Original: H IGHER - ORDER S TRUCTURE P REDICTION IN E VOLVING G RAPH S IMPLICIAL C OMPLEXES
Revised: H IGHER - ORDER S TRUCTURE P REDICTION IN E VOLVING G RAPH S IMPLICIAL C OMPLEXES

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: Dynamic graphs are rife with higher-order interactions, such as co-authorship relationships and protein-protein interactions in biological networks, that naturally arise between more than two nodes at once.
Revised: Dynamic graphs are rife with higher-order interactions, such as co-authorship relationships and protein-protein interactions in biological networks, that naturally arise between more than two nodes at once.

Original: In spite of the ubiquitous presence of such higher-order interactions, limited attention has been paid to the higher-order counterpart of the popular pairwise link prediction problem.
Revised: In spite of the ubiquitous presence of such higher-order interactions, limited attention has been paid to the higher-order counterpart of the popular pairwise link prediction problem.

Original: Existing higher-order structure prediction methods are mostly based on heuristic feature extraction procedures, which work well in practice but lack theoretical guarantees.
Revised: Existing higher-order structure prediction methods are mostly based on heuristic feature extraction procedures, which work well in practice but lack theoretical guarantees.

Original: Such heuristics are primarily focused on predicting links in a static snapshot of the graph.
Revised: Such heuristics are primarily focused on predicting links in a static snapshot of the graph.

Original: Moreover, these heuristic-based methods fail to effectively utilize and beneﬁt from the knowledge of latent substructures already present within the higher-order structures.
Revised: Moreover, these heuristic-based methods fail to effectively utilize and beneﬁt from the knowledge of latent substructures already present within the higher-order structures.

Original: In this paper, we overcome these obstacles by capturing higher-order interactions succinctly as simplices , model their neighborhood by face-vectors, and develop a nonparametric kernel estimator for simplices that views the evolving graph from the perspective of a time process (i.e., a sequence of graph snapshots).
Revised: In this paper, we overcome these obstacles by capturing higher-order interactions succinctly as simplices , model their neighborhood by face-vectors, and develop a nonparametric kernel estimator for simplices that views the evolving graph from the perspective of a time process (i.e., a sequence of graph snapshots).

Original: Our method substantially outperforms several baseline higherorder prediction methods.
Revised: Our method substantially outperforms several baseline higherorder prediction methods.

Original: As a theoretical achievement, we prove the consistency and asymptotic normality in terms of Wasserstein distance of our estimator using
Revised: As a theoretical achievement, we prove the consistency and asymptotic normality in terms of Wasserstein distance of our estimator using

Original: Stein’s method.
Revised: Stein’s method.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Numerous types of networks like social (Liben-Nowell & Kleinberg, 2007a), biological (Airoldi et al., 2006), and chemical reaction networks (Wegscheider, 1911) are highly dynamic, as they evolve and grow rapidly via the appearance of new interactions, represented as the introduction of new links / edges between the nodes of a network.
Revised: Numerous types of networks like social (Liben-Nowell & Kleinberg, 2007a), biological (Airoldi et al., 2006), and chemical reaction networks (Wegscheider, 1911) are highly dynamic, as they evolve and grow rapidly via the appearance of new interactions, represented as the introduction of new links / edges between the nodes of a network.

Original: Identifying the underlying mechanisms by which such networks evolve over time is a fundamental question that is not yet fully understood.
Revised: Identifying the underlying mechanisms by which such networks evolve over time is a fundamental question that is not yet fully understood.

Original: Typically, insight into the temporal evolution of networks has been obtained via a classical inferential problem called link prediction , where given a snapshot of the network at time t along with its linkage pattern, the task is to assess whether a pair of nodes will be linked at a later time t (cid:48) > t .
Revised: Typically, insight into the temporal evolution of networks has been obtained via a classical inferential problem called link prediction , where given a snapshot of the network at time t along with its linkage pattern, the task is to assess whether a pair of nodes will be linked at a later time t (cid:48) > t .

Original: While inferring pairwise links is an important problem, it is oftentimes observed that most of the real-world graphs exhibit higher-order group-wise interactions that involve more than two nodes at once.
Revised: While inferring pairwise links is an important problem, it is oftentimes observed that most of the real-world graphs exhibit higher-order group-wise interactions that involve more than two nodes at once.

Original: Examples illustrating human group behavior involve a co-author relationship on a single paper and a network of e-mails to multiple recipients.
Revised: Examples illustrating human group behavior involve a co-author relationship on a single paper and a network of e-mails to multiple recipients.

Original: In nature too, one can observe several proteins interacting together in a biological network simultaneously.
Revised: In nature too, one can observe several proteins interacting together in a biological network simultaneously.

Original: In spite of their signiﬁcance, in comparison to single edge inference, relatively fewer works have studied the problem of predicting higher-order group-wise interactions.
Revised: In spite of their signiﬁcance, in comparison to single edge inference, relatively fewer works have studied the problem of predicting higher-order group-wise interactions.

Original: Benson et al.
Revised: Benson et al.

Original: (2018) originally introduced a simplex to model group-wise interactions between nodes in a graph.
Revised: (2018) originally introduced a simplex to model group-wise interactions between nodes in a graph.

Original: They proposed predicting a simplicial closure event , whereby an open simplex (with just pairwise interactions between member vertices) transitions to a closed simplex (where all member vertices participate in the higher-order relationship simultaneously), in the near future.
Revised: They proposed predicting a simplicial closure event , whereby an open simplex (with just pairwise interactions between member vertices) transitions to a closed simplex (where all member vertices participate in the higher-order relationship simultaneously), in the near future.

Original: Figure 1 (Middle) shows an example of such a transition from an open triangle to a closed one.
Revised: Figure 1 (Middle) shows an example of such a transition from an open triangle to a closed one.

Original: Recently, several works have proposed modeling higher-order interactions as hyperedges in a hypergraph (Xu et al., 2013; Zhang et al., 2018; Yoon et al., 2020; Patil et al., 2020).
Revised: Recently, several works have proposed modeling higher-order interactions as hyperedges in a hypergraph (Xu et al., 2013; Zhang et al., 2018; Yoon et al., 2020; Patil et al., 2020).

Original: Given a hyperedge h t at time t , the inference task is to predict the future arrival of a new hyperedge h t (cid:48) , which covers a larger set of vertices than h t and contains all the vertices in h t .
Revised: Given a hyperedge h t at time t , the inference task is to predict the future arrival of a new hyperedge h t (cid:48) , which covers a larger set of vertices than h t and contains all the vertices in h t .

Original: Figure 1 (Right) illustrates this hyperedge prediction task.
Revised: Figure 1 (Right) illustrates this hyperedge prediction task.

Original: Although prediction models based on either simplicial closure event prediction or hyperedge arrival, deal with higher-order structures, they both fail to capture the highly complex and non-linear evolution of higher-order structures over time.
Revised: Although prediction models based on either simplicial closure event prediction or hyperedge arrival, deal with higher-order structures, they both fail to capture the highly complex and non-linear evolution of higher-order structures over time.

Original: Both these kinds of models have two major limitations.
Revised: Both these kinds of models have two major limitations.

Original: First, they predict structures from a single static snapshot of the graph, thus not viewing the evolution process of adding new edges as a time process.
Revised: First, they predict structures from a single static snapshot of the graph, thus not viewing the evolution process of adding new edges as a time process.

Original: Second, their feature extraction is mostly based on popular heuristics (Adamic & Adar, 2003; Brin & Page, 2012; Jeh & Widom, 2002; Zhou et al., 2009a; Barab ´ asi & Albert, 1999; Bhatia et al., 2019) that work well in practice but are not accompanied by strong theoretical guarantees.
Revised: Second, their feature extraction is mostly based on popular heuristics (Adamic & Adar, 2003; Brin & Page, 2012; Jeh & Widom, 2002; Zhou et al., 2009a; Barab ´ asi & Albert, 1999; Bhatia et al., 2019) that work well in practice but are not accompanied by strong theoretical guarantees.

Original: In addition to the aforementioned shortcomings, hypergraph based methods model higher-order structures as hyperedges, which omit lower-dimensional substructures present within a single hyperedge.
Revised: In addition to the aforementioned shortcomings, hypergraph based methods model higher-order structures as hyperedges, which omit lower-dimensional substructures present within a single hyperedge.

Original: As a consequence, they cannot distinguish between various substructure relationships.
Revised: As a consequence, they cannot distinguish between various substructure relationships.

Original: For example, hyperedge [ A, B, C ] in Figure 1 (Right) cannot distinguish between group relationships like [[ A, B ] , [ B, C ] , [ A, C ]] (a set of pairwise interactions) versus [ A, B, C ] (all A , B and C simultaneously in a relationship).
Revised: For example, hyperedge [ A, B, C ] in Figure 1 (Right) cannot distinguish between group relationships like [[ A, B ] , [ B, C ] , [ A, C ]] (a set of pairwise interactions) versus [ A, B, C ] (all A , B and C simultaneously in a relationship).

Original: This problem is remedied by the use of simplices because they naturally model these substructures as a collection of subsets (i.e., faces) of the simplex.
Revised: This problem is remedied by the use of simplices because they naturally model these substructures as a collection of subsets (i.e., faces) of the simplex.

Revised: We provide real-world examples of where our simplicial complex based approach can play a significant role.

Revised: (i) Organic Chemistry:

Revised: It is quite common to have the same set of elements interacting with each other in different conﬁgurations, which result in very different functioning compounds (Ma et al., 2011).

Original: To combat these challenges, our approach views the evolving graph 1 as a time process under the framework of nonparametric time series prediction, which models the evolution of higher-order structures (as simplices) and their local neighborhoods (spatial dimension) over a moving time window (temporal dimension).
Revised: Speciﬁcally, R-thalidomide and S-thalidomide are two different conﬁgurations of thalidomide, where the R-form was meant to help sedate pregnant women, while the S-form unfortunately resulted in birth defects. This is a famous example in stereo chemistry to show the consequences of mistaking two extremely close conﬁgurations (differing by a single bond) as being the same. Structure prediction to avoid such phenomenon in drug synthesis allows chemists to achieve a much higher yield and avoid wastage of expensive resources. (ii) Gene expression networks : Gene networks have nodes that represent genes and edges connect genes which have similar expression patterns (Zhang & Horvath, 2005).

Revised: Subgraphs called modules are tightly connected genes in such a gene expression network.

Revised: Genomics research provides evidence that higher-order gene expression relationships (like second and third-order) and their measurements can have very important implications for cancer prognosis.

Revised: When making structural predictions in these aforementioned examples, our simplicial complex based approach provides much more ﬁne-grained control over competing methods by capturing subtler differences in conﬁgurations.

Revised: To combat these challenges, our approach views the evolving graph 1 as a time process under the framework of nonparametric time series prediction, which models the evolution of higher-order structures (as simplices) and their local neighborhoods (spatial dimension) over a moving time window (temporal dimension).

Original: Our inference problem is then modeled as predicting the evolution to a higher-dimensional simplex at time t (cid:48) > t , given a simplex at time t .
Revised: Our inference problem is then modeled as predicting the evolution to a higher-dimensional simplex at time t (cid:48) > t , given a simplex at time t .

Original: It is important to note that this task is more general and greatly diverges from the task proposed by Benson et al.
Revised: It is important to note that this task is more general and greatly diverges from the task proposed by Benson et al.

Original: Our task requires just a single simplex σ in order to predict a higher-dimensional simplex τ , whose face / subset is σ , whereas Benson et al.
Revised: Our task requires just a single simplex σ in order to predict a higher-dimensional simplex τ , whose face / subset is σ , whereas Benson et al.

Original: (2018) requires the presence of all constituent σ faces in order to predict τ .
Revised: (2018) requires the presence of all constituent σ faces in order to predict τ .

Original: For example, in Benson et al.
Revised: For example, in Benson et al.

Original: (2018) (also shown in Figure 1 (Middle))
Revised: (2018) (also shown in Figure 1 (Middle))

Original: all faces [ A, B ] , [ B, C ] and [ A, C ] (open triangle) need to be present in order to predict a closed triangle [ A, B, C ] .
Revised: all faces [ A, B ] , [ B, C ] and [ A, C ] (open triangle) need to be present in order to predict a closed triangle [ A, B, C ] .

Original: Contrastingly, in our approach, just a single face/edge like [ A, B ] or [ B, C ] or [ A, C ] , sufﬁces to predict its evolution to [ A, B, C ] .
Revised: Contrastingly, in our approach, just a single face/edge like [ A, B ] or [ B, C ] or [ A, C ] , sufﬁces to predict its evolution to [ A, B, C ] .

Original: Figure 1 (Left) illustrates an additional example of our proposal to predict a 3 -simplex [ A, B, C, D ] given only one of its faces [ A, B, C ] .
Revised: Figure 1 (Left) illustrates an additional example of our proposal to predict a 3 -simplex [ A, B, C, D ] given only one of its faces [ A, B, C ] .

Original: To this effect, we succinctly capture the features characterizing the local neighborhood of a simplex as a combination of a face-vector (Bj¨orner & Kalai, 2006), which is a well-established vector signature in combinatorial topology literature and a novel scoring function which infers the afﬁnity of sub-simplices based on the strength of their past interactions.
Revised: To this effect, we succinctly capture the features characterizing the local neighborhood of a simplex as a combination of a face-vector (Bj¨orner & Kalai, 2006), which is a well-established vector signature in combinatorial topology literature and a novel scoring function which infers the afﬁnity of sub-simplices based on the strength of their past interactions.

Original: Based on these features, we design a kernel estimator to infer future evolution to higher-dimensional simplices and prove both the consistency and asymptotic normality of our estimator.
Revised: Based on these features, we design a kernel estimator to infer future evolution to higher-dimensional simplices and prove both the consistency and asymptotic normality of our estimator.

Original: Our contributions: (a) We propose a kernel estimator that predicts higher-order interactions in an evolving network.
Revised: Our contributions: (a) We propose a kernel estimator that predicts higher-order interactions in an evolving network.

Original: (b) We prove the consistency and asymptotic normality of our kernel estimator.
Revised: (b) We prove the consistency and asymptotic normality of our kernel estimator.

Original: (c) We evaluate our method on real-world dynamic networks by proposing higher-order link prediction baselines and observe signiﬁcant gains in prediction accuracy in comparison to the baselines.
Revised: (c) We evaluate our method on real-world dynamic networks by proposing higher-order link prediction baselines and observe signiﬁcant gains in prediction accuracy in comparison to the baselines.

Original: 1. R ELATED S TUDIES
Revised: 1. R ELATED S TUDIES

Original: Single link prediction: Most literature that predicts a single edge/link can be broadly classiﬁed as based on: (i) heuristics, (ii) random-walks, or (iii) graph neural networks (GNNs).
Revised: Single link prediction: Most literature that predicts a single edge/link can be broadly classiﬁed as based on: (i) heuristics, (ii) random-walks, or (iii) graph neural networks (GNNs).

Original: (i)
Revised: (i)

Original: Heuristic methods comprise of Common neighbors, Adamic-adar (Adamic & Adar, 2003), PageRank (Brin &
Revised: Heuristic methods comprise of Common neighbors, Adamic-adar (Adamic & Adar, 2003), PageRank (Brin &

Original: Page, 2012), SimRank (Jeh & Widom, 2002), resource allocation (Zhou et al., 2009a), preferential attachment (Barab´asi & Albert, 1999), persistence homology based ranking (Bhatia et al., 2019), and similarity-based methods (Liben-Nowell & Kleinberg, 2007b; L¨u & Zhou, 2011).
Revised: Page, 2012), SimRank (Jeh & Widom, 2002), resource allocation (Zhou et al., 2009a), preferential attachment (Barab ´ asi & Albert, 1999), persistence homology based ranking (Bhatia et al., 2019), and similarity-based methods (Liben-Nowell & Kleinberg, 2007b; L ¨ u & Zhou, 2011).

Original: (ii) Random walk based methods consist of DeepWalk (Perozzi et al., 2014) , Node2Vec (Grover & Leskovec, 2016b) and SpectralWalk (Sharma et al., 2020).
Revised: (ii) Random walk based methods consist of DeepWalk (Perozzi et al., 2014) , Node2Vec (Grover & Leskovec, 2016b) and SpectralWalk (Sharma et al., 2020).

Original: (iii) Finally, for both link prediction and node classiﬁcation tasks, recent works are mainly GNN-based methods such as VGAE (Kipf & Welling, 2016), WYS (Abu-El-Haija et al., 2018), and SEAL (Zhang & Chen, 2018b).
Revised: (iii) Finally, for both link prediction and node classiﬁcation tasks, recent works are mainly GNN-based methods such as VGAE (Kipf & Welling, 2016), WYS (Abu-El-Haija et al., 2018), and SEAL (Zhang & Chen, 2018b).

Original: Higher-order link prediction: Benson et al.
Revised: Higher-order link prediction: Benson et al.

Original: (2018) are the ﬁrst to introduce a higher-order link prediction problem where they study the likelihoods of future higher-order group interactions as simplicial closure events (explained earlier).
Revised: (2018) are the ﬁrst to introduce a higher-order link prediction problem where they study the likelihoods of future higher-order group interactions as simplicial closure events (explained earlier).

Original: Furthermore, there are studies using hypergraphs which also help naturally represent group relations (Xu et al., 2013; Zhang et al., 2018; Yoon et al., 2020;
Revised: Furthermore, there are studies using hypergraphs which also help naturally represent group relations (Xu et al., 2013; Zhang et al., 2018; Yoon et al., 2020;

Original: Patil et al., 2020).
Revised: Patil et al., 2020).

Original: Especially, to represent higher-order relationships, Yoon et al.
Revised: Especially, to represent higher-order relationships, Yoon et al.

Original: (2020) proposed n -projected graphs.
Revised: (2020) proposed n -projected graphs.

Original: For larger n , i.e., higher-order groups, the enumeration of subsets, and keeping track of node co-occurrences quickly becomes infeasible.
Revised: For larger n , i.e., higher-order groups, the enumeration of subsets, and keeping track of node co-occurrences quickly becomes infeasible.

Original: In comparison to a hypergraph, our graph simplicial complex is closed under taking subsets, which enables us to better encode more information for improved inference.
Revised: In comparison to a hypergraph, our graph simplicial complex is closed under taking subsets, which enables us to better encode more information for improved inference.

Original:  P RELIMINARY : G RAPH S IMPLICIAL C OMPLEX (GSC)
Revised:  P RELIMINARY : G RAPH S IMPLICIAL C OMPLEX (GSC)

Original: + [ 10, 7]
Revised: + [ 10, 7]

Original: We start with a general notion of an abstract simplicial complex (ASC), then deﬁne a simplex using ASCs.
Revised: We start with a general notion of an abstract simplicial complex (ASC), then deﬁne a simplex using ASCs.

Original: We specialize this deﬁnition to graphs and deﬁne a graph simplicial complex (GSC).
Revised: We specialize this deﬁnition to graphs and deﬁne a graph simplicial complex (GSC).

Original: Deﬁnition 1 (Abstract simplicial complex and simplex) .
Revised: Deﬁnition 1 (Abstract simplicial complex and simplex) .

Original: An abstract simplicial complex (ASC) is a collection
Revised: An abstract simplicial complex (ASC) is a collection

Original: A of ﬁnite non-empty sets, such that if σ is an element of A , then so is every nonempty subset of σ .
Revised: A of ﬁnite non-empty sets, such that if σ is an element of A , then so is every nonempty subset of σ .

Original: The element σ of A is called a simplex of A ; its dimension is one less than the number of its elements.
Revised: The element σ of A is called a simplex of A ; its dimension is one less than the number of its elements.

Original: Now, we analyze graphs using the deﬁnition of
Revised: Now, we analyze graphs using the deﬁnition of

Original: ASCs.
Revised: ASCs.

Original: Let G =
Revised: Let G =

Original: ( V, E ) be a ﬁnite graph with vertex set V and edge set E .
Revised: ( V, E ) be a ﬁnite graph with vertex set V and edge set E .

Original: A graph simpicial complex (GSC) G on G is an ASC consisting of subsets of V .
Revised: A graph simpicial complex (GSC) G on G is an ASC consisting of subsets of V .

Original: In particular, G is a collection of subgraphs of G .
Revised: In particular, G is a collection of subgraphs of G .

Original: With graphs, we denote a d -dimensional simplex (or d -simplex) of a GSC by σ ( d ) =
Revised: With graphs, we denote a d -dimensional simplex (or d -simplex) of a GSC by σ ( d ) =

Original: [ v 0 , v 1 , . .
Revised: [ v 0 , v 1 , . .

Original: . , v d ] .
Revised: . , v d ] .

Original: Each non-empty subset of σ ( d ) is called a face of σ ( d ) .
Revised: Each non-empty subset of σ ( d ) is called a face of σ ( d ) .

Original: We deﬁne several notions related to GSCs, that are useful for describing the evolution of graphs.
Revised: We deﬁne several notions related to GSCs, that are useful for describing the evolution of graphs.

Original: Deﬁnition 2 (Filtered GSC) .
Revised: Deﬁnition 2 (Filtered GSC) .

Original: For I ⊂ N , a ﬁltered GSC indexed over I is a family ( G t )
Revised: For I ⊂ N , a ﬁltered GSC indexed over I is a family ( G t )

Original: t ∈ I of GSCs such that for every t ≤ t (cid:48) in I , G t ⊂ G t (cid:48) holds.
Revised: t ∈ I of GSCs such that for every t ≤ t (cid:48) in I , G t ⊂ G t (cid:48) holds.

Original: Obviously, G t 0 ⊂ G t 1 ⊂ . . .
Revised: Obviously, G t 0 ⊂ G t 1 ⊂ . . .

Original: G t n is a discrete ﬁltration induced by the arrival times of the simplices:
Revised: G t n is a discrete ﬁltration induced by the arrival times of the simplices:

Original: G t i \ G t i − 1 =
Revised: G t i \ G t i − 1 =

Original: σ t i .
Revised: σ t i .

Original: This depicts a higher-order analogue of an evolving graph (incremental model), which allows attaching new simplices at each time-step to an existing GSC to build a new GSC.
Revised: This depicts a higher-order analogue of an evolving graph (incremental model), which allows attaching new simplices at each time-step to an existing GSC to build a new GSC.

Original: A ﬁltered GSC G t,p for the last p discrete time steps is deﬁned as G t,p :=
Revised: A ﬁltered GSC G t,p for the last p discrete time steps is deﬁned as G t,p :=

Original: ( G t (cid:48) ) tt (cid:48)
Revised: ( G t (cid:48) ) tt (cid:48)

Original: =
Revised: =

Original: t − p =
Revised: t − p =

Original: ( G t − p , . .
Revised: ( G t − p , . .

Original: . , G t ) , where G t (cid:48)
Revised: . , G t ) , where G t (cid:48)

Original: ⊃ G t (cid:48) − 1 .
Revised: ⊃ G t (cid:48) − 1 .

Original: We deﬁne a notion for dealing with the neighborhood around a given simplex σ ( d ) ∈ G .
Revised: We deﬁne a notion for dealing with the neighborhood around a given simplex σ ( d ) ∈ G .

Original: We introduce a set of all simplices of dimension d (cid:48) or less from G , i.e., G ( d (cid:48) )
Revised: We introduce a set of all simplices of dimension d (cid:48) or less from G , i.e., G ( d (cid:48) )

Original: − :=
Revised: − :=

Original: { σ ( d ) ∈ G | d ≤ d (cid:48) } .
Revised: { σ ( d ) ∈ G | d ≤ d (cid:48) } .

Original: G (0) − is a vertex set and G (1) − is the set of edges and vertices.
Revised: G (0) − is a vertex set and G (1) − is the set of edges and vertices.

Original: We also write i ∼ j whenever vertices i and j are adjacent in G (1) − , and write i ∼ k j to indicate that vertex j is k -reachable from i , i.e., there exists a path of length at most k , connecting i and j in G (1) − .
Revised: We also write i ∼ j whenever vertices i and j are adjacent in G (1) − , and write i ∼ k j to indicate that vertex j is k -reachable from i , i.e., there exists a path of length at most k , connecting i and j in G (1) − .

Original: Then, we deﬁne a ball around σ ( d ) .
Revised: Then, we deﬁne a ball around σ ( d ) .

Original: Deﬁnition 3 ( k -ball centered at vertex and simplex) .
Revised: Deﬁnition 3 ( k -ball centered at vertex and simplex) .

Original: At time t , we deﬁne a k - ball centered at vertex i by B t,k
Revised: At time t , we deﬁne a k - ball centered at vertex i by B t,k

Original: ( i ) :
Revised: ( i ) :

Original: = { j : i ∼ k j and i, j ∈ G (0) t − } .
Revised: = { j : i ∼ k j and i, j ∈ G (0) t − } .

Original: We also deﬁne a k - ball centered at a simplex σ ( d ) as
Revised: We also deﬁne a k - ball centered at a simplex σ ( d ) as

Original: B t,k ( σ ( d ) ) :
Revised: B t,k ( σ ( d ) ) :

Original: = (cid:83) i :
Revised: = (cid:83) i :

Original: Vert( σ ( d ) )
Revised: Vert( σ ( d ) )

Original: B t,k ( i ) , where Vert( σ ( d ) ) denotes all vertexes in σ ( d ) .
Revised: B t,k ( i ) , where Vert( σ ( d ) ) denotes all vertexes in σ ( d ) .

Original: Now, we deﬁne a sub-complex G (cid:48) t ( σ ( d ) ) ⊆
Revised: Now, we deﬁne a sub-complex G (cid:48) t ( σ ( d ) ) ⊆

Original: G t as the GSC that contains all the simplices in G t spanned by the vertices in the k -ball B t,k
Revised: G t as the GSC that contains all the simplices in G t spanned by the vertices in the k -ball B t,k

Original: ( σ ( d ) ) .
Revised: ( σ ( d ) ) .

Original:  P REDICTING THE A RRIVAL OF H IGHER - ORDER S IMPLICES
Revised:  P REDICTING THE A RRIVAL OF H IGHER - ORDER S IMPLICES

Original: We consider the prediction of a simplex’s arrival in the setting described below.
Revised: We consider the prediction of a simplex’s arrival in the setting described below.

Original: Consider a ﬁltered
Revised: Consider a ﬁltered

Original: GSC G t,p .
Revised: GSC G t,p .

Original: At time t , given a d -dimensional simplex σ ( d ) =
Revised: At time t , given a d -dimensional simplex σ ( d ) =

Original: [ v 0 , · · · , v d ] , we predict the formation of a ( d + 1) -simplex τ
Revised: [ v 0 , · · · , v d ] , we predict the formation of a ( d + 1) -simplex τ

Original: ( d +1) =
Revised: ( d +1) =

Original: [ v 0 , · · · , v d , (cid:101) v ] with a new vertex (cid:101) v ∈ B t,k ( σ ( d ) ) .
Revised: [ v 0 , · · · , v d , (cid:101) v ] with a new vertex (cid:101) v ∈ B t,k ( σ ( d ) ) .

Original: Here, we restrict (cid:101) v to be k -reachable from σ ( d ) .
Revised: Here, we restrict (cid:101) v to be k -reachable from σ ( d ) .

Original: In order to ﬁnd out which simplices and vertices are most likely to appear in τ ( d +1) at time t + 1 , we need to 
Revised: In order to ﬁnd out which simplices and vertices are most likely to appear in τ ( d +1) at time t + 1 , we need to design features for σ ( d ) and (cid:101) v .

Original: 3. F EATURE D ESIGN FOR S IMPLEX
Revised: 3. F EATURE D ESIGN FOR S IMPLEX

Original: We develop a feature design of a simplex associated with the notion of k -balls.
Revised: We develop a feature design of a simplex associated with the notion of k -balls.

Original: The design is organized into two main elements: (i) a face-vector with a sub-complex, and (ii) a scoring function.
Revised: The design is organized into two main elements: (i) a face-vector with a sub-complex, and (ii) a scoring function.

Original: (i) Face vector of sub-complex : We ﬁrst deﬁne a face-vector of a ﬁxed GSC.
Revised: (i) Face vector of sub-complex : We ﬁrst deﬁne a face-vector of a ﬁxed GSC.

Original: The face-vector is an important topological invariant 2 of the GSC.
Revised: The face-vector is an important topological invariant 2 of the GSC.

Original: Deﬁnition 4 (face-vector) .
Revised: Deﬁnition 4 (face-vector) .

Original: Abasic combinatorial statistic ofGSC G is the face-vector (or f -vector) 
Revised: A combinatorial statistic of G is the face-vector (or f -vector) as f ( G ) =

Original: [Equation]
Revised: ( f − 1 , f 0 , . . .

Original:  where f k = f k ( G ) records the number of k -dimensional faces σ ( k ) ∈ G .
Revised: , f d − 1 ) , where f k = f k ( G ) records the number of k -dimensional faces σ ( k ) ∈ G .

Original: We then deﬁne the feature of a simplex σ ( d ) at time t , denoted by 
Revised: We then deﬁne the feature of a simplex σ ( d ) at time t , denoted by N t ( σ ( d ) )

Original: [Equation]
Revised: =

Original:  In words, the feature is compactly represented as the face-vector of sub-complex G (cid:48) t ( σ ( d ) ) .
Revised: f ( G (cid:48) t ( σ ( d ) )) . In words, the feature is compactly represented as the face-vector of sub-complex G (cid:48) t ( σ ( d ) ) .

Original: Our facevector representation of a node’s neighborhood can be considered as a higher-order analogue of the Weisfeiler-Lehman (WL) kernel Shervashidze et al.
Revised: Our facevector representation of a node’s neighborhood can be considered as a higher-order analogue of the Weisfeiler-Lehman (WL) kernel Shervashidze et al.

Original: (2011) on unlabeled graphs, which for each vertex, iteratively aggregates the vertex degrees of its immediate neighbors to compute a unique vector of the target vertex that captures the structure of its extended neighborhood.
Revised: (2011) on unlabeled graphs, which for each vertex, iteratively aggregates the vertex degrees of its immediate neighbors to compute a unique vector of the target vertex that captures the structure of its extended neighborhood.

Original: (ii) Scoring function: The purpose of this function is to extract the features of (cid:101) v using its proximity to σ ( d ) .
Revised: (ii) Scoring function: The purpose of this function is to extract the features of (cid:101) v using its proximity to σ ( d ) .

Original: To this end, we begin by describing afﬁnity between two vertices. 
Revised: To this end, we begin by describing afﬁnity between two vertices. Given two vertices [Table] [Equation]

Revised: feature vector

Original: Given two vertices v, v (cid:48) ∈ G (0) − , we denote by s ( v, v (cid:48) )
Revised: P [Equation] set of σ ( d ) , (cid:101) v with corresponding feature F v, v (cid:48) ∈ G (0) − , we denote by s ( v, v (cid:48) )

Original: the weighted sum of all past co-occurrences of vertices v and v (cid:48) in σ ( d ) , where the weight is d from σ ( d ) .
Revised: the weighted sum of all past co-occurrences of vertices v and v (cid:48) in σ ( d ) , where the weight is d from σ ( d ) .

Original: We then devise a scoring function h ( · , · ) that assigns an integral score to the possible introduction of a vertex (cid:101) v to a d -simplex σ ( d ) =
Revised: We then devise a scoring function h ( · , · ) that assigns an integral score to the possible introduction of a vertex (cid:101) v to a d -simplex σ ( d ) =

Original: [ v 0 , · · · , v d ] as
Revised: [ v 0 , · · · , v d ] as

Original: [Equation]
Revised: [Equation]

Original: Intuitively, the scoring function describes higher co-occurrence between σ ( d ) and (cid:101) v at time t , indicates a higher likelihood of forming a ( d + 1) -simplex τ ( d +1) =
Revised: Intuitively, the scoring function describes higher co-occurrence between σ ( d ) and (cid:101) v at time t , indicates a higher likelihood of forming a ( d + 1) -simplex τ ( d +1) =

Original: [ σ ( d ) , (cid:101) v ] together at a future time t +1 .
Revised: [ σ ( d ) , (cid:101) v ] together at a future time t +1 .

Original: We give a higher score to past co-occurrences of vertex pairs in higher dimensional simplices.
Revised: We give a higher score to past co-occurrences of vertex pairs in higher dimensional simplices.

Original: Feature vector:
Revised: Feature vector:

Original: Finally, for a given d -simplex σ ( d ) at time t , and a possible introduction of a new vertex (cid:101) v ∈ B t,k ( σ [Equation] d ) ) , we assign a feature vector
Revised: Finally, for a given d -simplex σ ( d ) at time t , and a possible introduction of a new vertex (cid:101) v ∈ B t,k ( σ ( d ) ) , we assign a feature vector

Original: [Equation]
Revised: [Equation]

Original: We denote the set of all such possible ( σ ( d ) , (cid:101) v ) pairs with their corresponding feature vectors equal to F as P t ( σ ( d ) , F ) .
Revised: We denote the set of all such possible ( σ ( d ) , (cid:101) v ) pairs with their corresponding feature vectors equal to F as P t ( σ ( d ) , F ) .

Original: Furthermore, among the pairs in P t ( σ ( d ) , F ) , we denote by P τt ( σ ( d ) , F ) those set of pairs with feature vectors equal to F that actually form τ ( d +1) =
Revised: Furthermore, among the pairs in P t ( σ ( d ) , F ) , we denote by P τt ( σ ( d ) , F ) those set of pairs with feature vectors equal to F that actually form τ ( d +1) =

Original: [ σ ( d ) , (cid:101) v ] at time t , i.e., σ ( d ) appears as a face in τ ( d +1) at time t .
Revised: [ σ ( d ) , (cid:101) v ] at time t , i.e., σ ( d ) appears as a face in τ ( d +1) at time t .

Original: Note the distinction that not all d -simplices counted in P t ( σ ( d ) , F ) end up being promoted to higher ( d + 1) -simplices in the next time step. 
Revised: Note the distinction that not all d -simplices counted in P t ( σ ( d ) , F ) end up being promoted to higher ( d + 1) -simplices in the next time step. Tableprovides a list of notations.

Original: 3. P REDICTION M ODEL AND K ERNEL E STIMATOR
Revised: 3. P REDICTION M ODEL AND K ERNEL E STIMATOR

Original: For the prediction, we deﬁne an indicator variable that displays the appearance of a new simplex.
Revised: For the prediction, we deﬁne an indicator variable that displays the appearance of a new simplex.

Original: Given a d -simplex σ ( d ) =
Revised: Given a d -simplex σ ( d ) =

Original: [ v 0 , · · · , v d ] ∈ G t , the arrival at time t + 1 of a ( d + 1) -simplex
Revised: [ v 0 , · · · , v d ] ∈ G t , the arrival at time t + 1 of a ( d + 1) -simplex

Original: τ ( d +1) =
Revised: τ ( d +1) =

Original: [ v 0 , · · · , v d , (cid:101) v ] with a new vertex (cid:101) v ∈ B t,k ( σ ( d ) ) is captured by the following variableY t [Equation] 1 ( τ
Revised: [ v 0 , · · · , v d , (cid:101) v ] with a new vertex (cid:101) v ∈ B t,k ( σ ( d ) ) is captured by the following variable

Original: [Equation]

Original: d

Original: [Equation]
Revised: [Equation]

Original: = (cid:26) 1 if σ ( d ) is a face of τ ( d +1) , 0 otherwise .

Original: Prediction model : Our approach for the prediction is to model the indicator variable.
Revised: Prediction model : Our approach for the prediction is to model the indicator variable.

Original: Namely, we assume that the indicator variable follows the following distribution:
Revised: Namely, we assume that the indicator variable follows the following distribution:

Original: [Equation]
Revised: [Equation]

Original: where 0 ≤ g ( · ) ≤ 1 is a function of the feature vector F t ( σ ( d ) , (cid:101) v ) .
Revised: where 0 ≤ g ( · ) ≤ 1 is a function of the feature vector F t ( σ ( d ) , (cid:101) v ) .

Original: In words, the indicator variable Y t +1 ( τ ( d +1) ) is Bernoulli distributed with success probability given by function g ( F t ( σ ( d ) , (cid:101) v )) , conditioned on having seen the last p states of GSC G t .
Revised: In words, the indicator variable Y t +1 ( τ ( d +1) ) is Bernoulli distributed with success probability given by function g ( F t ( σ ( d ) , (cid:101) v )) , conditioned on having seen the last p states of GSC G t .

Original: This model describes that the appearance probabilities for two simplices σ i and σ j are likely to be similar, if their feature vector is also similar.
Revised: This model describes that the appearance probabilities for two simplices σ i and σ j are likely to be similar, if their feature vector is also similar.

Original: Estimator with Kernels : We utilize a kernel method to estimate the success probability of the model (Equation 4) based on observed simplices at time t .
Revised: Estimator with Kernels : We utilize a kernel method to estimate the success probability of the model (Equation 4) based on observed simplices at time t .

Original: Let G t ( d ) be a set of d -dimensional simplices at time t from G t,p .
Revised: Let G t ( d ) be a set of d -dimensional simplices at time t from G t,p .

Original: Also, for brevity, let F represent a feature F t ( σ ( d ) , (cid:101) v m ) with some t, σ ( d ) and (cid:101) v m subject to (cid:101) v m ∈ B t,k ( σ ( d ) i ) .
Revised: Also, for brevity, let F represent a feature F t ( σ ( d ) , (cid:101) v m ) with some t, σ ( d ) and (cid:101) v m subject to (cid:101) v m ∈ B t,k ( σ ( d ) i ) .

Original: Let (cid:107) F − F (cid:48) (cid:107) 1 denote the L 1 -distance between two feature vectors, and also deﬁne a L 1 -ball Γ( F, δ ) :
Revised: Let (cid:107) F − F (cid:48) (cid:107) 1 denote the L 1 -distance between two feature vectors, and also deﬁne a L 1 -ball Γ( F, δ ) :

Original: = { F (cid:48) : (cid:107) F − F (cid:48) (cid:107) 1 ≤ δ } .
Revised: = { F (cid:48) : (cid:107) F − F (cid:48) (cid:107) 1 ≤ δ } .

Original: We deﬁne our kernel function K ( · , · ) as follows.
Revised: We deﬁne our kernel function K ( · , · ) as follows.

Original: With two features F and F (cid:48) , we deﬁne it as
Revised: With two features F and F (cid:48) , we deﬁne it as

Original: [Equation]
Revised: [Equation]

Original: where β > 0 is the bandwidth parameter, and I is the indicator function.
Revised: where β > 0 is the bandwidth parameter, and I is the indicator function.

Original: Given the feature F with only integer components, we are interested in only those close by feature vectors that are either exactly the same as f or lie within an L 1 -ball of radius δ centered on f .
Revised: Given the feature F with only integer components, we are interested in only those close by feature vectors that are either exactly the same as f or lie within an L 1 -ball of radius δ centered on f .

Original: This explains the choice of our kernel function with discrete indicator variables.
Revised: This explains the choice of our kernel function with discrete indicator variables.

Original: Now, we deﬁne our estimator.
Revised: Now, we deﬁne our estimator.

Original: At time T , we ﬁx σ ( d ) and (cid:101) v and set F = F T ( σ ( d ) , (cid:101) v ) , our estimator of g ( F ) in (Equation 4) is written as follows:
Revised: At time T , we ﬁx σ ( d ) and (cid:101) v and set F = F T ( σ ( d ) , (cid:101) v ) , our estimator of g ( F ) in (Equation 4) is written as follows:

Original: [Equation]
Revised: [Equation]

Original: Time-complexity of our estimator : As both the face-vectors and counts P t +1 / P ( τ ) t +1 (updated in a data cube) are computed simultaneously, they incur the same time overhead.
Revised: Time-complexity of our estimator : As both the face-vectors and counts P t +1 / P ( τ ) t +1 (updated in a data cube) are computed simultaneously, they incur the same time overhead.

Original: At time t , for a dsimplex σ ( d ) it takes O ( | V k | + | E k | ) time to compute a k -ball around σ ( d ) , where V k and E k are the set of vertices and edges in the k -hop subgraph of a vertex.
Revised: At time t , for a dsimplex σ ( d ) it takes O ( | V k | + | E k | ) time to compute a k -ball around σ ( d ) , where V k and E k are the set of vertices and edges in the k -hop subgraph of a vertex.

Original: We must check this k -ball of σ ( d ) against p G ( d ) t −  number of simplices (with dimension at most d ) from the previous p time steps, by intersecting against them to get counts for the face vector and data cube.
Revised: We must check this k -ball of σ ( d ) against p |G ( d ) t − | number of simplices (with dimension at most d ) from the previous p time steps, by intersecting against them to get counts for the face vector and data cube.

Original: Each intersection test takes
Revised: Each intersection test takes

Original: O ( | V k | ) time.
Revised: O ( | V k | ) time.

Original: Recall that f d ( G t ) denoted the total number of d -simplices in G t .
Revised: Recall that f d ( G t ) denoted the total number of d -simplices in G t .

Original: V k and E k are the set of vertices and edges in the k -hop subgraph of a vertex.
Revised: V k and E k are the set of vertices and edges in the k -hop subgraph of a vertex.

Original: We must check this k -ball of σ ( d ) against p G ( d ) t −  number of simplices (with dimension at most d ) from the previous p time steps.
Revised: We must check this k -ball of σ ( d ) against p |G ( d ) t − | number of simplices (with dimension at most d ) from the previous p time steps.

Original: So, the entire computation across T time chunks has a time complexity of O ( Tpf d ( G t )( | V k | + | E k | ))
Revised: So, the entire computation across T time chunks has a time complexity of O ( Tpf d ( G t )( | V k | + | E k | ))

Original: G ( d ) t −  ) .
Revised: |G ( d ) t − | ) .

Revised: Storage-complexity of our estimator : For a single simplex, our estimator requires storing: (i) a pair of integer counts, namely ( P t ( · , · ) , P τt ( · , · )) , in a datacube, which costs O (1) and (ii) a d + 1dimensional face vector which takes storage

Revised: O ( d + 1) .

Revised: As the total number of simplices is denoted by f d ( G t ) , we arrive at a total storage cost of O ( df d ( G t )) .

Original:  T HEORETICAL P ROPERTY OF THE E STIMATOR4. C ONSISTENCY
Revised:  T HEORETICAL P ROPERTY OF THE E STIMATOR

Revised: [Equation] Consistency Asymptotic

Revised: Normality

Original: We study the consistency of our estimator.
Revised: We show that our estimator has theoretical validity: (i) consistency and (ii) asymptotic normality. (i) The consistency guarantees (cid:101) g T achieves zero error as T increases by converging to g . (ii)

Revised: The asymptotic normality implies that the error (cid:101) g T − g converges to a normal distribution, which is useful to evaluate the size of the error and can be applied to statistical tests and conﬁdence analysis.

Revised: Both properties are very important in statistics (Van der Vaart, 2000).

Revised: 4. C ONSISTENCY

Revised: We study the consistency of our estimator.

Original: To discuss the property of the estimators with GSCs, it is necessary to organize the Markov property , that the GSC evolution process clearly exhibits.
Revised: To discuss the property of the estimators with GSCs, it is necessary to organize the Markov property , that the GSC evolution process clearly exhibits.

Original: It is well-known that there exists a set of irreducible closed communication classes C in the state space
Revised: It is well-known that there exists a set of irreducible closed communication classes C in the state space

Original: S .
Revised: S .

Original: We denote the time of entering class C by T C and the event as E ( T C ) .
Revised: We denote the time of entering class C by T C and the event as E ( T C ) .

Original: Let S C denote the event
Revised: Let S C denote the event

Original: S t ∈ C , where S t is the state of the Markov chain at time t .
Revised: S t ∈ C , where S t is the state of the Markov chain at time t .

Original: Then, E ( T C ) ∩ S C is the event that the chain enters class C at time T C and remains in that communication class indeﬁnitely.
Revised: Then, E ( T C ) ∩ S C is the event that the chain enters class C at time T C and remains in that communication class indeﬁnitely.

Original: With the event S C , we provide the bias-variance decomposition of (cid:101) g T − g .
Revised: With the event S C , we provide the bias-variance decomposition of (cid:101) g T − g , which is common for theoretical analysis of estimators.

Revised: The bias represents an error due to the expressive power of the model and the variance represents the over-ﬁtting error due to algorithm uncertainty.

Revised: By analyzing these terms separately, we can analyze the overall prediction error.

Original: We deﬁne two functions as (cid:98) h T ( F ) =
Revised: We deﬁne two functions as (cid:98) h T ( F ) =

Original: ( T − p ) − 1 (cid:80) T − 1 t = p (cid:80) |G
Revised: ( T − p ) − 1 (cid:80) T − 1 t = p (cid:80) |G

Original: t ( d ) | j = |G t ( d ) | − 1 | P τt +1 ( σ ( d ) j , F ) | and (cid:98) d T ( F ) =
Revised: t ( d ) | j = |G t ( d ) | − 1 | P τt +1 ( σ ( d ) j , F ) | and (cid:98) d T ( F ) =

Original: ( T − p ) − 1 (cid:80) T − 1 t = p (cid:80) |G
Revised: ( T − p ) − 1 (cid:80) T − 1 t = p (cid:80) |G

Original: t ( d ) | j = |G t ( d ) | − 1 | P t +1 ( σ ( d ) j , F ) | .
Revised: t ( d ) | j = |G t ( d ) | − 1 | P t +1 ( σ ( d ) j , F ) | .

Original: For the sake of brevity, we ﬁx F and denote (cid:101) g T ( F ) by (cid:101) g T .
Revised: For the sake of brevity, we ﬁx F and denote (cid:101) g T ( F ) by (cid:101) g T .

Original: Similarly, g, (cid:98) h T and (cid:98) d T are used.
Revised: Similarly, g, (cid:98) h T and (cid:98) d T are used.

Original: Also, we deﬁne a term B T ( F, C ) =
Revised: Also, we deﬁne a term B T ( F, C ) =

Original: E [ (cid:98) h T | S C ] / E
Revised: E [ (cid:98) h T | S C ] / E

Original: [ (cid:98) d T |
Revised: [ (cid:98) d T |

Original: S C ] − g .
Revised: S C ] − g .

Original: Then, by Proposition 2 in the supplementary material, we decompose (cid:101) g T − g as
Revised: Then, by Proposition 2 in the supplementary material, we decompose (cid:101) g T − g as

Original: [Equation]
Revised: [Equation]

Original: g

Original: [Equation] [ (cid:98) [Equation] To bound the variance term V T , we make an assumption that our Markov chain X t exhibits a αmixing property which describes a dependent property of the dynamic process.
Revised: To bound the variance term V T , we make an assumption that our Markov chain X t exhibits a αmixing property which describes a dependent property of the dynamic process.

Original: It is one of the most common and well-used assumptions describing time-dependent processes including dynamic graphs (Sarkar et al., 2014b).
Revised: It is one of the most common and well-used assumptions describing time-dependent processes including dynamic graphs (Sarkar et al., 2014b).

Original: Precisely, we present the deﬁnition of α -mixing :
Revised: Precisely, we present the deﬁnition of α -mixing :

Original: Deﬁnition 5 ( α -mixing) .
Revised: Deﬁnition 5 ( α -mixing) .

Original: A stochastic process X t is α -mixing , if a coefﬁcient α ( r ) , deﬁned as
Revised: A stochastic process X t is α -mixing , if a coefﬁcient α ( r ) , deﬁned as

Original: [Equation]
Revised: [Equation]

Original: satisﬁes α ( r ) → 0 as r → ∞ .
Revised: satisﬁes α ( r ) → 0 as r → ∞ .

Original: Here, Σ( X − t 1 ) and Σ( X + t 2 ) are the sigma-algebras of past and future events of the stochastic process up to and including t 1 .
Revised: Here, Σ( X − t 1 ) and Σ( X + t 2 ) are the sigma-algebras of past and future events of the stochastic process up to and including t 1 .

Revised: This deﬁnition implies that time-dependent processes get close to independent as time passes.

Revised: That is, events at time t and t + 10000 are close to independent, while events at time t and t + 1 can be correlated.

Revised: The simplest example is the evolution of stock prices in ﬁnancial markets: the movement of stock prices today does not correlate with the movement of stock prices 10 years ago.

Original: To bound the bias term B T , we impose a smoothness condition on g .
Revised: To bound the bias term B T , we impose a smoothness condition on g .

Original: A similar assumption is often used in the problem of predicting links (e.g. Assumption 1 in Sarkar et al.
Revised: A similar assumption is often used in the problem of predicting links (e.g. Assumption 1 in Sarkar et al.

Original: (2014a)).
Revised: (2014a)).

Original: Our assumption is a general and weaker version of the common assumption.
Revised: Our assumption is a general and weaker version of the common assumption.

Original: Assumption 1 (Smoothness on g ) .
Revised: Assumption 1 (Smoothness on g ) .

Original: There exists a function κ : R → R in the Schwartz space (i.e. it is inﬁnitely differentiable and converging to zero faster than any polynomial as x → ±∞ ) with a parameter b > 0 such that 
Revised: There exists a function κ : R → R in the Schwartz space (i.e. it is inﬁnitely differentiable and converging to zero faster than any polynomial as x → ±∞ ) with a parameter b > 0 such that | g ( F ) − g ( F (cid:48) )

Original: [Equation]
Revised: | = O ( κ ( −(cid:107) F − F (cid:48) (cid:107) 1 /b )) , as b → 0 , ∀ F, F (cid:48) .

Original: We utilize all the assumptions, then prove the consistency of (cid:101) g T .
Revised: We utilize all the assumptions, then prove the consistency of (cid:101) g T .

Original: Theorem 1 (Consistency) .
Revised: Theorem 1 (Consistency) .

Original: Suppose that the GSC ﬁltration process is α -mixing,
Revised: Suppose that the GSC ﬁltration process is α -mixing,

Original: β
Revised: β

Original: = o (1) , and
Revised: = o (1) , and

Original: Assumption 1 holds.
Revised: Assumption 1 holds.

Original: Then, for any F and conditional on S C , our estimator (cid:101) g T ( F ) is well-deﬁned with probability tending to 1 , and | (cid:101) g T ( F ) − g
Revised: Then, for any F and conditional on S C , our estimator (cid:101) g T ( F ) is well-deﬁned with probability tending to 1 , and | (cid:101) g T ( F ) − g

Original: ( F )
Revised: ( F )

Original: | p → 0 holds as T → ∞ .
Revised: | p → 0 holds as T → ∞ .

Original: 4. A SYMPTOTIC N ORMALITY
Revised: 4. A SYMPTOTIC N ORMALITY

Original: We show the asymptotic normality of the proposed estimator.
Revised: We show the asymptotic normality of the proposed estimator.

Original: That is, we prove that the error of the estimator converges weakly to a normal distribution.
Revised: That is, we prove that the error of the estimator converges weakly to a normal distribution.

Original: This property allows for more detailed investigations, such as correcting for errors in estimators or performing statistical tests.
Revised: This property allows for more detailed investigations, such as correcting for errors in estimators or performing statistical tests.

Original: Technically speaking, we develop a distribution approximation result with Wasserstein distance (Villani, 2008) and Stein’s method (Stein et al., 1986) to handle the dependency property of GSCs.
Revised: Technically speaking, we develop a distribution approximation result with Wasserstein distance (Villani, 2008) and Stein’s method (Stein et al., 1986) to handle the dependency property of GSCs.

Original: We are interested in approximating a random variable Z n by a Gaussian random variable, where Z n is a sum of n mean-centered random variables { A i } ni =0 , where A i corresponds to a random variable which depends on the i -th d -simplex σ ( d ) i in our GSC, which is dependent on other d -simplices whose neighborhoods largely overlap with that of σ ( d ) i .
Revised: We are interested in approximating a random variable Z n by a Gaussian random variable, where Z n is a sum of n mean-centered random variables { A i } ni =0 , where A i corresponds to a random variable which depends on the i -th d -simplex σ ( d ) i in our GSC, which is dependent on other d -simplices whose neighborhoods largely overlap with that of σ ( d ) i .

Original: Here, let d w be the Wasserstein distance between the underlying distributions of the random variables, and N is a standard Gaussian variable.
Revised: Here, let d w be the Wasserstein distance between the underlying distributions of the random variables, and N is a standard Gaussian variable.

Original: Then, we develop the general results for Z n .
Revised: Then, we develop the general results for Z n .

Revised: We provide the following theoretical result.

Revised: Its formal statement is Theorem 3, which is deferred to the supplementary material due to its complexity.

Original: Proposition 1 (Gaussian approximation for dependent variables; Simple version of Theorem 3) .
Revised: Proposition 1 (Gaussian approximation for dependent variables; Simple version of Theorem 3) .

Original: Suppose Z n = (cid:80) ni
Revised: Suppose Z n = (cid:80) ni

Original: =

Revised: =

Original: A i , where { A i } ni =0 is generated by zero-mean random variables
Revised: A i , where { A i } ni =0 is generated by zero-mean random variables

Original: X 0 , X 1 , · · · , X n satisfying the α -mixing condition, such as A i
Revised: X 0 , X 1 , · · · , X n satisfying the α -mixing condition, such as A i

Original: = X i /B n with B n = E [ (cid:80) ni
Revised: = X i /B n with B n = E [ (cid:80) ni

Original: =

Revised: =

Original: X 2 i ] .
Revised: X 2 i ] .

Original: Also, suppose that Pr {| X i | ≤ L } = 1 holds for i
Revised: Also, suppose that Pr {| X i | ≤ L } = 1 holds for i

Original: = 1 , · · · , n with some constant L > 0 .
Revised: = 1 , · · · , n with some constant L > 0 .

Original: Then, with an existing ﬁnite constant C > 0 , we have 
Revised: Then, with an existing ﬁnite constant C > 0 , we have d w ( Z n , N ) ≤ C (cid:16)(cid:80) ni

Original: [Equation]
Revised: =1 E | A i

Revised: | 3 + TL 3 B 3 n (cid:80) n − 1 i =1 nα ( n ) (cid:17) .

Original: This result extends Sunklodas (2007) in the sense of Markov chains on GSCs that satisfy the αmixing condition.
Revised: This result extends Sunklodas (2007) in the sense of Markov chains on GSCs that satisfy the αmixing condition.

Original: This extension makes it possible to study the contributing effect of neighboring d -simplices (represented as a set of weakly dependent r.v.’s) on a central d -simplex.
Revised: This extension makes it possible to study the contributing effect of neighboring d -simplices (represented as a set of weakly dependent r.v.’s) on a central d -simplex.

Original: We provide the asymptotic normality of our estimator.
Revised: We provide the asymptotic normality of our estimator.

Original: It shows that our estimator converges to a normal distribution in terms of the Wasserstein distance, which leads to weak convergence.
Revised: It shows that our estimator converges to a normal distribution in terms of the Wasserstein distance, which leads to weak convergence.

Original: To achieve the result, we utilize the decomposed terms V T and B T from (7).
Revised: To achieve the result, we utilize the decomposed terms V T and B T from (7).

Original: Then, we regard V T as a sum of dependent random variables and apply the result developed in Proposition 1.
Revised: Then, we regard V T as a sum of dependent random variables and apply the result developed in Proposition 1.

Original: Let σ 2 c be a limit of variance of the numerator in T − 1 / 2 V T as T → ∞ .
Revised: Let σ 2 c be a limit of variance of the numerator in T − 1 / 2 V T as T → ∞ .

Revised: We recall that S C denotes the event

Revised: S t ∈ C , where S t is the state of the Markov chain at time t .

Original: Theorem 2 (Asymptotic Normality) .
Revised: Theorem 2 (Asymptotic Normality) .

Original: Suppose that Assumption 1 holds, the GSC ﬁltration process is α -mixing, and σ c > 0 .
Revised: Suppose that Assumption 1 holds, the GSC ﬁltration process is α -mixing, and σ c > 0 .

Original: If β = o ( T − 1 / 2 ) and b =
Revised: If β = o ( T − 1 / 2 ) and b =

Original: o ( T − 1 / 2 ) , then, for any F and conditioned on
Revised: o ( T − 1 / 2 ) , then, for any F and conditioned on

Original: S C , the following holds:
Revised: S C , the following holds:

Original: [Equation]
Revised: [Equation]

Original: By using this property, we can make detailed inferences based on the distribution of the estimation error.
Revised: By using this property, we can make detailed inferences based on the distribution of the estimation error.

Original: For example, it is possible to create conﬁdence intervals for predictions and perform statistical tests to rigorously test hypotheses about simplex arrivals.
Revised: For example, it is possible to create conﬁdence intervals for predictions and perform statistical tests to rigorously test hypotheses about simplex arrivals.

Original:  R EAL -W ORLD D ATA E XPERIMENTS
Revised:  R EAL -W ORLD D ATA E XPERIMENTS

Original: We empirically evaluate the performance of our proposed estimator on real-world dynamic graphs compared to baselines.
Revised: We empirically evaluate the performance of our proposed estimator on real-world dynamic graphs compared to baselines.

Original: The basic premise in our experiments is to capture local and higher-order properties surrounding a d -simplex up to time t to predict the appearance of a new ( d + 1) -simplex at time t (cid:48) > t , which contains σ ( d ) as its face .
Revised: The basic premise in our experiments is to capture local and higher-order properties surrounding a d -simplex up to time t to predict the appearance of a new ( d + 1) -simplex at time t (cid:48) > t , which contains σ ( d ) as its face .

Revised: Note that we compare our method to other closely related methods that were designed to solve different structure prediction tasks.

Original: Datasets : We report results on real-world dynamic graph datasets sourced from Benson et al.
Revised: Datasets : We report results on real-world dynamic graph datasets sourced from Benson et al.

Original: Each dataset contains n nodes, m formed edges, and x timestamped simplices (represented as a set of nodes).
Revised: Each dataset contains n nodes, m formed edges, and x timestamped simplices (represented as a set of nodes).

Original: There are four datasets named Enron ( n = 143 , m = 1 .
Revised: There are four datasets named Enron ( n = 143 , m = 1 .

Original: 8 K, x = 5 K), EU ( n = 998 , m = 29 .
Revised: 8 K, x = 5 K), EU ( n = 998 , m = 29 .

Original: 3 K, x = 8 K), Contact ( n = 327 , m = 5 .
Revised: 3 K, x = 8 K), Contact ( n = 327 , m = 5 .

Original: 8 K, x = 10 K), and NDC (National Drug Code) ( n = 1 .
Revised: 8 K, x = 10 K), and NDC (National Drug Code) ( n = 1 .

Original: 1 K, m = 6 .
Revised: 1 K, m = 6 .

Original: 2 K, x = 12 K).
Revised: 2 K, x = 12 K).

Original: Experimental setup :
Revised: Experimental setup :

Original: We ﬁrst ordered by arrival times and grouped the timestamped simplices into
Revised: We ﬁrst ordered by arrival times and grouped the timestamped simplices into

Original: T time slices.
Revised: T time slices.

Original: For most of our experiments, T was set to 20 , except for d = 2 , where T was set to 6 and 12 for EU and NDC , respectively.
Revised: For most of our experiments, T was set to 20 , except for d = 2 , where T was set to 6 and 12 for EU and NDC , respectively.

Original: Then, we randomly sampled a set of d -simplices from the time slices in the range [1 , T − 1] .
Revised: Then, we randomly sampled a set of d -simplices from the time slices in the range [1 , T − 1] .

Original: Those d -simplices paired with a vertex that successfully formed a face in a ( d + 1) -simplex in the T -th time slice were classiﬁed as positive samples , while the rest were deemed as negative samples .
Revised: Those d -simplices paired with a vertex that successfully formed a face in a ( d + 1) -simplex in the T -th time slice were classiﬁed as positive samples , while the rest were deemed as negative samples .

Original: We picked an equal number of positive and negative samples for evaluation.
Revised: We picked an equal number of positive and negative samples for evaluation.

Original: For K -fold cross-validation for β , we swapped the T -th time slice with one of the K slices preceding the T -th time slice for each fold.
Revised: For K -fold cross-validation for β , we swapped the T -th time slice with one of the K slices preceding the T -th time slice for each fold.

Original: K was set to 3 .
Revised: K was set to 3 .

Original: All experiments where repeated 10 times and average AUC scores and runtimes are reported.
Revised: All experiments where repeated 10 times and average AUC scores and runtimes are reported.

Original: [Table] ( β =0.01) ( β =10) Compared methods : As naive baselines, we averaged the results of single-edge prediction methods, where a new edge would form between each node in the d -simplex and the vertex to be paired with.
Revised: Compared methods : As naive baselines, we averaged the results of single-edge prediction methods, where a new edge would form between each node in the d -simplex and the vertex to be paired with.

Original: Speciﬁcally, we compare our estimator with both heuristic ( Adamic-Adar (AA) (Adamic & Adar, 2001), Jaccard Coefﬁcient (JC) (Salton & McGill, 1986), and Preferential attachment (PA) (Mitzenmacher, 2004) and deep-learning based ( Node2vec (NV) (Grover & Leskovec, 2016a), and SEAL (SL) (Zhang & Chen, 2018a)) link prediction methods.
Revised: Speciﬁcally, we compare our estimator with both heuristic ( Adamic-Adar (AA) (Adamic & Adar, 2001), Jaccard Coefﬁcient (JC) (Salton & McGill, 1986), and Preferential attachment (PA) (Mitzenmacher, 2004) and deep-learning based ( Node2vec (NV) (Grover & Leskovec, 2016a), and SEAL (SL) (Zhang & Chen, 2018a)) link prediction methods.

Original: We note that (Benson et al., 2018) for predicting a “simplicial closure” has the closest motivation to our method, yet has divergent objectives,  therefore we omit comparison to their work.
Revised: We note that (Benson et al., 2018) for predicting a “simplicial closure” has the closest motivation to our method, yet has divergent objectives, [Table] ( β =0.01) ( β =10) therefore we omit comparison to their work.

Original: For hyper-edge prediction (HP), we picked the recent most representative work by Yoon et al.
Revised: For hyper-edge prediction (HP), we picked the recent most representative work by Yoon et al.

Original: (2020) to compare against, although this work only works for static non-evolving hypergraphs.
Revised: (2020) to compare against, although this work only works for static non-evolving hypergraphs.

Original: The details of the comparative methods and the simplicial closure method are provided in Section B.2 in the supplementary material.
Revised: The details of the comparative methods and the simplicial closure method are provided in Section B.2 in the supplementary material.

Original: 5. R ESULTS AND D ISCUSSION
Revised: 5. R ESULTS AND D ISCUSSION

Original: We averaged the classiﬁcation accuracy and runtimes of our estimator and the baselines.
Revised: We averaged the classiﬁcation accuracy and runtimes of our estimator and the baselines.

Original: We performed two sets of experiments on the arrival of a ( d + 1) -simplex and summarize it in Table 1 for d = { 1 , 2 } .
Revised: We performed two sets of experiments on the arrival of a ( d + 1) -simplex and summarize it in Table 2 for d = { 1 , 2 } .

Original: We also report the bandwidth β for our estimator selected by cross-validation.
Revised: We also report the bandwidth β for our estimator selected by cross-validation.

Original: Predicting 2 -simplex ( d = 1 ) : We observe that our method is nearly two orders of magnitude faster than the deep learning based methods (NV and SL) and nearly an order of magnitude faster than the hypergraph prediction method (HP).
Revised: Predicting 2 -simplex ( d = 1 ) : We observe that our method is nearly two orders of magnitude faster than the deep learning based methods (NV and SL) and nearly an order of magnitude faster than the hypergraph prediction method (HP).

Original: While the single edge heuristic methods are relatively faster, their AUC scores are not comparable to our method’s AUC scores.
Revised: While the single edge heuristic methods are relatively faster, their AUC scores are not comparable to our method’s AUC scores.

Original: Also, we achieve nearly 30% improvement (in Enron) over the next best performing prediction method.
Revised: Also, we achieve nearly 30% improvement (in Enron) over the next best performing prediction method.

Original: In comparison to HP, we achieve very signiﬁcant improvements.
Revised: In comparison to HP, we achieve very signiﬁcant improvements.

Original:  Predicting 3 -simplex ( d = 2 ) : We notice that the gap in AUC scores between our method and the baselines becomes far more pronounced.
Revised: [Figure] Predicting 3 -simplex ( d = 2 ) : We notice that the gap in AUC scores between our method and the baselines becomes far more pronounced.

Original: Our runtimes also improve signiﬁcantly due to the far fewer number of simplices with dimensions exceeding 3 .
Revised: Our runtimes also improve signiﬁcantly due to the far fewer number of simplices with dimensions exceeding 3 .

Original: In agreement with observations made in Yoon et al.
Revised: In agreement with observations made in Yoon et al.

Original: (2020) about slight drops in accuracy for higher-dimensional hyper-edges, we also note that in HP, the AUC score remains the same or drops slightly compared to prediction at d = 1 .
Revised: (2020) about slight drops in accuracy for higher-dimensional hyper-edges, we also note that in HP, the AUC score remains the same or drops slightly compared to prediction at d = 1 .

Revised: Advantage of higher dimensional simplices : We perform additional experiments by increasing d from 1 toand show that handling high-dimensional simplices exhibits high prediction accuracy.

Revised: In Figure 3, the AUC score of prediction basically improves as d increases.

Original: Empirical summary : Traditional estimators fail to accurately capture the rich latent information present in higher-order structures (and their sub-structures) that evolve over time.
Revised: Empirical summary : Traditional estimators fail to accurately capture the rich latent information present in higher-order structures (and their sub-structures) that evolve over time.

Original: Our estimator succinctly captures this information via the f -vector and weighted scoring of ( σ ( d ) , v ) pair formation depending on the dimension of the simplex in which the pair co-occur in the past.
Revised: Our estimator succinctly captures this information via the f -vector and weighted scoring of ( σ ( d ) , v ) pair formation depending on the dimension of the simplex in which the pair co-occur in the past.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In our study, we develop a prediction method for  higher-order  group-wise interactions that can involve more than two nodes at once.
Revised: In our study, we develop a prediction method for a higher-order link prediction task which involves predicting group-wise interactions.

Original: A higher-order link prediction task involves predicting whether a given group-wise interaction at time t , evolves to participate in a larger group-wise interaction at time t (cid:48) > t .

Original: We modeled the higher-order interaction as a simplex and demonstrated a novel kernel estimator to solve the higher-order structure prediction problem.
Revised: We modeled the higher-order interaction as a simplex and demonstrated a novel kernel estimator to solve the higher-order structure prediction problem.

Original: From a theoretical standpoint, we proved the consistency and asymptotic normality of our estimator.
Revised: From a theoretical standpoint, we proved the consistency and asymptotic normality of our estimator.

Original: We empirically argue that our estimator outperforms hypergraph based and higher-order link prediction baselines from both heuristic and deep-learning based pairwise link prediction methods.
Revised: We empirically argue that our estimator outperforms hypergraph based and higher-order link prediction baselines from both heuristic and deep-learning based pairwise link prediction methods.

Original: A E XAMPLE OF S IMPLEX AND R ELATED N OTIONS
Revised: A E XAMPLE OF S IMPLEX AND R ELATED N OTIONS

Original: Example 1.
Revised: Example 1.

Original: We begin by computing the k -balls centered at 1 -simplex [9 , 10] in G t and G t − 1 , respectively.
Revised: We begin by computing the k -balls centered at 1 -simplex [9 , 10] in G t and G t − 1 , respectively.

Original: The k -ball at time t for k = 1 (i.e., 1-hop vertices only) centered at [9 , 10] is:
Revised: The k -ball at time t for k = 1 (i.e., 1-hop vertices only) centered at [9 , 10] is:

Original: B t, 1 ([9 , 10])
Revised: B t, 1 ([9 , 10])

Original: =
Revised: =

Original: B t, 1 ([9]) ∪ B t, 1 ([10]) .
Revised: B t, 1 ([9]) ∪ B t, 1 ([10]) .

Original: This is the union of k -balls at underlying vertices 9 andaccording to Deﬁnition 3.
Revised: This is the union of k -balls at underlying vertices 9 andaccording to Deﬁnition 3.

Original: [Equation]
Revised: [Equation]

Original: Similarly, The k -ball at previous time step t − 1 for k = 1 (i.e., 1-hop vertices only) centered at [9 , 10] is:
Revised: Similarly, The k -ball at previous time step t − 1 for k = 1 (i.e., 1-hop vertices only) centered at [9 , 10] is:

Original: B t − 1 , 1 ([9 , 10])
Revised: B t − 1 , 1 ([9 , 10])

Original: =
Revised: =

Original: B t − 1 , 1 ([9]) ∪ B t − 1 , 1 ([10]) = { 9 , 13 , 8 , 5 , 6 , 10 } ∪ { 10 , 14 , 13 , 9 , 6 , 11 , 15 } = { 9 , 13 , 8 , 5 , 6 , 10 , 14 , 11 , 15 }
Revised: B t − 1 , 1 ([9]) ∪ B t − 1 , 1 ([10]) = { 9 , 13 , 8 , 5 , 6 , 10 } ∪ { 10 , 14 , 13 , 9 , 6 , 11 , 15 } = { 9 , 13 , 8 , 5 , 6 , 10 , 14 , 11 , 15 }

Original: Notice that there is only a difference of vertex 7 missing from set B t − 1 , 1 ([9 , 10]) as compared to set
Revised: Notice that there is only a difference of vertex 7 missing from set B t − 1 , 1 ([9 , 10]) as compared to set

Original: B t, 1 ([9 , 10]) at time t .
Revised: B t, 1 ([9 , 10]) at time t .

Original: Now, we calculate the subcomplex spanned by B t, 1 ([9 , 10])
Revised: Now, we calculate the subcomplex spanned by B t, 1 ([9 , 10])

Original: Then 3 ,
Revised: Then 3 ,

Original: G (cid:48) t ([9 , 10]) =
Revised: G (cid:48) t ([9 , 10]) =

Original: Now, we compute the compressed f -vector notation of G (cid:48)
Revised: Now, we compute the compressed f -vector notation of G (cid:48)

Original: t
Revised: t

Original: ([9 , 10]) to get f ( G (cid:48) t ([9 , 10]))
Revised: ([9 , 10]) to get f ( G (cid:48) t ([9 , 10]))

Original: = (1 , 10 , 11 , 2)
Revised: = (1 , 10 , 11 , 2)

Original: Finally, the neighborhood N t ([9 , 10]) =
Revised: Finally, the neighborhood N t ([9 , 10]) =

Original: B F URTHER DETAILS OF THE EXPERIMENT B. D ATASETS
Revised: B F URTHER DETAILS OF THE EXPERIMENT B. D ATASETS

Original: We report results on real-world dynamic graph datasets sourced from Benson et.
Revised: We report results on real-world dynamic graph datasets sourced from Benson et.

Original: al.
Revised: al.

Original: Benson et al.
Revised: Benson et al.

Original: Each dataset is a set of timestamped simplices (represented as a set of nodes).
Revised: Each dataset is a set of timestamped simplices (represented as a set of nodes).

Original: In each dataset, let n , m , and x denote the number of nodes, edges formed, and timestamped simplices, respectively.
Revised: In each dataset, let n , m , and x denote the number of nodes, edges formed, and timestamped simplices, respectively.

Original: Enron ( n = 143 , m = 1 . 8 K, x = 5 K) and EU ( n = 998 , m = 29 .
Revised: Enron ( n = 143 , m = 1 . 8 K, x = 5 K) and EU ( n = 998 , m = 29 .

Original: 3 K, x = 8 K) model email networks where nodes are email addresses and all recipients of an email form a simplex in the network.
Revised: 3 K, x = 8 K) model email networks where nodes are email addresses and all recipients of an email form a simplex in the network.

Original: Contact ( n = 327 , m = 5 .
Revised: Contact ( n = 327 , m = 5 .

Original: 8 K, x = 10 K) is a proximity graph where nodes represent persons and a simplex is a set of persons in close proximity to each other.
Revised: 8 K, x = 10 K) is a proximity graph where nodes represent persons and a simplex is a set of persons in close proximity to each other.

Original: NDC ( n = 1 .
Revised: NDC ( n = 1 .

Original: 1 K, m = 6 .
Revised: 1 K, m = 6 .

Original: 2 K, x = 12 K) is a drug network from the National Drug Code directory, where nodes are class labels and a simplex is formed when a set of class labels appear together on a single drug.
Revised: 2 K, x = 12 K) is a drug network from the National Drug Code directory, where nodes are class labels and a simplex is formed when a set of class labels appear together on a single drug.

Original: B. C OMPARED METHODS
Revised: B. C OMPARED METHODS

Original: Adamic-Adar Adamic & Adar (2001) and the Jaccard Coefﬁcient Salton & McGill (1986) measure link probability between two nodes based on the closeness of their respective feature vectors.
Revised: Adamic-Adar Adamic & Adar (2001) and the Jaccard Coefﬁcient Salton & McGill (1986) measure link probability between two nodes based on the closeness of their respective feature vectors.

Original: Preferential attachment Mitzenmacher (2004) has received considerable attention as a model of growth of networks as they model future link probability as the product of the current number of neighbors of the two nodes.
Revised: Preferential attachment Mitzenmacher (2004) has received considerable attention as a model of growth of networks as they model future link probability as the product of the current number of neighbors of the two nodes.

Original: Motivated by resource allocation in transportation networks (much alike the Optimal
Revised: Motivated by resource allocation in transportation networks (much alike the Optimal

Original: Transport (OT) problem), Resource allocation index Zhou et al.
Revised: Transport (OT) problem), Resource allocation index Zhou et al.

Original: (2009b) proposes a node x tries to transmit a unit resource to node y via common neighbors that play the role of transmitters and similarity is measured by the amount of the resource y received from x . Node2vec Grover & Leskovec (2016a) and SEAL Zhang & Chen (2018a) are deep-learning based graph embedding methods that are used in link prediction.
Revised: (2009b) proposes a node x tries to transmit a unit resource to node y via common neighbors that play the role of transmitters and similarity is measured by the amount of the resource y received from x . Node2vec Grover & Leskovec (2016a) and SEAL Zhang & Chen (2018a) are deep-learning based graph embedding methods that are used in link prediction.

Original: Remark 1 (Difference between our setting and simplical closure ) .
Revised: Remark 1 (Difference between our setting and simplical closure ) .

Original: The closest work Benson et al.
Revised: The closest work Benson et al.

Original: (2018) proposed predicting a “simplicial closure” event where at time t there exists a set of nodes which are pairwise edge connected and the task is to predict whether at time t + 1 there will arrive a simplex which covers all these nodes.
Revised: (2018) proposed predicting a “simplicial closure” event where at time t there exists a set of nodes which are pairwise edge connected and the task is to predict whether at time t + 1 there will arrive a simplex which covers all these nodes.

Original: This phenomenon was termed as simplicial closure .
Revised: This phenomenon was termed as simplicial closure .

Original: For example, authors A , B and C have all co-authored in pairs (i.e., { A, B } , { A, C } and { B, C } ) and a simplicial closure event would take place at t + 1 , if a simplex { A, B, C } arrives, implying that all three authors co-author on a single paper.
Revised: For example, authors A , B and C have all co-authored in pairs (i.e., { A, B } , { A, C } and { B, C } ) and a simplicial closure event would take place at t + 1 , if a simplex { A, B, C } arrives, implying that all three authors co-author on a single paper.

Original: Our prediction task signiﬁcantly diverges and aims to solve a different problem.
Revised: Our prediction task signiﬁcantly diverges and aims to solve a different problem.

Original: Considering our previous example, we are given a single co-authorship relationship between say A and B at time t , we predict whether authors A and B will co-author with a third author C (ternary co-authorship relationship) on a single paper at time t + 1 in the future.
Revised: Considering our previous example, we are given a single co-authorship relationship between say A and B at time t , we predict whether authors A and B will co-author with a third author C (ternary co-authorship relationship) on a single paper at time t + 1 in the future.

Original: C P ROOF FOR C ONSISTENCY
Revised: C P ROOF FOR C ONSISTENCY

Original: For preparation, we rewrite the estimator (cid:98) g T .
Revised: For preparation, we rewrite the estimator (cid:98) g T .

Original: Plugging in the deﬁnition of our kernel (Equation 5) into the equation of our estimator (Equation 6) along with the deﬁnitions of P t ( · , · ) and P τt ( · , · ) to replace the indicator variables with actual counts, we obtain the following simpliﬁcation of Equation 6.
Revised: Plugging in the deﬁnition of our kernel (Equation 5) into the equation of our estimator (Equation 6) along with the deﬁnitions of P t ( · , · ) and P τt ( · , · ) to replace the indicator variables with actual counts, we obtain the following simpliﬁcation of Equation 6.

Original: Then, it is reformulated as
Revised: Then, it is reformulated as

Original: [Equation]
Revised: [Equation]

Original: When we set β = 0 , we look for other pairs whose feature corresponds to F and we calculate a fraction of how many such close by pairs actually form a ( d + 1) -simplex at time t (cid:48)
Revised: When we set β = 0 , we look for other pairs whose feature corresponds to F and we calculate a fraction of how many such close by pairs actually form a ( d + 1) -simplex at time t (cid:48)

Original: + 1 .
Revised: + 1 .

Original: This fraction is summed across various d -simplices σ ( d ) j by varying j and also across various discrete time steps by varying t (cid:48) .
Revised: This fraction is summed across various d -simplices σ ( d ) j by varying j and also across various discrete time steps by varying t (cid:48) .

Original: Setting β > 0 allows our estimator to smooth over close by features.
Revised: Setting β > 0 allows our estimator to smooth over close by features.

Original: It turns out simpler to study a proxy estimator (cid:98) g T , which omits the smoothing of the original one.
Revised: It turns out simpler to study a proxy estimator (cid:98) g T , which omits the smoothing of the original one.

Original: We will show that it is asymptotically equivalent to (cid:101) g .
Revised: We will show that it is asymptotically equivalent to (cid:101) g .

Original: Let |G t ( d ) | denote the total number of d -simplices in GSC G t at time t .
Revised: Let |G t ( d ) | denote the total number of d -simplices in GSC G t at time t .

Original: Then, for a feature F , we deﬁne the proxy estimator as
Revised: Then, for a feature F , we deﬁne the proxy estimator as

Original: [Equation]
Revised: [Equation]

Original: where the terms are deﬁned as
Revised: where the terms are deﬁned as

Original: [Equation]
Revised: [Equation]

Original: 1 (cid:88) t = p |G t ( d ) | (cid:88)
Revised: 1 (cid:88) t = p |G t ( d ) | (cid:88)

Original: j = | P τt +1 ( σ ( d ) j , F ) |
Revised: j = | P τt +1 ( σ ( d ) j , F ) |

Original: |G t ( d ) | , and (cid:98) d T ( F ) =
Revised: |G t ( d ) | , and (cid:98) d T ( F ) =

Original: T − p
Revised: T − p

Original: T − 1 (cid:88) t = p |G t ( d ) | (cid:88)
Revised: T − 1 (cid:88) t = p |G t ( d ) | (cid:88)

Original: j = | P t +1 ( σ ( d ) j , F ) | |G t ( d ) | .
Revised: j = | P t +1 ( σ ( d ) j , F ) | |G t ( d ) | .

Original: Observe that (cid:80) |G t ( d ) | i = ranges over the total number of d -simplices in G t 4 and note that |G t ( d ) | changes with time step t .
Revised: Observe that (cid:80) |G t ( d ) | i = ranges over the total number of d -simplices in G t 4 and note that |G t ( d ) | changes with time step t .

Original: Recall that the terms P τt +1 ( · , · ) and P t +1 ( · , · ) count all actual and possible formation of [ σ ( d ) i , (cid:101) v m ] ( ( d + 1) -simplex) that have the same feature vector F .
Revised: Recall that the terms P τt +1 ( · , · ) and P t +1 ( · , · ) count all actual and possible formation of [ σ ( d ) i , (cid:101) v m ] ( ( d + 1) -simplex) that have the same feature vector F .

Original: Lemma 1 in the supplementary material proves | (cid:101) g T ( F ) − (cid:98) g T ( F ) | → 0 as β → 0 .
Revised: Lemma 1 in the supplementary material proves | (cid:101) g T ( F ) − (cid:98) g T ( F ) | → 0 as β → 0 .

Original: First of all, we show the validity of the proxy estimator (cid:98) g T .
Revised: First of all, we show the validity of the proxy estimator (cid:98) g T .

Original: Lemma 1 (Approximation by proxy) .
Revised: Lemma 1 (Approximation by proxy) .

Original: We obtain
Revised: We obtain

Original: [Equation]
Revised: [Equation]

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

Original: Recall that Γ( F, δ ) denotes the set of features at L 1 -distance at most δ from F .
Revised: Recall that Γ( F, δ ) denotes the set of features at L 1 -distance at most δ from F .

Original: We denote by | Γ( F, δ ) | the cardinality of this set.
Revised: We denote by | Γ( F, δ ) | the cardinality of this set.

Original: [Equation]
Revised: [Equation]

Original: where C 1 = β (cid:80) j,t (cid:80) s ∈ Γ( F,δ ) | P τt +1 ( σ ( d ) j , s ) | and C 

Revised: where C 1 = β (cid:80) j,t (cid:80) s ∈ Γ( F,δ ) | P τt +1 ( σ ( d ) j , s ) | and C

Original: = β (cid:80) j,t (cid:80) s ∈ Γ( F,δ ) | P t +1 ( σ ( d ) j , s ) | .
Revised: = β (cid:80) j,t (cid:80) s ∈ Γ( F,δ ) | P t +1 ( σ ( d ) j , s ) | .

Original: Due to the ﬁniteness of features in P τt +1 ( · , · ) and P t +1 ( · , · ) , we have that C 1 = C 2 =
Revised: Due to the ﬁniteness of features in P τt +1 ( · , · ) and P t +1 ( · , · ) , we have that C 1 = C 2 =

Original: O ( β ) .
Revised: O ( β ) .

Original: Both C 1 and
Revised: Both C 1 and

Original: C 2 are non-negative integers.
Revised: C 2 are non-negative integers.

Original: So,
Revised: So,

Original: [Equation]
Revised: [Equation]

Original: In the last step, the second fraction is a positive constant and can thus be ignored from our asymptotic analysis because both (cid:98) h T and (cid:98) d T are bounded.
Revised: In the last step, the second fraction is a positive constant and can thus be ignored from our asymptotic analysis because both (cid:98) h T and (cid:98) d T are bounded.

Original: Next, we prove the convergence of the proxy estimator (cid:98) g T ( F ) .
Revised: Next, we prove the convergence of the proxy estimator (cid:98) g T ( F ) .

Original: As the ﬁrst step, we describe the detail of its decomposition in (7).
Revised: As the ﬁrst step, we describe the detail of its decomposition in (7).

Original: To simplify the notation, we will drop F from all estimator notations.
Revised: To simplify the notation, we will drop F from all estimator notations.

Original: Proposition 2.
Revised: Proposition 2.

Original: As written in Equation 7, we obtain
Revised: As written in Equation 7, we obtain

Original: [Equation]
Revised: [Equation]

Original: B T .
Revised: B T .

Original: Furthermore, with the event S C , there exist a stochastic terms q t for t such as
Revised: Furthermore, with the event S C , there exist a stochastic terms q t for t such as

Original: [Equation]
Revised: [Equation]

Original: Proof of Proposition 2.
Revised: Proof of Proposition 2.

Original: With the deﬁnition of B T ( F, C )
Revised: With the deﬁnition of B T ( F, C )

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: We are interested in the asymptotic behavior of the Markov chain at time T → ∞ .
Revised: We are interested in the asymptotic behavior of the Markov chain at time T → ∞ .

Original: Let F denote
Revised: Let F denote

Original: F T ( σ ( d ) i , (cid:101) v m ) .
Revised: F T ( σ ( d ) i , (cid:101) v m ) .

Original: Recall that the terms | P τT +1 ( · , · ) | and | P T +1 ( · , · ) | count all actual and possible formation of [ σ ( d ) i , (cid:101) v m ] ( ( d +1) -simplex) that result in the same feature vector F at time T +1 .
Revised: Recall that the terms | P τT +1 ( · , · ) | and | P T +1 ( · , · ) | count all actual and possible formation of [ σ ( d ) i , (cid:101) v m ] ( ( d +1) -simplex) that result in the same feature vector F at time T +1 .

Original: Our Markov chain has a ﬁnite state space and hence belongs to a closed communication class with probability approaching to 1 .
Revised: Our Markov chain has a ﬁnite state space and hence belongs to a closed communication class with probability approaching to 1 .

Original: We provide a statistical consistency conditional on S C for any communication class C .
Revised: We provide a statistical consistency conditional on S C for any communication class C .

Original: For a given time step t , we deﬁne
Revised: For a given time step t , we deﬁne

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

Original: Note that (cid:98) h T = 1 T − p (cid:80) T − 1 t
Revised: Note that (cid:98) h T = 1 T − p (cid:80) T − 1 t

Original: [Equation]
Revised: [Equation]

Original: (cid:98) h T
Revised: (cid:98) h T

Original: [Equation]
Revised: [Equation]

Original: t
Revised: t

Original: [Equation]
Revised: [Equation]

Original: and (cid:98) d T = 1 T −
Revised: and (cid:98) d T = 1 T −

Original: [Equation]
Revised: [Equation]

Original: (cid:80)
Revised: (cid:80)

Original: [Equation]
Revised: [Equation]

Original: (cid:98)
Revised: (cid:98)

Original: [Equation]
Revised: [Equation]

Original: t )
Revised: t )

Original: [Equation]
Revised: [Equation]

Original: Let us set
Revised: Let us set

Original: [Equation]
Revised: [Equation]

Original: Note that q t is the numerator of the stochastic term in Equation 7 and a bounded deterministic function of S C at a given time step t .
Revised: Note that q t is the numerator of the stochastic term in Equation 7 and a bounded deterministic function of S C at a given time step t .

Original: For the stochastic term (cid:98) d T which appears in the denominator, we show its convergence.
Revised: For the stochastic term (cid:98) d T which appears in the denominator, we show its convergence.

Original: The following two lemmas provide the result.
Revised: The following two lemmas provide the result.

Original: Lemma 2.
Revised: Lemma 2.

Original: If the GSC process is α -mixing, then, as T → ∞ , we obtain
Revised: If the GSC process is α -mixing, then, as T → ∞ , we obtain

Original: [Equation]
Revised: [Equation]

Original: , and
Revised: , and

Original: [Equation]
Revised: [Equation]

Original: 0 , for any F .
Revised: 0 , for any F .

Original: Proof of Lemma 2.
Revised: Proof of Lemma 2.

Original: We show that variance divided by T converges to a non-negative constant.
Revised: We show that variance divided by T converges to a non-negative constant.

Original: Let
Revised: Let

Original: U T := (cid:80) t q t / √ T , where q t (as shown in Equation 14) is a bounded deterministic function of the state of X t at time t .
Revised: U T := (cid:80) t q t / √ T , where q t (as shown in Equation 14) is a bounded deterministic function of the state of X t at time t .

Original: As demonstrated in Sarkar et.
Revised: As demonstrated in Sarkar et.

Original: al.
Revised: al.

Original: Sarkar et al.
Revised: Sarkar et al.

Original: (2014a), we too break our weighted sum U T across three time intervals: (i) [1 , T C − 1] , (ii)
Revised: (2014a), we too break our weighted sum U T across three time intervals: (i) [1 , T C − 1] , (ii)

Original: [ T C , T C + M − 1] , and (iii)
Revised: [ T C , T C + M − 1] , and (iii)

Original: [ T C + M, T ] , where M is a constant.
Revised: [ T C + M, T ] , where M is a constant.

Original: Now, from Sarkar et al.
Revised: Now, from Sarkar et al.

Original: (2014a)
Revised: (2014a)

Original: , we simply apply Lemma 5 .to get that E
Revised: , we simply apply Lemma 5 .to get that E

Original: [Var( U T | E ( T C ) , S C ) | S C ]
Revised: [Var( U T | E ( T C ) , S C ) | S C ]

Original: → σ c , for some σ c ≥ 0 and from Lemma 5 . 8 , we have that Var( E [ U T | E ( T C ) , S C ] | S C )
Revised: → σ c , for some σ c ≥ 0 and from Lemma 5 . 8 , we have that Var( E [ U T | E ( T C ) , S C ] | S C )

Original: =
Revised: =

Original: o (1) .
Revised: o (1) .

Original: Now, since the law of total variance provides
Revised: Now, since the law of total variance provides

Original: [Equation]
Revised: [Equation]

Original: we use the previous results from Lemmas 5 .
Revised: we use the previous results from Lemmas 5 .

Original: 7 and 5 . 

Revised: 7 and 5 .

Original: in Sarkar et al.
Revised: in Sarkar et al.

Original: (2014a) to get
Revised: (2014a) to get

Original: Var( U T | S C ) →
Revised: Var( U T | S C ) →

Original: σ c as T → 0 , for some constant σ c ≥
Revised: σ c as T → 0 , for some constant σ c ≥

Original: Plugging in the deﬁnition of q t into U T and calculating Var( U T | S C ) , it follows trivially that
Revised: Plugging in the deﬁnition of q t into U T and calculating Var( U T | S C ) , it follows trivially that

Original: Var( (cid:98) h T | S C ) → 0 and Var( (cid:98) d T | S C )
Revised: Var( (cid:98) h T | S C ) → 0 and Var( (cid:98) d T | S C )

Original: → 0 as T → ∞ .
Revised: → 0 as T → ∞ .

Original: We refer readers to Remark 5 .
Revised: We refer readers to Remark 5 .

Original: 10 in Sarkar et al.
Revised: 10 in Sarkar et al.

Original: (2014a) to see how these results also hold in the case when C is aperiodic .
Revised: (2014a) to see how these results also hold in the case when C is aperiodic .

Original: Lemma 3.
Revised: Lemma 3.

Original: If the GSC process is α -mixing, then there exist a function R ( C ) with a deterministic function of class C denote, such as
Revised: If the GSC process is α -mixing, then there exist a function R ( C ) with a deterministic function of class C denote, such as

Original: [Equation]
Revised: [Equation]

Original: Proof of Lemma 3.
Revised: Proof of Lemma 3.

Original: We know by deﬁnition that
Revised: We know by deﬁnition that

Original: [Equation]
Revised: [Equation]

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: (cid:35) (15)
Revised: (cid:35) (15)

Original: This is an average of terms E (cid:20) | P t +1 ( σ ( d ) j ,F ) | |G t ( d ) | (cid:12)(cid:12)(cid:12)(cid:12)
Revised: This is an average of terms E (cid:20) | P t +1 ( σ ( d ) j ,F ) | |G t ( d ) | (cid:12)(cid:12)(cid:12)(cid:12)

Original: E
Revised: E

Original: ( T C ) , S C (cid:21) spanning across d -simplices with indices j ∈ { 1 , · · · , |G t ( d ) |} and discrete time steps t ∈ { p, · · · , T − 1 } .
Revised: ( T C ) , S C (cid:21) spanning across d -simplices with indices j ∈ { 1 , · · · , |G t ( d ) |} and discrete time steps t ∈ { p, · · · , T − 1 } .

Original: For ease of notation, let
Revised: For ease of notation, let

Original: [Equation]
Revised: [Equation]

Original: X j denotes the total number of possible ( d + 1) -simplices with a d -face as σ ( d ) j divided by the total number of d -simplices in G t .
Revised: X j denotes the total number of possible ( d + 1) -simplices with a d -face as σ ( d ) j divided by the total number of d -simplices in G t .

Original: In the R.H.S. of Equation 15, the term inside the summation is simpliﬁed as
Revised: In the R.H.S. of Equation 15, the term inside the summation is simpliﬁed as

Original: [Equation]
Revised: [Equation]

Original: We know that both P t +1 ( σ ( d ) j , F ) and G t ( d ) are fully determined given the current state S t of the
Revised: We know that both P t +1 ( σ ( d ) j , F ) and G t ( d ) are fully determined given the current state S t of the

Original: Markov chain.
Revised: Markov chain.

Original: Let I S ( Y ) denote an indicator variable of whether “ Y is in state S ” or not.
Revised: Let I S ( Y ) denote an indicator variable of whether “ Y is in state S ” or not.

Original: We have,
Revised: We have,

Original: [Equation]
Revised: [Equation]

Original: As a result, the R.H.S. of Equation 15 becomes
Revised: As a result, the R.H.S. of Equation 15 becomes

Original: [Equation]
Revised: [Equation]

Original: Let λ ( S ) =
Revised: Let λ ( S ) =

Original: (cid:80) j,x x I S
Revised: (cid:80) j,x x I S

Original: ( X j = x ) as this term is fully determined by state S .
Revised: ( X j = x ) as this term is fully determined by state S .

Original: Then, Equation 16 can be rewritten as
Revised: Then, Equation 16 can be rewritten as

Original: [Equation]
Revised: [Equation]

Original: Due to stationarity, the average (cid:80) t Pr [ S t = S | E
Revised: Due to stationarity, the average (cid:80) t Pr [ S t = S | E

Original: ( T C ) , S C ] /T will converge to a constant function of state S , denoted by R ( S ) .
Revised: ( T C ) , S C ] /T will converge to a constant function of state S , denoted by R ( S ) .

Original: Given that λ ( S ) is bounded and the average term converges to a constant R ( S ) , we say that Equation 17 converges to some constant R ( C ) > 0 , where R ( C ) is a deterministic function of communication class C .
Revised: Given that λ ( S ) is bounded and the average term converges to a constant R ( S ) , we say that Equation 17 converges to some constant R ( C ) > 0 , where R ( C ) is a deterministic function of communication class C .

Original: This proves the ﬁrst part.
Revised: This proves the ﬁrst part.

Original: A simple application of the tower property of expectation followed by the dominated convergence theorem shows that lim T →∞ E [
Revised: A simple application of the tower property of expectation followed by the dominated convergence theorem shows that lim T →∞ E [

Original: [Equation]
Revised: [Equation]

Original: F ) | S C ]
Revised: F ) | S C ]

Original: = R C .
Revised: = R C .

Original: This completes the proof for the result.
Revised: This completes the proof for the result.

Original: Then, we are ready to prove the convergence of the variance term.
Revised: Then, we are ready to prove the convergence of the variance term.

Original: Proposition 3 (Variance) .
Revised: Proposition 3 (Variance) .

Original: If the GSC ﬁltration process is α -mixing, then, conditional on S C , we obtain V T p → 0 as T → ∞ .
Revised: If the GSC ﬁltration process is α -mixing, then, conditional on S C , we obtain V T p → 0 as T → ∞ .

Original: Proof of Proposition 3.
Revised: Proof of Proposition 3.

Original: By Proposition 2, the term V T is written as (
Revised: By Proposition 2, the term V T is written as (

Original: [Equation]
Revised: [Equation]

Original: For the denominator, Lemma 3 shows that E [ (cid:98) d T | S C ]
Revised: For the denominator, Lemma 3 shows that E [ (cid:98) d T | S C ]

Original: → R ( C ) , where R ( C ) is a positive deterministic function of class C .
Revised: → R ( C ) , where R ( C ) is a positive deterministic function of class C .

Original: Also, Lemma 2 states that V ar ( (cid:98) d T | S C ) → 0 holds as T → ∞ .
Revised: Also, Lemma 2 states that V ar ( (cid:98) d T | S C ) → 0 holds as T → ∞ .

Original: Thus, (cid:98) d T p → R ( C ) > 0 holds.
Revised: Thus, (cid:98) d T p → R ( C ) > 0 holds.

Original: Also, V T is asymptotically well deﬁned for class C .
Revised: Also, V T is asymptotically well deﬁned for class C .

Original: For the nominator, Lemma 2 also shows that lim T →∞ V ar ( (cid:80) t q t /T | S C ) = 0 as T → ∞ and
Revised: For the nominator, Lemma 2 also shows that lim T →∞ V ar ( (cid:80) t q t /T | S C ) = 0 as T → ∞ and

Original: E [ q t | S C ] = 0 , therefore we have lim T →∞ 1 /T
Revised: E [ q t | S C ] = 0 , therefore we have lim T →∞ 1 /T

Original: (cid:80) t q t qm → 0 conditioned on S C .
Revised: (cid:80) t q t qm → 0 conditioned on S C .

Original: By the continuous mapping theorem, we obtain the statement.
Revised: By the continuous mapping theorem, we obtain the statement.

Original: Next, we discuss the bias term B T .
Revised: Next, we discuss the bias term B T .

Original: To this aim, we rewrite the term B T ( F, C ) as follows:
Revised: To this aim, we rewrite the term B T ( F, C ) as follows:

Original: [Equation]
Revised: [Equation]

Original: Lemma 4.
Revised: Lemma 4.

Original: Given that Assumption 1 holds and that when T → ∞ , the bandwidth parameter b → 0 .
Revised: Given that Assumption 1 holds and that when T → ∞ , the bandwidth parameter b → 0 .

Original: Then, we have that B T ( F, C )
Revised: Then, we have that B T ( F, C )

Original: =
Revised: =

Original: O ( b ) =
Revised: O ( b ) =

Original: o (1) as T → ∞ .
Revised: o (1) as T → ∞ .

Original: Proof of Lemma 4.
Revised: Proof of Lemma 4.

Original: For t ∈ [ p, T − 1] , j ∈ [1 , |G t ( d ) | ] and a ﬁxed feature vector F , the numerator of B T ( F, C ) can be expressed as an average of terms of the form
Revised: For t ∈ [ p, T − 1] , j ∈ [1 , |G t ( d ) | ] and a ﬁxed feature vector F , the numerator of B T ( F, C ) can be expressed as an average of terms of the form

Original: [Equation]
Revised: [Equation]

Original: The ﬁrst term in Equation 18 can be rewritten using the tower property as
Revised: The ﬁrst term in Equation 18 can be rewritten using the tower property as

Original: [Equation]
Revised: [Equation]

Original: When we condition on E ( T C ) , it makes | P τt +1 ( σ ( d ) j ,F )
Revised: When we condition on E ( T C ) , it makes | P τt +1 ( σ ( d ) j ,F )

Original: | |G t ( d ) | conditionally independent of S C , if t > T C .
Revised: | |G t ( d ) | conditionally independent of S C , if t > T C .

Original: Also, for t ≥ T C , we have
Revised: Also, for t ≥ T C , we have

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

Original: where (cid:101) v n ∈ B t − 1 ,k ( σ ( d ) j ) .
Revised: where (cid:101) v n ∈ B t − 1 ,k ( σ ( d ) j ) .

Original: Given the result in Equation 19 and the fact that the term | P τt +1 ( σ
Revised: Given the result in Equation 19 and the fact that the term | P τt +1 ( σ

Original: [Equation]
Revised: [Equation]

Original: d ) j ,F ) | |G t ( d ) | is bounded results in
Revised: d ) j ,F ) | |G t ( d ) | is bounded results in

Original: [Equation]
Revised: [Equation]

Original: where c > 0 an existing constant.
Revised: where c > 0 an existing constant.

Original: Now, the numerator of B T ( F, C ) can be upper bounded as
Revised: Now, the numerator of B T ( F, C ) can be upper bounded as

Original: [Equation]
Revised: [Equation]

Original: The second term in Equation 21 vanishes as T → ∞ because it is of order O ( E [ T C ] /T ) .
Revised: The second term in Equation 21 vanishes as T → ∞ because it is of order O ( E [ T C ] /T ) .

Original: Thus, the numerator of B T ( F, C ) is an average of terms of the form
Revised: Thus, the numerator of B T ( F, C ) is an average of terms of the form

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Our feature vector counts and simplex neighborhoods are ﬁnite because |G t ( d ) | is bounded.
Revised: Our feature vector counts and simplex neighborhoods are ﬁnite because |G t ( d ) | is bounded.

Original: The expectation in Equation 22 is just a summation of ﬁnite terms.
Revised: The expectation in Equation 22 is just a summation of ﬁnite terms.

Original: We set F (cid:48)
Revised: We set F (cid:48)

Original: =
Revised: =

Original: F t ( σ ( d ) j , (cid:101) v n ) , and make use of our smoothness assumption 1, so that
Revised: F t ( σ ( d ) j , (cid:101) v n ) , and make use of our smoothness assumption 1, so that

Original: [Equation]
Revised: [Equation]

Original: We also use Lemma 3 to say that the denominator of our bias term converges to a constant R ( C ) .
Revised: We also use Lemma 3 to say that the denominator of our bias term converges to a constant R ( C ) .

Original: So,
Revised: So,

Original: [Equation]
Revised: [Equation]

Original: The last equality follows the property of κ in the Schwartz space.
Revised: The last equality follows the property of κ in the Schwartz space.

Original: Now, since B T ( F, C ) =
Revised: Now, since B T ( F, C ) =

Original: O ( b ) and b → 0 as T → ∞ , then we have that B T ( F, C )
Revised: O ( b ) and b → 0 as T → ∞ , then we have that B T ( F, C )

Original: =
Revised: =

Original: o (1) .
Revised: o (1) .

Original: This completes the proof.
Revised: This completes the proof.

Original: Then, we prove the convergence of the bias.
Revised: Then, we prove the convergence of the bias.

Original: Then, we have the result:
Revised: Then, we have the result:

Original: Proposition 4 (Bias) .
Revised: Proposition 4 (Bias) .

Original: If Assumption 1 holds, then, conditional on S C , B T p → 0 holds as T → ∞ .
Revised: If Assumption 1 holds, then, conditional on S C , B T p → 0 holds as T → ∞ .

Original: Proof of Proposition 4.
Revised: Proof of Proposition 4.

Original: Proposition 2 shows that B T ( F ) =
Revised: Proposition 2 shows that B T ( F ) =

Original: B T ( F, C ) / (cid:98) d T ( F ) .
Revised: B T ( F, C ) / (cid:98) d T ( F ) .

Original: By Lemma 2 and 3, we obtain (cid:98) d T → R ( C ) > 0 as T → ∞ , as similarly shown in the proof of
Revised: By Lemma 2 and 3, we obtain (cid:98) d T → R ( C ) > 0 as T → ∞ , as similarly shown in the proof of

Original: Proposition 3.
Revised: Proposition 3.

Original: Further, Lemma 4 states that B T ( F, C )
Revised: Further, Lemma 4 states that B T ( F, C )

Original: =
Revised: =

Original: o ( b ) .
Revised: o ( b ) .

Original: Combining the results, we obtain the statement.
Revised: Combining the results, we obtain the statement.

Original: Now, we can prove the consistency (Theorem 1).
Revised: Now, we can prove the consistency (Theorem 1).

Original: Proof of Theorem 1.
Revised: Proof of Theorem 1.

Original: For the result of (cid:98) g T , we apply the results of Proposition 3 and 4 to the decomposition in the equation 7, then obtain the statement.
Revised: For the result of (cid:98) g T , we apply the results of Proposition 3 and 4 to the decomposition in the equation 7, then obtain the statement.

Original: For the result of (cid:101) g T , we additionally combine the result of Lemma 1, then obtain the statement.
Revised: For the result of (cid:101) g T , we additionally combine the result of Lemma 1, then obtain the statement.

Original: D P ROOF FOR A SYMPTOTIC N ORMALITY D. I NTRODUCTION TO W ASSERSTEIN DISTANCE AND APPROXIMATION TECHNIQUE
Revised: D P ROOF FOR A SYMPTOTIC N ORMALITY D. I NTRODUCTION TO W ASSERSTEIN DISTANCE AND APPROXIMATION TECHNIQUE

Original: We denote by BL ( R ) the space of such bounded functions h that are 1 -Lipschitz.
Revised: We denote by BL ( R ) the space of such bounded functions h that are 1 -Lipschitz.

Original: More formally, (cid:107)
Revised: More formally, (cid:107)

Original: [Equation]
Revised: [Equation]

Original: ∞ =
Revised: ∞ =

Original: sup x ∈ R | h ( x ) | < ∞ and Lip ( h )
Revised: sup x ∈ R | h ( x ) | < ∞ and Lip ( h )

Original: = 1 , where Lip ( h ) =
Revised: = 1 , where Lip ( h ) =

Original: sup x (cid:54) =
Revised: sup x (cid:54) =

Original: y | h ( x ) − h
Revised: y | h ( x ) − h

Original: ( y ) | |
Revised: ( y ) | |

Original: [Equation]
Revised: [Equation]

Original: y |
Revised: y |

Original: So, h ∈ BL ( R ) .
Revised: So, h ∈ BL ( R ) .

Original: We now make use of the Wasserstein metric to measure the distance between distributions.
Revised: We now make use of the Wasserstein metric to measure the distance between distributions.

Original: Therefore, our estimator represented as W can be shown to converge to Z , when the Wasserstein distance between W and Z ’s underlying distributions converges to zero.
Revised: Therefore, our estimator represented as W can be shown to converge to Z , when the Wasserstein distance between W and Z ’s underlying distributions converges to zero.

Original: We have that
Revised: We have that

Original: [Equation]
Revised: [Equation]

Original: D.1. I NTRODUCTION TO S TEIN ’ S M ETHOD FOR NORMAL APPROXIMATION
Revised: D.1. I NTRODUCTION TO S TEIN ’ S M ETHOD FOR NORMAL APPROXIMATION

Original: Stein Stein et al.
Revised: Stein Stein et al.

Original: (1986) introduced a powerful technique to estimate the rate of convergence of sums of weakly dependent r.v.s to the standard normal distribution.
Revised: (1986) introduced a powerful technique to estimate the rate of convergence of sums of weakly dependent r.v.s to the standard normal distribution.

Original: A remarkable feature of Stein’s method is that it can be applied in many circumstances where dependence plays a role, therefore we propose an adaptation of Stein’s method to our setting of dynamic GSCs.
Revised: A remarkable feature of Stein’s method is that it can be applied in many circumstances where dependence plays a role, therefore we propose an adaptation of Stein’s method to our setting of dynamic GSCs.

Original: Given a standard normal r.v. Z , Stein’s lemma (stated below) provides a characterization of Z ’s distribution.
Revised: Given a standard normal r.v. Z , Stein’s lemma (stated below) provides a characterization of Z ’s distribution.

Original: Lemma 5 (Stein’s Lemma Chen et al.
Revised: Lemma 5 (Stein’s Lemma Chen et al.

Original: If W has a standard normal distribution, then
Revised: If W has a standard normal distribution, then

Original: [Equation]
Revised: [Equation]

Original: 24) for all absolutely continuous functions f : R → R with E | f (cid:48)
Revised: 24) for all absolutely continuous functions f : R → R with E | f (cid:48)

Original: ( Z ) | < ∞ .
Revised: ( Z ) | < ∞ .

Original: Conversely, if Equation 24 holds for all bounded, continuous, and piecewise continuously differentiable functions f with E | f (cid:48)
Revised: Conversely, if Equation 24 holds for all bounded, continuous, and piecewise continuously differentiable functions f with E | f (cid:48)

Original: ( Z ) | < ∞ , then W has a standard normal distribution.
Revised: ( Z ) | < ∞ , then W has a standard normal distribution.

Original: In order to show that a r.v. W has a distribution close to that of a target distribution of Z , one must compare the values of expectations of the two distributions on some collection of bounded functions h : R → R .
Revised: In order to show that a r.v. W has a distribution close to that of a target distribution of Z , one must compare the values of expectations of the two distributions on some collection of bounded functions h : R → R .

Original: Here, Stein’s lemma (Lemma 5) shows that W d = Z , if
Revised: Here, Stein’s lemma (Lemma 5) shows that W d = Z , if

Original: [Equation]
Revised: [Equation]

Original: holds.
Revised: holds.

Original: Observe that if the distribution of W is close to that of Z ’s distribution, then evaluating the L.H.S of Equation 25 when W is replaced by Z would result in a small value.
Revised: Observe that if the distribution of W is close to that of Z ’s distribution, then evaluating the L.H.S of Equation 25 when W is replaced by Z would result in a small value.

Original: Putting these difference equations together, the following linear differential equation known as Stein’s equation is arrived at
Revised: Putting these difference equations together, the following linear differential equation known as Stein’s equation is arrived at

Original: [Equation]
Revised: [Equation]

Original: The f that satisfy Equation 26 with h ∈ BL ( R ) must satisfy the following conditions for all y, z ∈ R
Revised: The f that satisfy Equation 26 with h ∈ BL ( R ) must satisfy the following conditions for all y, z ∈ R

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where h 0 ( y ) =
Revised: where h 0 ( y ) =

Original: h ( y ) − E h
Revised: h ( y ) − E h

Original: ( Z ) , c 1 = sup x ≥ 0 ξ ( x ) , c
Revised: ( Z ) , c 1 = sup x ≥ 0 ξ ( x ) , c

Original: 2 = sup x ≥ 0 x (1 − xξ ( x )) , and ξ ( x ) =
Revised: 2 = sup x ≥ 0 x (1 − xξ ( x )) , and ξ ( x ) =

Original: (1 − Φ)
Revised: (1 − Φ)

Original: /φ
Revised: /φ

Original: (where Φ( x ) is the distribution function and φ ( x ) =
Revised: (where Φ( x ) is the distribution function and φ ( x ) =

Original: Φ (cid:48)
Revised: Φ (cid:48)

Original: ( x ) ).
Revised: ( x ) ).

Original: Then, D = ( c 1 + c 2 ) (cid:107) h 0 (cid:107) ∞ + 2 is a constant.
Revised: Then, D = ( c 1 + c 2 ) (cid:107) h 0 (cid:107) ∞ + 2 is a constant.

Original: Additionally, we have a bound on the covariance, given the dependent r.v.’s are also bounded.
Revised: Additionally, we have a bound on the covariance, given the dependent r.v.’s are also bounded.

Original: [Equation]
Revised: [Equation]

Original: We take a similar approach to Sarkar et.
Revised: We take a similar approach to Sarkar et.

Original: al.
Revised: al.

Original: Sarkar et al.
Revised: Sarkar et al.

Original: (2014a)
Revised: (2014a)

Original: in terms of using the Wasserstein distance to bound the normal approximation.
Revised: in terms of using the Wasserstein distance to bound the normal approximation.

Original: We ﬁrst deﬁne the dependency in our GSCs and then propose a notion of α mixing in our context of GSCs.
Revised: We ﬁrst deﬁne the dependency in our GSCs and then propose a notion of α mixing in our context of GSCs.

Original: We obtain a tighter bound than the bound proposed in Sarkar et al.
Revised: We obtain a tighter bound than the bound proposed in Sarkar et al.

Original: (2014a), by instead following an approach proposed by Sunklodas Sunklodas (2007).
Revised: (2014a), by instead following an approach proposed by Sunklodas Sunklodas (2007).

Original: D. G AUSSIAN APPROXIMATION FOR DEPENDENT VARIABLES WITH GSC
Revised: D. G AUSSIAN APPROXIMATION FOR DEPENDENT VARIABLES WITH GSC

Original: In our model, we assume the r.v.
Revised: In our model, we assume the r.v.

Original: A i to represent a d -simplex σ ( d ) i in a GSC G .
Revised: A i to represent a d -simplex σ ( d ) i in a GSC G .

Original: In order to have a notion of α -mixing in our setting, we must ﬁrst deﬁne a distance between two d -simplices σ i and σ j .
Revised: In order to have a notion of α -mixing in our setting, we must ﬁrst deﬁne a distance between two d -simplices σ i and σ j .

Original: We drop the ( d ) superscript for brevity and ease of notation.
Revised: We drop the ( d ) superscript for brevity and ease of notation.

Original: We deﬁne this distance as the
Revised: We deﬁne this distance as the

Original: Hausdorff distance between simplices as
Revised: Hausdorff distance between simplices as

Original: [Equation]
Revised: [Equation]

Original: where d g ( v, v (cid:48) ) counts the number of edges in the geodesic connecting vertices v and v (cid:48) in G (1) − .
Revised: where d g ( v, v (cid:48) ) counts the number of edges in the geodesic connecting vertices v and v (cid:48) in G (1) − .

Original: In Stein’s method, the sum of dependent r.v.’s is studied by breaking the sum Z n into two sets based on the r in mixing coefﬁcient α ( r ) .
Revised: In Stein’s method, the sum of dependent r.v.’s is studied by breaking the sum Z n into two sets based on the r in mixing coefﬁcient α ( r ) .

Original: In our setting, given a ﬁxed d -simplex σ i , we study two partial sums pertaining to: 1) all d -simplices that are at most r -apart from A i and 2) the remaining partial sum after removing the variables pertaining to 1) from Z n .
Revised: In our setting, given a ﬁxed d -simplex σ i , we study two partial sums pertaining to: 1) all d -simplices that are at most r -apart from A i and 2) the remaining partial sum after removing the variables pertaining to 1) from Z n .

Original: With this notion of distance between sets of r.v.’s, we modify with slight deviations from proposition 4 in Sunklodas et.
Revised: With this notion of distance between sets of r.v.’s, we modify with slight deviations from proposition 4 in Sunklodas et.

Original: al.
Revised: al.

Original: Sunklodas (2007) to accommodate our α -mixing in Markov chains based on
Revised: Sunklodas (2007) to accommodate our α -mixing in Markov chains based on

Original: GSCs.
Revised: GSCs.

Original: For a sequence of r.v.’s
Revised: For a sequence of r.v.’s

Original: X 1 , X 2 , · · · satisfying the α -mixing condition, we write
Revised: X 1 , X 2 , · · · satisfying the α -mixing condition, we write

Original: [Equation]
Revised: [Equation]

Original: We assume B n > 0 .
Revised: We assume B n > 0 .

Original: T ( m ) i denotes the contribution of d -simplices that are further than m away from σ i and x ( σ i , r ) denotes the partial sum of r.v.’s representing simplices that are exactly r away from A i .
Revised: T ( m ) i denotes the contribution of d -simplices that are further than m away from σ i and x ( σ i , r ) denotes the partial sum of r.v.’s representing simplices that are exactly r away from A i .

Original: Therefore, (cid:80) mr =0 x ( σ i , r ) gets us all those d -simplices that are greater than or equal to m away from σ i .
Revised: Therefore, (cid:80) mr =0 x ( σ i , r ) gets us all those d -simplices that are greater than or equal to m away from σ i .

Original: We are interested in the contribution of simplices r away from σ i as we vary r from 0 to m .
Revised: We are interested in the contribution of simplices r away from σ i as we vary r from 0 to m .

Original: With
Revised: With

Original: Proposition 5, we proceed to derive an upper bound on d w ( Z n , N ) (i.e., the Wasserstein distance between Z n and N ).
Revised: Proposition 5, we proceed to derive an upper bound on d w ( Z n , N ) (i.e., the Wasserstein distance between Z n and N ).

Original: Proposition 5.
Revised: Proposition 5.

Original: Let S ( σ i , r ) denote the set of d -simplices whose Hausdorff distance equals r .
Revised: Let S ( σ i , r ) denote the set of d -simplices whose Hausdorff distance equals r .

Original: More formally,
Revised: More formally,

Original: [Equation]
Revised: [Equation]

Original: Additionally, let (cid:98) X denote a mean-centered version of r.v. X .
Revised: Additionally, let (cid:98) X denote a mean-centered version of r.v. X .

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: Suppose that E Z n = 0 , E Z 2 n = 1 , and E A 2 i < ∞ for all i = 1 , · · · , n .
Revised: Suppose that E Z n = 0 , E Z 2 n = 1 , and E A 2 i < ∞ for all i = 1 , · · · , n .

Original: Let (cid:15) be a r.v. uniformly distributed in [0 , 1] and independent of other r.v.’s.
Revised: Let (cid:15) be a r.v. uniformly distributed in [0 , 1] and independent of other r.v.’s.

Original: Let f : R → R be a differentiable function such that sup x ∈ R | f (cid:48)
Revised: Let f : R → R be a differentiable function such that sup x ∈ R | f (cid:48)

Original: ( x ) | < ∞ .
Revised: ( x ) | < ∞ .

Original: Then we have
Revised: Then we have

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: Theorem 3.
Revised: Theorem 3.

Original: Consider a sequence of r
Revised: Consider a sequence of r

Original: [Equation]
Revised: [Equation]

Original: v.’s X 1 , X 2 , · · · that satisfy α -mixing condition (Deﬁnition 5).
Revised: v.’s X 1 , X 2 , · · · that satisfy α -mixing condition (Deﬁnition 5).

Original: Let E X i = 0 , Pr {| X i | ≤ L } = 1 , for i = 1 , · · · , n , for some constant L > 0 .
Revised: Let E X i = 0 , Pr {| X i | ≤ L } = 1 , for i = 1 , · · · , n , for some constant L > 0 .

Original: Then, for every h ∈ BL ( R ) ,
Revised: Then, for every h ∈ BL ( R ) ,

Original: [Equation]
Revised: [Equation]

Original: where C ( S, D ) is a ﬁnite constant which depends on D and max i | S ( σ i , r ) | .
Revised: where C ( S, D ) is a ﬁnite constant which depends on D and max i | S ( σ i , r ) | .

Original: Proof of Theorem 3.
Revised: Proof of Theorem 3.

Original: Given a d -simplex σ i and its corresponding r.v.
Revised: Given a d -simplex σ i and its corresponding r.v.

Original: A i , recall that S ( σ i , r ) denotes the set of d -simplices that are at Hausdorff distance r away from σ i .
Revised: A i , recall that S ( σ i , r ) denotes the set of d -simplices that are at Hausdorff distance r away from σ i .

Original: We additionally deﬁne S m to denote the maximum cardinality of S ( σ i , r ) for all i and a ﬁxed r , i.e.,
Revised: We additionally deﬁne S m to denote the maximum cardinality of S ( σ i , r ) for all i and a ﬁxed r , i.e.,

Original: [Equation]
Revised: [Equation]

Original: In order to upper bound the Wasserstein distance between Z n and N , we estimate the difference E h
Revised: In order to upper bound the Wasserstein distance between Z n and N , we estimate the difference E h

Original: ( Z n ) − E h ( N ) using Proposition 5.
Revised: ( Z n ) − E h ( N ) using Proposition 5.

Original: It was shown in Proposition 5 that this difference is a sum of terms E 1 , · · · , E 7 .
Revised: It was shown in Proposition 5 that this difference is a sum of terms E 1 , · · · , E 7 .

Original: We will proceed by individually bounding each term.
Revised: We will proceed by individually bounding each term.

Original: Bounding E 1 :
Revised: Bounding E 1 :

Original: [Equation]
Revised: [Equation]

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

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

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

Original: [Equation]
Revised: [Equation]

Original: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
Revised: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: We can upper bound term ( i ) in Equation 30 using Equation 27 by
Revised: We can upper bound term ( i ) in Equation 30 using Equation 27 by

Original: [Equation]
Revised: [Equation]

Original: (since | (cid:15) | ≤ 1 )
Revised: (since | (cid:15) | ≤ 1 )

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: Now, we upper bound x ( σ i , r ) as
Revised: Now, we upper bound x ( σ i , r ) as

Original: [Equation]
Revised: [Equation]

Original: (32) since each normalized r.v. in Z n is upper bounded by L/B n .
Revised: (32) since each normalized r.v. in Z n is upper bounded by L/B n .

Original: The L.H.S. of Equation 31 is upper bounded by DS 2 m (cid:16) L 2 B 2 n (cid:17) .
Revised: The L.H.S. of Equation 31 is upper bounded by DS 2 m (cid:16) L 2 B 2 n (cid:17) .

Original: We know that
Revised: We know that

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

Original: Notice that term ( ii ) is nothing but the summand in Equation 30.
Revised: Notice that term ( ii ) is nothing but the summand in Equation 30.

Original: We apply the covariance bounds (Equation 28), to obtain
Revised: We apply the covariance bounds (Equation 28), to obtain

Original: [Equation]
Revised: [Equation]

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: we have that
Revised: we have that

Original: [Equation]
Revised: [Equation]

Original: Here, the summation (cid:80) n − 1 r =1 appears, because there are only n variables that should measure the dependence between each other.
Revised: Here, the summation (cid:80) n − 1 r =1 appears, because there are only n variables that should measure the dependence between each other.

Original: Bounding E 2 :
Revised: Bounding E 2 :

Original: [Equation]
Revised: [Equation]

Original: Using Equation
Revised: Using Equation

Original: [Equation]
Revised: [Equation]

Original: we have that term ( iii ) ≤ D |
Revised: we have that term ( iii ) ≤ D |

Original: A i | .
Revised: A i | .

Original: Then, | E 2 | can simply be bounded as | E
Revised: Then, | E 2 | can simply be bounded as | E

Original: [Equation]
Revised: [Equation]

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

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

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

Original: Bounding E 3 :
Revised: Bounding E 3 :

Original: [Equation]
Revised: [Equation]

Original: Let us focus on bounding terms ( iv ) and ( v ) , separately.
Revised: Let us focus on bounding terms ( iv ) and ( v ) , separately.

Original: For term ( iv ) : We can further split it as
Revised: For term ( iv ) : We can further split it as

Original: [Equation]
Revised: [Equation]

Original: We have previously worked out the bounds for terms A i and x ( σ i , r ) .
Revised: We have previously worked out the bounds for terms A i and x ( σ i , r ) .

Original: Therefore, we now focus our attention on bounding δ ( q ) i .
Revised: Therefore, we now focus our attention on bounding δ ( q ) i .

Original: [Equation]
Revised: [Equation]

Original: (using Equation 27)
Revised: (using Equation 27)

Original: [Equation]
Revised: [Equation]

Original: Therefore, term b (in term ( iv ) ) is upper bounded by DS 2 m L 2 B 2 n .
Revised: Therefore, term b (in term ( iv ) ) is upper bounded by DS 2 m L 2 B 2 n .

Original: Applying the covariance bound (Equation 28), we have that
Revised: Applying the covariance bound (Equation 28), we have that

Original: [Equation]
Revised: [Equation]

Original: For term ( v ) :
Revised: For term ( v ) :

Original: We have calculated some bounds previously, so we can again split ( v ) as
Revised: We have calculated some bounds previously, so we can again split ( v ) as

Original: [Equation]
Revised: [Equation]

Original: Now, combining the inequalities for terms ( iv ) and ( v )
Revised: Now, combining the inequalities for terms ( iv ) and ( v )

Original: [Equation]
Revised: [Equation]

Original: we have that
Revised: we have that

Original: [Equation]
Revised: [Equation]

Original: Bounding E 4 : We have that
Revised: Bounding E 4 : We have that

Original: [Equation]
Revised: [Equation]

Original: ( r
Revised: ( r

Original: [Equation]
Revised: [Equation]

Original: because (cid:80) r ≥ 1 (cid:80) q ≥ 2 r
Revised: because (cid:80) r ≥ 1 (cid:80) q ≥ 2 r

Original: +

Revised: +

Original: α
Revised: α

Original: ( q − r ) < (cid:80)
Revised: ( q − r ) < (cid:80)

Original: n − 1 r =1 rα ( r ) .
Revised: n − 1 r =1 rα ( r ) .

Original: Now, using our previous bounds, the remaining terms, i.e., | E 5 | , | E 6 | , and | E 7 | are bounded as follows:
Revised: Now, using our previous bounds, the remaining terms, i.e., | E 5 | , | E 6 | , and | E 7 | are bounded as follows:

Original: [Equation]
Revised: [Equation]

Original: Finally, we combine these bounds to arrive at our ﬁnal upper bound as
Revised: Finally, we combine these bounds to arrive at our ﬁnal upper bound as

Original: [Equation]
Revised: [Equation]

Original: where C ( S m , D ) is a constant term depending on constants K m and D .
Revised: where C ( S m , D ) is a constant term depending on constants K m and D .

Original: This completes our proof.
Revised: This completes our proof.

Original: D. D EFERRED P ROOF
Revised: D. D EFERRED P ROOF

Original: In this section, we establish our estimator’s result on the asymptotic normality.
Revised: In this section, we establish our estimator’s result on the asymptotic normality.

Original: Let n = |G t ( d ) | denote the total number of d -simplices in GSC G t at time t .
Revised: Let n = |G t ( d ) | denote the total number of d -simplices in GSC G t at time t .

Original: Recall from Equationthat
Revised: Recall from Equationthat

Original: [Equation]
Revised: [Equation]

Original: Also, from Equation 14, we have
Revised: Also, from Equation 14, we have

Original: [Equation]
Revised: [Equation]

Original: Additionally, let us deﬁne a variable p t for convenience as follows
Revised: Additionally, let us deﬁne a variable p t for convenience as follows

Original: [Equation]
Revised: [Equation]

Original: Note that p t is conditioned on both E ( T C ) and S C , while q t is just conditioned on S C .
Revised: Note that p t is conditioned on both E ( T C ) and S C , while q t is just conditioned on S C .

Original: Keeping these expressions in mind, we begin by showing the weak convergence with p t .
Revised: Keeping these expressions in mind, we begin by showing the weak convergence with p t .

Original: Lemma 6.
Revised: Lemma 6.

Original: Under Assumption 1, given p t for any ﬁnite T C and σ c >
Revised: Under Assumption 1, given p t for any ﬁnite T C and σ c >

Original: [Equation]
Revised: [Equation]

Original: Proof of Lemma 6.
Revised: Proof of Lemma 6.

Original: Given a normalized r.v. W T which is a sum of weakly dependent r.v.’s as
Revised: Given a normalized r.v. W T which is a sum of weakly dependent r.v.’s as

Original: [Equation]
Revised: [Equation]

Original: where p t is already bounded and mean-centered .
Revised: where p t is already bounded and mean-centered .

Original: We have to show that our upper bound in Theorem 3 converges to zero, so that according to Stein’s lemma 5, we have
Revised: We have to show that our upper bound in Theorem 3 converges to zero, so that according to Stein’s lemma 5, we have

Original: [Equation]
Revised: [Equation]

Original: conditioned on E
Revised: conditioned on E

Original: [Equation]
Revised: [Equation]

Original: T C )
Revised: T C )

Original: [Equation]
Revised: [Equation]

Original: Recall that by Lemma 5 . 7 in Sarkar et al.
Revised: Recall that by Lemma 5 . 7 in Sarkar et al.

Original: (2014a), we have
Revised: (2014a), we have

Original: [Equation]
Revised: [Equation]

Original: Now, we show that conditioned on event E ( T C ) ∩ S C , our bound in Theorem 3 has a convergence rate of O
Revised: Now, we show that conditioned on event E ( T C ) ∩ S C , our bound in Theorem 3 has a convergence rate of O

Original: ( T − 1 / 2 ) .
Revised: ( T − 1 / 2 ) .

Original: Note that our p t corresponds to A i and T to n in Theorem 3.
Revised: Note that our p t corresponds to A i and T to n in Theorem 3.

Original: As p t is a function of S t , it involves p +
Revised: As p t is a function of S t , it involves p +

Original: GSCs ( G t − p +1 , · · · , G t +1 ) and the distance between the i and j -th GSC is deﬁned as dist ( i, j ) =
Revised: GSCs ( G t − p +1 , · · · , G t +1 ) and the distance between the i and j -th GSC is deﬁned as dist ( i, j ) =

Original: max( | i − j | − ( p + 1) , 0) .
Revised: max( | i − j | − ( p + 1) , 0) .

Original: Therefore, you will observe that we have for x ( σ i , r ) only 2 states that are at distance r apart from σ i , i.e., A i − r
Revised: Therefore, you will observe that we have for x ( σ i , r ) only 2 states that are at distance r apart from σ i , i.e., A i − r

Original: and A i + r .
Revised: and A i + r .

Original: Therefore, S m =
Revised: Therefore, S m =

Original: O (1) .
Revised: O (1) .

Original: Given that Pr {| p t | ≤ L } = 1 , for t = 1 , · · · , T and for some constant bound L > 0 , we have
Revised: Given that Pr {| p t | ≤ L } = 1 , for t = 1 , · · · , T and for some constant bound L > 0 , we have

Original: T
Revised: T

Original: [Equation]
Revised: [Equation]

Original: (ignoring some constant terms)
Revised: (ignoring some constant terms)

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

Original: We additionally impose that (cid:80) ∞ r =1 rα ( r ) < ∞ because we use a decaying function for α ( · ) and
Revised: We additionally impose that (cid:80) ∞ r =1 rα ( r ) < ∞ because we use a decaying function for α ( · ) and

Original: B 2 T ≥ c 0 T with a positive constant c 0 .
Revised: B 2 T ≥ c 0 T with a positive constant c 0 .

Original: We choose
Revised: We choose

Original: [Equation]
Revised: [Equation]

Original: for r ≥ 2 and ﬁxed p > 1 , where 0 < C 1 < ∞ is a constant.
Revised: for r ≥ 2 and ﬁxed p > 1 , where 0 < C 1 < ∞ is a constant.

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: where K ( L, C 1 , c 0 ) is a positive and ﬁnite constant only depending on the quantities within the parenthesis.
Revised: where K ( L, C 1 , c 0 ) is a positive and ﬁnite constant only depending on the quantities within the parenthesis.

Original: The inequality follows the fact B 2 T ≥ c 0 T .
Revised: The inequality follows the fact B 2 T ≥ c 0 T .

Original: Thus, we achieve an asymptotic bound of
Revised: Thus, we achieve an asymptotic bound of

Original: O ( T − 1 / 2 ) .
Revised: O ( T − 1 / 2 ) .

Original: This completes our proof.
Revised: This completes our proof.

Original: The weak convergence with p t implies the weak convergence with q t , by a simple application of
Revised: The weak convergence with p t implies the weak convergence with q t , by a simple application of

Original: Lemma 7.2 in Sarkar et al.
Revised: Lemma 7.2 in Sarkar et al.

Original: (2014a).
Revised: (2014a).

Original: Lemma 7 (Lemma 7.2 in Sarkar et al.
Revised: Lemma 7 (Lemma 7.2 in Sarkar et al.

Original: (2014a)) .
Revised: (2014a)) .

Original: Suppose Lemma 6 holds.
Revised: Suppose Lemma 6 holds.

Original: Then, under Assumptionand assuming σ c > 0 ,
Revised: Then, under Assumptionand assuming σ c > 0 ,

Original: Conditioned on S C , (cid:88)
Revised: Conditioned on S C , (cid:88)

Original: [Equation]
Revised: [Equation]

Original: We now prove the weak convergence of our estimator.
Revised: We now prove the weak convergence of our estimator.

Original: Proof of Theorem 2.
Revised: Proof of Theorem 2.

Original: By the deﬁnition of (cid:98) g T − g in Theorem 1, we achieve
Revised: By the deﬁnition of (cid:98) g T − g in Theorem 1, we achieve

Original: [Equation]
Revised: [Equation]

Original: We are able to assess the convergence of each item.
Revised: We are able to assess the convergence of each item.

Original: Since Lemma 3 shows E[ (cid:98) d T | S C ] → R ( C ) and
Revised: Since Lemma 3 shows E[ (cid:98) d T | S C ] → R ( C ) and

Original: Lemma 2 shows Var( (cid:98) d T | S C ) → 0 , we obtain (cid:98) d
Revised: Lemma 2 shows Var( (cid:98) d T | S C ) → 0 , we obtain (cid:98) d

Original: T p → R ( C ) .
Revised: T p → R ( C ) .

Original: By Lemma 7, T − 1 / 2 (cid:80) t q t converges to N (0 , σ 2 c ) .
Revised: By Lemma 7, T − 1 / 2 (cid:80) t q t converges to N (0 , σ 2 c ) .

Original: Further, Lemma 4 proves B T
Revised: Further, Lemma 4 proves B T

Original: = O ( b ) .
Revised: = O ( b ) .

Original: Combining these results with the Slutsky’s lemma, we get the following results:
Revised: Combining these results with the Slutsky’s lemma, we get the following results:

Original: [Equation]
Revised: [Equation]

Original: where (cid:102) W T is a random variable such as
Revised: where (cid:102) W T is a random variable such as

Original: [Equation]
Revised: [Equation]

Original: With the settings β
Revised: With the settings β

Original: = o ( T − 1 / 2 ) and b = o ( T − 1 / 2 ) , we obtain the statement.
Revised: = o ( T − 1 / 2 ) and b = o ( T − 1 / 2 ) , we obtain the statement.