Original: S PECTRAL
Revised: S PECTRAL

Original: G RAPH
Revised: G RAPH

Original: W AVELETS
Revised: W AVELETS

Original: FOR
Revised: FOR

Original: S TRUCTURAL
Revised: S TRUCTURAL

Original: R OLE S IMILARITY IN N ETWORKS
Revised: R OLE S IMILARITY IN N ETWORKS

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: Nodes residing in different parts of a graph can have similar structural roles within their local network topology.
Revised: Nodes residing in different parts of a graph can have similar structural roles within their local network topology.

Original: The identiﬁcation of such roles provides key insight into the organization of networks and can also be used to inform machine learning on graphs.
Revised: The identiﬁcation of such roles provides key insight into the organization of networks and can also be used to inform machine learning on graphs.

Original: However, learning structural representations of nodes is a challenging unsupervised-learning task, which typically involves manually specifying and tailoring topological features for each node.
Revised: However, learning structural representations of nodes is a challenging unsupervised-learning task, which typically involves manually specifying and tailoring topological features for each node.

Original: Here we develop
Revised: Here we develop

Original: G RAPH W AVE , a method that represents each node’s local network neighborhood via a low-dimensional embedding by leveraging spectral graph wavelet diffusion patterns.
Revised: G RAPH W AVE , a method that represents each node’s local network neighborhood via a low-dimensional embedding by leveraging spectral graph wavelet diffusion patterns.

Original: We prove that nodes with similar local network neighborhoods will have similar G RAPH W AVE embeddings even though these nodes may reside in very different parts of the network.
Revised: We prove that nodes with similar local network neighborhoods will have similar G RAPH W AVE embeddings even though these nodes may reside in very different parts of the network.

Original: Our method scales linearly with the number of edges and does not require any hand-tailoring of topological features.
Revised: Our method scales linearly with the number of edges and does not require any hand-tailoring of topological features.

Original: We evaluate performance on both synthetic and real-world datasets, obtaining improvements of up to 71% over state-of-the-art baselines.
Revised: We evaluate performance on both synthetic and real-world datasets, obtaining improvements of up to 71% over state-of-the-art baselines.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Structural role discovery in graphs focuses on identifying nodes which have topologically similar local neighborhoods ( i.e. , similar local structural roles) while residing in potentially distant areas of the network (Figure 1).
Revised: Structural role discovery in graphs focuses on identifying nodes which have topologically similar local neighborhoods ( i.e. , similar local structural roles) while residing in potentially distant areas of the network (Figure 1).

Original: Such alternative deﬁnition of node similarity is very different than more traditional notions (Perozzi et al., 2014; Grover and Leskovec, 2016; Yang et al., 2016; Monti et al., 2016; Kipf and Welling, 2017; Hamilton et al., 2017a;b; Garcia-Duran and Niepert, 2017), which all assume some notion of “smoothness” over the graph and thus consider nodes residing in close network proximity to be similar.
Revised: Such alternative deﬁnition of node similarity is very different than more traditional notions (Perozzi et al., 2014; Grover and Leskovec, 2016; Yang et al., 2016; Monti et al., 2016; Kipf and Welling, 2017; Hamilton et al., 2017a;b; Garcia-Duran and Niepert, 2017), which all assume some notion of “smoothness” over the graph and thus consider nodes residing in close network proximity to be similar.

Original: Such structural role information about the nodes can be used for a variety of tasks, including as input to machine learning problems, or even to identify key nodes in a system (principal “inﬂuencers” in a social network, critical hubs in contagion graphs, etc.).
Revised: Such structural role information about the nodes can be used for a variety of tasks, including as input to machine learning problems, or even to identify key nodes in a system (principal “inﬂuencers” in a social network, critical hubs in contagion graphs, etc.).

Original: When structural roles of nodes are deﬁned over a discrete space, they correspond to different topologies of local network neighborhoods ( e.g. , edge of a chain, center of a star, a bridge between two clusters).
Revised: When structural roles of nodes are deﬁned over a discrete space, they correspond to different topologies of local network neighborhoods ( e.g. , edge of a chain, center of a star, a bridge between two clusters).

Original: However, such discrete roles must be pre-deﬁned, requiring domain expertise and manual inspection of the graph structure.
Revised: However, such discrete roles must be pre-deﬁned, requiring domain expertise and manual inspection of the graph structure.

Original: A more powerful and robust method for identifying structural similarity involves learning a continuous vector-valued structural signature χ a of each node a in an unsupervised way.
Revised: A more powerful and robust method for identifying structural similarity involves learning a continuous vector-valued structural signature χ a of each node a in an unsupervised way.

Original: This motivates a natural deﬁnition of structural similarity in terms of closeness of topological signatures: For any (cid:15) > 0 , nodes a and b are deﬁned to be (cid:15) -structurally similar with respect to a given distance if: dist ( χ a , χ b ) ≤ (cid:15) .
Revised: This motivates a natural deﬁnition of structural similarity in terms of closeness of topological signatures: For any (cid:15) > 0 , nodes a and b are deﬁned to be (cid:15) -structurally similar with respect to a given distance if: dist ( χ a , χ b ) ≤ (cid:15) .

Original: Thus, a robust structural similarity metric must introduce both an appropriate signature and an adequate distance metric.
Revised: Thus, a robust structural similarity metric must introduce both an appropriate signature and an adequate distance metric.

Original: While several methods have been proposed for structural role discovery in graphs, existing approaches are extremely sensitive to small perturbations in the topology and typically lack one or more desirable properties.
Revised: While several methods have been proposed for structural role discovery in graphs, existing approaches are extremely sensitive to small perturbations in the topology and typically lack one or more desirable properties.

Original: They often require manually hand-labeling topological features (Henderson et al., 2012), rely on non-scalable heuristics (Ribeiro et al., 2017), and/or return a single similarity score instead of a multidimensional structural signature (Jin et al., 2011; 2014).
Revised: They often require manually hand-labeling topological features (Henderson et al., 2012), rely on non-scalable heuristics (Ribeiro et al., 2017), and/or return a single similarity score instead of a multidimensional structural signature (Jin et al., 2011; 2014).

Original: Here we address the problem of structure learning on graphs by developing G RAPH W AVE .
Revised: Here we address the problem of structure learning on graphs by developing G RAPH W AVE .

Original: Building upon techniques from graph signal processing (Coifman et al., 2006; Hammond et al., 2011; Shuman et al., 2013), our approach learns a structural embedding for each node based on the diffusion of a spectral graph wavelet centered at that node.
Revised: Building upon techniques from graph signal processing (Coifman et al., 2006; Hammond et al., 2011; Shuman et al., 2013), our approach learns a structural embedding for each node based on the diffusion of a spectral graph wavelet centered at that node.

Original: Intuitively, each node propagates a unit of energy over the graph and characterizes its neighboring topology based on the response of the network to
Revised: Intuitively, each node propagates a unit of energy over the graph and characterizes its neighboring topology based on the response of the network to

Original: [Figure]
Revised: [Figure]

Original: While raw spectral graph wavelet signatures/coefﬁcients Ψ of a and b might be very different, we treat them as probability distributions and show that the coefﬁcient distributions are indeed similar.
Revised: While raw spectral graph wavelet signatures/coefﬁcients Ψ of a and b might be very different, we treat them as probability distributions and show that the coefﬁcient distributions are indeed similar.

Original: this probe.
Revised: this probe.

Original: In contrast to prior work that characterizes the wavelet diffusion as a function of the wavelet scaling parameter, we study how the wavelet diffuses through the network at a given scale as a function of the initial source node.
Revised: In contrast to prior work that characterizes the wavelet diffusion as a function of the wavelet scaling parameter, we study how the wavelet diffuses through the network at a given scale as a function of the initial source node.

Original: We prove that the coefﬁcients of this wavelet directly relate to graph topological properties.
Revised: We prove that the coefﬁcients of this wavelet directly relate to graph topological properties.

Original: Hence, these coefﬁcients contain all the necessary information to recover structurally similar nodes, without requiring the hand-labeling of features.
Revised: Hence, these coefﬁcients contain all the necessary information to recover structurally similar nodes, without requiring the hand-labeling of features.

Original: However, the wavelets are, by design, localized on the graph.
Revised: However, the wavelets are, by design, localized on the graph.

Original: Therefore to compare structural signatures for nodes that are far away from each other, typical graph signal processing methods (using metrics like correlation between wavelets or (cid:96) 2 distance) cannot be used without specifying an exact one-to-one mapping between nodes for every pairwise comparison, a computationally intractable task.
Revised: Therefore to compare structural signatures for nodes that are far away from each other, typical graph signal processing methods (using metrics like correlation between wavelets or (cid:96) 2 distance) cannot be used without specifying an exact one-to-one mapping between nodes for every pairwise comparison, a computationally intractable task.

Original: To overcome this challenge, we propose a novel way of treating the wavelets as probability distributions over the graph.
Revised: To overcome this challenge, we propose a novel way of treating the wavelets as probability distributions over the graph.

Original: This way the structural information is contained in how the diffusion spreads over the network rather than where it spreads.
Revised: This way the structural information is contained in how the diffusion spreads over the network rather than where it spreads.

Original: In order to provide vector-valued signatures which can then be used as input to any machine learning algorithm, we embed these wavelet distributions using the empirical characteristic function (Lukacs, 1970).
Revised: In order to provide vector-valued signatures which can then be used as input to any machine learning algorithm, we embed these wavelet distributions using the empirical characteristic function (Lukacs, 1970).

Original: The advantage of empirical characteristic functions is that they capture all the moments of a given distribution.
Revised: The advantage of empirical characteristic functions is that they capture all the moments of a given distribution.

Original: This allows G RAPH W AVE to be robust to small perturbations in the local edge structure, as we prove mathematically.
Revised: This allows G RAPH W AVE to be robust to small perturbations in the local edge structure, as we prove mathematically.

Original: Computational complexity of G RAPH W AVE is linear in the number of edges, thus allowing it to scale to large (sparse) networks.
Revised: Computational complexity of G RAPH W AVE is linear in the number of edges, thus allowing it to scale to large (sparse) networks.

Original: Finally, we compare G RAPH W AVE to several state-of-the-art baselines on both real and synthetic datasets, obtaining improvements of up to 71% and demonstrating how our approach is a useful tool for characterizing structural signatures in graphs.
Revised: Finally, we compare G RAPH W AVE to several state-of-the-art baselines on both real and synthetic datasets, obtaining improvements of up to 71% and demonstrating how our approach is a useful tool for characterizing structural signatures in graphs.

Original: Summary of contributions.
Revised: Summary of contributions.

Original: The main contributions of our paper are as follows:
Revised: The main contributions of our paper are as follows:

Original: • We develop a novel use of spectral graph wavelets by treating them as probability distributions and characterizing the distributions using empirical characteristic functions.
Revised: • We develop a novel use of spectral graph wavelets by treating them as probability distributions and characterizing the distributions using empirical characteristic functions.

Original: • We leverage these insights to develop a scalable method (G RAPH W AVE ) for learning node embeddings based on structural similarity in graphs.
Revised: • We leverage these insights to develop a scalable method (G RAPH W AVE ) for learning node embeddings based on structural similarity in graphs.

Original: • We prove that G RAPH W AVE accurately recovers structurally similar nodes.
Revised: • We prove that G RAPH W AVE accurately recovers structurally similar nodes.

Original: Further related work.
Revised: Further related work.

Original: Prior work on discovering nodes with similar structural roles has typically relied on explicit featurization of nodes.
Revised: Prior work on discovering nodes with similar structural roles has typically relied on explicit featurization of nodes.

Original: These methods generate an exhaustive listing of each node’s local topological properties ( e.g. , node degree, number of triangles it participates in, number of k -cliques, its PageRank score) before computing node similarities based on such heuristic representations.
Revised: These methods generate an exhaustive listing of each node’s local topological properties ( e.g. , node degree, number of triangles it participates in, number of k -cliques, its PageRank score) before computing node similarities based on such heuristic representations.

Original: A notable example of such approaches is RolX (Henderson et al., 2012), which aims to recover a soft-clustering of nodes into a predetermined number of K distinct roles using recursive feature extraction (Henderson et al., 2011).
Revised: A notable example of such approaches is RolX (Henderson et al., 2012), which aims to recover a soft-clustering of nodes into a predetermined number of K distinct roles using recursive feature extraction (Henderson et al., 2011).

Original: Similarly, struc2vec (Ribeiro et al., 2017) uses a heuristic to construct a multilayered graph based on topological metrics and simulates random walks on the graph to capture structural information.
Revised: Similarly, struc2vec (Ribeiro et al., 2017) uses a heuristic to construct a multilayered graph based on topological metrics and simulates random walks on the graph to capture structural information.

Original: In contrast, our approach does not rely on heuristics (we mathematically prove its efﬁcacy) and does not require explicit manual feature engineering or hand-tuning of parameters.
Revised: In contrast, our approach does not rely on heuristics (we mathematically prove its efﬁcacy) and does not require explicit manual feature engineering or hand-tuning of parameters.

Original: Another line of related work are graph diffusion kernels (Coifman et al., 2006) which have been utilized for various graph modeling purposes (Kondor and Lafferty, 2002; Chung, 2007; Rustamov and Guibas, 2013; Tremblay et al., 2014).
Revised: Another line of related work are graph diffusion kernels (Coifman et al., 2006) which have been utilized for various graph modeling purposes (Kondor and Lafferty, 2002; Chung, 2007; Rustamov and Guibas, 2013; Tremblay et al., 2014).

Original: However, to the best of our knowledge, our paper is the ﬁrst to apply graph diffusion kernels for determining structural roles in graphs.
Revised: However, to the best of our knowledge, our paper is the ﬁrst to apply graph diffusion kernels for determining structural roles in graphs.

Original: Kernels have been shown to efﬁciently capture geometrical properties and have been successfully used for shape detection in the image processing community (Sun et al., 2009; Ovsjanikov et al., 2010; et al.,
Revised: Kernels have been shown to efﬁciently capture geometrical properties and have been successfully used for shape detection in the image processing community (Sun et al., 2009; Ovsjanikov et al., 2010; et al.,

Original: However, in contrast to shape-matching problems, G RAPH W AVE considers these kernels as probability distributions over real-world graphs.
Revised: However, in contrast to shape-matching problems, G RAPH W AVE considers these kernels as probability distributions over real-world graphs.

Original: This is because the graphs that we consider are highly irregular (as opposed to the Euclidean and manifold graphs).
Revised: This is because the graphs that we consider are highly irregular (as opposed to the Euclidean and manifold graphs).

Original: Therefore, traditional wavelet methods, which typically analyze node diffusions across speciﬁc nodes that occur in regular and predictable patterns, do not apply.
Revised: Therefore, traditional wavelet methods, which typically analyze node diffusions across speciﬁc nodes that occur in regular and predictable patterns, do not apply.

Original: Instead, by treating wavelets as distributions, G RAPH W AVE characterizes the shape of the diffusion, rather than the speciﬁc nodes where the diffusion occurs.
Revised: Instead, by treating wavelets as distributions, G RAPH W AVE characterizes the shape of the diffusion, rather than the speciﬁc nodes where the diffusion occurs.

Original: This key insight allows us to uncover structural signatures and to discover structurally similar nodes.
Revised: This key insight allows us to uncover structural signatures and to discover structurally similar nodes.

Original:  L EARNING STRUCTURAL SIGNATURES
Revised:  L EARNING STRUCTURAL SIGNATURES

Original: Given an undirected connected graph G =
Revised: Given an undirected connected graph G =

Original: ( V , E ) with N nodes V = { a 1 , . . .
Revised: ( V , E ) with N nodes V = { a 1 , . . .

Original: , a N } , edges E , an adjacency matrix A (binary or weighted), and a degree matrix D ii = (cid:80)
Revised: , a N } , edges E , an adjacency matrix A (binary or weighted), and a degree matrix D ii = (cid:80)

Original: j A ij , we consider the problem of learning, for every node a i , a structural signature representing a i ’s position in a continuous multidimensional space of structural roles.
Revised: j A ij , we consider the problem of learning, for every node a i , a structural signature representing a i ’s position in a continuous multidimensional space of structural roles.

Original: We frame this as an unsupervised learning problem based on spectral graph wavelets (Hammond et al., 2011) and develop an approach called G RAPH W AVE that provides mathematical guarantees on the optimality of learned structural signatures.
Revised: We frame this as an unsupervised learning problem based on spectral graph wavelets (Hammond et al., 2011) and develop an approach called G RAPH W AVE that provides mathematical guarantees on the optimality of learned structural signatures.

Original: 2. S PECTRAL GRAPH WAVELETS
Revised: 2. S PECTRAL GRAPH WAVELETS

Original: In this section, we provide background on a spectral graph wavelet-based model (Hammond et al., 2011; Shuman et al., 2013) that we will use in the rest of the paper.
Revised: In this section, we provide background on a spectral graph wavelet-based model (Hammond et al., 2011; Shuman et al., 2013) that we will use in the rest of the paper.

Original: Let U be the eigenvector decomposition of the unnormalized graph Laplacian L = D − A and let λ 1 < λ 2 ≤ · · · ≤ λ N ( Λ = Diag ( λ 1 , . . .
Revised: Let U be the eigenvector decomposition of the unnormalized graph Laplacian L = D − A and let λ 1 < λ 2 ≤ · · · ≤ λ N ( Λ = Diag ( λ 1 , . . .

Original: , λ N ) ) denote the eigenvalues of L .
Revised: , λ N ) ) denote the eigenvalues of L .

Original: Let g s be a ﬁlter kernel with scaling parameter s .
Revised: Let g s be a ﬁlter kernel with scaling parameter s .

Original: For simplicity, we use the heat kernel g s ( λ ) =
Revised: For simplicity, we use the heat kernel g s ( λ ) =

Original: e − λs throughout this paper, but our results apply to any low-pass ﬁlter kernel (Shuman et al., 2016).
Revised: e − λs throughout this paper, but our results apply to any low-pass ﬁlter kernel (Shuman et al., 2016).

Original: For now, we assume that s is given; we develop a method for selecting an appropriate value of s in Appendix C.
Revised: For now, we assume that s is given; we develop a method for selecting an appropriate value of s in Appendix C.

Original: Graph signal processing (Hammond et al., 2011; Shuman et al., 2013) deﬁnes the spectral graph wavelet associated with g s as the signal resulting from the modulation in the spectral domain of a
Revised: Graph signal processing (Hammond et al., 2011; Shuman et al., 2013) deﬁnes the spectral graph wavelet associated with g s as the signal resulting from the modulation in the spectral domain of a

Original: Dirac signal centered around node a .
Revised: Dirac signal centered around node a .

Original: The spectral graph wavelet Ψ a is given by an N -dimensional vector:
Revised: The spectral graph wavelet Ψ a is given by an N -dimensional vector:

Original: [Equation]
Revised: [Equation]

Original: where δ a = 1 ( a ) is the one-hot vector for node a .
Revised: where δ a = 1 ( a ) is the one-hot vector for node a .

Original: For notational simplicity, we drop the explicit dependency of spectral graph wavelet Ψ a on s .
Revised: For notational simplicity, we drop the explicit dependency of spectral graph wavelet Ψ a on s .

Original: The m -th wavelet coefﬁcient of this column vector is thus given by Ψ ma =
Revised: The m -th wavelet coefﬁcient of this column vector is thus given by Ψ ma =

Original: [Equation]
Revised: [Equation]

Original: In spectral graph wavelets, the kernel g s modulates the eigenspectrum such that the resulting signal is typically localized on the graph and in the spectral domain (Shuman et al., 2013).
Revised: In spectral graph wavelets, the kernel g s modulates the eigenspectrum such that the resulting signal is typically localized on the graph and in the spectral domain (Shuman et al., 2013).

Original: Spectral graph wavelets are based on an analogy between temporal frequencies of a signal and the Laplacian’s eigenvalues.
Revised: Spectral graph wavelets are based on an analogy between temporal frequencies of a signal and the Laplacian’s eigenvalues.

Original: Eigenvectors associated with smaller eigenvalues carry slow varying signal, encouraging nodes that are geographically close in the graph to share similar values.
Revised: Eigenvectors associated with smaller eigenvalues carry slow varying signal, encouraging nodes that are geographically close in the graph to share similar values.

Original: In contrast, eigenvectors associated with larger eigenvalues carry faster-varying signal across edges.
Revised: In contrast, eigenvectors associated with larger eigenvalues carry faster-varying signal across edges.

Original: The low-pass ﬁlter kernel g s can thus be seen as a modulation operator that discounts higher eigenvalues and enforces smoothness in the signal variation on the graph.
Revised: The low-pass ﬁlter kernel g s can thus be seen as a modulation operator that discounts higher eigenvalues and enforces smoothness in the signal variation on the graph.

Original: 2. G RAPH W AVE ALGORITHM
Revised: 2. G RAPH W AVE ALGORITHM

Original: First we describe the G RAPH W AVE algorithm (Alg. 1) and then analyze it in the next section.
Revised: First we describe the G RAPH W AVE algorithm (Alg. 1) and then analyze it in the next section.

Original: For every node a , G RAPH W AVE returns a 2 d -dimensional vector
Revised: For every node a , G RAPH W AVE returns a 2 d -dimensional vector

Original: χ
Revised: χ

Original: a representing its structural signature , where nodes with structurally similar local network neighborhoods will have similar signatures.
Revised: a representing its structural signature , where nodes with structurally similar local network neighborhoods will have similar signatures.

Original: We ﬁrst apply spectral graph wavelets to obtain a diffusion pattern for every node (Line 3), which we gather in a matrix Ψ .
Revised: We ﬁrst apply spectral graph wavelets to obtain a diffusion pattern for every node (Line 3), which we gather in a matrix Ψ .

Original: Here, Ψ is a N × N matrix, where a -th column vector is the spectral graph wavelet for a heat kernel centered at node a .
Revised: Here, Ψ is a N × N matrix, where a -th column vector is the spectral graph wavelet for a heat kernel centered at node a .

Original: In contrast to prior work that studies wavelet coefﬁcients as a function of the scaling parameter s , we study them as a function of the network ( i.e. , how the coefﬁcients vary across the local network neighborhood around the node a ).
Revised: In contrast to prior work that studies wavelet coefﬁcients as a function of the scaling parameter s , we study them as a function of the network ( i.e. , how the coefﬁcients vary across the local network neighborhood around the node a ).

Original: In particular, coefﬁcients in each wavelet are identiﬁed with the nodes and Ψ ma represents the amount of energy that node a
Revised: In particular, coefﬁcients in each wavelet are identiﬁed with the nodes and Ψ ma represents the amount of energy that node a

Original: Append Re ( φ a ( t )) and Im ( φ a ( t )) to χ a has received from node m .
Revised: Append Re ( φ a ( t )) and Im ( φ a ( t )) to χ a has received from node m .

Original: As we will later show nodes a and b with similar network neighborhoods have similar spectral wavelet coefﬁcients Ψ (assuming that we know how to solve the “isomorphism” problem and ﬁnd the explicit one-to-one mapping of the nodes from a ’s neighborhood to the nodes of the b ’s neighborhood).
Revised: As we will later show nodes a and b with similar network neighborhoods have similar spectral wavelet coefﬁcients Ψ (assuming that we know how to solve the “isomorphism” problem and ﬁnd the explicit one-to-one mapping of the nodes from a ’s neighborhood to the nodes of the b ’s neighborhood).

Original: To resolve the node mapping problem G RAPH W AVE treats the wavelet coefﬁcients as a probability distribution and characterizes the distribution via empirical characteristic functions.
Revised: To resolve the node mapping problem G RAPH W AVE treats the wavelet coefﬁcients as a probability distribution and characterizes the distribution via empirical characteristic functions.

Original: This is the key insight that makes it possible for G RAPH W AVE to learn nodes’ structural signatures via spectral graph wavelets.
Revised: This is the key insight that makes it possible for G RAPH W AVE to learn nodes’ structural signatures via spectral graph wavelets.

Original: More precisely, we embed spectral graph wavelet coefﬁcient distributions into 2 d -dimensional space (Line 4-7) by calculating the characteristic function for each node’s coefﬁcients Ψ a and sample it at d evenly spaced points.
Revised: More precisely, we embed spectral graph wavelet coefﬁcient distributions into 2 d -dimensional space (Line 4-7) by calculating the characteristic function for each node’s coefﬁcients Ψ a and sample it at d evenly spaced points.

Original: The characteristic function of a probability distribution X is deﬁned as: φ X ( t ) =
Revised: The characteristic function of a probability distribution X is deﬁned as: φ X ( t ) =

Original: E [ e itX ] , t ∈ R (Lukacs, 1970).
Revised: E [ e itX ] , t ∈ R (Lukacs, 1970).

Original: The function φ X ( t ) fully characterizes the distribution of X because it captures information about all the moments of probability distribution X (Lukacs, 1970).
Revised: The function φ X ( t ) fully characterizes the distribution of X because it captures information about all the moments of probability distribution X (Lukacs, 1970).

Original: For a given node a and scale s , the empirical characteristic function of Ψ a is deﬁned as:
Revised: For a given node a and scale s , the empirical characteristic function of Ψ a is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: Finally, structural signature χ a of node a is obtained by sampling the 2-dimensional parametric function (Eq.
Revised: Finally, structural signature χ a of node a is obtained by sampling the 2-dimensional parametric function (Eq.

Original: at d evenly spaced points t 1 , . . .
Revised: at d evenly spaced points t 1 , . . .

Original: t d and concatenating the values:
Revised: t d and concatenating the values:

Original: [Equation]
Revised: [Equation]

Original: Note that we sample/evaluate the empirical characteristic function φ a ( t ) at d points and this creates a structural signature of size 2 d .
Revised: Note that we sample/evaluate the empirical characteristic function φ a ( t ) at d points and this creates a structural signature of size 2 d .

Original: This means that the dimensionality of the structural signature is independent of the graph size.
Revised: This means that the dimensionality of the structural signature is independent of the graph size.

Original: Furthermore, nodes from different graphs can be embedded into the same space and their structural roles can be compared across different graphs.
Revised: Furthermore, nodes from different graphs can be embedded into the same space and their structural roles can be compared across different graphs.

Original: Distance between structural signatures.
Revised: Distance between structural signatures.

Original: The ﬁnal output of G RAPH W AVE is a structural signature χ a for each node a in the graph.
Revised: The ﬁnal output of G RAPH W AVE is a structural signature χ a for each node a in the graph.

Original: We can explore distances between the signatures through the use of the (cid:96) 2 distance on χ a .
Revised: We can explore distances between the signatures through the use of the (cid:96) 2 distance on χ a .

Original: The structural distance between nodes a and b is then deﬁned as: dist ( a, b )
Revised: The structural distance between nodes a and b is then deﬁned as: dist ( a, b )

Original: =
Revised: =

Original: (cid:107) χ a − χ b (cid:107) 2 .
Revised: (cid:107) χ a − χ b (cid:107) 2 .

Original: By deﬁnition of the characteristic function, this technique amounts to comparing moments of different orders deﬁned on wavelet coefﬁcient distributions.
Revised: By deﬁnition of the characteristic function, this technique amounts to comparing moments of different orders deﬁned on wavelet coefﬁcient distributions.

Original: Scaling parameter.
Revised: Scaling parameter.

Original: The scaling parameter s determines the radius of network neighborhood around each node a (Tremblay et al.
Revised: The scaling parameter s determines the radius of network neighborhood around each node a (Tremblay et al.

Original: Hammond et al.
Revised: Hammond et al.

Original: A small value of s determines node signatures based on similarity of nodes’ immediate neighborhoods.
Revised: A small value of s determines node signatures based on similarity of nodes’ immediate neighborhoods.

Original: In contrast, a larger value of s allows the diffusion process to spread farther in the network, resulting in signatures based on neighborhoods with greater radii.
Revised: In contrast, a larger value of s allows the diffusion process to spread farther in the network, resulting in signatures based on neighborhoods with greater radii.

Original: G RAPH W AVE can also integrate information across different radii of neighborhoods by jointly considering many different values of s .
Revised: G RAPH W AVE can also integrate information across different radii of neighborhoods by jointly considering many different values of s .

Original: This is achieved by concatenating J representations χ ( s j ) a , each associated with a scale s j , where s j ∈ [ s min , s max ] .
Revised: This is achieved by concatenating J representations χ ( s j ) a , each associated with a scale s j , where s j ∈ [ s min , s max ] .

Original: We provide a theoretically justiﬁed method for ﬁnding an appropriate range s min and s max in Appendix C. In this multiscale version of
Revised: We provide a theoretically justiﬁed method for ﬁnding an appropriate range s min and s max in Appendix C. In this multiscale version of

Original: G RAPH W AVE , the ﬁnal aggregated structural signature for node a is a vector χ a ∈ R 2 dJ with the following form: χ a =
Revised: G RAPH W AVE , the ﬁnal aggregated structural signature for node a is a vector χ a ∈ R 2 dJ with the following form: χ a =

Original: [ Re ( φ ( s j ) a ( t i )) , Im ( φ ( s j ) a ( t i )]
Revised: [ Re ( φ ( s j ) a ( t i )) , Im ( φ ( s j ) a ( t i )]

Original: t i ,s j .
Revised: t i ,s j .

Original: Computational complexity.
Revised: Computational complexity.

Original: We use Chebyshev polynomials (Shuman et al., 2011) to compute Line 3 in Algorithm 1.
Revised: We use Chebyshev polynomials (Shuman et al., 2011) to compute Line 3 in Algorithm 1.

Original: As in Defferrard et al.
Revised: As in Defferrard et al.

Original: (2016), each power of the Laplacian has a computational cost of O ( |E| ) , yielding an overall complexity of O
Revised: (2016), each power of the Laplacian has a computational cost of O ( |E| ) , yielding an overall complexity of O

Original: ( K |E| ) , where K denotes the order Chebyshev polynomial approximation.
Revised: ( K |E| ) , where K denotes the order Chebyshev polynomial approximation.

Original: The overall complexity of G RAPH W AVE is linear in the number of edges, which allows G RAPH W AVE to scale to large sparse networks.
Revised: The overall complexity of G RAPH W AVE is linear in the number of edges, which allows G RAPH W AVE to scale to large sparse networks.

Original:  A NALYSIS OF G RAPH W AVE
Revised:  A NALYSIS OF G RAPH W AVE

Original: In this section, we provide theoretical motivation for our spectral graph wavelet-based model (Shuman et al., 2013).
Revised: In this section, we provide theoretical motivation for our spectral graph wavelet-based model (Shuman et al., 2013).

Original: First we analytically show that spectral graph wavelet coefﬁcients characterize the topological structure of local network neighborhoods (Section 3.1).
Revised: First we analytically show that spectral graph wavelet coefﬁcients characterize the topological structure of local network neighborhoods (Section 3.1).

Original: Then we show that structurally equivalent/similar nodes have near-identical/similar signatures (Sections 3.2 and 3.3), thereby providing a mathematical guarantee on the optimality of G RAPH W AVE .
Revised: Then we show that structurally equivalent/similar nodes have near-identical/similar signatures (Sections 3.2 and 3.3), thereby providing a mathematical guarantee on the optimality of G RAPH W AVE .

Original: 3. S PECTRAL GRAPH WAVELETS AS A MEASURE OF NETWORK STRUCTURE
Revised: 3. S PECTRAL GRAPH WAVELETS AS A MEASURE OF NETWORK STRUCTURE

Original: We start by establishing the relationship between the spectral graph wavelet of a given node a and the topological properties of local network neighborhood centered at a .
Revised: We start by establishing the relationship between the spectral graph wavelet of a given node a and the topological properties of local network neighborhood centered at a .

Original: In particular, we prove that a wavelet coefﬁcient Ψ ma provides a measure of network connectivity between nodes a and m .
Revised: In particular, we prove that a wavelet coefﬁcient Ψ ma provides a measure of network connectivity between nodes a and m .

Original: We use the fact that the spectrum of the graph Laplacian is discrete and contained in the compact set [0 , λ N ] .
Revised: We use the fact that the spectrum of the graph Laplacian is discrete and contained in the compact set [0 , λ N ] .

Original: It then follows from the Stone-Weierstrass theorem that the restriction of kernel g s to the interval [0 , λ N ] can be approximated by a polynomial.
Revised: It then follows from the Stone-Weierstrass theorem that the restriction of kernel g s to the interval [0 , λ N ] can be approximated by a polynomial.

Original: This polynomial approximation, denoted as
Revised: This polynomial approximation, denoted as

Original: P , is tight and its error can be uniformly bounded.
Revised: P , is tight and its error can be uniformly bounded.

Original: Formally, this means:
Revised: Formally, this means:

Original: [Equation]
Revised: [Equation]

Original: such that
Revised: such that

Original: [Equation]
Revised: [Equation]

Original: where K is the order of polynomial approximation, α k are coefﬁcients of the polynomial, and r ( λ )
Revised: where K is the order of polynomial approximation, α k are coefﬁcients of the polynomial, and r ( λ )

Original: =
Revised: =

Original: g s ( λ ) − P ( λ ) is the residual.
Revised: g s ( λ ) − P ( λ ) is the residual.

Original: We can now express the spectral graph wavelet for node a in terms of the polynomial approximation as:
Revised: We can now express the spectral graph wavelet for node a in terms of the polynomial approximation as:

Original: [Equation]
Revised: [Equation]

Original: We note that Ψ a is a function of L k =
Revised: We note that Ψ a is a function of L k =

Original: ( D − A ) k
Revised: ( D − A ) k

Original: and thus can be interpreted using graph theory.
Revised: and thus can be interpreted using graph theory.

Original: In particular, it contains terms of the form D k (capturing the degree), A k (capturing the number of k -length paths that node a participates in), and terms containing both A and D , which denote paths of length up to k going from node a to every other node m .
Revised: In particular, it contains terms of the form D k (capturing the degree), A k (capturing the number of k -length paths that node a participates in), and terms containing both A and D , which denote paths of length up to k going from node a to every other node m .

Original: Using the Cauchy-Schwartz’s inequality and the facts that U is unitary and r ( λ ) is uniformly bounded (Eq.
Revised: Using the Cauchy-Schwartz’s inequality and the facts that U is unitary and r ( λ ) is uniformly bounded (Eq.

Original: (4)), we can bound the second term on the right-hand side of Eq.
Revised: (4)), we can bound the second term on the right-hand side of Eq.

Original: (5) by:
Revised: (5) by:

Original: [Equation]
Revised: [Equation]

Original: As a consequence, each wavelet Ψ a can be approximated by a K -th order polynomial that captures information about the K -hop neighborhood of node a .
Revised: As a consequence, each wavelet Ψ a can be approximated by a K -th order polynomial that captures information about the K -hop neighborhood of node a .

Original: The analysis of Eq.
Revised: The analysis of Eq.

Original: (5), where we show that the second term is limited by (cid:15) , indicates that spectral graph wavelets are predominately governed by topological features (speciﬁcally, degrees, cycles and paths) according to the speciﬁed heat kernel.
Revised: (5), where we show that the second term is limited by (cid:15) , indicates that spectral graph wavelets are predominately governed by topological features (speciﬁcally, degrees, cycles and paths) according to the speciﬁed heat kernel.

Original: The wavelets thus contain the information necessary to generate structural signatures of nodes.
Revised: The wavelets thus contain the information necessary to generate structural signatures of nodes.

Original: 3. S IGNATURES OF STRUCTURALLY EQUIVALENT NODES
Revised: 3. S IGNATURES OF STRUCTURALLY EQUIVALENT NODES

Original: Let us consider nodes a and b whose K -hop
Revised: Let us consider nodes a and b whose K -hop

Original: neighborhoods are identical (where K is an integer less than the diameter of the graph), meaning that nodes a and b are structurally equivalent.
Revised: neighborhoods are identical (where K is an integer less than the diameter of the graph), meaning that nodes a and b are structurally equivalent.

Original: We now show that a and b have (cid:15) -structurally similar signatures in G RAPH W AVE .
Revised: We now show that a and b have (cid:15) -structurally similar signatures in G RAPH W AVE .

Original: First, we use the Taylor expansion to obtain an explicit K -th order polynomial approximation of g s as: P ( λ, s ) =
Revised: First, we use the Taylor expansion to obtain an explicit K -th order polynomial approximation of g s as: P ( λ, s ) =

Original: (cid:80) Kk =

Revised: (cid:80) Kk =

Original: ( − 1)
Revised: ( − 1)

Original: k ( sλ ) k /k ! .
Revised: k ( sλ ) k /k ! .

Original: Then, for each eigenvalue λ , we use the Taylor-Lagrange equality to ensure the existence of c λ ∈ [0 , s ] such that:
Revised: Then, for each eigenvalue λ , we use the Taylor-Lagrange equality to ensure the existence of c λ ∈ [0 , s ] such that:

Original: [Equation]
Revised: [Equation]

Original: If we take any s such that it satisﬁes: s ≤ (( K + 1)!
Revised: If we take any s such that it satisﬁes: s ≤ (( K + 1)!

Original: (cid:15) ))
Revised: (cid:15) ))

Original: 1 / ( K +1) /λ 2 , then the absolute residual | r ( λ ) | in Eq.
Revised: 1 / ( K +1) /λ 2 , then the absolute residual | r ( λ ) | in Eq.

Original: (7) can be bounded by (cid:15) for each eigenvalue λ .
Revised: (7) can be bounded by (cid:15) for each eigenvalue λ .

Original: Here, (cid:15) is a parameter that we can specify depending on how close we want the signatures of structurally equivalent nodes to be (note that smaller values of the scale s lead to smaller values of (cid:15) and thus tighter bounds).
Revised: Here, (cid:15) is a parameter that we can specify depending on how close we want the signatures of structurally equivalent nodes to be (note that smaller values of the scale s lead to smaller values of (cid:15) and thus tighter bounds).

Original: Because a and b are structurally equivalent, there exists a one-to-one mapping π from the Khop neighborhood of a ( i
Revised: Because a and b are structurally equivalent, there exists a one-to-one mapping π from the Khop neighborhood of a ( i

Original: [Equation]
Revised: [Equation]

Original: e. , N K ( a ) ) to the K -hop neighborhood of b ( i
Revised: e. , N K ( a ) ) to the K -hop neighborhood of b ( i

Original: [Equation]
Revised: [Equation]

Original: e. , N K ( b ) ), such that:
Revised: e. , N K ( b ) ), such that:

Original: N K ( b ) =
Revised: N K ( b ) =

Original: π ( N K ( a )) .
Revised: π ( N K ( a )) .

Original: We extend the mapping π to the whole graph G by randomly mapping the remaining nodes.
Revised: We extend the mapping π to the whole graph G by randomly mapping the remaining nodes.

Original: Following Eq.
Revised: Following Eq.

Original: (5), we write the difference between each pair of mapped coefﬁcients Ψ ma and Ψ π ( m ) b in terms of the K -th order approximation of the graph Laplacian:
Revised: (5), we write the difference between each pair of mapped coefﬁcients Ψ ma and Ψ π ( m ) b in terms of the K -th order approximation of the graph Laplacian:

Original: [Equation]
Revised: [Equation]

Original: Here, we analyze the ﬁrst term on the second line in Eq.
Revised: Here, we analyze the ﬁrst term on the second line in Eq.

Original: Since the K -hop neighborhoods around a and b are identical and by the localization properties of the k -th power of the Laplacian ( k -length paths, Section 3.1), the following holds:
Revised: Since the K -hop neighborhoods around a and b are identical and by the localization properties of the k -th power of the Laplacian ( k -length paths, Section 3.1), the following holds:

Original: [Equation]
Revised: [Equation]

Original: meaning that this term cancels out in Eq.
Revised: meaning that this term cancels out in Eq.

Original: To analyze the second and third terms on the second line of Eq.
Revised: To analyze the second and third terms on the second line of Eq.

Original: (8), we use bound for the residual term in the spectral graph wavelet (Eq.
Revised: (8), we use bound for the residual term in the spectral graph wavelet (Eq.

Original: to uniformly bound entries in matrix Ur (Λ) U T by (cid:15) .
Revised: to uniformly bound entries in matrix Ur (Λ) U T by (cid:15) .

Original: Therefore, each wavelet coefﬁcient in Ψ a is within 2 (cid:15) of its corresponding wavelet coefﬁcient in Ψ b , i.e. , | Ψ ma − Ψ π ( m ) b | ≤ 2 (cid:15).
Revised: Therefore, each wavelet coefﬁcient in Ψ a is within 2 (cid:15) of its corresponding wavelet coefﬁcient in Ψ b , i.e. , | Ψ ma − Ψ π ( m ) b | ≤ 2 (cid:15).

Original: As a result, because similarity in distributions translates to similarity in the resulting characteristic functions (Lévy’s continuity theorem), then assuming the appropriate selection of scale, structurally equivalent nodes have (cid:15) -structurally similar signatures.
Revised: As a result, because similarity in distributions translates to similarity in the resulting characteristic functions (Lévy’s continuity theorem), then assuming the appropriate selection of scale, structurally equivalent nodes have (cid:15) -structurally similar signatures.

Original: 3. S IGNATURES OF STRUCTURALLY SIMILAR NODES
Revised: 3. S IGNATURES OF STRUCTURALLY SIMILAR NODES

Original: We now analyze structurally similar nodes, or nodes whose K -hop neighborhoods are identical up to a small perturbation of the edges.
Revised: We now analyze structurally similar nodes, or nodes whose K -hop neighborhoods are identical up to a small perturbation of the edges.

Original: We show that such nodes have similar G RAPH W AVE signatures.
Revised: We show that such nodes have similar G RAPH W AVE signatures.

Original: Let (cid:101) N K ( a ) denote a perturbed K -hop neighborhood of node a obtained by rewiring edges in the original K -hop neighborhood N K ( a ) .
Revised: Let (cid:101) N K ( a ) denote a perturbed K -hop neighborhood of node a obtained by rewiring edges in the original K -hop neighborhood N K ( a ) .

Original: We denote by ˜ L the graph Laplacian associated with that perturbation.
Revised: We denote by ˜ L the graph Laplacian associated with that perturbation.

Original: We next show that when perturbation of a node neighborhood is small, the changes in the wavelet coefﬁcients for that node are small as well.
Revised: We next show that when perturbation of a node neighborhood is small, the changes in the wavelet coefﬁcients for that node are small as well.

Original: Formally, assuming a small perturbation of the graph structure ( i.e. , sup ||
Revised: Formally, assuming a small perturbation of the graph structure ( i.e. , sup ||

Original: L k − ˜
Revised: L k − ˜

Original: L k || F ≤ (cid:15), for all k ≤ K ), we use K -th order Taylor expansion of kernel g s to express the wavelet coefﬁcients in the perturbed graph as:
Revised: L k || F ≤ (cid:15), for all k ≤ K ), we use K -th order Taylor expansion of kernel g s to express the wavelet coefﬁcients in the perturbed graph as:

Original: [Equation]
Revised: [Equation]

Original: We then use the Weyl’s theorem (Coburn et al., 1966) to relate perturbations in the graph structure to the change in the eigenvalues of the graph Laplacian.
Revised: We then use the Weyl’s theorem (Coburn et al., 1966) to relate perturbations in the graph structure to the change in the eigenvalues of the graph Laplacian.

Original: In particular, a small perturbation of the graph yields small perturbations of the eigenvalues.
Revised: In particular, a small perturbation of the graph yields small perturbations of the eigenvalues.

Original: That is, for each ˜ λ , r (˜ λ ) is close its original value r ( λ ) : r (˜ λ )
Revised: That is, for each ˜ λ , r (˜ λ ) is close its original value r ( λ ) : r (˜ λ )

Original: =
Revised: =

Original: r ( λ ) + o ( (cid:15) )
Revised: r ( λ ) + o ( (cid:15) )

Original: ≤ C(cid:15) , where C is a constant.
Revised: ≤ C(cid:15) , where C is a constant.

Original: Taking everything together, we get:
Revised: Taking everything together, we get:

Original: [Equation]
Revised: [Equation]

Original: indicating that structurally similar nodes have similar signatures in G RAPH W AVE .
Revised: indicating that structurally similar nodes have similar signatures in G RAPH W AVE .

Original:  E XPERIMENTS ON SYNTHETIC GRAPHS
Revised:  E XPERIMENTS ON SYNTHETIC GRAPHS

Original: Baselines.
Revised: Baselines.

Original: We compare our G RAPH W AVE method against two state-of-the-art baselines, struc2vec
Revised: We compare our G RAPH W AVE method against two state-of-the-art baselines, struc2vec

Original: (Ribeiro et al., 2017) and RolX (Henderson et al., 2012).
Revised: (Ribeiro et al., 2017) and RolX (Henderson et al., 2012).

Original: We note that RolX requires the number of desired structural classes as input, whereas the two other methods learn embeddings that capture a continuous spectrum of roles rather than discrete classes.
Revised: We note that RolX requires the number of desired structural classes as input, whereas the two other methods learn embeddings that capture a continuous spectrum of roles rather than discrete classes.

Original: We thus use RolX as an oracle estimator, providing it with the correct number of classes 1 .
Revised: We thus use RolX as an oracle estimator, providing it with the correct number of classes 1 .

Original: We also note that homophily-based methods (Kipf and Welling (2017); Hamilton et al.
Revised: We also note that homophily-based methods (Kipf and Welling (2017); Hamilton et al.

Original: (2017a), etc.) are unable to recover structural similarities.
Revised: (2017a), etc.) are unable to recover structural similarities.

Original: [Figure]
Revised: [Figure]

Original: B ARBELL GRAPH
Revised: B ARBELL GRAPH

Original: We consider a barbell graph consisting of two dense cliques connected by a long chain (Figure 2A).
Revised: We consider a barbell graph consisting of two dense cliques connected by a long chain (Figure 2A).

Original: We run G RAPH W AVE , RolX , and struc2vec and plot a 2D PCA representation of learned structural signatures in Figure 2B-D.
Revised: We run G RAPH W AVE , RolX , and struc2vec and plot a 2D PCA representation of learned structural signatures in Figure 2B-D.

Original: G RAPH W AVE correctly learns identical representations for structurally equivalent nodes, providing empirical evidence for our theoretical result in Section 3.2.
Revised: G RAPH W AVE correctly learns identical representations for structurally equivalent nodes, providing empirical evidence for our theoretical result in Section 3.2.

Original: This can be seen by structurally equivalent nodes in Figure 2A (nodes of the same color) having identical projections in the PCA plot (Figure 2D).
Revised: This can be seen by structurally equivalent nodes in Figure 2A (nodes of the same color) having identical projections in the PCA plot (Figure 2D).

Original: In contrast, both RolX and struc2vec fail to recover the exact structural equivalences.
Revised: In contrast, both RolX and struc2vec fail to recover the exact structural equivalences.

Original: All three methods correctly group the clique nodes (purple) together.
Revised: All three methods correctly group the clique nodes (purple) together.

Original: However, only G RAPH W AVE correctly differentiates between nodes connecting the two dense cliques in the barbell graph, providing empirical evidence for our theoretical result in Section 3.3.
Revised: However, only G RAPH W AVE correctly differentiates between nodes connecting the two dense cliques in the barbell graph, providing empirical evidence for our theoretical result in Section 3.3.

Original: G RAPH W AVE represents those nodes in a gradient-like pattern that captures the spectrum of structural roles of those nodes (Figure 2D).
Revised: G RAPH W AVE represents those nodes in a gradient-like pattern that captures the spectrum of structural roles of those nodes (Figure 2D).

Original: 4. C OMPLEX AND NOISY GRAPH STRUCTURES
Revised: 4. C OMPLEX AND NOISY GRAPH STRUCTURES

Original: Graphs.
Revised: Graphs.

Original: We next consider four types of synthetic graphs where the structural role of each node is known and used as ground truth information to evaluate performance.
Revised: We next consider four types of synthetic graphs where the structural role of each node is known and used as ground truth information to evaluate performance.

Original: The graphs are given by basic shapes of one of different types (“house”, “fan”, “star”) that are regularly placed along a cycle (Table 1 and Figure 3A).
Revised: The graphs are given by basic shapes of one of different types (“house”, “fan”, “star”) that are regularly placed along a cycle (Table 1 and Figure 3A).

Original: In the “varied” setup, we mix the three basic shapes when placing them along a cycle, thus generating synthetic graphs with richer and more complex structural role patterns.
Revised: In the “varied” setup, we mix the three basic shapes when placing them along a cycle, thus generating synthetic graphs with richer and more complex structural role patterns.

Original: Additional graphs are generated by placing these shapes irregularly along the cycle followed by adding a number of edges uniformly at random.
Revised: Additional graphs are generated by placing these shapes irregularly along the cycle followed by adding a number of edges uniformly at random.

Original: In our experiments, we set this number to be around 5 % of the edges in the original structure.
Revised: In our experiments, we set this number to be around 5 % of the edges in the original structure.

Original: This setup is designed to assess the robustness of the methods to data perturbations (“house perturbed”, “varied perturbed”).
Revised: This setup is designed to assess the robustness of the methods to data perturbations (“house perturbed”, “varied perturbed”).

Original: Experimental setup.
Revised: Experimental setup.

Original: For each graph, we run RolX , struc2vec , and G RAPH W AVE to learn the signatures.
Revised: For each graph, we run RolX , struc2vec , and G RAPH W AVE to learn the signatures.

Original: We choose to use a multiscale version of G RAPH W AVE where the scale was set as explained in Appendix C. We then use k -means to cluster the learned signatures and use three standard metrics to evaluate the clustering quality.
Revised: We choose to use a multiscale version of G RAPH W AVE where the scale was set as explained in Appendix C. We then use k -means to cluster the learned signatures and use three standard metrics to evaluate the clustering quality.

Original: (1) Cluster homogeneity is the conditional entropy of the ground-truth structural roles given the proposed clustering (Rosenberg and Hirschberg, 2007).
Revised: (1) Cluster homogeneity is the conditional entropy of the ground-truth structural roles given the proposed clustering (Rosenberg and Hirschberg, 2007).

Original: (2) Cluster completeness (Rosenberg and Hirschberg, 2007) evaluates whether nodes with the same structural role are in the same cluster.
Revised: (2) Cluster completeness (Rosenberg and Hirschberg, 2007) evaluates whether nodes with the same structural role are in the same cluster.

Original: (3) Silhouette score compares the mean intra-cluster distance to the mean between-cluster distance, assessing the density of the recovered clusters.
Revised: (3) Silhouette score compares the mean intra-cluster distance to the mean between-cluster distance, assessing the density of the recovered clusters.

Original: This score takes a value in [-1,1] (higher is better).
Revised: This score takes a value in [-1,1] (higher is better).

Original: Results.
Revised: Results.

Original: G RAPH W AVE consistently outperforms struc2vec , yielding improvements for the homogeneity of up to 50%, and completeness up to 69% in the “varied” setting (Table 1).
Revised: G RAPH W AVE consistently outperforms struc2vec , yielding improvements for the homogeneity of up to 50%, and completeness up to 69% in the “varied” setting (Table 1).

Original: Both G RAPH W AVE and RolX achieved perfect performance in the noise-free “house” setting, however, G RAPH W AVE outperformed RolX by up to 4% (completeness) in the more complex “varied” setting.
Revised: Both G RAPH W AVE and RolX achieved perfect performance in the noise-free “house” setting, however, G RAPH W AVE outperformed RolX by up to 4% (completeness) in the more complex “varied” setting.

Original: We evaluated methods on graphs in the presence of noise (“perturbed” in Table 1): G RAPH W AVE outperformed
Revised: We evaluated methods on graphs in the presence of noise (“perturbed” in Table 1): G RAPH W AVE outperformed

Original: RolX and struc2vec
Revised: RolX and struc2vec

Original: by 4% and 50% (homogeneity), respectively, providing empirical evidence for our analytical result that G RAPH W AVE is robust to noise in the edge structure.
Revised: by 4% and 50% (homogeneity), respectively, providing empirical evidence for our analytical result that G RAPH W AVE is robust to noise in the edge structure.

Original: The silhouette scores also show that the clusters recovered by G RAPH W AVE are denser and better separated than for the other methods.
Revised: The silhouette scores also show that the clusters recovered by G RAPH W AVE are denser and better separated than for the other methods.

Original: As an example, we show a cycle graph with attached “house” shapes (Figure 3A).
Revised: As an example, we show a cycle graph with attached “house” shapes (Figure 3A).

Original: We plot 2D PCA projections of G RAPH W AVE ’s signatures in Figure 3B, conﬁrming that G RAPH W AVE accurately
Revised: We plot 2D PCA projections of G RAPH W AVE ’s signatures in Figure 3B, conﬁrming that G RAPH W AVE accurately

Original: [Figure]
Revised: [Figure]

Original: Method
Revised: Method

Original: Homogeneity Completeness Silhouette
Revised: Homogeneity Completeness Silhouette

Original: [Table]
Revised: [Table]

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

Original: House
Revised: House

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

Original: [Table]
Revised: [Table]

Original: House
Revised: House

Original: [Table]
Revised: [Table]

Original: perturbed
Revised: perturbed

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: Varied struc2vec
Revised: Varied struc2vec

Original: [Table]
Revised: [Table]

Original: Varied struc2vec
Revised: Varied struc2vec

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: perturbed
Revised: perturbed

Original: [Table]
Revised: [Table]

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

Original: distinguishes between nodes with distinct structural roles.
Revised: distinguishes between nodes with distinct structural roles.

Original: We also visualize the functions (Eq.
Revised: We also visualize the functions (Eq.

Original: (2)) in Figure 3C.
Revised: (2)) in Figure 3C.

Original: In general, their interpretation is as follows (Appendix D):
Revised: In general, their interpretation is as follows (Appendix D):

Original: • Nodes located in the periphery of the graph struggle to diffuse the signal over the graph, and thus span wavelets that are characterized by a smaller number of non-zero coefﬁcients.
Revised: • Nodes located in the periphery of the graph struggle to diffuse the signal over the graph, and thus span wavelets that are characterized by a smaller number of non-zero coefﬁcients.

Original: Characteristic functions of such nodes thus span a small loop-like 2D curve.
Revised: Characteristic functions of such nodes thus span a small loop-like 2D curve.

Original: • Nodes located in the core (dense region) of the graph tend to diffuse the signal farther away and reach farther nodes for the same value of t .
Revised: • Nodes located in the core (dense region) of the graph tend to diffuse the signal farther away and reach farther nodes for the same value of t .

Original: Characteristic functions of such nodes thus have a farther projection on the x and y axis.
Revised: Characteristic functions of such nodes thus have a farther projection on the x and y axis.

Original: In Figure 3C, different shapes of the characteristic functions capture different structural roles.
Revised: In Figure 3C, different shapes of the characteristic functions capture different structural roles.

Original: We note the visual proximity between the roles of the blue, light green and red nodes that these curves carry, as well as their clear difference with the core dark green and purple nodes.
Revised: We note the visual proximity between the roles of the blue, light green and red nodes that these curves carry, as well as their clear difference with the core dark green and purple nodes.

Original:  E XPERIMENTS ON TWO REAL - WORLD GRAPHS 5. T HE E NRON EMAIL GRAPH
Revised:  E XPERIMENTS ON TWO REAL - WORLD GRAPHS 5. T HE E NRON EMAIL GRAPH

Original: Data and setup.
Revised: Data and setup.

Original: Nodes represent Enron employees and edges correspond to email communication between the employees (Klimt and Yang, 2004).
Revised: Nodes represent Enron employees and edges correspond to email communication between the employees (Klimt and Yang, 2004).

Original: An employee has one of seven functions in the company ( e.g. , CEO, president, manager).
Revised: An employee has one of seven functions in the company ( e.g. , CEO, president, manager).

Original: These functions provide ground-truth information about roles of the corresponding nodes in the network.
Revised: These functions provide ground-truth information about roles of the corresponding nodes in the network.

Original: We use G RAPH W AVE to learn a structural signature for every Enron employee.
Revised: We use G RAPH W AVE to learn a structural signature for every Enron employee.

Original: We then use these signatures to compute the average (cid:96) 22 distance between every two categories of employees.
Revised: We then use these signatures to compute the average (cid:96) 22 distance between every two categories of employees.

Original: Results.
Revised: Results.

Original: G RAPH W AVE captures intricate organizational structure of Enron (Figure 4).
Revised: G RAPH W AVE captures intricate organizational structure of Enron (Figure 4).

Original: For example,
Revised: For example,

Original: CEOs and presidents are structurally distant from all other job titles.
Revised: CEOs and presidents are structurally distant from all other job titles.

Original: This indicates their unique position in the email exchange graph, which can be explained by their local graph connectivity patterns standing out from the others.
Revised: This indicates their unique position in the email exchange graph, which can be explained by their local graph connectivity patterns standing out from the others.

Original: Traders, on the other hand, appear very far from presidents
Revised: Traders, on the other hand, appear very far from presidents

Original: Method
Revised: Method

Original: Hom.
Revised: Hom.

Original: Compl.
Revised: Compl.

Original: Silhouette
Revised: Silhouette

Original: RolX 0.
Revised: RolX 0.

Original: struc2vec 0.
Revised: struc2vec 0.

Original: G RAPH W AVE 0.
Revised: G RAPH W AVE 0.

Original: [Figure]
Revised: [Figure]

Original: and are closer to directors.
Revised: and are closer to directors.

Original: In contrast, struc2vec is less successful at revealing intricate relationships between the job titles, yielding an almost uniform distribution of distances between every class.
Revised: In contrast, struc2vec is less successful at revealing intricate relationships between the job titles, yielding an almost uniform distribution of distances between every class.

Original: We assess the separation between “top” job titles (CEO and President) and lower levels in the job title hierarchy.
Revised: We assess the separation between “top” job titles (CEO and President) and lower levels in the job title hierarchy.

Original: G RAPH W AVE achieves 28% better homogeneity and 139% better completeness than RolX .
Revised: G RAPH W AVE achieves 28% better homogeneity and 139% better completeness than RolX .

Original: We also note that the variability within each cluster of struc2vec is higher than the average distance between clusters (dark green colors on the diagonal, and lighter colors on the off-diagonal).
Revised: We also note that the variability within each cluster of struc2vec is higher than the average distance between clusters (dark green colors on the diagonal, and lighter colors on the off-diagonal).

Original: 5. E UROPEAN AIRLINE GRAPHS
Revised: 5. E UROPEAN AIRLINE GRAPHS

Original: Data and setup.
Revised: Data and setup.

Original: The airline graphs are taken from the list of airlines operating ﬂights between
Revised: The airline graphs are taken from the list of airlines operating ﬂights between

Original: European airports (Cardillo et al., 2013).
Revised: European airports (Cardillo et al., 2013).

Original: Each airline is represented with a graph, where nodes represent airports and links stand for direct ﬂights between two airports.
Revised: Each airline is represented with a graph, where nodes represent airports and links stand for direct ﬂights between two airports.

Original: Given an airline graph, we use G RAPH W AVE to learn structural signature of every airport.
Revised: Given an airline graph, we use G RAPH W AVE to learn structural signature of every airport.

Original: We then create a visualization using
Revised: We then create a visualization using

Original: PCA on the learned signatures to layout the graph on a two-dimensional structural space.
Revised: PCA on the learned signatures to layout the graph on a two-dimensional structural space.

Original: Results.
Revised: Results.

Original: Figure 5 shows graph visualizations of two very different airlines.
Revised: Figure 5 shows graph visualizations of two very different airlines.

Original: Air France is a national
Revised: Air France is a national

Original: French airline, whose graph has the so-called hub and spoke structure, because the airline is designed to provide an almost complete coverage of the airports in France.
Revised: French airline, whose graph has the so-called hub and spoke structure, because the airline is designed to provide an almost complete coverage of the airports in France.

Original: We note that the signature of CDG (Charles De Gaulle, which is Air France’s central airport) clearly stands out in this 2D projection, indicating its unique role in the network.
Revised: We note that the signature of CDG (Charles De Gaulle, which is Air France’s central airport) clearly stands out in this 2D projection, indicating its unique role in the network.

Original: In contrast to Air France, Ryanair is a low-cost airline whose graph avoids the overly centralized structure.
Revised: In contrast to Air France, Ryanair is a low-cost airline whose graph avoids the overly centralized structure.

Original: Ryanair’s network has near-continuous spectrum of structural roles that range from regional French airports (Lille (LIL), Brest Bretagne (BES))
Revised: Ryanair’s network has near-continuous spectrum of structural roles that range from regional French airports (Lille (LIL), Brest Bretagne (BES))

Original: all the way to London Stansted (STN) and Dublin (DUB).
Revised: all the way to London Stansted (STN) and Dublin (DUB).

Original: These airlines have thus developed according to different structural and commercial constraints, which is clearly reﬂected in their visualizations.
Revised: These airlines have thus developed according to different structural and commercial constraints, which is clearly reﬂected in their visualizations.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: We have developed a new method for learning structural signatures in graphs.
Revised: We have developed a new method for learning structural signatures in graphs.

Original: Our approach,
Revised: Our approach,

Original: G RAPH W AVE , uses spectral graph wavelets to generate a structural embedding for each node, which we accomplish by treating the wavelets as a distributions and evaluating the resulting characteristic functions.
Revised: G RAPH W AVE , uses spectral graph wavelets to generate a structural embedding for each node, which we accomplish by treating the wavelets as a distributions and evaluating the resulting characteristic functions.

Original: Considering the wavelets as distributions instead of vectors is a key insight needed to capture structural similarity in graphs.
Revised: Considering the wavelets as distributions instead of vectors is a key insight needed to capture structural similarity in graphs.

Original: Our method provides mathematical guarantees on the optimality of learned structural signatures.
Revised: Our method provides mathematical guarantees on the optimality of learned structural signatures.

Original: Using spectral graph theory, we prove that structurally equivalent/similar nodes have nearidentical/similar structural signatures in G RAPH W AVE .
Revised: Using spectral graph theory, we prove that structurally equivalent/similar nodes have nearidentical/similar structural signatures in G RAPH W AVE .

Original: Experiments on real and synthetic networks provide empirical evidence for our analytical results and yield large gains in performance over stateof-the-art baselines.
Revised: Experiments on real and synthetic networks provide empirical evidence for our analytical results and yield large gains in performance over stateof-the-art baselines.

Original: For future work, these signatures could be used for transfer learning, leveraging data from a well-explored region of the graph to infer knowledge about less-explored regions.
Revised: For future work, these signatures could be used for transfer learning, leveraging data from a well-explored region of the graph to infer knowledge about less-explored regions.

Original: N OTATION
Revised: N OTATION

Original: In the appendix, we use the same notation as in the main paper.
Revised: In the appendix, we use the same notation as in the main paper.

Original: However, in the main paper, we dropped the explicit dependency of the heat kernel wavelet Ψ on the scaling parameter s .
Revised: However, in the main paper, we dropped the explicit dependency of the heat kernel wavelet Ψ on the scaling parameter s .

Original: We note that we explicitly keep this dependency throughout the appendix as we study the relationship between heat kernel wavelets and the scaling parameter.
Revised: We note that we explicitly keep this dependency throughout the appendix as we study the relationship between heat kernel wavelets and the scaling parameter.

Original: A K NOWN PROPERTIES OF HEAT KERNEL WAVELETS
Revised: A K NOWN PROPERTIES OF HEAT KERNEL WAVELETS

Original: We here list ﬁve known properties of heat kernel wavelets that we later use to derive a method for automatic selection of the scaling parameter in Appendix C.
Revised: We here list ﬁve known properties of heat kernel wavelets that we later use to derive a method for automatic selection of the scaling parameter in Appendix C.

Original: P1.
Revised: P1.

Original: The heat kernel wavelet matrix Ψ ( s ) is symmetric :
Revised: The heat kernel wavelet matrix Ψ ( s ) is symmetric :

Original: [Equation]
Revised: [Equation]

Original: P2.
Revised: P2.

Original: First eigenvector: By deﬁnition of the graph Laplacian L = D − A , the vector 1 is an eigenvector of L , corresponding to the smallest eigenvalue λ 1 = 0 .
Revised: First eigenvector: By deﬁnition of the graph Laplacian L = D − A , the vector 1 is an eigenvector of L , corresponding to the smallest eigenvalue λ 1 = 0 .

Original: This means that:
Revised: This means that:

Original: [Equation]
Revised: [Equation]

Original: where the last equality follows from the fact that eigenvectors in U are orthogonal
Revised: where the last equality follows from the fact that eigenvectors in U are orthogonal

Original: P3.
Revised: P3.

Original: Scaling inequality: Using the Cauchy Schwartz inequality, we get:
Revised: Scaling inequality: Using the Cauchy Schwartz inequality, we get:

Original: [Equation]
Revised: [Equation]

Original: It thus follows that: Ψ ( s ) ma ≤ max(Ψ ( s ) aa , Ψ ( s ) mm ) .
Revised: It thus follows that: Ψ ( s ) ma ≤ max(Ψ ( s ) aa , Ψ ( s ) mm ) .

Original: P4.
Revised: P4.

Original: Convergence: We use the fact that when s → ∞ , the value of e − sλ converges to 0 for any non-zero eigenvalue λ of the graph Laplacian.
Revised: Convergence: We use the fact that when s → ∞ , the value of e − sλ converges to 0 for any non-zero eigenvalue λ of the graph Laplacian.

Original: Combining this fact with the spectral graph wavelet deﬁnition in Eq.
Revised: Combining this fact with the spectral graph wavelet deﬁnition in Eq.

Original: we get:
Revised: we get:

Original: [Equation]
Revised: [Equation]

Original: since it follows from Eq.
Revised: since it follows from Eq.

Original: (**) that U a 1 = U 1 a = 1 / √ N for every a .
Revised: (**) that U a 1 = U 1 a = 1 / √ N for every a .

Original: The diffusion over the network thus converges to a state where all nodes have an identical temperature 1 /N .
Revised: The diffusion over the network thus converges to a state where all nodes have an identical temperature 1 /N .

Original: P5.
Revised: P5.

Original: Convergence rate: To analyze the convergence rate of heat kernel wavelets as a function of s , we expand | Ψ
Revised: Convergence rate: To analyze the convergence rate of heat kernel wavelets as a function of s , we expand | Ψ

Original: ( s +1)
Revised: ( s +1)

Original: aa − 1 N | using the spectral graph wavelet deﬁnition in Eq.
Revised: aa − 1 N | using the spectral graph wavelet deﬁnition in Eq.

Original: [Equation]
Revised: [Equation]

Original: where:
Revised: where:

Original: [Equation]
Revised: [Equation]

Original: Taking everything together, we get: e − λ
Revised: Taking everything together, we get: e − λ

Original: 2 s U 2 a 2 ≤ | Ψ ( s ) aa − 1 N | ≤ e − λ 2 s (1 − 1 N ) .
Revised: 2 s U 2 a 2 ≤ | Ψ ( s ) aa − 1 N | ≤ e − λ 2 s (1 − 1 N ) .

Original: It thus follows that: | Ψ ( s +1) aa
Revised: It thus follows that: | Ψ ( s +1) aa

Original: [Equation]
Revised: [Equation]

Original: B P ROPERTIES OF HEAT KERNEL WAVELET DISTRIBUTIONS
Revised: B P ROPERTIES OF HEAT KERNEL WAVELET DISTRIBUTIONS

Original: We prove three propositions about distributions generated by heat kernel wavelets that will be used in
Revised: We prove three propositions about distributions generated by heat kernel wavelets that will be used in

Original: Appendix C.
Revised: Appendix C.

Original: Proposition 1.
Revised: Proposition 1.

Original: The mean of heat kernel wavelet Ψ ( s ) a is equal to µ ( s ) a = 1 N (cid:80)
Revised: The mean of heat kernel wavelet Ψ ( s ) a is equal to µ ( s ) a = 1 N (cid:80)

Original: Nm =1 Ψ ( s ) ma
Revised: Nm =1 Ψ ( s ) ma

Original: = 1 N .
Revised: = 1 N .

Original: The mean µ ( s ) a is thus independent of the value of scaling parameter s and node a .
Revised: The mean µ ( s ) a is thus independent of the value of scaling parameter s and node a .

Original: Proof.
Revised: Proof.

Original: We expand the mean of heat kernel wavelet Ψ ( s ) a using spectral graph wavelet deﬁnition in
Revised: We expand the mean of heat kernel wavelet Ψ ( s ) a using spectral graph wavelet deﬁnition in

Original: Eq.
Revised: Eq.

Original: (1) and Property 2 (P2) from Appendix A:
Revised: (1) and Property 2 (P2) from Appendix A:

Original: [Equation]
Revised: [Equation]

Original: Proposition 2.
Revised: Proposition 2.

Original: The heat kernel wavelet coefﬁcient Ψ ( s ) aa at the initial source node a is a monotonically decreasing function of scaling parameter s .
Revised: The heat kernel wavelet coefﬁcient Ψ ( s ) aa at the initial source node a is a monotonically decreasing function of scaling parameter s .

Original: Its value is bounded by: 1 N ≤ Ψ ( s ) aa ≤ 1 .
Revised: Its value is bounded by: 1 N ≤ Ψ ( s ) aa ≤ 1 .

Original: Proof.
Revised: Proof.

Original: This follows directly using deﬁnition Ψ ( s ) aa
Revised: This follows directly using deﬁnition Ψ ( s ) aa

Original: = (cid:80) mj
Revised: = (cid:80) mj

Original: =1 e − λ j s U 2 ja and the scaling inequality in
Revised: =1 e − λ j s U 2 ja and the scaling inequality in

Original: Property 3 (P3).
Revised: Property 3 (P3).

Original: This way, we get:
Revised: This way, we get:

Original: [Equation]
Revised: [Equation]

Original: Additionally, for any m (cid:54) = a , the wavelet coefﬁcient Ψ ( s ) ma is non-negative and bounded.
Revised: Additionally, for any m (cid:54) = a , the wavelet coefﬁcient Ψ ( s ) ma is non-negative and bounded.

Original: Speciﬁcally, the wavelet coefﬁcient can be written as: Ψ ( s ) ma =[Equation] =
Revised: Speciﬁcally, the wavelet coefﬁcient can be written as: Ψ ( s ) ma =

Original: [Equation]
Revised: [Equation]

Original: ja
Revised: ja

Original: [Equation]

Original: =

Original: [Equation]

Original: N + (cid:80) Nj =

Original: [Equation]
Revised: [Equation]

Original: e − λ j s U ja U jm .

Original: It can be bounded by: 0 ≤ Ψ ( s ) ma ≤ max(Ψ ( s ) aa , Ψ ( s ) mm ) (Property 4).
Revised: It can be bounded by: 0 ≤ Ψ ( s ) ma ≤ max(Ψ ( s ) aa , Ψ ( s ) mm ) (Property 4 (P4)).

Original: Proposition 3.
Revised: Proposition 3.

Original: The variance of heat kernel wavelet Var [Ψ ( s ) a ] is a strictly decreasing function of scaling parameter s .
Revised: The variance of heat kernel wavelet Var [Ψ ( s ) a ] is a strictly decreasing function of scaling parameter s .

Original: Proof.
Revised: Proof.

Original: We use the deﬁnition of variance to get:
Revised: We use the deﬁnition of variance to get:

Original: [Equation]
Revised: [Equation]

Original: We rewrite the sum on the far right hand-side using the symmetry of wavelet matrix Ψ (Property 1):
Revised: We rewrite the sum on the far right hand-side using the symmetry of wavelet matrix Ψ (Property 1):

Original: N (cid:88)
Revised: N (cid:88)

Original: [Equation]
Revised: [Equation]

Original: concluding that variance Var [Ψ ( s ) a ] is decreasing, since it is sum of functions, all of which decrease as s gets larger.
Revised: concluding that variance Var [Ψ ( s ) a ] is decreasing, since it is sum of functions, all of which decrease as s gets larger.

Original: C S CALING PARAMETER OF HEAT KERNEL
Revised: C S CALING PARAMETER OF HEAT KERNEL

Original: We here develop a method that automatically ﬁnds an appropriate range of values for the scaling parameter s in heat kernel g s , which we use in the multiscale version of G RAPH W AVE (Section 2.2)
Revised: We here develop a method that automatically ﬁnds an appropriate range of values for the scaling parameter s in heat kernel g s , which we use in the multiscale version of G RAPH W AVE (Section 2.2)

Original: We ﬁnd the appropriate range of values for s by specifying an interval bounded by s min and s max through the analysis of variance in heat kernel wavelets.
Revised: We ﬁnd the appropriate range of values for s by specifying an interval bounded by s min and s max through the analysis of variance in heat kernel wavelets.

Original: Intuitively, whether or not a given value for s is appropriate for structural signature learning depends on the relationship between the scaling parameter and the temporal aspects of heat equation.
Revised: Intuitively, whether or not a given value for s is appropriate for structural signature learning depends on the relationship between the scaling parameter and the temporal aspects of heat equation.

Original: In particular, small values of s allow little time for the heat to propagate, yielding diffusion distributions ( i.e. , heat kernel wavelet distributions) that are trivial in the sense that only a few coefﬁcients have non-zero values and are thus unﬁt for comparison.
Revised: In particular, small values of s allow little time for the heat to propagate, yielding diffusion distributions ( i.e. , heat kernel wavelet distributions) that are trivial in the sense that only a few coefﬁcients have non-zero values and are thus unﬁt for comparison.

Original: For larger values of s the network converges to a state in which all nodes have an identical temperature equal to 1 /N
Revised: For larger values of s the network converges to a state in which all nodes have an identical temperature equal to 1 /N

Original: (Property 4), meaning that diffusion distributions are data-independent, hence non-informative.
Revised: (Property 4 (P4)), meaning that diffusion distributions are data-independent, hence non-informative.

Original: Next we prove Propositions 4 and 5 to provide new insights into the variance and convergence rate of heat kernel wavelets.
Revised: Next we prove Propositions 4 and 5 to provide new insights into the variance and convergence rate of heat kernel wavelets.

Original: We then use these results to select s min and s max .
Revised: We then use these results to select s min and s max .

Original: Proposition 4.
Revised: Proposition 4.

Original: Given the scaling parameter s , the variance of off-diagonal coefﬁcients in heat kernel wavelet Ψ a is proportional to:
Revised: Given the scaling parameter s , the variance of off-diagonal coefﬁcients in heat kernel wavelet Ψ a is proportional to:

Original: [Equation]
Revised: [Equation]

Original: where ∆ ( s ) a = | Ψ ( s ) aa − 1 N | is a monotonically decreasing function of s .
Revised: where ∆ ( s ) a = | Ψ ( s ) aa − 1 N | is a monotonically decreasing function of s .

Original: Proof.
Revised: Proof.

Original: Let us denote the mean of off-diagonal coefﬁcients in wavelet Ψ a by: ˜ µ
Revised: Let us denote the mean of off-diagonal coefﬁcients in wavelet Ψ a by: ˜ µ

Original: [Equation]
Revised: [Equation]

Original: s )
Revised: s )

Original: a =
Revised: a =

Original: [Equation]
Revised: [Equation]

Original: m
Revised: m

Original: [Equation]
Revised: [Equation]

Original: = a Ψ ( s ) ma / ( N − 1) .
Revised: = a Ψ ( s ) ma / ( N − 1) .

Original: We use the fact that (cid:80) m (cid:54) =
Revised: We use the fact that (cid:80) m (cid:54) =

Original: a Ψ ( s ) ma
Revised: a Ψ ( s ) ma

Original: = 1 − Ψ ( s ) aa , along with the deﬁnition of the variance, to obtain:
Revised: = 1 − Ψ ( s ) aa , along with the deﬁnition of the variance, to obtain:

Original: [Equation]
Revised: [Equation]

Original: Proposition 4 proves that the variance is a function of ∆ ( s ) a .
Revised: Proposition 4 proves that the variance is a function of ∆ ( s ) a .

Original: Therefore, to maximize the variance, we must analyze the behavior of ∆ ( s ) a .
Revised: Therefore, to maximize the variance, we must analyze the behavior of ∆ ( s ) a .

Original: To ensure sufﬁcient variability in the distribution of wavelet coefﬁcients, we need to select a range [ s min , s max ] that bounds the ∆ ( s ) a .
Revised: To ensure sufﬁcient variability in the distribution of wavelet coefﬁcients, we need to select a range [ s min , s max ] that bounds the ∆ ( s ) a .

Original: Our goal thus becomes establishing that ∆ ( s ) a is large enough that the diffusion has had time to spread, while remaining sufﬁciently small to ensure that the diffusion is far from its converged state.
Revised: Our goal thus becomes establishing that ∆ ( s ) a is large enough that the diffusion has had time to spread, while remaining sufﬁciently small to ensure that the diffusion is far from its converged state.

Original: Proposition 5.
Revised: Proposition 5.

Original: The convergence of heat kernel wavelet coefﬁcient Ψ ( s ) am is bounded by:
Revised: The convergence of heat kernel wavelet coefﬁcient Ψ ( s ) am is bounded by:

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: We use Property 5 from Appendix A and induction over s to complete this proof.
Revised: We use Property 5 from Appendix A and induction over s to complete this proof.

Original: For a given s ≥ 0 we analyze:
Revised: For a given s ≥ 0 we analyze:

Original: [Equation]
Revised: [Equation]

Original: and conclude that: e − λ N ≤ | Ψ
Revised: and conclude that: e − λ N ≤ | Ψ

Original: [Equation]
Revised: [Equation]

Original: aa
Revised: aa

Original: [Equation]
Revised: [Equation]

Original: Given any s ∈ N , we use the induction principle to get:
Revised: Given any s ∈ N , we use the induction principle to get:

Original: [Equation]
Revised: [Equation]

Original: which immediately yields the desired bound: e − λ N s ∆ (0) a ≤ ∆ ( s
Revised: which immediately yields the desired bound: e − λ N s ∆ (0) a ≤ ∆ ( s

Original: [Equation]
Revised: [Equation]

Original: ≤ e
Revised: ≤ e

Original: [Equation]
Revised: [Equation]

Original: 2 s ∆ (0) a .
Revised: 2 s ∆ (0) a .

Original: Since ∆ ( s ) a is smooth increasing function of s , we can take the ﬂoor/ceiling of any non-integer s ≥ 0 and this proposition must hold.
Revised: Since ∆ ( s ) a is smooth increasing function of s , we can take the ﬂoor/ceiling of any non-integer s ≥ 0 and this proposition must hold.

Original: C. S ELECTION OF s max
Revised: C. S ELECTION OF s max

Original: We select s max such that wavelet coefﬁcients are localized in the network.
Revised: We select s max such that wavelet coefﬁcients are localized in the network.

Original: To do so, we use
Revised: To do so, we use

Original: Proposition 5 and bound ∆ ( s ) a by the graph Laplacian’s eigenvalues.
Revised: Proposition 5 and bound ∆ ( s ) a by the graph Laplacian’s eigenvalues.

Original: When the bulk of the eigenvalues leans towards λ N , ∆ ( s ) a is closer to e − λ N ( i.e. , lower bound in Proposition 5).
Revised: When the bulk of the eigenvalues leans towards λ N , ∆ ( s ) a is closer to e − λ N ( i.e. , lower bound in Proposition 5).

Original: When the bulk of the eigenvalues is closer to λ 2 , ∆ ( s ) a will lean towards e − λ 2 ( i.e. , upper bound in Proposition 5).
Revised: When the bulk of the eigenvalues is closer to λ 2 , ∆ ( s ) a will lean towards e − λ 2 ( i.e. , upper bound in Proposition 5).

Original: In each case, the diffusion is localized if ∆ ( s ) a is above a given threshold η < 1 .
Revised: In each case, the diffusion is localized if ∆ ( s ) a is above a given threshold η < 1 .

Original: Indeed, this ensures that
Revised: Indeed, this ensures that

Original: ∆ ( s ) a has shrunk to at most η ∗ 100 % of its initial value at s = 0 , and yields a bound of the form: ∆ ( s ) a / ∆ (0) a ≥ η .
Revised: ∆ ( s ) a has shrunk to at most η ∗ 100 % of its initial value at s = 0 , and yields a bound of the form: ∆ ( s ) a / ∆ (0) a ≥ η .

Original: The bound implies that: e − λs ≥ η , or s ≤ − log( η ) /λ.
Revised: The bound implies that: e − λs ≥ η , or s ≤ − log( η ) /λ.

Original: To ﬁnd a middle ground between the two convergence scenarios, we take λ to be the geometric mean of λ 2 and λ N .
Revised: To ﬁnd a middle ground between the two convergence scenarios, we take λ to be the geometric mean of λ 2 and λ N .

Original: Indeed, as opposed to the arithmetic mean, the geometric mean maintains an equal weighting across the range [ λ 2 , λ N ], and a change of (cid:15) % in λ 2 has the same effect as a change in (cid:15) % of λ N .
Revised: Indeed, as opposed to the arithmetic mean, the geometric mean maintains an equal weighting across the range [ λ 2 , λ N ], and a change of (cid:15) % in λ 2 has the same effect as a change in (cid:15) % of λ N .

Original: We thus select s max as: s max = − log( η ) / √ λ 2 λ N .
Revised: We thus select s max as: s max = − log( η ) / √ λ 2 λ N .

Original: C. S ELECTION OF s min
Revised: C. S ELECTION OF s min

Original: We select s min to ensure the adequate diffusion resolution.
Revised: We select s min to ensure the adequate diffusion resolution.

Original: In particular, we select a minimum value s min such that each wavelet has sufﬁcient time to spread.
Revised: In particular, we select a minimum value s min such that each wavelet has sufﬁcient time to spread.

Original: That is, ∆ ( s ) a / ∆ (0) a ≤ γ .
Revised: That is, ∆ ( s ) a / ∆ (0) a ≤ γ .

Original: As in the case of s max above, we obtain a bound of s ≥ − log( γ ) /λ .
Revised: As in the case of s max above, we obtain a bound of s ≥ − log( γ ) /λ .

Original: Hence, we set s min to: s min = − log( γ ) / √ λ 2 λ N .
Revised: Hence, we set s min to: s min = − log( γ ) / √ λ 2 λ N .

Original: To cover an appropriate range of scales, we suggest setting η = 0 .
Revised: To cover an appropriate range of scales, we suggest setting η = 0 .

Original: 90 and γ = 0 .
Revised: 90 and γ = 0 .

Original: D V ISUALIZATION OF CHARACTERISTIC FUNCTIONS
Revised: D V ISUALIZATION OF CHARACTERISTIC FUNCTIONS

Original: Here, we study the properties of characteristic functions in G RAPH W AVE .
Revised: Here, we study the properties of characteristic functions in G RAPH W AVE .

Original: Our goal is to provide intuition to understand the behavior of these functions and how their resulting 2D parametric curves reﬂect nodes’ local topology (see Figure 3C).
Revised: Our goal is to provide intuition to understand the behavior of these functions and how their resulting 2D parametric curves reﬂect nodes’ local topology (see Figure 3C).

Original: We begin by reviewing the deﬁnition of the characteristic function.
Revised: We begin by reviewing the deﬁnition of the characteristic function.

Original: Deﬁnition 1.
Revised: Deﬁnition 1.

Original: The empirical characteristic function (Lukacs, 1970) of wavelet Ψ ( s ) a is function φ ( s ) a deﬁned as:
Revised: The empirical characteristic function (Lukacs, 1970) of wavelet Ψ ( s ) a is function φ ( s ) a deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: In the phase plot, a given value of t thus yields the following set of coordinates:
Revised: In the phase plot, a given value of t thus yields the following set of coordinates:

Original: [Equation]
Revised: [Equation]

Original: By varying t , we get a characteric curve in this 2D plane.
Revised: By varying t , we get a characteric curve in this 2D plane.

Original: Here, we note several properties of this curve:
Revised: Here, we note several properties of this curve:

Original: Value at t = 0 : ∀ s, φ ( s ) a (0) =
Revised: Value at t = 0 : ∀ s, φ ( s ) a (0) =

Original: (cid:18) 10 (cid:19) , independent of the scale or initial source node.
Revised: (cid:18) 10 (cid:19) , independent of the scale or initial source node.

Original: Behavior for s = 0 :
Revised: Behavior for s = 0 :

Original: Only the coefﬁcient corresponding to the initial source node has non-zero value ( i.e. , Ψ (0) aa
Revised: Only the coefﬁcient corresponding to the initial source node has non-zero value ( i.e. , Ψ (0) aa

Original: = 1 , and ∀ m (cid:54)
Revised: = 1 , and ∀ m (cid:54)

Original: = a, Ψ (0) ma = 0 ).
Revised: = a, Ψ (0) ma = 0 ).

Original: Hence, ∀ t, φ (0) a ( t ) =
Revised: Hence, ∀ t, φ (0) a ( t ) =

Original: 1 N e it + N − 1 N .
Revised: 1 N e it + N − 1 N .

Original: The phase plot of the characteristic curve is a circle of radius 1 N with period 2 π centered at N −N .
Revised: The phase plot of the characteristic curve is a circle of radius 1 N with period 2 π centered at N −N .

Original: Behavior as s → ∞ : From Property 4 in Appendix A, the coefﬁcients all converge to the same limit: ∀ m, a, lim s →∞ Ψ ( s ) ma
Revised: Behavior as s → ∞ : From Property 4 in Appendix A, the coefﬁcients all converge to the same limit: ∀ m, a, lim s →∞ Ψ ( s ) ma

Original: = 1 N .
Revised: = 1 N .

Original: This in turn implies that: ∀ t, lim s →∞ φ ( s ) a ( t ) =
Revised: This in turn implies that: ∀ t, lim s →∞ φ ( s ) a ( t ) =

Original: e i tN .
Revised: e i tN .

Original: Hence, the curve converges to a circle centered at 0, with radius 1 and period 2 πN .
Revised: Hence, the curve converges to a circle centered at 0, with radius 1 and period 2 πN .

Original: Gradient of the curve: ∇ t φ ( s ) a ( t ) = iN (cid:80) Nm =1 Ψ ( s ) am e it Ψ
Revised: Gradient of the curve: ∇ t φ ( s ) a ( t ) = iN (cid:80) Nm =1 Ψ ( s ) am e it Ψ

Original: ( s ) am , or equivalently, in the phase plot:
Revised: ( s ) am , or equivalently, in the phase plot:

Original: [Equation]
Revised: [Equation]

Original: Since all the wavelet coefﬁcients are non-negative, the curve is thus directed counterclockwise.
Revised: Since all the wavelet coefﬁcients are non-negative, the curve is thus directed counterclockwise.