Original: U NVEILING THE SAMPLING DENSITY IN NONUNIFORM GEOMETRIC GRAPHS
Revised: U NVEILING THE SAMPLING DENSITY IN NONUNIFORM GEOMETRIC GRAPHS

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: A powerful framework for studying graphs is to consider them as geometric graphs: nodes are randomly sampled from an underlying metric space, and any pair of nodes is connected if their distance is less than a specified neighborhood radius.
Revised: A powerful framework for studying graphs is to consider them as geometric graphs: nodes are randomly sampled from an underlying metric space, and any pair of nodes is connected if their distance is less than a specified neighborhood radius.

Original: Currently, the literature mostly focuses on uniform sampling and constant neighborhood radius.
Revised: Currently, the literature mostly focuses on uniform sampling and constant neighborhood radius.

Original: However, real-world graphs are likely to be better represented by a model in which the sampling density and the neighborhood radius can both vary over the latent space.
Revised: However, real-world graphs are likely to be better represented by a model in which the sampling density and the neighborhood radius can both vary over the latent space.

Original: For instance, in a social network communities can be modeled as densely sampled areas, and hubs as nodes with larger neighborhood radius.
Revised: For instance, in a social network communities can be modeled as densely sampled areas, and hubs as nodes with larger neighborhood radius.

Original: In this work, we first perform a rigorous mathematical analysis of this (more general) class of models, including derivations of the resulting graph shift operators.
Revised: In this work, we first perform a rigorous mathematical analysis of this (more general) class of models, including derivations of the resulting graph shift operators.

Original: The key insight is that graph shift operators should be corrected in order to avoid potential distortions introduced by the non-uniform sampling.
Revised: The key insight is that graph shift operators should be corrected in order to avoid potential distortions introduced by the non-uniform sampling.

Original: Then, we develop methods to estimate the unknown sampling density in a selfsupervised fashion.
Revised: Then, we develop methods to estimate the unknown sampling density in a selfsupervised fashion.

Original: Finally, we present exemplary applications in which the learnt density is used to 1) correct the graph shift operator and improve performance on a variety of tasks, 2) improve pooling, and 3) extract knowledge from networks.
Revised: Finally, we present exemplary applications in which the learnt density is used to 1) correct the graph shift operator and improve performance on a variety of tasks, 2) improve pooling, and 3) extract knowledge from networks.

Original: Our experimental findings support our theory and provide strong evidence for our model.
Revised: Our experimental findings support our theory and provide strong evidence for our model.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Graphs are mathematical objects used to represent relationships among entities.
Revised: Graphs are mathematical objects used to represent relationships among entities.

Original: Their use is ubiquitous, ranging from social networks to recommender systems, from protein-protein interactions to functional brain networks.
Revised: Their use is ubiquitous, ranging from social networks to recommender systems, from protein-protein interactions to functional brain networks.

Original: Despite their versatility, their non-euclidean nature makes graphs hard to analyze.
Revised: Despite their versatility, their non-euclidean nature makes graphs hard to analyze.

Original: For instance, the indexing of the nodes is arbitrary, there is no natural definition of orientation, and neighborhoods can vary in size and topology.
Revised: For instance, the indexing of the nodes is arbitrary, there is no natural definition of orientation, and neighborhoods can vary in size and topology.

Original: Moreover, it is not clear how to compare a general pair of graphs since they can have a different number of nodes.
Revised: Moreover, it is not clear how to compare a general pair of graphs since they can have a different number of nodes.

Original: Therefore, new ways of thinking about graphs were developed by the community.
Revised: Therefore, new ways of thinking about graphs were developed by the community.

Original: One approach is proposed in graphon theory (Lov ´ asz, 2012): graphs are sampled from continuous graph models called graphons , and any two graphs of any size and topology can be compared using certain metrics defined in the space of graphons.
Revised: One approach is proposed in graphon theory (Lov ´ asz, 2012): graphs are sampled from continuous graph models called graphons , and any two graphs of any size and topology can be compared using certain metrics defined in the space of graphons.

Original: A geometric graph is an important case of a graph sampled from a graphon.
Revised: A geometric graph is an important case of a graph sampled from a graphon.

Original: In a geometric graph, a set of points is uniformly sampled from a metric-measure space, and every pair of points is linked if their distance is less than a specified neighborhood radius.
Revised: In a geometric graph, a set of points is uniformly sampled from a metric-measure space, and every pair of points is linked if their distance is less than a specified neighborhood radius.

Original: Therefore, a geometric graph inherits a geometric structure from its latent space that can be leveraged to perform rigorous mathematical analysis and to derive computational methods.
Revised: Therefore, a geometric graph inherits a geometric structure from its latent space that can be leveraged to perform rigorous mathematical analysis and to derive computational methods.

Original: Geometric graphs have a long history, dating back to the 60s (Gilbert, 1961).
Revised: Geometric graphs have a long history, dating back to the 60s (Gilbert, 1961).

Original: They have been extensively used to model complex spatial networks (Barthelemy, 2011).
Revised: They have been extensively used to model complex spatial networks (Barthelemy, 2011).

Original: One of the first models of geometric graphs is the random geometric graph (Penrose, 2003), where the latent space is a
Revised: One of the first models of geometric graphs is the random geometric graph (Penrose, 2003), where the latent space is a

Original: Euclidean unit square.
Revised: Euclidean unit square.

Original: Various generalizations and modifications of this model have been proposed in the literature, such as random rectangular graphs (Estrada & Sheerin, 2015), random spherical graphs (Allen-Perkins, 2018), and random hyperbolic graphs (Krioukov et al., 2010).
Revised: Various generalizations and modifications of this model have been proposed in the literature, such as random rectangular graphs (Estrada & Sheerin, 2015), random spherical graphs (Allen-Perkins, 2018), and random hyperbolic graphs (Krioukov et al., 2010).

Original: Geometric graphs are particularly useful since they share properties with real-world networks.
Revised: Geometric graphs are particularly useful since they share properties with real-world networks.

Original: For instance, random hyperbolic graphs are small-world , scale-free , with high clustering (Papadopoulos et al., 2010; Gugelmann et al., 2012).
Revised: For instance, random hyperbolic graphs are small-world , scale-free , with high clustering (Papadopoulos et al., 2010; Gugelmann et al., 2012).

Original: The small-world property asserts that the distance between any two nodes is small, even if the graph is large.
Revised: The small-world property asserts that the distance between any two nodes is small, even if the graph is large.

Original: The scale-free property is the description of the degree sequence as a heavy-tailed distribution: a small number of nodes have many connections, while the rest have small neighborhoods.
Revised: The scale-free property is the description of the degree sequence as a heavy-tailed distribution: a small number of nodes have many connections, while the rest have small neighborhoods.

Original: These two properties are intimately related to the presence of hubs – nodes with a considerable number of neighbors – while the high clustering is related to the network’s community structure.
Revised: These two properties are intimately related to the presence of hubs – nodes with a considerable number of neighbors – while the high clustering is related to the network’s community structure.

Original: However, standard geometric graph models focus mainly on uniform sampling, which does not describewell real-world networks.
Revised: However, standard geometric graph models focus mainly on uniform sampling, which does not describe real-world networks well.

Original: For instance, in location-based social networks, the spatial distribution of nodes is rarely uniform because people tend to congregate around the city centers (Cho et al., 2011; Wang & Gonz´alez, 2009).
Revised: For instance, in location-based social networks, the spatial distribution of nodes is rarely uniform because people tend to congregate around the city centers (Cho et al., 2011; Wang & Gonz´alez, 2009).

Original: In online communities such as the LiveJournal social network, nonuniformity arises since the probability of befriending a particular person is inversely proportional to the number of closer people (Hu et al., 2011;
Revised: In online communities such as the LiveJournal social network, nonuniformity arises since the probability of befriending a particular person is inversely proportional to the number of closer people (Hu et al., 2011;

Original: Liben-Nowell et al., 2005).
Revised: Liben-Nowell et al., 2005).

Original: In a WWW network, there are more pages (and thus nodes) for popular topics than obscure ones.
Revised: In a WWW network, there are more pages (and thus nodes) for popular topics than obscure ones.

Original: In social networks, different demographics (age, gender, ethnicity, etc.) may join a social media platform at different rates.
Revised: In social networks, different demographics (age, gender, ethnicity, etc.) may join a social media platform at different rates.

Original: For surface meshes, specific locations may be sampled more finely, depending on the required level of detail.
Revised: For surface meshes, specific locations may be sampled more finely, depending on the required level of detail.

Original: The imbalance caused by non-uniform sampling could affect the analysis and lead to biased results.
Revised: The imbalance caused by non-uniform sampling could affect the analysis and lead to biased results.

Original: For instance, Janssen et al.
Revised: For instance, Janssen et al.

Original: (2016) show that incorrectly assuming uniform density consistently overestimates the node distances while using the (estimated) density gives more accurate results.
Revised: (2016) show that incorrectly assuming uniform density consistently overestimates the node distances while using the (estimated) density gives more accurate results.

Original: Therefore, it is essential to assess the sampling density, which is one of the main goals of this paper.
Revised: Therefore, it is essential to assess the sampling density, which is one of the main goals of this paper.

Original: Barring a few exceptions, non-uniformity is rarely considered in geometric graphs.
Revised: Barring a few exceptions, non-uniformity is rarely considered in geometric graphs.

Original: Iyer & Thacker (2012) study a class of non-uniform random geometric graphs where the radii depend on the location.
Revised: Iyer & Thacker (2012) study a class of non-uniform random geometric graphs where the radii depend on the location.

Original: Mart ´ ınez-Mart
Revised: Mart ´ ınez-Mart

Original: ´ ınez et al.
Revised: ´ ınez et al.

Original: (2022) study non-uniform graphs on the plane with the density functions specified in polar coordinates.
Revised: (2022) study non-uniform graphs on the plane with the density functions specified in polar coordinates.

Original: Pratt et al.
Revised: Pratt et al.

Original: (2018) consider temporal connectivity in finite networks with non-uniform measures.
Revised: (2018) consider temporal connectivity in finite networks with non-uniform measures.

Original: In all of these works, the focus is on (asymptotic) statistical properties of the graphs, such as the average degree and the number of isolated nodes.
Revised: In all of these works, the focus is on (asymptotic) statistical properties of the graphs, such as the average degree and the number of isolated nodes.

Original: 1. O UR C ONTRIBUTION
Revised: 1. O UR C ONTRIBUTION

Original: While traditional Laplacian approximation approaches solve the direct problem – approximating a known continuous Laplacian with a graph Laplacian – in this paper we solve the inverse problem – constructing a graph Laplacian from an observed graph that is guaranteed to approximate an unknown continuous Laplacian.
Revised: While traditional Laplacian approximation approaches solve the direct problem – approximating a known continuous Laplacian with a graph Laplacian – in this paper we solve the inverse problem – constructing a graph Laplacian from an observed graph that is guaranteed to approximate an unknown continuous Laplacian.

Original: We believe that our approach has high practical significance, as in practical data science on graphs, the graph is typically given, but the underlying continuous model is unknown.
Revised: We believe that our approach has high practical significance, as in practical data science on graphs, the graph is typically given, but the underlying continuous model is unknown.

Original: To be able to solve this inverse problem, we introduce the non-uniform geometric graph (NuG) model.
Revised: To be able to solve this inverse problem, we introduce the non-uniform geometric graph (NuG) model.

Original: Unlike the standard geometric graph model, a NuG is generated by a non-uniform sampling density and a non-constant neighborhood radius.
Revised: Unlike the standard geometric graph model, a NuG is generated by a non-uniform sampling density and a non-constant neighborhood radius.

Original: In this setting, we propose a class of graph shift operators (GSOs), called non-uniform geometric GSOs , that are computed solely from the topology of the graph and the node/edge features while guaranteeing that these GSOs approximate corresponding latent continuous operators defined on the underlying geometric spaces.
Revised: In this setting, we propose a class of graph shift operators (GSOs), called non-uniform geometric GSOs , that are computed solely from the topology of the graph and the node/edge features while guaranteeing that these GSOs approximate corresponding latent continuous operators defined on the underlying geometric spaces.

Original: Together with
Revised: Together with

Original: Dasoulas et al.
Revised: Dasoulas et al.

Original: (2021) and Sahbi (2021), our work can be listed as a theoretically grounded way to learn the GSO.
Revised: (2021) and Sahbi (2021), our work can be listed as a theoretically grounded way to learn the GSO.

Original: Justified by formulas grounded in Monte-Carlo analysis, we show how to compensate for the nonuniformity in the sampling when computing non-uniform geometric GSOs.
Revised: Justified by formulas grounded in Monte-Carlo analysis, we show how to compensate for the nonuniformity in the sampling when computing non-uniform geometric GSOs.

Original: This requires having estimates both of the sampling density and the neighborhood radii.
Revised: This requires having estimates both of the sampling density and the neighborhood radii.

Original: Estimating these by only observing the graph is a hard task.
Revised: Estimating these by only observing the graph is a hard task.

Original: For example, graph quantities like the node degrees are affected both by the density and the radius, and hence, it is hard to decouple the density from the radius by only observing the graph.
Revised: For example, graph quantities like the node degrees are affected both by the density and the radius, and hence, it is hard to decouple the density from the radius by only observing the graph.

Original: We hence propose methods for estimating the density (and radius) using a self-supervision approach.
Revised: We hence propose methods for estimating the density (and radius) using a self-supervision approach.

Original: The idea is to train, against some arbitrary task, a spectral graph neural network, where the GSOs underlying the convolution operators are taken as a non-uniform geometric GSO with learnable density.
Revised: The idea is to train, against some arbitrary task, a spectral graph neural network, where the GSOs underlying the convolution operators are taken as a non-uniform geometric GSO with learnable density.

Original: For the model to perform well, it learns to estimate the underlying sampling density, even though it is not directly supervised to do so.
Revised: For the model to perform well, it learns to estimate the underlying sampling density, even though it is not directly supervised to do so.

Original: We explain heuristically the feasibility of the self-supervision approach on a sub-class of non-uniform geometric graphs that we call geometric graphs with hubs .
Revised: We explain heuristically the feasibility of the self-supervision approach on a sub-class of non-uniform geometric graphs that we call geometric graphs with hubs .

Original: This is a class of geometric graphs, motivated by properties of real-world networks, where the radius is roughly piece-wise constant, and the sampling density is smooth.
Revised: This is a class of geometric graphs, motivated by properties of real-world networks, where the radius is roughly piece-wise constant, and the sampling density is smooth.

Original: We show experimentally that the NuG model can effectively model real-world graphs by training a graph autoencoder, where the encoder embeds the nodes in an underlying geometric space, and the decoder produces edges according to the NuG model.
Revised: We show experimentally that the NuG model can effectively model real-world graphs by training a graph autoencoder, where the encoder embeds the nodes in an underlying geometric space, and the decoder produces edges according to the NuG model.

Original: Moreover, we show that using our nonuniform geometric GSOs with learned sampling density in spectral graph neural networks improves downstream tasks.
Revised: Moreover, we show that using our nonuniform geometric GSOs with learned sampling density in spectral graph neural networks improves downstream tasks.

Original: Finally, we present proof-of-concept applications in which we use the learned density to improve pooling and extract knowledge from graphs.
Revised: Finally, we present proof-of-concept applications in which we use the learned density to improve pooling and extract knowledge from graphs.

Original:  N ON -U NIFORM G EOMETRIC M ODELS
Revised:  N ON -U NIFORM G EOMETRIC M ODELS

Original: In this section, we define non-uniform geometric GSOs, and a subclass of such GSOs called geometric graphs with hubs.
Revised: In this section, we define non-uniform geometric GSOs, and a subclass of such GSOs called geometric graphs with hubs.

Original: To compute such GSOs from the data, we show how to estimate the sampling density from a given graph using self-supervision.
Revised: To compute such GSOs from the data, we show how to estimate the sampling density from a given graph using self-supervision.

Original: 2. G RAPH S HIFT O PERATORS AND K ERNEL O PERATORS
Revised: 2. G RAPH S HIFT O PERATORS AND K ERNEL O PERATORS

Original: We denote graphs by G =
Revised: We denote graphs by G =

Original: ( V , E ) , where V is the set of nodes, |V| is the number of nodes, and E is the set of edges.
Revised: ( V , E ) , where V is the set of nodes, |V| is the number of nodes, and E is the set of edges.

Original: A one-dimensional graph signal is a mapping u : V → R .
Revised: A one-dimensional graph signal is a mapping u : V → R .

Original: For a higher feature dimension F ∈ N , a signal is a mapping u : V → R F .
Revised: For a higher feature dimension F ∈ N , a signal is a mapping u : V → R F .

Original: In graph data science, typically, the data comprises only the graph structure G and node/edge features u , and the practitioner has the freedom to design a graph shift operator (GSO).
Revised: In graph data science, typically, the data comprises only the graph structure G and node/edge features u , and the practitioner has the freedom to design a graph shift operator (GSO).

Original: Loosely speaking, given a graph G
Revised: Loosely speaking, given a graph G

Original: =
Revised: =

Original: ( V , E ) , a GSO is any matrix L ∈ R |V|×|V| that respects the connectivity of the graph, i.e., L i,j = 

Revised: ( V , E ) , a GSO is any matrix L ∈ R |V|×|V| that respects the connectivity of the graph, i.e., L i,j =

Original: whenever ( i, j ) / ∈ E , i ̸
Revised: whenever ( i, j ) / ∈ E , i ̸

Original: = j (Mateos et al., 2019).
Revised: = j (Mateos et al., 2019).

Original: GSOs are used in graph signal processing to define filters, as functions of the GSO of the form f ( L ) , where f : R → R is, e.g., a polynomial (Defferrard et al., 2016) or a rational function (Levie et al., 2019).
Revised: GSOs are used in graph signal processing to define filters, as functions of the GSO of the form f ( L ) , where f : R → R is, e.g., a polynomial (Defferrard et al., 2016) or a rational function (Levie et al., 2019).

Original: The filters operate on graph signals u by f ( L ) u .
Revised: The filters operate on graph signals u by f ( L ) u .

Original: Spectral graph convolutional networks are the class of graph neural networks that implement convolutions as filters.
Revised: Spectral graph convolutional networks are the class of graph neural networks that implement convolutions as filters.

Original: When a spectral graph convolutional network is trained, only the filters f : R → R are learned.
Revised: When a spectral graph convolutional network is trained, only the filters f : R → R are learned.

Original: One significant advantage of the spectral approach is that the convolution network is not tied to a specific graph, but can rather be transferred between different graphs of different sizes and topologies.
Revised: One significant advantage of the spectral approach is that the convolution network is not tied to a specific graph, but can rather be transferred between different graphs of different sizes and topologies.

Original: In this work, we see GSOs as randomly sampled from kernel operators defined on underlying geometric spaces.
Revised: In this work, we see GSOs as randomly sampled from kernel operators defined on underlying geometric spaces.

Original: The underlying spaces are modelled as metric spaces.
Revised: The underlying spaces are modelled as metric spaces.

Original: To allow modeling the random sampling of points, each metric space is also assumed to be a probability space.
Revised: To allow modeling the random sampling of points, each metric space is also assumed to be a probability space.

Original: Definition 1.
Revised: Definition 1.

Original: Let ( S , d, µ ) be a metric-probability space 1 with probability measure µ and metric d ; let m ∈ L ∞ ( S ) 2 ; let K ∈ L ∞ ( S × S ) .
Revised: Let ( S , d, µ ) be a metric-probability space 1 with probability measure µ and metric d ; let m ∈ L ∞ ( S ) 2 ; let K ∈ L ∞ ( S × S ) .

Original: The metric-probability Laplacian L = L K,m is defined as
Revised: The metric-probability Laplacian L = L K,m is defined as

Original: [Equation]
Revised: [Equation]

Original: For example, let S be a Riemannian manifold, and take K ( x, y )
Revised: For example, let S be a Riemannian manifold, and take K ( x, y )

Original: =
Revised: =

Original: 1 B α ( x ) ( y ) /µ
Revised: 1 B α ( x ) ( y ) /µ

Original: (B α ( x )) and m ( x ) = 1 , where B α ( x ) is the ball or radius α about x .
Revised: (B α ( x )) and m ( x ) = 1 , where B α ( x ) is the ball or radius α about x .

Original: In this case, the operator L K,m approximates the Laplace-Beltrami operator when α is small (Burago et al., 2019).
Revised: In this case, the operator L K,m approximates the Laplace-Beltrami operator when α is small (Burago et al., 2019).

Original: A random graph is generated by randomly sampling points from the metric-probability space ( S , d, µ ) .
Revised: A random graph is generated by randomly sampling points from the metric-probability space ( S , d, µ ) .

Original: As a modeling assumption, we suppose the sampling is performed according to a measure ν .
Revised: As a modeling assumption, we suppose the sampling is performed according to a measure ν .

Original: We assume ν is a weighted measure with respect to µ , i.e., there exists a density function ρ : S → (0 , ∞ ) such that d ν ( y ) =
Revised: We assume ν is a weighted measure with respect to µ , i.e., there exists a density function ρ : S → (0 , ∞ ) such that d ν ( y ) =

Original: ρ ( y ) d µ ( y ) 3 .
Revised: ρ ( y ) d µ ( y ) 3 .

Original: We assume that ρ is bounded away from zero and infinity.
Revised: We assume that ρ is bounded away from zero and infinity.

Original: Using a change of variable, it is easy to see that
Revised: Using a change of variable, it is easy to see that

Original: [Equation]
Revised: [Equation]

Original: Let x = { x i } Ni =1 be a random independent sample from S according to the distribution ν .
Revised: Let x = { x i } Ni =1 be a random independent sample from S according to the distribution ν .

Original: The corresponding sampled GSO L is defined by
Revised: The corresponding sampled GSO L is defined by

Original: [Equation]
Revised: [Equation]

Original: Given a signal u ∈ L ∞ ( S ) , and its sampled version u = { u ( x i ) } Ni =1 , it is well known that ( Lu ) i approximates L u ( x i ) for every i ∈ { 1 , . . .
Revised: Given a signal u ∈ L ∞ ( S ) , and its sampled version u = { u ( x i ) } Ni =1 , it is well known that ( Lu ) i approximates L u ( x i ) for every i ∈ { 1 , . . .

Original: , N } (Hein et al., 2007; von Luxburg et al., 2008).
Revised: , N } (Hein et al., 2007; von Luxburg et al., 2008).

Original: 2. N ON -U NIFORM G EOMETRIC GSO S
Revised: 2. N ON -U NIFORM G EOMETRIC GSO S

Original: According to (2), a GSO L can be directly sampled from the metric-probability Laplacian L .
Revised: According to (2), a GSO L can be directly sampled from the metric-probability Laplacian L .

Original: However, such an approach would violate our motivating guidelines, since we are interested in GSOs that can be computed directly from the graph structure, without explicitly knowing the underlying continuous kernel and density.
Revised: However, such an approach would violate our motivating guidelines, since we are interested in GSOs that can be computed directly from the graph structure, without explicitly knowing the underlying continuous kernel and density.

Original: In this subsection, we define a class of metric-probability Laplacians that allow such direct sampling.
Revised: In this subsection, we define a class of metric-probability Laplacians that allow such direct sampling.

Original: For that, we first define a model of adjacency in the metric space.
Revised: For that, we first define a model of adjacency in the metric space.

Original: Definition 2.
Revised: Definition 2.

Original: Let ( S , d, µ ) be a metric-probability space.
Revised: Let ( S , d, µ ) be a metric-probability space.

Original: Let α : S → (0 , + ∞ ) be a non-negative measurable function, referred to as neighborhood radius .
Revised: Let α : S → (0 , + ∞ ) be a non-negative measurable function, referred to as neighborhood radius .

Original: The neighborhood model N is defined as the set-valued function that assigns to each x ∈ S the ball
Revised: The neighborhood model N is defined as the set-valued function that assigns to each x ∈ S the ball

Original: [Equation]
Revised: [Equation]

Original: Since y ∈ N ( x ) implies x ∈ N ( y ) for all x , y ∈ S , Def. 2 models only symmetric graphs.
Revised: Since y ∈ N ( x ) implies x ∈ N ( y ) for all x , y ∈ S , Def. 2 models only symmetric graphs.

Original: Next, we define a class of continuous Laplacians based on neighborhood models.
Revised: Next, we define a class of continuous Laplacians based on neighborhood models.

Original: Definition 3.
Revised: Definition 3.

Original: Let ( S , d, µ ) be a metric-probability space, and N a neighborhood model as in Def. 2.
Revised: Let ( S , d, µ ) be a metric-probability space, and N a neighborhood model as in Def. 2.

Original: Let m ( i ) :
Revised: Let m ( i ) :

Original: R → R be a continuous function for every i ∈ { 1 , . .
Revised: R → R be a continuous function for every i ∈ { 1 , . .

Original: The metric-probability
Revised: The metric-probability

Original: Laplacian model is the kernel operator L N that operates on signals u : S → R by
Revised: Laplacian model is the kernel operator L N that operates on signals u : S → R by

Original: [Equation]
Revised: [Equation]

Original: In order to give a concrete example, suppose the neighborhood radius α ( x ) =
Revised: In order to give a concrete example, suppose the neighborhood radius α ( x ) =

Original: α is a constant, m (1) ( x ) = m (3) ( x ) = x − 1 , and m (2) ( x ) = m (4) ( x ) = 1 , then (3) gives
Revised: α is a constant, m (1) ( x ) = m (3) ( x ) = x − 1 , and m (2) ( x ) = m (4) ( x ) = 1 , then (3) gives

Original: [Equation]
Revised: [Equation]

Original: which is an approximation of the Laplace-Beltrami operator.
Revised: which is an approximation of the Laplace-Beltrami operator.

Original: Since the neighborhood model of S represents adjacency in the metric space, we make the modeling assumption that graphs are sampled from neighborhood models, as follows.
Revised: Since the neighborhood model of S represents adjacency in the metric space, we make the modeling assumption that graphs are sampled from neighborhood models, as follows.

Original: First, random independent points x = { x i } Ni =1 are sampled from S according to the “non-uniform” distribution ν as before.
Revised: First, random independent points x = { x i } Ni =1 are sampled from S according to the “non-uniform” distribution ν as before.

Original: Then, an edge is created between each pair x i and x j if x j ∈ N ( x i ) , to form the graph G .
Revised: Then, an edge is created between each pair x i and x j if x j ∈ N ( x i ) , to form the graph G .

Original: Now, a GSO can be sampled from a metric-probability Laplacian model L N by (2), if the underlying continuous model is known.
Revised: Now, a GSO can be sampled from a metric-probability Laplacian model L N by (2), if the underlying continuous model is known.

Original: However, such knowledge is not required, since the special structure of the metric-probability Laplacian model allows deriving the GSO directly from the sampled graph G and the sampled density { ρ ( x i ) } Ni =1 .
Revised: However, such knowledge is not required, since the special structure of the metric-probability Laplacian model allows deriving the GSO directly from the sampled graph G and the sampled density { ρ ( x i ) } Ni =1 .

Original: Def. 4 below gives such a construction of GSO.
Revised: Def. 4 below gives such a construction of GSO.

Original: In the following, given a vector u ∈ R N and a function m : R → R , we denote by m ( u ) the vector { m ( u i ) } Ni =1 , and by diag( u ) ∈ R N × N the diagonal matrix with diagonal u .
Revised: In the following, given a vector u ∈ R N and a function m : R → R , we denote by m ( u ) the vector { m ( u i ) } Ni =1 , and by diag( u ) ∈ R N × N the diagonal matrix with diagonal u .

Original: Definition 4.
Revised: Definition 4.

Original: Let G =
Revised: Let G =

Original: ( V , E ) be a graph with adjacency matrix A ; let ρ ρ ρ : V → (0 , ∞ ) be a graph signal, referred to as graph density .
Revised: ( V , E ) be a graph with adjacency matrix A ; let ρ ρ ρ : V → (0 , ∞ ) be a graph signal, referred to as graph density .

Original: The non-uniform geometric GSO is defined to be
Revised: The non-uniform geometric GSO is defined to be

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where A ρ = A diag( ρρρ ) − 1 and D ( i ) ρ = diag (cid:16) m ( i ) (cid:0) N − 1 A ρ 111 (cid:1)(cid:17) .
Revised: where A ρ = A diag( ρρρ ) − 1 and D ( i ) ρ = diag (cid:16) m ( i ) (cid:0) N − 1 A ρ 111 (cid:1)(cid:17) .

Original: Def. 4 can retrieve, as particular cases, the usual GSOs, as shown in Tab. 3 in Appendix C. For example, in case of m (1) ( x ) = m (3) ( x ) = x − 1 , m (2) ( x ) = m (4) ( x ) = 1 , and uniform sampling ρ = 1 , (4) leads to the random-walk Laplacian L G ,
Revised: Def. 4 can retrieve, as particular cases, the usual GSOs, as shown in Tab. 3 in Appendix C. For example, in case of m (1) ( x ) = m (3) ( x ) = x − 1 , m (2) ( x ) = m (4) ( x ) = 1 , and uniform sampling ρ = 1 , (4) leads to the random-walk Laplacian L G ,

Original: 1 = D − 1 A − I .
Revised: 1 = D − 1 A − I .

Original: The non-uniform geometric
Revised: The non-uniform geometric

Original: GSO in Def. 4 is the Monte-Carlo approximation of the metric-probability Laplacian in Def. 3.
Revised: GSO in Def. 4 is the Monte-Carlo approximation of the metric-probability Laplacian in Def. 3.

Original: This is shown in the following proposition, whose proof can be found in Appendix D.
Revised: This is shown in the following proposition, whose proof can be found in Appendix D.

Original: Proposition 1.
Revised: Proposition 1.

Original: Let G =
Revised: Let G =

Original: ( V , E ) be a random graph with i.i.d.
Revised: ( V , E ) be a random graph with i.i.d.

Original: sample x = { x i } Ni =1 from the metricprobability space ( S , d, µ ) with neighborhood structure N .
Revised: sample x = { x i } Ni =1 from the metricprobability space ( S , d, µ ) with neighborhood structure N .

Original: Let L G ,ρρρ be the non-uniform geometric
Revised: Let L G ,ρρρ be the non-uniform geometric

Original: GSO as in Def. 4.
Revised: GSO as in Def. 4.

Original: Let u ∈ L ∞ ( S ) and u = { u ( x i ) } Ni =1 .
Revised: Let u ∈ L ∞ ( S ) and u = { u ( x i ) } Ni =1 .

Original: Then, for every i = 1 , . .
Revised: Then, for every i = 1 , . .

Original: , N ,
Revised: , N ,

Original: [Equation]
Revised: [Equation]

Original: In Appendix D we also show that, in probability at least 1 − p , it holds
Revised: In Appendix D we also show that, in probability at least 1 − p , it holds

Original: [Equation]
Revised: [Equation]

Original: Prop. 1 means that if we are given  graph that was sampled from a neighborhood model, and we know (or have an estimate of) the sampling density at every node of the graph, then we can compute a
Revised: Prop. 1 means that if we are given a graph that was sampled from a neighborhood model, and we know (or have an estimate of) the sampling density at every node of the graph, then we can compute a

Original: GSO according to (4) that is guaranteed to approximate a corresponding unknown metric-probability
Revised: GSO according to (4) that is guaranteed to approximate a corresponding unknown metric-probability

Original: Laplacian.
Revised: Laplacian.

Original: The next goal is hence to estimate the sampling density from a given graph.
Revised: The next goal is hence to estimate the sampling density from a given graph.

Original: 2. I NFERRING THE S AMPLING D ENSITY
Revised: 2. I NFERRING THE S AMPLING D ENSITY

Original: In real-world scenarios, the true value of the sampling density is not known.
Revised: In real-world scenarios, the true value of the sampling density is not known.

Original: The following result gives a first rough estimate of the sampling density in a special case.
Revised: The following result gives a first rough estimate of the sampling density in a special case.

Original: Lemma 1.
Revised: Lemma 1.

Original: Let ( S , d, µ ) be a metric-probability space; let N be a neighborhood model; let ν be a weighted measure with respect to µ with continuous density ρ bounded away from zero and infinity.
Revised: Let ( S , d, µ ) be a metric-probability space; let N be a neighborhood model; let ν be a weighted measure with respect to µ with continuous density ρ bounded away from zero and infinity.

Original: There exists a function c : S → S such that c ( x ) ∈ N ( x ) and ( ρ ◦ c )( x ) =
Revised: There exists a function c : S → S such that c ( x ) ∈ N ( x ) and ( ρ ◦ c )( x ) =

Original: ν (cid:0) N ( x ) (cid:1) /µ
Revised: ν (cid:0) N ( x ) (cid:1) /µ

Original: (cid:0) N ( x ) (cid:1) .
Revised: (cid:0) N ( x ) (cid:1) .

Original: The proof can be found in Appendix D.
Revised: The proof can be found in Appendix D.

Original: In light of Lemma 1, if the neighborhood radius of x is small enough, if the volumes µ ( N ( x )) are approximately constant, and if ρ does not vary too fast, the sampling density at x is roughly proportional to ν ( N ( x )) , that is, the likelihood a point is drawn from N ( x ) .
Revised: In light of Lemma 1, if the neighborhood radius of x is small enough, if the volumes µ ( N ( x )) are approximately constant, and if ρ does not vary too fast, the sampling density at x is roughly proportional to ν ( N ( x )) , that is, the likelihood a point is drawn from N ( x ) .

Original: Therefore, in this situation, the sampling density ρ ( x ) can be approximated by the degree of the node x .
Revised: Therefore, in this situation, the sampling density ρ ( x ) can be approximated by the degree of the node x .

Original: In practice, we are interested in graphs where the volumes of the neighborhoods µ ( N ( x )) are not constant.
Revised: In practice, we are interested in graphs where the volumes of the neighborhoods µ ( N ( x )) are not constant.

Original: Still, a normalization of the GSO by the degree can soften the distortion introduced by non-uniform sampling, at least locally in areas where µ ( N ( x )) is slowly varying.
Revised: Still, a normalization of the GSO by the degree can soften the distortion introduced by non-uniform sampling, at least locally in areas where µ ( N ( x )) is slowly varying.

Original: This suggests that the degree of a node is a good input feature for a method that learns the sampling density from the graph structure and the node features.
Revised: This suggests that the degree of a node is a good input feature for a method that learns the sampling density from the graph structure and the node features.

Original: Such a method is developed next.
Revised: Such a method is developed next.

Original: 2. G EOMETRIC G RAPHS WITH H UBS
Revised: 2. G EOMETRIC G RAPHS WITH H UBS

Original: When designing a method to estimate the sampling density from the graph, the degree is not a sufficient input parameter.
Revised: When designing a method to estimate the sampling density from the graph, the degree is not a sufficient input parameter.

Original: The reason is that the degree of a node has two main contributions: the sampling density and the neighborhood radius.
Revised: The reason is that the degree of a node has two main contributions: the sampling density and the neighborhood radius.

Original: The problem of decoupling the two contributions is difficult in the general case.
Revised: The problem of decoupling the two contributions is difficult in the general case.

Original: However, if the sampling density is slowly varying, and if the neighborhood radius is piecewise constant, the problem becomes easier.
Revised: However, if the sampling density is slowly varying, and if the neighborhood radius is piecewise constant, the problem becomes easier.

Original: Intuitively, a slowly varying sampling density causes a slight change in the degree of adjacent nodes.
Revised: Intuitively, a slowly varying sampling density causes a slight change in the degree of adjacent nodes.

Original: In contrast, a sudden change in the degree is caused by a radius jump.
Revised: In contrast, a sudden change in the degree is caused by a radius jump.

Original: In time-frequency analysis and compressed sensing, various results guarantee the ability to separate a signal into its different components, e.g., piecewise constant and smooth components (Do et al., 2022; Donoho & Kutyniok, 2013; Gribonval & Bacry, 2003).
Revised: In time-frequency analysis and compressed sensing, various results guarantee the ability to separate a signal into its different components, e.g., piecewise constant and smooth components (Do et al., 2022; Donoho & Kutyniok, 2013; Gribonval & Bacry, 2003).

Original: This motivates our model of geometric graphs with hubs .
Revised: This motivates our model of geometric graphs with hubs .

Original: Definition 5.
Revised: Definition 5.

Original: A geometric graph with hubs is a random graph with non-uniform geometric GSO, sampled from a metric-probability space ( S , d, µ ) with neighborhood model N , where the sampling density ρ is Lipschitz continuous in S and µ ( N ( x )) is piecewise constant.
Revised: A geometric graph with hubs is a random graph with non-uniform geometric GSO, sampled from a metric-probability space ( S , d, µ ) with neighborhood model N , where the sampling density ρ is Lipschitz continuous in S and µ ( N ( x )) is piecewise constant.

Original: We call this model a geometric graph with hubs since we typically assume that µ ( N ( x )) has a low value for most points x ∈ S , while only a few small regions, called hubs , have large neighborhoods.
Revised: We call this model a geometric graph with hubs since we typically assume that µ ( N ( x )) has a low value for most points x ∈ S , while only a few small regions, called hubs , have large neighborhoods.

Original: In Section 3.1, we exhibit that geometric graphs with hubs can model real-world graphs.
Revised: In Section 3.1, we exhibit that geometric graphs with hubs can model real-world graphs.

Original: To validate this, we train a graph auto-encoder on real-world networks, where the decoder is restricted to be a geometric graph with hubs.
Revised: To validate this, we train a graph auto-encoder on real-world networks, where the decoder is restricted to be a geometric graph with hubs.

Original: The fact that such a decoder can achieve low error rates suggests that real-world graphs can often be modeled as geometric graphs with hubs.
Revised: The fact that such a decoder can achieve low error rates suggests that real-world graphs can often be modeled as geometric graphs with hubs.

Original: Geometric graphs with hubs are also reasonable from a modeling point of view.
Revised: Geometric graphs with hubs are also reasonable from a modeling point of view.

Original: For example, it is reasonable to assume that different demographics join a social media platform at different rates.
Revised: For example, it is reasonable to assume that different demographics join a social media platform at different rates.

Original: Since the demographic is directly related to the node features, and the graph roughly exhibits homophily, the features are slowly varying over the graph, and hence, so is the sampling density.
Revised: Since the demographic is directly related to the node features, and the graph roughly exhibits homophily, the features are slowly varying over the graph, and hence, so is the sampling density.

Original: On the other hand, hubs in social networks are associated with influencers.
Revised: On the other hand, hubs in social networks are associated with influencers.

Original: The conditions that make a certain user an influencer are not directly related to the features.
Revised: The conditions that make a certain user an influencer are not directly related to the features.

Original: Indeed, if the node features in a social network are user interests, users that follow an influencer tend to share their features with the influencer, so the features themselves are not enough to determine if a node is deemed to be a center of a hub or not.
Revised: Indeed, if the node features in a social network are user interests, users that follow an influencer tend to share their features with the influencer, so the features themselves are not enough to determine if a node is deemed to be a center of a hub or not.

Original: Hence, the radius does not tend to be continuous over the graph, and, instead, is roughly constant and small over most of the graph (non-influencers), except for a number of narrow and sharp peaks (influencers).
Revised: Hence, the radius does not tend to be continuous over the graph, and, instead, is roughly constant and small over most of the graph (non-influencers), except for a number of narrow and sharp peaks (influencers).

Original: 2. L EARNING THE S AMPLING D ENSITY
Revised: 2. L EARNING THE S AMPLING D ENSITY

Original: In the current section, we propose a strategy to assess the sampling density ρ ρ ρ .
Revised: In the current section, we propose a strategy to assess the sampling density ρ ρ ρ .

Original: As suggested by the above discussion, the local changes in the degree of the graph give us a lot of information about the local changes in the sampling density and neighborhood radius of geometric graphs with hubs.
Revised: As suggested by the above discussion, the local changes in the degree of the graph give us a lot of information about the local changes in the sampling density and neighborhood radius of geometric graphs with hubs.

Original: Hence, we implement the density estimator as a message-passing graph neural network (MPNN) Θ because it performs local computations and it is equivariant to node indexing, a property that both the density and the degree satisfy.
Revised: Hence, we implement the density estimator as a message-passing graph neural network (MPNN) Θ because it performs local computations and it is equivariant to node indexing, a property that both the density and the degree satisfy.

Original: Since we are mainly interested in estimating the inverse of the sampling density, Θ takes as input the inverse of the degree and the inverse of the mean degree of the one-hop neighborhood for all nodes in the graph as two input channels.
Revised: Since we are mainly interested in estimating the inverse of the sampling density, Θ takes as input the inverse of the degree and the inverse of the mean degree of the one-hop neighborhood for all nodes in the graph as two input channels.

Original: However, it is not yet clear how to train Θ .
Revised: However, it is not yet clear how to train Θ .

Original: Since in real-world scenarios the ground-truth density is not known, we train Θ in a self-supervised manner.
Revised: Since in real-world scenarios the ground-truth density is not known, we train Θ in a self-supervised manner.

Original: In this context, we choose a task (link prediction, node or graph classification, etc.) on a real-world graph G
Revised: In this context, we choose a task (link prediction, node or graph classification, etc.) on a real-world graph G

Original: and we solve it by means of a graph neural network Ψ , referred to as task network .
Revised: and we solve it by means of a graph neural network Ψ , referred to as task network .

Original: Since we want Ψ to depend on the sampling density estimator Θ , we define Ψ as a spectral graph convolution network based on the non-uniform geometric GSO L G , Θ( G ) , e.g., GCN (Kipf & Welling, 2017), ChebNet (Defferrard et al., 2016) or
Revised: Since we want Ψ to depend on the sampling density estimator Θ , we define Ψ as a spectral graph convolution network based on the non-uniform geometric GSO L G , Θ( G ) , e.g., GCN (Kipf & Welling, 2017), ChebNet (Defferrard et al., 2016) or

Original: CayleyNet (Levie et al., 2019).
Revised: CayleyNet (Levie et al., 2019).

Original: We, therefore, train Ψ end-to-end on the given task.
Revised: We, therefore, train Ψ end-to-end on the given task.

Original: The idea behind the proposed method is that the task depends mostly on the underlying continuous model.
Revised: The idea behind the proposed method is that the task depends mostly on the underlying continuous model.

Original: For example, in shape classification, the label of each graph depends on the surface from which the graph is sampled, rather than the specific intricate structure of the discretization.
Revised: For example, in shape classification, the label of each graph depends on the surface from which the graph is sampled, rather than the specific intricate structure of the discretization.

Original: Therefore, the task network Ψ can perform well if it learns to ignore the particular fine details of the discretization, and focus on the underlying space.
Revised: Therefore, the task network Ψ can perform well if it learns to ignore the particular fine details of the discretization, and focus on the underlying space.

Original: The correction of the GSO via the estimated sampling density ((4)) gives the network exactly such power.
Revised: The correction of the GSO via the estimated sampling density ((4)) gives the network exactly such power.

Original: Therefore, we conjecture that Θ will indeed learn how to estimate the sampling density for graphs that exhibit homophily.
Revised: Therefore, we conjecture that Θ will indeed learn how to estimate the sampling density for graphs that exhibit homophily.

Original: In order to verify the previous claim, and to validate our model, we focus on link prediction on synthetic datasets (see Appendix B), for which the ground-truth sampling density is known.
Revised: In order to verify the previous claim, and to validate our model, we focus on link prediction on synthetic datasets (see Appendix B), for which the ground-truth sampling density is known.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 1, the MPNN Θ is able to correctly identify hubs, and correctly predict the ground-truth density in a self-supervised manner.
Revised: 1, the MPNN Θ is able to correctly identify hubs, and correctly predict the ground-truth density in a self-supervised manner.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: In the following, we validate the NuG model experimentally.
Revised: In the following, we validate the NuG model experimentally.

Original: Moreover, we verify the validity of our method first on synthetic datasets, then on real-world graphs in a transductive (node classification) and inductive (graph classification) setting.
Revised: Moreover, we verify the validity of our method first on synthetic datasets, then on real-world graphs in a transductive (node classification) and inductive (graph classification) setting.

Original: Finally, we propose proof-of-concept applications in explainability, learning GSOs, and differentiable pooling
Revised: Finally, we propose proof-of-concept applications in explainability, learning GSOs, and differentiable pooling

Original: . 3. L INK P REDICTION
Revised: . 3. L INK P REDICTION

Original: The method proposed in Section 2.5 is applied on synthetic datasets of geometric graphs with hubs (see for details Appendices A.1 to A.2).
Revised: The method proposed in Section 2.5 is applied on synthetic datasets of geometric graphs with hubs (see for details Appendices A.1 to A.2).

Original: In Fig.
Revised: In Fig.

Original: 1, it is shown that Θ is able to correctly predict the value of the sampling density.
Revised: 1, it is shown that Θ is able to correctly predict the value of the sampling density.

Original: The left plots of Figs.
Revised: The left plots of Figs.

Original: 1a and 1b show that the density is well
Revised: 1a and 1b show that the density is well

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: approximated both at hubs and non-hubs.
Revised: approximated both at hubs and non-hubs.

Original: Looking at the right plots, it is evident that the density cannot be predicted solely from the degree.
Revised: Looking at the right plots, it is evident that the density cannot be predicted solely from the degree.

Original: Fig. 2 and Fig.
Revised: Fig. 2 and Fig.

Original: 7 in Appendix A.3 show that the NuG model is able to effectively represent real-world graphs, outperforming other graph auto-encoder methods (see Tab. 1 for the number of parameters of each method).
Revised: 7 in Appendix A.3 show that the NuG model is able to effectively represent real-world graphs, outperforming other graph auto-encoder methods (see Tab. 1 for the number of parameters of each method).

Original: Here, we learn an auto-encoder with four types of decoders: inner product, MLP, constant neighborhood radius, and piecewise constant neighborhood radius corresponding to a geometric graph with hubs (see Appendix A.3 for more details).
Revised: Here, we learn an auto-encoder with four types of decoders: inner product, MLP, constant neighborhood radius, and piecewise constant neighborhood radius corresponding to a geometric graph with hubs (see Appendix A.3 for more details).

Original: Better performances are reached if the graph is allowed to be a geometric graph with hubs as in Def. 5.
Revised: Better performances are reached if the graph is allowed to be a geometric graph with hubs as in Def. 5.

Original: Moreover, the performances of distance and distance+hubs decoder seem to be consistent among different datasets, unlike the inner product and MLP decoders.
Revised: Moreover, the performances of distance and distance+hubs decoder seem to be consistent among different datasets, unlike the inner product and MLP decoders.

Original: This corroborates the claims that real-world graphs can be better modeled as geometric graphs with non-constant neighborhood radius.
Revised: This corroborates the claims that real-world graphs can be better modeled as geometric graphs with non-constant neighborhood radius.

Original: Fig.
Revised: Fig.

Original: 8 in Appendix A.3 shows the learned probabilities of being a hub, and the learned values of α and β , for the Pubmed graph.
Revised: 8 in Appendix A.3 shows the learned probabilities of being a hub, and the learned values of α and β , for the Pubmed graph.

Original: 3. N ODE C LASSIFICATION
Revised: 3. N ODE C LASSIFICATION

Original: Another exemplary application is to use a non-uniform geometric GSO L G ,ρ ( Def. 4) in a spectral graph convolution network for node classification tasks, where the density ρ i at each node i is computed by a different graph neural network, and the whole model is trained end-to-end on the task.
Revised: Another exemplary application is to use a non-uniform geometric GSO L G ,ρ ( Def. 4) in a spectral graph convolution network for node classification tasks, where the density ρ i at each node i is computed by a different graph neural network, and the whole model is trained end-to-end on the task.

Original: The details of the implementation are reported in Appendix A.2.
Revised: The details of the implementation are reported in Appendix A.2.

Original: In Fig.
Revised: In Fig.

Original: 3 we show the accuracy of the best-scoring GSO out of the ones reported in Tab. 3 when the density is ignored against the performances of the best-scoring GSO when the sampling density is learned.
Revised: 3 we show the accuracy of the best-scoring GSO out of the ones reported in Tab. 3 when the density is ignored against the performances of the best-scoring GSO when the sampling density is learned.

Original: For Citeseer and
Revised: For Citeseer and

Original: FacebookPagePage, the best GSOs are the symmetric normalized adjacency matrix.
Revised: FacebookPagePage, the best GSOs are the symmetric normalized adjacency matrix.

Original: For Cora and
Revised: For Cora and

Original: Pubmed, the best density-ignored GSO is the symmetric normalized adjacency matrix, while the best density-normalized GSO is the adjacency matrix.
Revised: Pubmed, the best density-ignored GSO is the symmetric normalized adjacency matrix, while the best density-normalized GSO is the adjacency matrix.

Original: For AmazonComputers and AmazonPhoto, the best-scoring GSOs are the symmetric normalized Laplacian.
Revised: For AmazonComputers and AmazonPhoto, the best-scoring GSOs are the symmetric normalized Laplacian.

Original: This validates our analysis: if the
Revised: This validates our analysis: if the

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original:  symmetric normalized adjacency  sampling density is ignored, the best choice is to normalize the Laplacian by the degree to soften the distortion of non-uniform sampling.
Revised: Laplacian the symmetric normalized adjacency averaged runs sampling density is ignored, the best choice is to normalize the Laplacian by the degree to soften the distortion of non-uniform sampling.

Original: 3. G RAPH C LASSIFICATION & D IFFERENTIABLE P OOLING
Revised: 3. G RAPH C LASSIFICATION & D IFFERENTIABLE P OOLING

Original: In this experiment we perform graph classification on the AIDS dataset (Riesen & Bunke, 2008), as explained in Appendix A.2.
Revised: In this experiment we perform graph classification on the AIDS dataset (Riesen & Bunke, 2008), as explained in Appendix A.2.

Original: Fig.
Revised: Fig.

Original: 4 shows that the classification performances of a spectral graph neural network are better if a quota of parameters is used to learn ρρρ which is used in a non-uniform geometric GSO (Def. 4).
Revised: 4 shows that the classification performances of a spectral graph neural network are better if a quota of parameters is used to learn ρρρ which is used in a non-uniform geometric GSO (Def. 4).

Original: The learnable ρρρ on the AIDS dataset can be used not only to correct the
Revised: The learnable ρρρ on the AIDS dataset can be used not only to correct the

Original: Laplacian but also to perform a better pooling (Appendix A.2 explains how this is implemented).
Revised: Laplacian but also to perform a better pooling (Appendix A.2 explains how this is implemented).

Original: Usually, a graph convolutional neural network is followed by a global pooling layer in order to extract a representation of the whole graph.
Revised: Usually, a graph convolutional neural network is followed by a global pooling layer in order to extract a representation of the whole graph.

Original: A vanilla pooling layer aggregates uniformly the contribution of all nodes.
Revised: A vanilla pooling layer aggregates uniformly the contribution of all nodes.

Original: We implemented a weighted pooling layer that takes into account the importance of each node.
Revised: We implemented a weighted pooling layer that takes into account the importance of each node.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 4, the weighted pooling layer can indeed improve performances on the graph classification task.
Revised: 4, the weighted pooling layer can indeed improve performances on the graph classification task.

Original: Fig.
Revised: Fig.

Original: 6 in Appendix A.2 shows a comparison between the degree, the density learned to correct the GSO and the density learned for pooling.
Revised: 6 in Appendix A.2 shows a comparison between the degree, the density learned to correct the GSO and the density learned for pooling.

Revised: From the plot it is clear that the degree cannot predict the density.

Revised: Indeed, the sampling density at nodes with the same degree can have different values .

Original: [Figure]
Revised: [Figure]

Original: RhodiumSeleniumNitrogenOxigenCarbonFluorineThalliumSulfurBoronGoldBismuthBromineChlorineArsenicMercuryNickelAntimonyRutheniumSiliconTelluriumPhosphorusGermaniumHolmiumTerbiumLeadZincIodineTinLithiumCopperGalliumPlatinumPalladiumTungstenCobaltMagnesiumSodiumPotassium
Revised: RhodiumSeleniumNitrogenOxigenCarbonFluorineThalliumSulfurBoronGoldBismuthBromineChlorineArsenicMercuryNickelAntimonyRutheniumSiliconTelluriumPhosphorusGermaniumHolmiumTerbiumLeadZincIodineTinLithiumCopperGalliumPlatinumPalladiumTungstenCobaltMagnesiumSodiumPotassium

Original: ClassActiveInactive
Revised: ClassActiveInactive

Original: [Figure]
Revised: [Figure]

Original: 3. E XPLAINABILITY IN G RAPH C LASSIFICATION
Revised: 3. E XPLAINABILITY IN G RAPH C LASSIFICATION

Original: In this experiment, we show how to use the density estimator for explainability.
Revised: In this experiment, we show how to use the density estimator for explainability.

Original: The inverse density vector ρρρ − 1 can be interpreted as a measure of importance of each node, relative to the task at hand, instead of seeing it as the sampling density.
Revised: The inverse density vector ρρρ − 1 can be interpreted as a measure of importance of each node, relative to the task at hand, instead of seeing it as the sampling density.

Original: Thinking about ρρρ − 1 as importance is useful when the graph is not naturally seen as randomly generated from a graphon model.
Revised: Thinking about ρρρ − 1 as importance is useful when the graph is not naturally seen as randomly generated from a graphon model.

Original: We applied this paradigm to the AIDS dataset, as explained in the previous subsection.
Revised: We applied this paradigm to the AIDS dataset, as explained in the previous subsection.

Original: The fact that the classification performances are better when ρρρ is learned demonstrates that ρρρ is an important feature for the classification task, and hence, it can be exploited to extract knowledge from the graph.
Revised: The fact that the classification performances are better when ρρρ is learned demonstrates that ρρρ is an important feature for the classification task, and hence, it can be exploited to extract knowledge from the graph.

Original: We define the mean importance of each chemical element e as the sum of all values of ρρρ − 1 corresponding to nodes labeled as e divided by the number of nodes labeled e .
Revised: We define the mean importance of each chemical element e as the sum of all values of ρρρ − 1 corresponding to nodes labeled as e divided by the number of nodes labeled e .

Original: Fig.
Revised: Fig.

Original: 5 shows the mean importance of each element, when ρρρ − 1 is estimated by using it as a module in the task network in two ways.
Revised: 5 shows the mean importance of each element, when ρρρ − 1 is estimated by using it as a module in the task network in two ways.

Original: The importance ρρρ −is used to correct the GSO.
Revised: The importance ρρρ −is used to correct the GSO.

Original: The importance ρρρ − 1 is used in a pooling layer, that maps the output of the graph neural network Ψ to one feature of the form (cid:80) |V| j =1 ρ j − 1 Ψ( X ) j , where X denotes the node features.
Revised: The importance ρρρ − 1 is used in a pooling layer, that maps the output of the graph neural network Ψ to one feature of the form (cid:80) |V| j =1 ρ j − 1 Ψ( X ) j , where X denotes the node features.

Original: In both cases, the most important elements are the same; therefore, the two methods seem to be consistent.
Revised: In both cases, the most important elements are the same; therefore, the two methods seem to be consistent.

Original: C ONCLUSIONS
Revised: C ONCLUSIONS

Original: In this paper, we addressed the problem of learning the latent sampling density by which graphs are sampled from their underlying continuous models.
Revised: In this paper, we addressed the problem of learning the latent sampling density by which graphs are sampled from their underlying continuous models.

Original: We developed formulas for representing graphs given their connectivity structure and sampling density using non-uniform geometric GSOs.
Revised: We developed formulas for representing graphs given their connectivity structure and sampling density using non-uniform geometric GSOs.

Original: We then showcased how the density of geometric graphs with hubs can be estimated using self-supervision, and validated our approach experimentally.
Revised: We then showcased how the density of geometric graphs with hubs can be estimated using self-supervision, and validated our approach experimentally.

Original: Last, we showed how knowing the sampling density can help with various tasks, e.g., improving spectral methods, improving pooling, and gaining knowledge from graphs.
Revised: Last, we showed how knowing the sampling density can help with various tasks, e.g., improving spectral methods, improving pooling, and gaining knowledge from graphs.

Original: One limitation of our methodology is the difficulty in validating that real-world graphs are indeed sampled from latent geometric spaces.
Revised: One limitation of our methodology is the difficulty in validating that real-world graphs are indeed sampled from latent geometric spaces.

Original: While we reported experiments that support this modeling assumption, an important future direction is to develop further experiments and tools to support our model.
Revised: While we reported experiments that support this modeling assumption, an important future direction is to develop further experiments and tools to support our model.

Original: For instance, can we learn a density estimator on one class of graphs and transfer it to another?
Revised: For instance, can we learn a density estimator on one class of graphs and transfer it to another?

Original: Can we use ground-truth demographic data to validate the estimated density in social networks?
Revised: Can we use ground-truth demographic data to validate the estimated density in social networks?

Original: We believe future research will shed light on those questions and find new ways to exploit the sampling density for various applications.
Revised: We believe future research will shed light on those questions and find new ways to exploit the sampling density for various applications.

Original: A I MPLEMENTATION D ETAILS A. S YNTHETIC D ATASET G ENERATION
Revised: A I MPLEMENTATION D ETAILS A. S YNTHETIC D ATASET G ENERATION

Original: This section explains how to generate a synthetic dataset of geometric graphs with hubs.
Revised: This section explains how to generate a synthetic dataset of geometric graphs with hubs.

Original: We first consider a metric space.
Revised: We first consider a metric space.

Original: For our experiments, we mainly focused on the unit-circle S 1 and on the unit-disk D (see Appendix B for more details).
Revised: For our experiments, we mainly focused on the unit-circle S 1 and on the unit-disk D (see Appendix B for more details).

Original: Each graph is generated as follows.
Revised: Each graph is generated as follows.

Original: First, a non-uniform distribution is randomly generated.
Revised: First, a non-uniform distribution is randomly generated.

Original: We considered an angular non-uniformity as described in Def. 6, where the number of oscillating terms, as well as the parameters ccc , nnn , µµµ , are chosen randomly.
Revised: We considered an angular non-uniformity as described in Def. 6, where the number of oscillating terms, as well as the parameters ccc , nnn , µµµ , are chosen randomly.

Original: In the case of 2 -dimensional spaces, the radial distribution is the one shown in Tab. 2.
Revised: In the case of 2 -dimensional spaces, the radial distribution is the one shown in Tab. 2.

Original: According to each generated probability density function, N points { x i } Ni =1 are drawn independently.
Revised: According to each generated probability density function, N points { x i } Ni =1 are drawn independently.

Original: Among them, m <
Revised: Among them, m <

Original: N are chosen randomly to be hubs, and any other node whose distance from a hub is less than some ε > 0 is also marked as a hub.
Revised: N are chosen randomly to be hubs, and any other node whose distance from a hub is less than some ε > 0 is also marked as a hub.

Original: We consider two parameters α , β > 0 .
Revised: We consider two parameters α , β > 0 .

Original: The neighborhood radius about non-hub (respectively, hubs) nodes is taken to be α (respectively α + β ).
Revised: The neighborhood radius about non-hub (respectively, hubs) nodes is taken to be α (respectively α + β ).

Original: Any two points are then connected if
Revised: Any two points are then connected if

Original: [Equation]
Revised: [Equation]

Original: In practical terms, α is computed such that the resulting graph is strongly connected, hence, it differs from graph to graph; β is set to be 3 α and ϵ to be α/
Revised: In practical terms, α is computed such that the resulting graph is strongly connected, hence, it differs from graph to graph; β is set to be 3 α and ϵ to be α/

Original: [Equation]
Revised: [Equation]

Original: A. D ENSITY E STIMATION WITH S ELF -S UPERVISION
Revised: A. D ENSITY E STIMATION WITH S ELF -S UPERVISION

Original: Density Estimation Network
Revised: Density Estimation Network

Original: In our experiments, the inverse of the sampling density, 1 /ρ , is learned by means of an EdgeConv neural network Θ (Wang et al., 2019), which is referred to as
Revised: In our experiments, the inverse of the sampling density, 1 /ρ , is learned by means of an EdgeConv neural network Θ (Wang et al., 2019), which is referred to as

Original: PNet in the following, where the message function is a multi-layer perceptron (MLP), and the aggregation function is max( · ) , followed by a abs( · ) non-linearity.
Revised: PNet in the following, where the message function is a multi-layer perceptron (MLP), and the aggregation function is max( · ) , followed by a abs( · ) non-linearity.

Original: The number of hidden layers, hidden channels, and output channels is 3 , 32 , and 1 , respectively.
Revised: The number of hidden layers, hidden channels, and output channels is 3 , 32 , and 1 , respectively.

Original: Since the degree is an approximation of the sampling density, as stated in Lemma 1, and since we are interested in computing its inverse to correct the GSO, the input of PNet is the inverse of the degree and the inverse of the mean degree of the one-hop neighborhood.
Revised: Since the degree is an approximation of the sampling density, as stated in Lemma 1, and since we are interested in computing its inverse to correct the GSO, the input of PNet is the inverse of the degree and the inverse of the mean degree of the one-hop neighborhood.

Original: Justified by the Monte-Carlo approximation
Revised: Justified by the Monte-Carlo approximation

Original: [Equation]
Revised: [Equation]

Original: the output of PNet is normalized by its mean.
Revised: the output of PNet is normalized by its mean.

Original: Self-Supervision of PNet via Link Prediction on Synthetic Dataset To train the PNet Θ , for each graph G , we use Θ( G ) to define a GSO L G , Θ( G ) .
Revised: Self-Supervision of PNet via Link Prediction on Synthetic Dataset To train the PNet Θ , for each graph G , we use Θ( G ) to define a GSO L G , Θ( G ) .

Original: Then, we define a graph auto-encoder, where the encoder is implemented as a spectral graph convolution network with GSO L G , Θ( G ) .
Revised: Then, we define a graph auto-encoder, where the encoder is implemented as a spectral graph convolution network with GSO L G , Θ( G ) .

Original: The decoder is the usual inner-product decoder.
Revised: The decoder is the usual inner-product decoder.

Original: The graph signal is a slice of 20 random columns of the adjacency matrix.
Revised: The graph signal is a slice of 20 random columns of the adjacency matrix.

Original: The number of hidden channels, hidden layers, and output channels is respectively 32 , 2 , and 2 .
Revised: The number of hidden channels, hidden layers, and output channels is respectively 32 , 2 , and 2 .

Original: For each node j , the network outputs a feature Θ( G ) j in R n .
Revised: For each node j , the network outputs a feature Θ( G ) j in R n .

Original: Here, R n is seen as the metric space underlying the NuG. In our experiments (Section 3.1), we choose n = 2 .
Revised: Here, R n is seen as the metric space underlying the NuG. In our experiments (Section 3.1), we choose n = 2 .

Original: Some results are shown in Fig.
Revised: Some results are shown in Fig.

Original: Node Classification Let G be the real-world graph.
Revised: Node Classification Let G be the real-world graph.

Original: In Section 3.2, we considered G to be one of the graphs reported in Tab. 1.
Revised: In Section 3.2, we considered G to be one of the graphs reported in Tab. 1.

Original: The task network Ψ is a polynomial convolutional neural network implementing a GSO L G , Θ( G ) , where Θ is the PNet; the order of the polynomial spectral filters is 1 , the number of hidden channels 32 , and the number of hidden layers 2 ; the GSOs used are the ones in
Revised: The task network Ψ is a polynomial convolutional neural network implementing a GSO L G , Θ( G ) , where Θ is the PNet; the order of the polynomial spectral filters is 1 , the number of hidden channels 32 , and the number of hidden layers 2 ; the GSOs used are the ones in

Original: Tab. 3.
Revised: Tab. 3.

Original: The optimizer is ADAM (Kingma & Ba, 2015) with learning rate 10 − 2 .
Revised: The optimizer is ADAM (Kingma & Ba, 2015) with learning rate 10 − 2 .

Original: We split the nodes in training ( 85% ), validation ( 5% ), and test ( 10% ) in a stratified fashion, and apply early stopping.
Revised: We split the nodes in training ( 85% ), validation ( 5% ), and test ( 10% ) in a stratified fashion, and apply early stopping.

Original: The performances of the method are shown in Fig.
Revised: The performances of the method are shown in Fig.

Original: Graph Classification Let G be the real-world graph.
Revised: Graph Classification Let G be the real-world graph.

Original: In Section 3.4, G is any compound in the
Revised: In Section 3.4, G is any compound in the

Original: AIDS dataset.
Revised: AIDS dataset.

Original: The task network Ψ is a polynomial convolutional neural network implementing a
Revised: The task network Ψ is a polynomial convolutional neural network implementing a

Original: GSO L G , Θ( G ) , where Θ is the PNet; the order of the spectral polynomial filters is 1 , the number of
Revised: GSO L G , Θ( G ) , where Θ is the PNet; the order of the spectral polynomial filters is 1 , the number of

Original: 7 Degree 0.
Revised: 7 Degree 0.

Original: L      7 Degree 0.
Revised: L      7 Degree 0.

Original: P hidden channels 128 and the number of hidden layers 2 .
Revised: P hidden channels 128 and the number of hidden layers 2 .

Original: The optimizer is ADAM with learning rate 10 − 2 .
Revised: The optimizer is ADAM with learning rate 10 − 2 .

Original: We perform a stratified splitting of the graphs in training ( 85% ), validation ( 5% ), and test ( 10% ), and applied early stopping.
Revised: We perform a stratified splitting of the graphs in training ( 85% ), validation ( 5% ), and test ( 10% ), and applied early stopping.

Original: The chosen batch size is 64 .
Revised: The chosen batch size is 64 .

Original: The pooling layer is a global add layer.
Revised: The pooling layer is a global add layer.

Original: In case of weighted pooling as in Section 3.3, the task network Ψ implements as GSO L G , 111 , while Θ is used to output the weights of the pooling layer.
Revised: In case of weighted pooling as in Section 3.3, the task network Ψ implements as GSO L G , 111 , while Θ is used to output the weights of the pooling layer.

Original: The performance metrics of both approaches are shown in Fig.
Revised: The performance metrics of both approaches are shown in Fig.

Original: A. G EOMETRIC G RAPHS WITH H UBS A UTO -E NCODER
Revised: A. G EOMETRIC G RAPHS WITH H UBS A UTO -E NCODER

Original: Here, we validate that real-world graphs can be modeled approximately as geometric graphs with hubs, as claimed in Section 3.1.
Revised: Here, we validate that real-world graphs can be modeled approximately as geometric graphs with hubs, as claimed in Section 3.1.

Original: We consider the datasets listed in Tab. 1.
Revised: We consider the datasets listed in Tab. 1.

Original: The auto-encoder is defined as follows.
Revised: The auto-encoder is defined as follows.

Original: Let G be the real-world graph with N nodes and F node features; let X ∈ R N × F be the feature matrix.
Revised: Let G be the real-world graph with N nodes and F node features; let X ∈ R N × F be the feature matrix.

Original: Let n be the dimension of the metric space in which nodes are embedded.
Revised: Let n be the dimension of the metric space in which nodes are embedded.

Original: Let Ψ be a spectral graph convolutional network, referred to as encoder .
Revised: Let Ψ be a spectral graph convolutional network, referred to as encoder .

Original: Let Ψ( X ) i and Ψ( X ) j ∈ R n be the embedding of nodes i and j respectively.
Revised: Let Ψ( X ) i and Ψ( X ) j ∈ R n be the embedding of nodes i and j respectively.

Original: A decoder is a mapping R n × R n → [0 , 1] that takes as input the embedding of two nodes i , j and returns the probability that the edge ( i, j ) exists.
Revised: A decoder is a mapping R n × R n → [0 , 1] that takes as input the embedding of two nodes i , j and returns the probability that the edge ( i, j ) exists.

Original: We use four types of decoders.
Revised: We use four types of decoders.

Original: (1) The inner product decoder from Kipf & Welling (2016) is defined as σ (cid:0) ⟨ Ψ( X ) i , Ψ( X ) j ⟩ (cid:1) , where σ ( · ) is the logistic sigmoid function.
Revised: (1) The inner product decoder from Kipf & Welling (2016) is defined as σ (cid:0) ⟨ Ψ( X ) i , Ψ( X ) j ⟩ (cid:1) , where σ ( · ) is the logistic sigmoid function.

Original: The MLP decoder is defined as σ (cid:0) MLP([Ψ
Revised: The MLP decoder is defined as σ (cid:0) MLP([Ψ

Original: ( X )
Revised: ( X )

Original: i , Ψ( X ) j ]) (cid:1) , where [Ψ( X ) i , Ψ( X ) j ] ∈ R 2 n denotes the concatenation of Ψ( X ) i and Ψ( X ) j , and MLP denotes a multi-layer perceptron.
Revised: i , Ψ( X ) j ]) (cid:1) , where [Ψ( X ) i , Ψ( X ) j ] ∈ R 2 n denotes the concatenation of Ψ( X ) i and Ψ( X ) j , and MLP denotes a multi-layer perceptron.

Original: (3) The distance decoder corresponds to geometric graphs.
Revised: (3) The distance decoder corresponds to geometric graphs.

Original: It is defined as σ ( α − ∥ Ψ( X )
Revised: It is defined as σ ( α − ∥ Ψ( X )

Original: i − Ψ
Revised: i − Ψ

Original: ( X ) j ∥ 2 ) , where α is the trainable neighborhood radius.
Revised: ( X ) j ∥ 2 ) , where α is the trainable neighborhood radius.

Original: (4) The distance+hubs decoder corresponds to geometric graphs with hubs.
Revised: (4) The distance+hubs decoder corresponds to geometric graphs with hubs.

Original: It is defined as σ ( α + max { Υ( ˜ D )
Revised: It is defined as σ ( α + max { Υ( ˜ D )

Original: i , Υ( ˜ D ) j } β − ∥ Ψ( X )
Revised: i , Υ( ˜ D ) j } β − ∥ Ψ( X )

Original: i − Ψ
Revised: i − Ψ

Original: ( X ) j ∥ 2 ) , where α , β are trainable parameters that describe the radii of hubs and non-hubs.
Revised: ( X ) j ∥ 2 ) , where α , β are trainable parameters that describe the radii of hubs and non-hubs.

Original: Υ is a message-passing graph neural network (with the same architecture of PNet) that takes as input a signal ˜ D computed from the node degrees (i.e., the inverse of the degree and the inverse of the mean degree of the one-hop neighborhood), and outputs the probability that each node is a hub.
Revised: Υ is a message-passing graph neural network (with the same architecture of PNet) that takes as input a signal ˜ D computed from the node degrees (i.e., the inverse of the degree and the inverse of the mean degree of the one-hop neighborhood), and outputs the probability that each node is a hub.

Original: Υ is learned end-to-end together with the rest of the auto-encoder.
Revised: Υ is learned end-to-end together with the rest of the auto-encoder.

Original: In order to guarantee that 0 ≤ Υ( G ) j ≤ 1 the network is followed by a min-max normalization.
Revised: In order to guarantee that 0 ≤ Υ( G ) j ≤ 1 the network is followed by a min-max normalization.

Original: The distance decoder is justified by the fact that the condition ∥ Ψ( X ) i − Ψ( X ) j ∥ 2 ≤ α can be rewritten as H( α − ∥ Ψ( X )
Revised: The distance decoder is justified by the fact that the condition ∥ Ψ( X ) i − Ψ( X ) j ∥ 2 ≤ α can be rewritten as H( α − ∥ Ψ( X )

Original: i − Ψ
Revised: i − Ψ

Original: ( X ) j ∥ 2 ) , where H( · ) is the Heaviside function.
Revised: ( X ) j ∥ 2 ) , where H( · ) is the Heaviside function.

Original: The Heaviside function is relaxed to the logistic sigmoid for differentiability.
Revised: The Heaviside function is relaxed to the logistic sigmoid for differentiability.

Original: Similar reasoning lies behind the formula of the distance+hubs decoder.
Revised: Similar reasoning lies behind the formula of the distance+hubs decoder.

Original: The encoder Ψ is a polynomial spectral graph convolutional neural network implementing as GSO the symmetric normalized adjacency matrix; the order of the polynomial filters is 1 , the number of hidden channels 32 and the number of hidden layers 2 .
Revised: The encoder Ψ is a polynomial spectral graph convolutional neural network implementing as GSO the symmetric normalized adjacency matrix; the order of the polynomial filters is 1 , the number of hidden channels 32 and the number of hidden layers 2 .

Original: In the case of inner-product, MLP and distance decoder, the loss is the cross entropy of existing and non-existing edges.
Revised: In the case of inner-product, MLP and distance decoder, the loss is the cross entropy of existing and non-existing edges.

Original: In the case of distance+hubs-decoder, we also add ∥
Revised: In the case of distance+hubs-decoder, we also add ∥

Original: Υ( G ) ∥ 1 /N to the loss, as a regularization term, since in our model we suppose the number of hubs is low.
Revised: Υ( G ) ∥ 1 /N to the loss, as a regularization term, since in our model we suppose the number of hubs is low.

Original: The optimizer is ADAM with learning rate 10 − 2 .
Revised: The optimizer is ADAM with learning rate 10 − 2 .

Original: We split the edges in training ( 85% ), validation ( 5% ) and test ( 10% ), and apply early stopping.
Revised: We split the edges in training ( 85% ), validation ( 5% ) and test ( 10% ), and apply early stopping.

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: The performances of both methods are shown in Figs.
Revised: The performances of both methods are shown in Figs.

Original: 2 and
Revised: 2 and

Original: [Figure]
Revised: [Figure]

Original: while examples of embeddings and learned α and β are shown in Fig.
Revised: while examples of embeddings and learned α and β are shown in Fig.

Original: The distance decoder has one learnable parameter more than the inner-product decoder.
Revised: The distance decoder has one learnable parameter more than the inner-product decoder.

Original: Since PNet has a fixed number of input channels, the distance+hubs decoder has 2 , 535 learnable parameters more than the inner-product one.
Revised: Since PNet has a fixed number of input channels, the distance+hubs decoder has 2 , 535 learnable parameters more than the inner-product one.

Original: On the contrary, the mlp decoder has a number of input channels that depends on the latent dimension; therefore, the number of hidden channels is chosen to guarantee that the number of learnable parameters of the mlp decoder is approximately 2 , 535 .
Revised: On the contrary, the mlp decoder has a number of input channels that depends on the latent dimension; therefore, the number of hidden channels is chosen to guarantee that the number of learnable parameters of the mlp decoder is approximately 2 , 535 .

Original: B S YNTHETIC D ATASETS - A B LUEPRINT
Revised: B S YNTHETIC D ATASETS - A B LUEPRINT

Original: In the following
Revised: In the following

Original: [Figure]
Revised: [Figure]

Original: we consider some simple latent metric spaces and construct methods for randomly generating non-uniform samples.
Revised: we consider some simple latent metric spaces and construct methods for randomly generating non-uniform samples.

Original: For each space, structural properties of the corresponding NuG are studied, such as the expected degree of a node and the expected average degree, in case the radius is fixed and the sampling is non-uniform.
Revised: For each space, structural properties of the corresponding NuG are studied, such as the expected degree of a node and the expected average degree, in case the radius is fixed and the sampling is non-uniform.

Original: All proofs can be found in Appendix D, if not otherwise stated.
Revised: All proofs can be found in Appendix D, if not otherwise stated.

Original: Three natural metric measure spaces are the euclidean, spherical, and hyperbolic spaces.
Revised: Three natural metric measure spaces are the euclidean, spherical, and hyperbolic spaces.

Original: If we restrict the attention to 2 -dimensional spaces, a way to uniformly sample is summarized in Tab. 2.
Revised: If we restrict the attention to 2 -dimensional spaces, a way to uniformly sample is summarized in Tab. 2.

Original: In all three cases, the radial component arises naturally from the measure of the space.
Revised: In all three cases, the radial component arises naturally from the measure of the space.

Original: A possible way to introduce non-uniformity is changing the angular distribution.
Revised: A possible way to introduce non-uniformity is changing the angular distribution.

Original: In this way, preferential directions will be identified, leading to an anisotropic model.
Revised: In this way, preferential directions will be identified, leading to an anisotropic model.

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

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

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: Definition 6.
Revised: Definition 6.

Original: Given a natural number C ∈ N , and vectors ccc ∈ R C , nnn ∈ N C , µµµ ∈ R C
Revised: Given a natural number C ∈ N , and vectors ccc ∈ R C , nnn ∈ N C , µµµ ∈ R C

Original: the function sb
Revised: the function sb

Original: [Equation]
Revised: [Equation]

Original: 
Revised: 

Original: [Equation]
Revised: [Equation]

Original: is a continuous, 2 π -periodic probability density function.
Revised: is a continuous, 2 π -periodic probability density function.

Original: It will be referred to as spectrally bounded .
Revised: It will be referred to as spectrally bounded .

Original: The cosine can be replaced by a generic 2 π -periodic function; the only change in the construction will be the offset and the normalization constant.
Revised: The cosine can be replaced by a generic 2 π -periodic function; the only change in the construction will be the offset and the normalization constant.

Original: Definition 7.
Revised: Definition 7.

Original: Given a natural number C ∈ N , and the vectors ccc ∈ R C , nnn ∈ N C , µµµ ∈ R C , κκκ ∈ R C ≥ 0 , the function
Revised: Given a natural number C ∈ N , and the vectors ccc ∈ R C , nnn ∈ N C , µµµ ∈ R C , κκκ ∈ R C ≥ 0 , the function

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: A = (cid:88)
Revised: A = (cid:88)

Original: i : c i <
Revised: i : c i <

Original: [Equation]
Revised: [Equation]

Original: is a continuous, 2 π -periodic probability density function.
Revised: is a continuous, 2 π -periodic probability density function.

Original: It will be referred to as multimodal von
Revised: It will be referred to as multimodal von

Original: Mises .
Revised: Mises .

Original: Both densities introduced previously can be thought of as functions over the unit circle.
Revised: Both densities introduced previously can be thought of as functions over the unit circle.

Original: Hence, the very first space to be studied is S 1 = { x ∈ R 2 :
Revised: Hence, the very first space to be studied is S 1 = { x ∈ R 2 :

Original: ∥ x ∥ = 1 } equipped with geodesic distance.
Revised: ∥ x ∥ = 1 } equipped with geodesic distance.

Original: As shown in the next proposition, the geodesic distance can be computed in a fairly easy way.
Revised: As shown in the next proposition, the geodesic distance can be computed in a fairly easy way.

Original: Proposition 2.
Revised: Proposition 2.

Original: Given two points xxx,yyy ∈ S 1 corresponding to the angles x, y ∈
Revised: Given two points xxx,yyy ∈ S 1 corresponding to the angles x, y ∈

Original: [ − π, π ) , their geodesic distance is equal to
Revised: [ − π, π ) , their geodesic distance is equal to

Original: [Equation]
Revised: [Equation]

Original: The next proposition computes the degree of a node in a non-uniform unit circle graph.
Revised: The next proposition computes the degree of a node in a non-uniform unit circle graph.

Original: Proposition 3.
Revised: Proposition 3.

Original: Given a spectrally bounded probability density function as in Def. 6, the expected degree of a node θ in a unit circle geometric graph with neighborhood radius α is
Revised: Given a spectrally bounded probability density function as in Def. 6, the expected degree of a node θ in a unit circle geometric graph with neighborhood radius α is

Original: [Equation]
Revised: [Equation]

Original: and the expected average degree of the whole graph is
Revised: and the expected average degree of the whole graph is

Original: [Equation]
Revised: [Equation]

Original: As a direct consequence
Revised: As a direct consequence

Original: [Equation]
Revised: [Equation]

Original: in the limit of r going to zero
Revised: in the limit of r going to zero

Original: [Equation]
Revised: [Equation]

Original: thus, for sufficiently small α , the probability of a ball centered at θ is proportional to the density computed in θ .
Revised: thus, for sufficiently small α , the probability of a ball centered at θ is proportional to the density computed in θ .

Original: Moreover, the error can be computed as
Revised: Moreover, the error can be computed as

Original: [Equation]
Revised: [Equation]

Original: which shows that the approximation worsens the more oscillatory terms there are.
Revised: which shows that the approximation worsens the more oscillatory terms there are.

Original: In the case of multimodal von Mises distribution, a closed formula for the probability of balls does not exist.
Revised: In the case of multimodal von Mises distribution, a closed formula for the probability of balls does not exist.

Original: The following proposition introduces an approximation based solely on cosine functions.
Revised: The following proposition introduces an approximation based solely on cosine functions.

Original: Proposition 4.
Revised: Proposition 4.

Original: A multimodal von Mises probability density function can be approximated by a spectrally bounded one.
Revised: A multimodal von Mises probability density function can be approximated by a spectrally bounded one.

Original: The previous result, combined with Prop. 3, gives a way to approximate the expected degree of spatial networks sampled accordingly to a multimodal von Mises angular distribution.
Revised: The previous result, combined with Prop. 3, gives a way to approximate the expected degree of spatial networks sampled accordingly to a multimodal von Mises angular distribution.

Original: However, the computation is straightforward when nnn is the constant vector n 111 , since the product of two von Mises pdf is the kernel of a von Mises pdf
Revised: However, the computation is straightforward when nnn is the constant vector n 111 , since the product of two von Mises pdf is the kernel of a von Mises pdf

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: The unit circle model is preparatory to the study of more complex spaces, for instance, the unit disk D = { x ∈ R 2 : ∥ x ∥ ≤ 1 } equipped with geodesic distance, as in Tab. 2.
Revised: The unit circle model is preparatory to the study of more complex spaces, for instance, the unit disk D = { x ∈ R 2 : ∥ x ∥ ≤ 1 } equipped with geodesic distance, as in Tab. 2.

Original: Proposition 5.
Revised: Proposition 5.

Original: Given a spectrally bounded angular distribution as in Def. 6, the degree of a node ( r, θ ) in a unit disk geometric graph with neighborhood radius α is
Revised: Given a spectrally bounded angular distribution as in Def. 6, the degree of a node ( r, θ ) in a unit disk geometric graph with neighborhood radius α is

Original: [Equation]
Revised: [Equation]

Original: and the average degree of the whole network is
Revised: and the average degree of the whole network is

Original: [Equation]
Revised: [Equation]

Original: Fig.
Revised: Fig.

Original: 9a shows some examples of non-uniform sampling of the unit disk.
Revised: 9a shows some examples of non-uniform sampling of the unit disk.

Original: The last example will be the hyperbolic disk with radius R ≫ 1 , equipped with geodesic distance as in Tab. 2.
Revised: The last example will be the hyperbolic disk with radius R ≫ 1 , equipped with geodesic distance as in Tab. 2.

Original: Proposition 6.
Revised: Proposition 6.

Original: Given a spectrally bounded angular distribution as in Def. 6, the degree of a node ( r, θ ) in a hyperbolic geometric graph with neighborhood radius α is deg
Revised: Given a spectrally bounded angular distribution as in Def. 6, the degree of a node ( r, θ ) in a hyperbolic geometric graph with neighborhood radius α is deg

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: and the average degree of the whole network is O
Revised: and the average degree of the whole network is O

Original: ( N e α − 2 R 2 ) .
Revised: ( N e α − 2 R 2 ) .

Original: The proof can be found in Appendix D.
Revised: The proof can be found in Appendix D.

Original: The computed approximation is in line with the findings of
Revised: The computed approximation is in line with the findings of

Original: Krioukov et al.
Revised: Krioukov et al.

Original: (2010), where a closed formula for the uniform case is provided when α = R .
Revised: (2010), where a closed formula for the uniform case is provided when α = R .

Original: To the best of our knowledge, this is the first work that considers α ̸ = R .
Revised: To the best of our knowledge, this is the first work that considers α ̸ = R .

Original: Examples of non-uniform sampling of the hyperbolic disk are shown in Fig. 9b.
Revised: Examples of non-uniform sampling of the hyperbolic disk are shown in Fig. 9b.

Original: C R ETRIEVING AND B UILDING GSO S
Revised: C R ETRIEVING AND B UILDING GSO S

Original: In the current section, we first show how to retrieve the usual definition of graph shift operators from
Revised: In the current section, we first show how to retrieve the usual definition of graph shift operators from

Original: Def. 4, and then how Def. 4 can be used to create novel GSOs.
Revised: Def. 4, and then how Def. 4 can be used to create novel GSOs.

Original: For simplicity, for both goals we suppose uniform sampling ρρρ = 1 ; (4) can be rewritten as
Revised: For simplicity, for both goals we suppose uniform sampling ρρρ = 1 ; (4) can be rewritten as

Original: [Equation]
Revised: [Equation]

Original: where A is the adjacency matrix and d is the degree vector.
Revised: where A is the adjacency matrix and d is the degree vector.

Original: Tab. 3 exhibit which choice of { m ( i ) } 4 i = correspond to which graph Laplacian.
Revised: Tab. 3 exhibit which choice of { m ( i ) } 4 i = correspond to which graph Laplacian.

Original: A question that may arise is whether the innermost diag( · ) in (7) can be factored out of the outermost one.
Revised: A question that may arise is whether the innermost diag( · ) in (7) can be factored out of the outermost one.

Original: As shown in the next proposition, it is not possible in general.
Revised: As shown in the next proposition, it is not possible in general.

Original: [Figure]
Revised: [Figure]

Original: Proposition 7.
Revised: Proposition 7.

Original: Let A ∈ R N × N , A
Revised: Let A ∈ R N × N , A

Original: = A T ; let v ∈ R N ≥ 

Revised: = A T ; let v ∈ R N ≥

Original: and V = diag( v ) , it holds
Revised: and V = diag( v ) , it holds

Original: [Equation]
Revised: [Equation]

Original: Moreover
Revised: Moreover

Original: [Equation]
Revised: [Equation]

Original: The proof of the statement can be found in Appendix D.
Revised: The proof of the statement can be found in Appendix D.

Original: An important consequence of Prop. 7 is that the graph Laplacian
Revised: An important consequence of Prop. 7 is that the graph Laplacian

Original: [Equation]
Revised: [Equation]

Original: obtained with m ( i ) ( x ) = x − 12 for every i ∈ { 1 , . .
Revised: obtained with m ( i ) ( x ) = x − 12 for every i ∈ { 1 , . .

Original: , 4 } , is in general different from the symmetric normalized Laplacian, since
Revised: , 4 } , is in general different from the symmetric normalized Laplacian, since

Original: [Equation]
Revised: [Equation]

Original: In light of Prop. 7, the two Laplacians are equivalent if every node is connected to nodes with the same degree, e.g., if the graph is k -regular.
Revised: In light of Prop. 7, the two Laplacians are equivalent if every node is connected to nodes with the same degree, e.g., if the graph is k -regular.

Original: The difference between the two Laplacians can be better seen by studying their spectrum.
Revised: The difference between the two Laplacians can be better seen by studying their spectrum.

Original: The next proposition introduces an upper bound on the eigenvalues of the Laplacian in (8).
Revised: The next proposition introduces an upper bound on the eigenvalues of the Laplacian in (8).

Original: Proposition 8.
Revised: Proposition 8.

Original: Let G =
Revised: Let G =

Original: ( V , E ) be an undirected graph with adjacency matrix A ∈ R N × N and degree matrix D = diag( A1 ) .
Revised: ( V , E ) be an undirected graph with adjacency matrix A ∈ R N × N and degree matrix D = diag( A1 ) .

Original: Let λ be an eigenvalue of the graph Laplacian
Revised: Let λ be an eigenvalue of the graph Laplacian

Original: [Equation]
Revised: [Equation]

Original: it holds | λ
Revised: it holds | λ

Original: [Equation]
Revised: [Equation]

Original: The proof of the proposition can be found in Appendix D.
Revised: The proof of the proposition can be found in Appendix D.

Original: It is well known that the spectral radius of the symmetric normalized Laplacian is less than or equal to 2 (Chung, 1997), with equality holding for bipartite graphs.
Revised: It is well known that the spectral radius of the symmetric normalized Laplacian is less than or equal to 2 (Chung, 1997), with equality holding for bipartite graphs.

Original: However, this is not the case for the Laplacian in (8), as shown in the next example.
Revised: However, this is not the case for the Laplacian in (8), as shown in the next example.

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: Example 1 (Complete Bipartite Graph) .
Revised: Example 1 (Complete Bipartite Graph) .

Original: Consider the complete bipartite graph with n nodes in the first part and m ≥ n nodes in the second part.
Revised: Consider the complete bipartite graph with n nodes in the first part and m ≥ n nodes in the second part.

Original: Its adjacency and degree matrix are
Revised: Its adjacency and degree matrix are

Original: [Equation]
Revised: [Equation]

Original: A simple computation leads to
Revised: A simple computation leads to

Original: [Equation]
Revised: [Equation]

Original: It can be noted that L has null eigenvalue λ 1 = 0 corresponding to the constant eigenvector 1 n + m .
Revised: It can be noted that L has null eigenvalue λ 1 = 0 corresponding to the constant eigenvector 1 n + m .

Original: The vector v i = − e 

Revised: The vector v i = − e

Original: + e i , i ∈ { 2 , . . . , n } is an eigenvector with eigenvalue λ 2 = − (cid:112) m/n , whose multiplicity is n − 1 .
Revised: + e i , i ∈ { 2 , . . . , n } is an eigenvector with eigenvalue λ 2 = − (cid:112) m/n , whose multiplicity is n − 1 .

Original: Analogously, v i = − e n +1 + e i +1 , i ∈ { n + 1 , . .
Revised: Analogously, v i = − e n +1 + e i +1 , i ∈ { n + 1 , . .

Original: . , n + m − 1 } is an eigenvector with eigenvalue λ 3 = − (cid:112) n/m , whose multiplicity is m − 1 .
Revised: . , n + m − 1 } is an eigenvector with eigenvalue λ 3 = − (cid:112) n/m , whose multiplicity is m − 1 .

Original: Finally, the vector v n + m = [ − m/n 1 T n , 1 T m ] T is eigenvector with eigenvalues λ
Revised: Finally, the vector v n + m = [ − m/n 1 T n , 1 T m ] T is eigenvector with eigenvalues λ

Original: 4 = λ 2 + λ 3 .
Revised: 4 = λ 2 + λ 3 .

Original: Therefore, the spectral radius of L is
Revised: Therefore, the spectral radius of L is

Original: [Equation]
Revised: [Equation]

Original: In the case of a balanced graph, n = m implies that the spectral radius is 2 .
Revised: In the case of a balanced graph, n = m implies that the spectral radius is 2 .

Original: In the case of a star graph, n = 1 and | λ 4 |
Revised: In the case of a star graph, n = 1 and | λ 4 |

Original: =
Revised: =

Original: O ( √ m ) as m → ∞ ; therefore, the asymptotic in Prop. 8 is tight.
Revised: O ( √ m ) as m → ∞ ; therefore, the asymptotic in Prop. 8 is tight.

Original: D P ROOFS
Revised: D P ROOFS

Original: Proof of Prop. 1 and concentration of error.
Revised: Proof of Prop. 1 and concentration of error.

Original: Let x = { x i } Ni =1 be an i.i.d.
Revised: Let x = { x i } Ni =1 be an i.i.d.

Original: random sample from ρ .
Revised: random sample from ρ .

Original: Let K and m be the kernel and diagonal parts corresponding to the metric-probability Laplacian
Revised: Let K and m be the kernel and diagonal parts corresponding to the metric-probability Laplacian

Original: L N .
Revised: L N .

Original: Let L , u be
Revised: Let L , u be

Original: [Equation]
Revised: [Equation]

Original: Note that the non-uniform geometric GSO L G ,ρρρ based on the graph G , which is randomly sampled from S with neighborhood model N via the sample points x , is exactly equal to L .
Revised: Note that the non-uniform geometric GSO L G ,ρρρ based on the graph G , which is randomly sampled from S with neighborhood model N via the sample points x , is exactly equal to L .

Original: Conditioned on x i = x , the expected value is
Revised: Conditioned on x i = x , the expected value is

Original: [Equation]
Revised: [Equation]

Original: Since the random variables { x j } Nj =1 are i.i.d.
Revised: Since the random variables { x j } Nj =1 are i.i.d.

Original: to y , then also the random variables
Revised: to y , then also the random variables

Original: [Equation]
Revised: [Equation]

Original: are i.i.d., hence,
Revised: are i.i.d., hence,

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

Original: which proves (5)
Revised: which proves (5)

Original: [Equation]
Revised: [Equation]

Original: Next, we prove the concentration of error result.
Revised: Next, we prove the concentration of error result.

Original: We know that there exist a, b > 0 such that almost everywhere K ( x, x j ) ρ ( x j ) − 1 u ( x j ) ∈ [ a, b ] , since K , 1 /ρ
Revised: We know that there exist a, b > 0 such that almost everywhere K ( x, x j ) ρ ( x j ) − 1 u ( x j ) ∈ [ a, b ] , since K , 1 /ρ

Original: and u are essentially bounded.
Revised: and u are essentially bounded.

Original: By
Revised: By

Original: Hoeffding’s inequality, for t > 0 ,
Revised: Hoeffding’s inequality, for t > 0 ,

Original: [Equation]
Revised: [Equation]

Original: Setting
Revised: Setting

Original: [Equation]
Revised: [Equation]

Original: solving for t
Revised: solving for t

Original: [Equation]
Revised: [Equation]

Original: we obtain that for every node there is an event with probability at least 1 − p/N such that
Revised: we obtain that for every node there is an event with probability at least 1 − p/N such that

Original: [Equation]
Revised: [Equation]

Original: We then intersect all of these events to obtain an event of probability at least 1 − p that satisfies (6).
Revised: We then intersect all of these events to obtain an event of probability at least 1 − p that satisfies (6).

Original: Proof of Lemma 1.
Revised: Proof of Lemma 1.

Original: By hypothesis, there exist m x , M x > 0 such that m x ≤ ρ ( y ) ≤ M x for all y ∈ N ( x ) .
Revised: By hypothesis, there exist m x , M x > 0 such that m x ≤ ρ ( y ) ≤ M x for all y ∈ N ( x ) .

Original: Therefore,
Revised: Therefore,

Original: [Equation]
Revised: [Equation]

Original: from which
Revised: from which

Original: [Equation]
Revised: [Equation]

Original: By the Intermediate Value Theorem, there exists c x ∈ N ( x ) such that
Revised: By the Intermediate Value Theorem, there exists c x ∈ N ( x ) such that

Original: [Equation]
Revised: [Equation]

Original: from which the thesis follows.
Revised: from which the thesis follows.

Original: Proof of Prop. 2.
Revised: Proof of Prop. 2.

Original: Consider the map
Revised: Consider the map

Original: [Equation]
Revised: [Equation]

Original: (cid:1)
Revised: (cid:1)

Original: [Equation]
Revised: [Equation]

Original: and the angles x, y, ∈ [ − π, π ) such that φ ( x ) = xxx , φ ( y ) =
Revised: and the angles x, y, ∈ [ − π, π ) such that φ ( x ) = xxx , φ ( y ) =

Original: yyy , it holds
Revised: yyy , it holds

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

Original: Proof of Prop. 3.
Revised: Proof of Prop. 3.

Original: The expected degree of a node θ is the probability of the ball centered at θ times the size N of the sample.
Revised: The expected degree of a node θ is the probability of the ball centered at θ times the size N of the sample.

Original: The probability of a ball can be computed by noting that
Revised: The probability of a ball can be computed by noting that

Original: [Equation]
Revised: [Equation]

Original: Therefore, the average degree can be computed as
Revised: Therefore, the average degree can be computed as

Original: [Equation]
Revised: [Equation]

Original: The inspection of sb( θ ; ccc,nnn,µµµ ) and P [B α ( θ )] shows that the only terms surviving integration are the constant term and the product of cosines with the same frequency
Revised: The inspection of sb( θ ; ccc,nnn,µµµ ) and P [B α ( θ )] shows that the only terms surviving integration are the constant term and the product of cosines with the same frequency

Original: [Equation]
Revised: [Equation]

Original: from which the thesis follows.
Revised: from which the thesis follows.

Original: Proof of Prop. 4.
Revised: Proof of Prop. 4.

Original: Using Taylor expansion, it holds
Revised: Using Taylor expansion, it holds

Original: [Equation]
Revised: [Equation]

Original: A first approximation can be made noting that cos( x ) 2 m ≤ cos( x ) 2 and cos( x ) 2 m − 1 ≈ cos( x ) for all m ≥ 1 , obtaining
Revised: A first approximation can be made noting that cos( x ) 2 m ≤ cos( x ) 2 and cos( x ) 2 m − 1 ≈ cos( x ) for all m ≥ 1 , obtaining

Original: [Equation]
Revised: [Equation]

Original: Such approximation deteriorates fast when κ i increases.
Revised: Such approximation deteriorates fast when κ i increases.

Original: A more refined approximation is obtained considering the power of cosine with higher coefficient in the Taylor expansion.
Revised: A more refined approximation is obtained considering the power of cosine with higher coefficient in the Taylor expansion.

Original: Using Stirling’s approximation of factorial, it can be shown that
Revised: Using Stirling’s approximation of factorial, it can be shown that

Original: [Equation]
Revised: [Equation]

Original: In order to make the computation easier, suppose κ i is an integer; When m = κ i + 1 , it holds
Revised: In order to make the computation easier, suppose κ i is an integer; When m = κ i + 1 , it holds

Original: [Equation]
Revised: [Equation]

Original: where the first inequality is justified by the fact that ( κ i / ( κ i +1))
Revised: where the first inequality is justified by the fact that ( κ i / ( κ i +1))

Original: κ i +1 is an increasing sequence that tends to 1 /e .
Revised: κ i +1 is an increasing sequence that tends to 1 /e .

Original: The previous formula shows that the coefficient with m = κ i + 1 is always smaller than the coefficient with m = κ i .
Revised: The previous formula shows that the coefficient with m = κ i + 1 is always smaller than the coefficient with m = κ i .

Original: The same reasoning can be applied to all the coefficients with m > κ i .
Revised: The same reasoning can be applied to all the coefficients with m > κ i .

Original: Suppose now κ
Revised: Suppose now κ

Original: i ≥ 3 , if m ≤ κ i − 2 the previous reasoning holds.
Revised: i ≥ 3 , if m ≤ κ i − 2 the previous reasoning holds.

Original: A peculiarity happens when m = κ i − 1 :
Revised: A peculiarity happens when m = κ i − 1 :

Original: [Equation]
Revised: [Equation]

Original: because the sequence ( κ i / ( κ i − 1)) κ i − 1 / 2 is decreasing; therefore m = κ
Revised: because the sequence ( κ i / ( κ i − 1)) κ i − 1 / 2 is decreasing; therefore m = κ

Original: i − 1 is the point of maximum
Revised: i − 1 is the point of maximum

Original: [Equation]
Revised: [Equation]

Original: and m = κ i is the second largest value.
Revised: and m = κ i is the second largest value.

Original: Therefore, the following approximation for exp( κ i cos( n i ( θ − µ i ))) holds:
Revised: Therefore, the following approximation for exp( κ i cos( n i ( θ − µ i ))) holds:

Original: [Equation]
Revised: [Equation]

Original: The thesis follows from the equality
Revised: The thesis follows from the equality

Original: [Equation]
Revised: [Equation]

Original: Proof of Prop. 5.
Revised: Proof of Prop. 5.

Original: The domain of integration can be parametrized as d D (( r, θ ) , ( r c , θ c )) ≤ α , leading to
Revised: The domain of integration can be parametrized as d D (( r, θ ) , ( r c , θ c )) ≤ α , leading to

Original: [Equation]
Revised: [Equation]

Original: Three cases must be discussed: (1) 0 ≤ r c − α ≤ r c + α ≤ 1 , (2) r
Revised: Three cases must be discussed: (1) 0 ≤ r c − α ≤ r c + α ≤ 1 , (2) r

Original: c
Revised: c

Original: − α < 0 , (3) r c + α > 1 .
Revised: − α < 0 , (3) r c + α > 1 .

Original: In scenario (1), the ball B α ( r c , θ c ) is contained in D .
Revised: In scenario (1), the ball B α ( r c , θ c ) is contained in D .

Original: The probability of the ball can be computed as
Revised: The probability of the ball can be computed as

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: where the last equality comes from Prop. 3.
Revised: where the last equality comes from Prop. 3.

Original: For simplicity
Revised: For simplicity

Original: [Equation]
Revised: [Equation]

Original: define
Revised: define

Original: [Equation]
Revised: [Equation]

Original: (cid:32)
Revised: (cid:32)

Original: r
Revised: r

Original: [Equation]
Revised: [Equation]

Original: + r
Revised: + r

Original: [Equation]
Revised: [Equation]

Original: c − α
Revised: c − α

Original: [Equation]
Revised: [Equation]

Original: 2 r r c (cid:33) , where U k is the k -th Chebyshev polynomial of second kind.
Revised: 2 r r c (cid:33) , where U k is the k -th Chebyshev polynomial of second kind.

Original: It is worthy to note that f n i − 1 ( r c + α ) = 0 , f n i − 1 ( r c − α )
Revised: It is worthy to note that f n i − 1 ( r c + α ) = 0 , f n i − 1 ( r c − α )

Original: = 0 , f n i − 1 ( α − r c )
Revised: = 0 , f n i − 1 ( α − r c )

Original: = 

Revised: =

Original: and f n i − 1 ( r c ) =
Revised: and f n i − 1 ( r c ) =

Original: α (cid:115) 1 − (cid:18)
Revised: α (cid:115) 1 − (cid:18)

Original: [Equation]
Revised: [Equation]

Original: (cid:19)
Revised: (cid:19)

Original: [Equation]
Revised: [Equation]

Original: (cid:114)
Revised: (cid:114)

Original: [Equation]
Revised: [Equation]

Original: (cid:18)
Revised: (cid:18)

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

Original: (cid:19) , while g ( r c + α ) = 0 , g ( r c − α )
Revised: (cid:19) , while g ( r c + α ) = 0 , g ( r c − α )

Original: = 0 , g ( α − r c ) =
Revised: = 0 , g ( α − r c ) =

Original: ( α − r c ) π and g ( r c )
Revised: ( α − r c ) π and g ( r c )

Original: =
Revised: =

Original: r c arccos (cid:32) 1 − α2 r 2 c (cid:33) , g ( α ) =
Revised: r c arccos (cid:32) 1 − α2 r 2 c (cid:33) , g ( α ) =

Original: α arccos (cid:18) r c 2 α (cid:19) .
Revised: α arccos (cid:18) r c 2 α (cid:19) .

Original: The integral in (9) can be approximated by the semi-area of an ellipse having α and f n i − 1 ( r c ) (respectively g ( r c ) ) as axes
Revised: The integral in (9) can be approximated by the semi-area of an ellipse having α and f n i − 1 ( r c ) (respectively g ( r c ) ) as axes

Original: [Equation]
Revised: [Equation]

Original: that can be seen as a modified version of Simpson’s rule since the latter would lead to a coefficient of 4 / 3 instead of π/ 2 .
Revised: that can be seen as a modified version of Simpson’s rule since the latter would lead to a coefficient of 4 / 3 instead of π/ 2 .

Original: A comparison between the two methods is shown in Appendix D. In scenario (2) the domain of integration contains the origin and the argument of arccos in Appendix D could be not well defined.
Revised: A comparison between the two methods is shown in Appendix D. In scenario (2) the domain of integration contains the origin and the argument of arccos in Appendix D could be not well defined.

Original: The singularity can be removed by decomposing the domain of integration as the union of a disk of radius α − r c around the origin and the remaining.
Revised: The singularity can be removed by decomposing the domain of integration as the union of a disk of radius α − r c around the origin and the remaining.

Original: Hence
Revised: Hence

Original: [Equation]
Revised: [Equation]

Original: The same reasoning as before leads to the approximations
Revised: The same reasoning as before leads to the approximations

Original: [Equation]
Revised: [Equation]

Original: In scenario (3) the domain of integration partially lies outside D .
Revised: In scenario (3) the domain of integration partially lies outside D .

Original: Hence
Revised: Hence

Original: [Equation]
Revised: [Equation]

Original: that can be approximated as
Revised: that can be approximated as

Original: [Equation]
Revised: [Equation]

Original: The three scenarios can be summarized in one big formula.
Revised: The three scenarios can be summarized in one big formula.

Original: For simplicity, define the operator
Revised: For simplicity, define the operator

Original: I [ f ]
Revised: I [ f ]

Original: [Equation]
Revised: [Equation]

Original: that given a function f returns the ellipse approximation of the integral over balls, it holds
Revised: that given a function f returns the ellipse approximation of the integral over balls, it holds

Original: [Equation]
Revised: [Equation]

Original: from which the thesis follows.
Revised: from which the thesis follows.

Original: To compute the average degree of a spatial network from the unit disk, the quantity
Revised: To compute the average degree of a spatial network from the unit disk, the quantity

Original: [Equation]
Revised: [Equation]

Original: r c 101 r c 3.3×10 3 3.4×103.5×10 3 3.6×10 3 3.7×103.8×10 3 3.9×10 3 4×104.1×

Revised: r c 101 r c 3.3×10 3 3.4×103.5×10 3 3.6×10 3 3.7×103.8×10 3 3.9×10 3 4×104.1×

Original: QuadratureEllipse approx.
Revised: QuadratureEllipse approx.

Original: Simpson's rule 1 r c 2×103×104×10 r c 10101 r c 3×10 3 3.2×103.4×10 3 3.6×103.8×

Revised: Simpson's rule 1 r c 2×103×104×10 r c 10101 r c 3×10 3 3.2×103.4×10 3 3.6×103.8×

Original: QuadratureEllipse approx.
Revised: QuadratureEllipse approx.

Original: Simpson's rule 1 r c 2×103×104×10 must be computed.
Revised: Simpson's rule 1 r c 2×103×104×10 must be computed.

Original: Using Prop. 3, the integral can be written in the form
Revised: Using Prop. 3, the integral can be written in the form

Original: [Equation]
Revised: [Equation]

Original: From U k (1) =
Revised: From U k (1) =

Original: k the following approximation can be derived
Revised: k the following approximation can be derived

Original: [Equation]
Revised: [Equation]

Original: , hence the integral boils down to
Revised: , hence the integral boils down to

Original: [Equation]
Revised: [Equation]

Original: The term
Revised: The term

Original: [Equation]
Revised: [Equation]

Original: from which the thesis follows.
Revised: from which the thesis follows.

Original: Proof of Prop. 6.
Revised: Proof of Prop. 6.

Original: Similarly to what has been done in Prop.
Revised: Similarly to what has been done in Prop.

Original: [Equation]
Revised: [Equation]

Original: the domain of integration can be parametrized as
Revised: the domain of integration can be parametrized as

Original: [Equation]
Revised: [Equation]

Original: In order to remove the singularity of the argument of arccos
Revised: In order to remove the singularity of the argument of arccos

Original: [Equation]
Revised: [Equation]

Original: the domain of integration can be decomposed as a ball containing the origin and the remaining, leading to
Revised: the domain of integration can be decomposed as a ball containing the origin and the remaining, leading to

Original: [Equation]
Revised: [Equation]

Original: +
Revised: +

Original: [Equation]
Revised: [Equation]

Original: c
Revised: c

Original: [Equation]
Revised: [Equation]

Original: θ c − θ r sinh
Revised: θ c − θ r sinh

Original: [Equation]
Revised: [Equation]

Original: r
Revised: r

Original: [Equation]
Revised: [Equation]

Original: θ ; ccc,nnn,µµµ )
Revised: θ ; ccc,nnn,µµµ )

Original: d θ d r
Revised: d θ d r

Original: [Equation]
Revised: [Equation]

Original: where l 1 = 0 , u 1 = max { α − r c , 0 } , l 2 = | α − r c | and u 2 = min { r c + α, R } .
Revised: where l 1 = 0 , u 1 = max { α − r c , 0 } , l 2 = | α − r c | and u 2 = min { r c + α, R } .

Original: [Equation]
Revised: [Equation]

Original: The approximations θ r ≈ √ 2 − 2 d r and d r ≈ 1 + 2 ( e − 2 r + e − 2 r c − e α − r c − r − e − α − r c − r ) as in
Revised: The approximations θ r ≈ √ 2 − 2 d r and d r ≈ 1 + 2 ( e − 2 r + e − 2 r c − e α − r c − r − e − α − r c − r ) as in

Original: Gugelmann et al.
Revised: Gugelmann et al.

Original: (2012) can be used to analyze the behaviors of both integrals.
Revised: (2012) can be used to analyze the behaviors of both integrals.

Original: For large R , it holds
Revised: For large R , it holds

Original: [Equation]
Revised: [Equation]

Original: where the last approximation is justified by √ 1 + x = 1 + O ( x )
Revised: where the last approximation is justified by √ 1 + x = 1 + O ( x )

Original: when | x | ≤ 1 .
Revised: when | x | ≤ 1 .

Original: Noting that − 1 ≤ d r ≤
Revised: Noting that − 1 ≤ d r ≤

Original: [Equation]
Revised: [Equation]

Original: one can get rid of the polynomial contribution
Revised: one can get rid of the polynomial contribution

Original: [Equation]
Revised: [Equation]

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: the probability of balls is approximately
Revised: the probability of balls is approximately

Original: [Equation]
Revised: [Equation]

Original: and the expected average degree of the network is
Revised: and the expected average degree of the network is

Original: [Equation]
Revised: [Equation]

Original: Proof of Prop. 7.
Revised: Proof of Prop. 7.

Original: Equality (2) is trivial, since diagonal matrices commutes; equality (1) follows from
Revised: Equality (2) is trivial, since diagonal matrices commutes; equality (1) follows from

Original: [Equation]
Revised: [Equation]

Original: k
Revised: k

Original: [Equation]
Revised: [Equation]

Original: n (cid:88) k
Revised: n (cid:88) k

Original: [Equation]
Revised: [Equation]

Original: A i
Revised: A i

Original: [Equation]
Revised: [Equation]

Original: k
Revised: k

Original: [Equation]
Revised: [Equation]

Original: ( V diag( A1 ))
Revised: ( V diag( A1 ))

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: i .
Revised: i .

Original: In order to prove (3) , we note that V can be decomposed as V = (cid:80) ni
Revised: In order to prove (3) , we note that V can be decomposed as V = (cid:80) ni

Original: =

Revised: =

Original: v i e ( i ) e
Revised: v i e ( i ) e

Original: ( i )T .
Revised: ( i )T .

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: (diag( AV1 ) − diag( VA1 ))
Revised: (diag( AV1 ) − diag( VA1 ))

Original: k
Revised: k

Original: [Equation]
Revised: [Equation]

Original: k = diag  
Revised: k = diag  

Original: [Equation]
Revised: [Equation]

Original: A
Revised: A

Original: [Equation]
Revised: [Equation]

Original: k
Revised: k

Original: [Equation]
Revised: [Equation]

Original: =   n (cid:88)
Revised: =   n (cid:88)

Original: i =
Revised: i =

Original: v i  
Revised: v i  

Original: [Equation]
Revised: [Equation]

Original: − n (cid:88) j
Revised: − n (cid:88) j

Original: =
Revised: =

Original: A
Revised: A

Original: [Equation]
Revised: [Equation]

Original: j e ( i ) 
Revised: j e ( i ) 

Original:  
Revised:  

Original:  k
Revised:  k

Original: = n (cid:88)
Revised: = n (cid:88)

Original: [Equation]
Revised: [Equation]

Original: A
Revised: A

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: must hold for all values of k .
Revised: must hold for all values of k .

Original: Consider the indices k 1 , k 2 , . .
Revised: Consider the indices k 1 , k 2 , . .

Original: , k n corresponding to the values v k 1 ≤ v k 2 ≤ · · · ≤ v k n , then
Revised: , k n corresponding to the values v k 1 ≤ v k 2 ≤ · · · ≤ v k n , then

Original: [Equation]
Revised: [Equation]

Original: then A k 1 ,i = 0 for each i such that v i > v k 1 .
Revised: then A k 1 ,i = 0 for each i such that v i > v k 1 .

Original: Take the index k 2 and consider
Revised: Take the index k 2 and consider

Original: [Equation]
Revised: [Equation]

Original: The second addend is 0 because v k 2 can be either equal to v k 1 , in which case the difference is null, or v k 2 > v k 1 , in which case from the previous step A k 2 ,k 1 = 0 .
Revised: The second addend is 0 because v k 2 can be either equal to v k 1 , in which case the difference is null, or v k 2 > v k 1 , in which case from the previous step A k 2 ,k 1 = 0 .

Original: Therefore A k 2 ,i = 

Revised: Therefore A k 2 ,i =

Original: for each i such that v i > v k 2 .
Revised: for each i such that v i > v k 2 .

Original: By finite induction, the thesis holds when A has null entries in position ( i, j ) whenever v i ̸
Revised: By finite induction, the thesis holds when A has null entries in position ( i, j ) whenever v i ̸

Original: = v j .
Revised: = v j .

Original: Proof of Prop. 8.
Revised: Proof of Prop. 8.

Original: The eigenvalues can be characterized via the Rayleigh quotient
Revised: The eigenvalues can be characterized via the Rayleigh quotient

Original: [Equation]
Revised: [Equation]

Original: u
Revised: u

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

Original: Using Prop.
Revised: Using Prop.

Original: [Equation]
Revised: [Equation]

Original: and considering u = D 12 v the previous formula can be rewritten as
Revised: and considering u = D 12 v the previous formula can be rewritten as

Original: [Equation]
Revised: [Equation]

Original: Let d i
Revised: Let d i

Original: = D
Revised: = D

Original: i,i the degree of the i -th node, using the symmetry of A , the numerator can be rewritten as
Revised: i,i the degree of the i -th node, using the symmetry of A , the numerator can be rewritten as

Original: [Equation]
Revised: [Equation]

Original: From the last equality follows that the eigenvalues are all positive.
Revised: From the last equality follows that the eigenvalues are all positive.

Original: From ( a − b ) 2 ≤ 2( a 2 + b 2 ) follows
Revised: From ( a − b ) 2 ≤ 2( a 2 + b 2 ) follows

Original: [Equation]
Revised: [Equation]

Original: from which the thesis follows.
Revised: from which the thesis follows.