTIONS

Original: While our theoretical results in Section 4 hold for any threshold-based loss function, we noticed that for less trivial datasets, where the perfect embedding is impossible, the choice of the loss function becomes important.
Revised: While our theoretical results in Section 4 hold for any threshold-based loss function, we noticed that for less trivial datasets, where the perfect embedding is impossible, the choice of the loss function becomes important.

Original: In most of the experiments we use zero-one loss function:
Revised: In most of the experiments we use zero-one loss function:

Original: (cid:0) n 2 (cid:1) (cid:88)
Revised: (cid:0) n 2 (cid:1) (cid:88)

Original: i,j I [ v i ∼ v j ] · I [ d ( f ( v i
Revised: i,j I [ v i ∼ v j ] · I [ d ( f ( v i

Original: [Equation]
Revised: [Equation]

Original: 1] +
Revised: 1] +

Original: [Equation]
Revised: [Equation]

Original: · I [ d ( f ( v i )
Revised: · I [ d ( f ( v i )

Original: [Equation]
Revised: [Equation]

Original: f ( v j )
Revised: f ( v j )

Original: [Equation]
Revised: [Equation]

Original: 1] which essentially measures how well the values I [ d ( f ( v i ) f ( v j )) > 1] approximate I [ v i ∼ v j ] .
Revised: 1] which essentially measures how well the values I [ d ( f ( v i ) f ( v j )) > 1] approximate I [ v i ∼ v j ] .

Original: However, due to the fact that the real-world graphs are usually sparse, optimizing zero-one loss may lead to algorithms biased towards sparse embeddings.
Revised: However, due to the fact that the real-world graphs are usually sparse, optimizing zero-one loss may lead to algorithms biased towards sparse embeddings.

Original: A standard solution to this is to consider balanced accuracy instead of accuracy, which leads to balanced zero-one loss:
Revised: A standard solution to this is to consider balanced accuracy instead of accuracy, which leads to balanced zero-one loss:

Original: [Equation]
Revised: [Equation]

Original: The recent study by G¨osgens et al.
Revised: The recent study by G¨osgens et al.

Original: (2019) analyses the problem of bias discussed above in detail.
Revised: (2019) analyses the problem of bias discussed above in detail.

Original: It formally deﬁnes and analyzes a constant baseline requirement and shows that it is satisﬁed, in particular, by Pearson correlation coefﬁcient between the vectors consisting of I [ v i ∼ v j ] and
Revised: It formally deﬁnes and analyzes a constant baseline requirement and shows that it is satisﬁed, in particular, by Pearson correlation coefﬁcient between the vectors consisting of I [ v i ∼ v j ] and

Original: I [ d ( f ( v i ) f ( v j )) > 1] .
Revised: I [ d ( f ( v i ) f ( v j )) > 1] .

Original: So, we add the corresponding Pearson loss (one minus correlation coefﬁcient) to our experiments (see (G¨osgens et al., 2019)).
Revised: So, we add the corresponding Pearson loss (one minus correlation coefﬁcient) to our experiments (see (G¨osgens et al., 2019)).

Original: E. E XPERIMENTAL SETUP
Revised: E. E XPERIMENTAL SETUP

Original: Our experiments are based on a publicly available implementation of graph embedding by Gu et al.
Revised: Our experiments are based on a publicly available implementation of graph embedding by Gu et al.

Original: The main advantage of this algorithm is that it works for any curvature: negative, positive or zero.
Revised: The main advantage of this algorithm is that it works for any curvature: negative, positive or zero.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: We modiﬁed the implementation for our task: in particular, we added a regime responsible for minimizing threshold-based loss functions.
Revised: We modiﬁed the implementation for our task: in particular, we added a regime responsible for minimizing threshold-based loss functions.

Original: E.g., zero-one loss is proportional to (cid:88)
Revised: E.g., zero-one loss is proportional to (cid:88)

Original: [Equation]
Revised: [Equation]

Original: This function is not differentiable, so we replace it by gradient descent friendly function which we call ReLuLoss:
Revised: This function is not differentiable, so we replace it by gradient descent friendly function which we call ReLuLoss:

Original: [Equation]
Revised: [Equation]

Original: , where ReLu( x ) = max(0 , x ) , ε = 0 .
Revised: , where ReLu( x ) = max(0 , x ) , ε = 0 .

Original: In our experiments we observed that such loss function outperforms other analogues (e.g., MSE and sigmoid-based smoothing) since it speeds up the convergence.
Revised: In our experiments we observed that such loss function outperforms other analogues (e.g., MSE and sigmoid-based smoothing) since it speeds up the convergence.

Original: To adapt to various threshold-based loss functions, we also reserve several iterations to gradient steps based on the corresponding sigmoid-based smoothing.
Revised: To adapt to various threshold-based loss functions, we also reserve several iterations to gradient steps based on the corresponding sigmoid-based smoothing.

Original: Note that in experiments we use only the more advanced Forman curvature ˆ F ( G ) .
Revised: Note that in experiments we use only the more advanced Forman curvature ˆ F ( G ) .

Original: For computing
Revised: For computing

Original: Ollivier and Forman curvature we use the open source package by Ni et al. (2015).
Revised: Ollivier and Forman curvature we use the open source package by Ni et al. (2015).

Original: For the embedding algorithm, in all experiments we ﬁx epochs = 

Revised: For the embedding algorithm, in all experiments we ﬁx epochs =

Original: and we also try several learning rates (usually from 0.01 to 10 4 ).
Revised: and we also try several learning rates (usually from 0.01 to 10 4 ).

Original: We noticed that learning rate signiﬁcantly affect the performance of the learning algorithm and also the optimal learning rate is usually smaller for smaller datasets and it is larger for spaces of negative curvature.
Revised: We noticed that learning rate signiﬁcantly affect the performance of the learning algorithm and also the optimal learning rate is usually smaller for smaller datasets and it is larger for spaces of negative curvature.

Original: We also use the option use adagrad =
Revised: We also use the option use adagrad =

Original: True since we noticed that it inproves the embedding into Euclidean spaces (otherwise there is a peak in distortion at c = 0 ).
Revised: True since we noticed that it inproves the embedding into Euclidean spaces (otherwise there is a peak in distortion at c = 0 ).

Original: Let us emphasize that in our experiments we do not care about overﬁtting, because we do not analyze the embedding algorithm, our aim is to ﬁnd a curvature which minimizes loss function.
Revised: Let us emphasize that in our experiments we do not care about overﬁtting, because we do not analyze the embedding algorithm, our aim is to ﬁnd a curvature which minimizes loss function.

Original: (It can be also thought of as though we focus on graph reconstruction task).
Revised: (It can be also thought of as though we focus on graph reconstruction task).

Original: d i s t o r t
Revised: d i s t o r t

Original: i o n curvature d = 2 d =0.04 0.
Revised: i o n curvature d = 2 d =0.04 0.

Original: z e r o - o n e l o ss curvature d = 2 d =
Revised: z e r o - o n e l o ss curvature d = 2 d =

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: p e a r s o n l o ss curvature d = 2 d = 0.
Revised: p e a r s o n l o ss curvature d = 2 d = 0.

Original: b a l a n c e d z e r o - o n e l o ss curvature d = 2 d =
Revised: b a l a n c e d z e r o - o n e l o ss curvature d = 2 d =

Original: To plot all the ﬁgures we try curvatures in some range (usually between -2 and 1) with step size 0.1.
Revised: To plot all the ﬁgures we try curvatures in some range (usually between -2 and 1) with step size 0.1.

Original: Our code is publicly available.
Revised: Our code is publicly available.

Original: E. C OMPARISON OF CURVATURES ON SYNTHETIC DATASETS
Revised: E. C OMPARISON OF CURVATURES ON SYNTHETIC DATASETS

Original: See the results in Tables 4 and 5.
Revised: See the results in Tables 4 and 5.

Original: E. E FFECT OF DIMENSION AND LOSS FUNCTION
Revised: E. E FFECT OF DIMENSION AND LOSS FUNCTION

Original: In this section, we visualize the dependence of the loss function on curvature (see Figures 8-18).
Revised: In this section, we visualize the dependence of the loss function on curvature (see Figures 8-18).

Original: In particular, we demonstrate that the optimal curvature signiﬁcantly depends on both loss function and dimension.
Revised: In particular, we demonstrate that the optimal curvature signiﬁcantly depends on both loss function and dimension.

Original: More precisely, while our theoretical results hold for any threshold-based loss function, for less trivial graphs the optimal curvature depends on the particular choice.
Revised: More precisely, while our theoretical results hold for any threshold-based loss function, for less trivial graphs the optimal curvature depends on the particular choice.

Original: The obtained results support our intuition (based on theoretical results) that while hyperbolic space helps in many cases for small dimensions, it becomes less pronounced in larger dimensions.
Revised: The obtained results support our intuition (based on theoretical results) that while hyperbolic space helps in many cases for small dimensions, it becomes less pronounced in larger dimensions.

Original: This is most clearly seen for distortion loss (see, e.g., Karate club, Football, CSPhDs, Euroroad, EuroSiS,
Revised: This is most clearly seen for distortion loss (see, e.g., Karate club, Football, CSPhDs, Euroroad, EuroSiS,

Original: Power).
Revised: Power).

Original: E. E MPIRICAL ANALYSIS OF VOLUME - BASED CURVATURE
Revised: E. E MPIRICAL ANALYSIS OF VOLUME - BASED CURVATURE

Original: As discussed in the previous section, optimizing distortion leads to more stable results than threshold-based losses, so we focus on this loss here.
Revised: As discussed in the previous section, optimizing distortion leads to more stable results than threshold-based losses, so we focus on this loss here.

Original: It turns out that the proposed volume-based estimator is able to reﬂect well whether it is reasonable to embed a network to a hyperbolic space for a given loss function if we care about distortion.
Revised: It turns out that the proposed volume-based estimator is able to reﬂect well whether it is reasonable to embed a network to a hyperbolic space for a given loss function if we care about distortion.

Original: Table 6 shows the estimated values of a curvature.
Revised: Table 6 shows the estimated values of a curvature.

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: p e a r s o n l o ss curvature d = 2 d =0.
Revised: p e a r s o n l o ss curvature d = 2 d =0.

Original: b a l a n c e d z e r o - o n e l o ss curvature
Revised: b a l a n c e d z e r o - o n e l o ss curvature

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original:  e a r s o n l o ss
Revised: p e a r s o n l o ss

Original: [Figure]
Revised: [Figure]

Revised: r

Original: [Figure]
Revised: [Figure]

Revised: n

Revised: [Figure]

Revised: l

Original: [Figure]
Revised: [Figure]

Revised: curvature

Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Original: i
Revised: i

Original: [Figure]
Revised: [Figure]

Original: t o r t
Revised: t o r t

Original: [Figure]
Revised: [Figure]

Original: z e r o - o n e l o ss curvature d = 2 d =0.p e a r s o n l o ss curvature d =
Revised: z e r o - o n e l o ss curvature d = 2 d =0.

Original: [Figure]

Original: [Figure] [Figure] [Figure] [Figure]
Revised: p e a r s o n l o ss curvature d = 2 d =0.

Original: [Figure]

Original: [Figure] [Figure] [Figure] b a l a n c e d z e r o - o n e l o ss curvature 
Revised: b a l a n c e d z e r o - o n e l o ss curvature d = 2 d =

Original: [Figure]
Revised: [Figure]

Original: [Figure]

Original: [Figure]
Revised: [Figure]

Original: p e a r s o n l o ss curvature
Revised: p e a r s o n l o ss curvature

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: d i s t o r t
Revised: d i s t o r t

Original: i o n
Revised: i o n

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Table]
Revised: [Table]

Original: One can see that for d = 2 in many cases signiﬁcantly negative curvature is predicted which is indeed conﬁrmed by our experiments, shown in Figures 8-18 (distortion loss).
Revised: One can see that for d = 2 in many cases signiﬁcantly negative curvature is predicted which is indeed conﬁrmed by our experiments, shown in Figures 8-18 (distortion loss).

Original: Non-negative curvature is predicted for Chicago, CSPhDs, Euroroad and Power and indeed for these datasets the optimal curvature is close to zero.
Revised: Non-negative curvature is predicted for Chicago, CSPhDs, Euroroad and Power and indeed for these datasets the optimal curvature is close to zero.

Original: In contrast, for d = 10 negative curvature is never predicted, which agrees with the experiment well.
Revised: In contrast, for d = 10 negative curvature is never predicted, which agrees with the experiment well.

Original: So, in these cases it is not reasonable to embed networks to a hyperbolic space.
Revised: So, in these cases it is not reasonable to embed networks to a hyperbolic space.