Original: Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunIT
Revised: Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunIT

Original: Abstract
Revised: Abstract

Original: Topological data analysis (TDA) delivers invaluable and complementary information on the intrinsic properties of data inaccessible to conventional methods.
Revised: Topological data analysis (TDA) delivers invaluable and complementary information on the intrinsic properties of data inaccessible to conventional methods.

Original: However, high computational costs remain the primary roadblock hindering the successful application of TDA in real-world studies, particularly with machine learning on large complex networks.
Revised: However, high computational costs remain the primary roadblock hindering the successful application of TDA in real-world studies, particularly with machine learning on large complex networks.

Original: Indeed, most modern networks such as citation, blockchain, and online social networks often have hundreds of thousands of vertices, making the application of existing TDA methods infeasible.
Revised: Indeed, most modern networks such as citation, blockchain, and online social networks often have hundreds of thousands of vertices, making the application of existing TDA methods infeasible.

Original: We develop two new, remarkably simple but effective algorithms to compute the exact persistence diagrams of large graphs to address this major TDA limitation.
Revised: We develop two new, remarkably simple but effective algorithms to compute the exact persistence diagrams of large graphs to address this major TDA limitation.

Original: First, we prove that ( k + 1) -core of a graph G sufﬁces to compute its k th persistence diagram, PD k ( G ) .
Revised: First, we prove that ( k + 1) -core of a graph G suffices to compute its k th persistence diagram, PD k ( G ) .

Original: Second, we introduce a pruning algorithm for graphs to compute their persistence diagrams by removing the dominated vertices.
Revised: Second, we introduce a pruning algorithm for graphs to compute their persistence diagrams by removing the dominated vertices.

Original: Our experiments on large networks show that our novel approach can achieve computational gains up to 95%.
Revised: Our experiments on large networks show that our novel approach can achieve computational gains up to 95%.

Original: The developed framework provides the ﬁrst bridge between the graph theory and
Revised: The developed framework provides the first bridge between the graph theory and

Original: TDA, with applications in machine learning of large complex networks.
Revised: TDA, with applications in machine learning of large complex networks.

Original: Our implementation is available at github.com/cakcora/PersistentHomologyWithCoralPrunit.
Revised: Our implementation is available at github.com/cakcora/PersistentHomologyWithCoralPrunit.

Original:  Introduction
Revised:  Introduction

Original: Topological data analysis (TDA) has emerged as powerful machinery in machine learning (ML), allowing us to extract complementary information on the observed objects, especially, from graphstructured data.
Revised: Topological data analysis (TDA) has emerged as powerful machinery in machine learning (ML), allowing us to extract complementary information on the observed objects, especially, from graphstructured data.

Original: In particular, TDA has become quite popular in various ML tasks, ranging from bioinformatics [38, 44], ﬁnance [39, 2] material science [33], biosurveillance [51, 18], network analysis [54, 16], as well as insurance and agriculture [56, 34] (see the literature overviews [4, 17] and the TDA applications library [27]).
Revised: In particular, TDA has become quite popular in various ML tasks, ranging from bioinformatics [38, 44], finance [39, 2] material science [33], biosurveillance [51, 19], network analysis [54, 16], as well as insurance and agriculture [56, 34] (see the literature overviews [4, 17] and the TDA applications library [27]).

Original: Recently there has emerged a highly active research area that combines the PH machinery with geometric deep learning (GDL) methods [30, 57, 31].
Revised: Recently there has emerged a highly active research area that combines the PH machinery with geometric deep learning (GDL) methods [30, 57, 31].

Original: Persistent homology (PH) is a key approach in TDA, allowing us to extract the evolution of subtler patterns in the data shape dynamics at multiple resolution scales, which are not accessible to more conventional, non-topological methods [15].
Revised: Persistent homology (PH) is a key approach in TDA, allowing us to extract the evolution of subtler patterns in the data shape dynamics at multiple resolution scales, which are not accessible to more conventional, non-topological methods [15].

Original: The main idea is to construct a nested sequence of topological spaces (ﬁltration) induced from the data, and record the evolution of topological features in this sequence.
Revised: The main idea is to construct a nested sequence of topological spaces (filtration) induced from the data, and record the evolution of topological features in this sequence.

Original: In other words, the extracted patterns, or homological features, along with how long such features persist throughout the considered ﬁltration of a scale parameter, convey a critical insight into salient graph characteristics and hidden mechanisms behind system organization.
Revised: In other words, the extracted patterns, or homological features, along with how long such features persist throughout the considered filtration of a scale parameter, convey a critical insight into salient graph characteristics and hidden mechanisms behind system organization.

Original: PH has been very effective in many graph machine learning tasks, such as graph and node classiﬁcation [49, 14, 58, 29], link prediction [6, 55] and anomaly detection [12, 47].
Revised: PH has been very effective in many graph machine learning tasks, such as graph and node classification [49, 14, 58, 29], link prediction [6, 55] and anomaly detection [12, 47].

Original: Nevertheless, while PH has shown promise in various graph learning applications, prohibitive computational costs of PH constrain its wider usage.
Revised: Nevertheless, while PH has shown promise in various graph learning applications, prohibitive computational costs of PH constrain its wider usage.

Original: Indeed, most PH studies are limited to small graphs with a few thousand vertices at most.
Revised: Indeed, most PH studies are limited to small graphs with a few thousand vertices at most.

Original: The problem is that the complexity of the standard PH algorithm is cubic in the number of simplices [48], so one needs to limit homology computations to 0 -th and 1 -th levels only.
Revised: The problem is that the complexity of the standard PH algorithm is cubic in the number of simplices [48], so one needs to limit homology computations to 0 -th and 1 -th levels only.

Original: Computation of higher-level persistence for relatively large graphs can take days or weeks.
Revised: Computation of higher-level persistence for relatively large graphs can take days or weeks.

Original: In this paper, we aim to address this fundamental bottleneck in the application of TDA to large networks by introducing two new efﬁcient algorithms which signiﬁcantly reduce the cost of computing persistence diagrams (PD) for large real-world networks: CoralTDA and PrunIT .
Revised: In this paper, we aim to address this fundamental bottleneck in the application of TDA to large networks by introducing two new efficient algorithms which significantly reduce the cost of computing persistence diagrams (PD) for large real-world networks: CoralTDA and PrunIT .

Original: CoralTDA Algorithm:
Revised: CoralTDA Algorithm:

Original: Based on our observation that many vertices in large real-world networks have low degrees and do not contribute to PDs in higher dimensions, we developed the CoralTDA algorithm (Theorem 2) where we prove that ( k + 1) -core
Revised: Based on our observation that many vertices in large real-world networks have low degrees and do not contribute to PDs in higher dimensions, we developed the CoralTDA algorithm (Theorem 2) where we prove that ( k + 1) -core

Original: G k +1 of a graph G is enough to compute the k th PD of the graph, i.e. PD k ( G ) =
Revised: G k +1 of a graph G is enough to compute the k th PD of the graph, i.e. PD k ( G ) =

Original: PD k ( G k +1 ) 
Revised: PD k ( G k +1 ) .

Original: Using this property, with a much smaller core graph G k +1 , we compute the exact higher persistence diagram PD k ( G ) losing no information.
Revised: Using this property, with a much smaller core graph G k +1 , we compute the exact higher persistence diagram PD k ( G ) losing no information.

Original: Our experiments show that even for lower dimensional topological features, such as k = 1 , we reduce the graph order by up to 73% for some datasets (See Figure 4).
Revised: Our experiments show that even for lower dimensional topological features, such as k = 1 , we reduce the graph order by up to 73% for some datasets (See Figure 4).

Original: Our ﬁndings show that many real-life data sets exhibit nontrivial second and third persistence diagrams, facilitating various classiﬁcation problems.
Revised: Our findings show that many real-life data sets exhibit nontrivial second and third persistence diagrams, facilitating various classification problems.

Original: On the other hand, our reduction reaches 100% for the third or higher dimensions in several networks, implying that higher PDs are trivial for these datasets.
Revised: On the other hand, our reduction reaches 100% for the third or higher dimensions in several networks, implying that higher PDs are trivial for these datasets.

Original: As a result, our reduction approach improves our understanding of the existence of higher-order dimensional holes and their role in the organization of complex networks.
Revised: As a result, our reduction approach improves our understanding of the existence of higher-order dimensional holes and their role in the organization of complex networks.

Original: PrunIT Algorithm: We further develop a topologically simple but highly efﬁcient algorithm to facilitate computations of PDs of graphs for any dimension.
Revised: PrunIT Algorithm: We further develop a topologically simple but highly efficient algorithm to facilitate computations of PDs of graphs for any dimension.

Original: In particular, for a graph G =
Revised: In particular, for a graph G =

Original: ( V , E ) and ﬁltration by clique complexes, we show that removing (pruning) a dominated vertex from the graph does not change PDs at any level, provided that the dominated vertex enters the ﬁltration after the dominating vertex (Theorem 7).
Revised: ( V , E ) and filtration by clique complexes, we show that removing (pruning) a dominated vertex from the graph does not change PDs at any level, provided that the dominated vertex enters the filtration after the dominating vertex (Theorem 7).

Original: Our experiments indicate that the new algorithm is highly efﬁcient in PD computations of a broad category of large graphs from 100K to 1M vertices, and it can reach 95% vertex reduction (see
Revised: Our experiments indicate that the new algorithm is highly efficient in PD computations of a broad category of large graphs from 100K to 1M vertices, and it can reach 95% vertex reduction (see

Original: Table 1).
Revised: Table 1).

Original: Further, when we combine CoralTDA and PrunIT algorithms, we can signiﬁcantly reduce the graph sizes for the computation of PDs (Figure 6).
Revised: Further, when we combine CoralTDA and PrunIT algorithms, we can significantly reduce the graph sizes for the computation of PDs (Figure 6).

Original: We summarize the key novelty of our contributions as follows:
Revised: We summarize the key novelty of our contributions as follows:

Original: • We show that the graphs’ ( k + 1 )st and higher persistence diagrams only depend on their k -cores.
Revised: • We show that the graphs’ ( k + 1 )th and higher persistence diagrams only depend on their k -cores.

Original: • We introduce a highly effective Pruning Algorithm that signiﬁcantly reduces the graph size without changing any persistence diagram of the original graph.
Revised: • We introduce a highly effective pruning algorithm that significantly reduces the graph size without changing any persistence diagram of the original graph.

Original: • Our experiments in large datasets and large graphs show up to 95% reduction in the graph size for the computation of persistence diagrams.
Revised: • Our experiments in large datasets and large graphs show up to 95% reduction in the graph size for the computation of persistence diagrams.

Original: • With our reduction algorithms, highly successful TDA methods can be applied to very large graphs and large datasets where previously its use was constrained by prohibitive computational costs.
Revised: • With our reduction algorithms, highly successful TDA methods can be applied to very large graphs and large datasets where previously its use was constrained by prohibitive computational costs.

Original:  Related Work
Revised:  Related Work

Original: There are mainly two settings in practice where we use PH to obtain a topological ﬁngerprint of a dataset.
Revised: There are mainly two settings in practice where we use PH to obtain a topological fingerprint of a dataset.

Original: The ﬁrst one is the point cloud setting , where the dataset comes as a point cloud in an ambient space R n .
Revised: The first one is the point cloud setting , where the dataset comes as a point cloud in an ambient space R n .

Original: Then, we deﬁne PH by constructing a sequence of simplicial complexes induced by the pair-wise distances of data points (Vietoris-Rips ﬁltration) and keeping track of the topological changes in this sequence [59, 23].
Revised: Then, we define PH by constructing a sequence of simplicial complexes induced by the pair-wise distances of data points (Vietoris-Rips filtration) and keeping track of the topological changes in this sequence [59, 23].

Original: The second one is the network setting where the typical PH construction uses a ﬁltering function on the network.
Revised: The second one is the network setting where the typical PH construction uses a filtering function on the network.

Original: By construction, while the principal identiﬁer to deﬁne PH in the point cloud setting is the pair-wise distances of points, the principal identiﬁer in the network setting is the ﬁltering function.
Revised: By construction, while the principal identifier to define PH in the point cloud setting is the pair-wise distances of points, the principal identifier in the network setting is the filtering function.

Original: Because of this, PH machinery works differently in a network setting, as explained in Section 3.
Revised: Because of this, PH machinery works differently in a network setting, as explained in Section 3.

Original: There are several works in the point cloud setting to reduce the computational costs and run-time of the persistence diagrams.
Revised: There are several works in the point cloud setting to reduce the computational costs and run-time of the persistence diagrams.

Original: Malott and Wilsey used the idea of data reduction and data partitioning [41].
Revised: Malott and Wilsey used the idea of data reduction and data partitioning [41].

Original: Mischaikow and Nanda brought the discrete Morse Theory of geometric topology to the combinatorial setting [43].
Revised: Mischaikow and Nanda brought the discrete Morse Theory of geometric topology to the combinatorial setting [43].

Original: In [46, 20, 21, 24], the authors studied the same problem with different approaches in the point cloud setting.
Revised: In [46, 20, 21, 24], the authors studied the same problem with different approaches in the point cloud setting.

Original: While several works improve the run-time of PH in the point cloud setting, only a few of them could reduce the computational costs of persistent homology in the network setting.
Revised: While several works improve the run-time of PH in the point cloud setting, only a few of them could reduce the computational costs of persistent homology in the network setting.

Original: An idea is to use discrete Morse Theory to capture the topological features occurring during the process [36] by applying the techniques developed in [43] to the network setting.
Revised: An idea is to use discrete Morse Theory to capture the topological features occurring during the process [36] by applying the techniques developed in [43] to the network setting.

Original: While the computational complexity of k th persistence diagram (PD) is O ( n 3 ) where n is the number of k -simplices [48], [43] achieves O ( m 2 × n log n ) where m is the number of critical k -simplices.
Revised: While the computational complexity of k th persistence diagram (PD) is O ( n 3 ) where n is the number of k -simplices [48], [43] achieves O ( m 2 × n log n ) where m is the number of critical k -simplices.

Original: With the additional time to ﬁnd the critical k -simplices in each ﬁltration step, the computational complexity O ( m 2 × n log n ) is not scalable for very large networks.
Revised: With the additional time to find the critical k -simplices in each filtration step, the computational complexity O ( m 2 × n log n ) is not scalable for very large networks.

Original:  Persistent Homology
Revised:  Persistent Homology

Original: This part provides a background on the theory of persistent homology.
Revised: This part provides a background on the theory of persistent homology.

Original: Homology H k
Revised: Homology H k

Original: ( X ) is an essential invariant in algebraic topology, which captures the information of the k -dimensional holes (connected components, loops, cavities) in a topological space X .
Revised: ( X ) is an essential invariant in algebraic topology, which captures the information of the k -dimensional holes (connected components, loops, cavities) in a topological space X .

Original: For example, a connected component in a graph is a zero-dimensional hole, whereas a graph loop is a 1-dimensional hole.
Revised: For example, a connected component in a graph is a zero-dimensional hole, whereas a graph loop is a 1-dimensional hole.

Original: Persistent homology is a way to use this invariant to keep track of the changes in a controlled topological space sequence induced by the original space X .
Revised: Persistent homology is a way to use this invariant to keep track of the changes in a controlled topological space sequence induced by the original space X .

Original: For basic background on persistent homology, see [23, 22].
Revised: For basic background on persistent homology, see [23, 22].

Original: There are several ways to use PH in a network setting, such as power ﬁltration or using different complexes (e.g., Vietoris-Rips, ˇ Cech complexes) to construct the ﬁltration for a given ﬁltering function [3].
Revised: There are several ways to use PH in a network setting, such as power filtration or using different complexes (e.g., Vietoris-Rips, ˇ Cech complexes) to construct the filtration for a given filtering function [3].

Original: We focus on the most common methods to deﬁne PH for graphs: sub/superlevel ﬁltrations obtained by a ﬁltering function and the clique (ﬂag) complexes.
Revised: We focus on the most common methods to define PH for graphs: sub/superlevel filtrations obtained by a filtering function and the clique (flag) complexes.

Original: Sub/superlevel ﬁltrations are the most common methods because one can inject domain information into the PH process if the chosen ﬁltering function comes from the network domain (e.g., atomic number in protein networks, transaction amount for blockchain networks).
Revised: Sub/superlevel filtrations are the most common methods because one can inject domain information into the PH process if the chosen filtering function comes from the network domain (e.g., atomic number in protein networks, transaction amount for blockchain networks).

Original: Note that our results can be generalized to the persistent homology deﬁned with a ﬁltering function for different complexes.
Revised: Note that our results can be generalized to the persistent homology defined with a filtering function for different complexes.

Original: Throughout the paper, we use the terms graph and network interchangeably.
Revised: Throughout the paper, we use the terms graph and network interchangeably.

Original: Let G be a graph with vertex set V = { v r } and edge set E = { e rs } , i.e. e rs ∈ E if there is an edge between the vertex v r and v s in G .
Revised: Let G be a graph with vertex set V = { v r } and edge set E = { e rs } , i.e. e rs ∈ E if there is an edge between the vertex v r and v s in G .

Original: Let f : V → R be a ﬁltering function deﬁned on the vertices of G .
Revised: Let f : V → R be a filtering function defined on the vertices of G .

Original: Let I =
Revised: Let I =

Original: { α i } be a threshold set with α 0 =
Revised: { α i } be a threshold set with α 0 =

Original: min v r ∈V f ( v r ) <
Revised: min v r ∈V f ( v r ) <

Original: α 1 < ... < α m = max v r ∈V f ( v r ) .
Revised: α 1 < ... < α m = max v r ∈V f ( v r ) .

Original: For α i ∈ I , let
Revised: For α i ∈ I , let

Original: V i = { v r ∈ V | f ( v r ) ≤ α i } .
Revised: V i = { v r ∈ V | f ( v r ) ≤ α i } .

Original: Let G i be the induced subgraph of G by V i , i.e. G i =
Revised: Let G i be the induced subgraph of G by V i , i.e. G i =

Original: ( V i , E i ) where
Revised: ( V i , E i ) where

Original: E i = { e rs ∈ E | v r , v s ∈ V i } .
Revised: E i = { e rs ∈ E | v r , v s ∈ V i } .

Original: Let (cid:98) G i be the clique complex of G i .
Revised: Let (cid:98) G i be the clique complex of G i .

Original: A clique complex is obtained by ﬁlling in all the ( k + 1) -complete
Revised: A clique complex is obtained by filling in all the ( k + 1) -complete

Original: subgraphs with k -simplices.
Revised: subgraphs with k -simplices.

Original: In other words, if the vertices { v r 0 , v r 1 , ..., v r k } ⊂ G i are pairwise connected by an edge in G , then the clique complex (cid:98) G i contains a k -simplex σ =
Revised: In other words, if the vertices { v r 0 , v r 1 , ..., v r k } ⊂ G i are pairwise connected by an edge in G , then the clique complex (cid:98) G i contains a k -simplex σ =

Original: [ v r 0 , v r 1 , ..., v r k ] .
Revised: [ v r 0 , v r 1 , ..., v r k ] .

Original: This simplicial complex (cid:98) G i obtained by ﬁlling in all complete subgraphs is called the clique complex of G i .
Revised: This simplicial complex (cid:98) G i obtained by filling in all complete subgraphs is called the clique complex of G i .

Original: This construction induces a nested sequence of high dimensional simplicial complexes:
Revised: This construction induces a nested sequence of high dimensional simplicial complexes:

Original: [Equation]
Revised: [Equation]

Original: This sequence of simplicial complexes is called the sublevel ﬁltration for G .
Revised: This sequence of simplicial complexes is called the sublevel filtration for G .

Original: Superlevel ﬁltrations can be deﬁned similarly by considering the generating sets { f ( v r ) ≥ α i } instead of { f ( v r ) ≤
Revised: Superlevel filtrations can be defined similarly by considering the generating sets { f ( v r ) ≥ α i } instead of { f ( v r ) ≤

Original: α i } above.
Revised: α i } above.

Original: Here, (cid:98) G i can be taken as the different simplicial complexes induced by G i which gives different types of ﬁltrations [3].
Revised: Here, (cid:98) G i can be taken as the different simplicial complexes induced by G i which gives different types of filtrations [3].

Original: After obtaining the ﬁltration, one considers the homology groups
Revised: After obtaining the filtration, one considers the homology groups

Original: H k
Revised: H k

Original: ( (cid:98) G i ) of each simplicial complex (cid:98) G i .
Revised: ( (cid:98) G i ) of each simplicial complex (cid:98) G i .

Original: The homology group H k
Revised: The homology group H k

Original: ( X ) keeps the information of k -dimensional topological features in the simplicial complex X .
Revised: ( X ) keeps the information of k -dimensional topological features in the simplicial complex X .

Original: Persistent homology keeps track of the topological changes in the sequence { (cid:98) G i } by using the homology groups { H k ( (cid:98) G i ) } .
Revised: Persistent homology keeps track of the topological changes in the sequence { (cid:98) G i } by using the homology groups { H k ( (cid:98) G i ) } .

Original: When a k -dimensional hole σ (a connected component, loop or cavity) appears in H k
Revised: When a k -dimensional hole σ (a connected component, loop or cavity) appears in H k

Original: ( (cid:98) G i ) , we mark b σ = α i as its birth time.
Revised: ( (cid:98) G i ) , we mark b σ = α i as its birth time.

Original: The feature σ can disappear at a later time in
Revised: The feature σ can disappear at a later time in

Original: H k ( (cid:98) G j ) by merging with another feature or by being ﬁlled in.
Revised: H k ( (cid:98) G j ) by merging with another feature or by being filled in.

Original: Then, we mark d σ = α j as its death time.
Revised: Then, we mark d σ = α j as its death time.

Original: Hence, we say that σ persists along the interval [ b σ , d σ ) , i.e. [ α i , α j ) .
Revised: Hence, we say that σ persists along the interval [ b σ , d σ ) , i.e. [ α i , α j ) .

Original: The longer the interval ( d σ − b σ ), the more persistent the feature σ .
Revised: The longer the interval ( d σ − b σ ), the more persistent the feature σ .

Original: The multi-set PD k ( G , f ) = { ( b σ , d σ ) | σ ∈ H k
Revised: The multi-set PD k ( G , f ) = { ( b σ , d σ ) | σ ∈ H k

Original: ( (cid:98) G i ) for b σ ≤ i < d σ } is called the k th persistence diagram of ( G , f ) which is the collection of 2 -tuples marking the birth and death times of kdimensional holes { σ } in { (cid:98) G i } .
Revised: ( (cid:98) G i ) for b σ ≤ i < d σ } is called the k th persistence diagram of ( G , f ) which is the collection of 2 -tuples marking the birth and death times of kdimensional holes { σ } in { (cid:98) G i } .

Original: In particular, PD k ( G , f ) represents the k th PD of the sublevel ﬁltration, induced by the ﬁltering function f : V → R .
Revised: In particular, PD k ( G , f ) represents the k th PD of the sublevel filtration, induced by the filtering function f : V → R .

Original: For brevity, we suppress f and use PD k ( G ) throughout the text.
Revised: For brevity, we suppress f and use PD k ( G ) throughout the text.

Original:  CoralTDA Reduction and Higher Persistence Diagrams
Revised:  CoralTDA Reduction and Higher Persistence Diagrams

Original: A k -core G k of a graph G is the subgraph of G obtained by iteratively deleting all vertices (and edges connected to it) with degree less than k
Revised: A k -core G k of a graph G is the subgraph of G obtained by iteratively deleting all vertices (and edges connected to it) with degree less than k

Original: In other words, G k is the largest subgraph of G where all the vertices have a degree of at least k .
Revised: In other words, G k is the largest subgraph of G where all the vertices have a degree of at least k .

Original: Figure 1 shows a graph with its core structure.
Revised: Figure 1 shows a graph with its core structure.

Original: Here, vertexbelongs to 0-core as it is disconnected from the graph.
Revised: Here, vertexbelongs to 0-core as it is disconnected from the graph.

Original: Vertex colors indicate shared coreness.
Revised: Vertex colors indicate shared coreness.

Original: When we use vertex degree as the ﬁltering function and allow graph cliques of size three at most, the only one-dimensional hole (shown with the red circle) appears at degree 4 for vertices 4, 6, 8, and 9.
Revised: When we use vertex degree as the filtering function and allow graph cliques of size three at most, the only one-dimensional hole (shown with the red circle) appears at degree 4 for vertices 4, 6, 8, and 9.

Original: Vertices 3, 5, and 7 can only contribute to 0-dimensional holes because their degree is 1.
Revised: Vertices 3, 5, and 7 can only contribute to 0-dimensional holes because their degree is 1.

Original: Similarly, 8 can only contribute to 0 and 1-dimensional holes because its degree is 2.
Revised: Similarly, 8 can only contribute to 0 and 1-dimensional holes because its degree is 2.

Original: The k -core decomposition is a fundamental operation in many areas such as graph similarity matching [45], graph clustering [25], network visualization [26], anomaly detection [53] and robustness analysis [13].
Revised: The k -core decomposition is a fundamental operation in many areas such as graph similarity matching [45], graph clustering [25], network visualization [26], anomaly detection [53] and robustness analysis [13].

Original: A naïve implementation of k -core iteratively deletes vertices whose degree falls below a k , until it deletes all vertices from the graph.
Revised: A naïve implementation of k -core iteratively deletes vertices whose degree falls below a k , until it deletes all vertices from the graph.

Original: The implementation has a computational complexity of O ( m log n ) , where m and n are the number of edges and vertices in the network, respectively.
Revised: The implementation has a computational complexity of O ( m log n ) , where m and n are the number of edges and vertices in the network, respectively.

Original: Batagelj and Zaversnik reduce the complexity to O ( m + n ) “by keeping an in-memory array of all possible degree values and keeping track of bin boundaries” [5].
Revised: Batagelj and Zaversnik reduce the complexity to O ( m + n ) “by keeping an in-memory array of all possible degree values and keeping track of bin boundaries” [5].

Original: 4.1 Relation between (cid:98) G i and (cid:98) G ki
Revised: 4.1 Relation between (cid:98) G i and (cid:98) G ki

Original: Our main idea is to compute high-dimensional persistence features on their associated graph cores.
Revised: Our main idea is to compute high-dimensional persistence features on their associated graph cores.

Original: Note that a k -clique in graph theory corresponds to a ( k − 1) -simplex in PH; a k -clique (complete subgraph of order k ) in G induces a ( k − 1)
Revised: Note that a k -clique in graph theory corresponds to a ( k − 1) -simplex in PH; a k -clique (complete subgraph of order k ) in G induces a ( k − 1)

Original: -simplex
Revised: -simplex

Original: in (cid:98) G .
Revised: in (cid:98) G .

Original: The clique complex (cid:98) G is a simplicial complex of dimension K − 1 where K denotes the degeneracy of G , i.e. K = max { k | G k (cid:54) = ∅} .
Revised: The clique complex (cid:98) G is a simplicial complex of dimension K − 1 where K denotes the degeneracy of G , i.e. K = max { k | G k ̸ = ∅} .

Original: That is, (cid:98) G contains a ( k − 1) -simplex if and only if its k -core G k is not empty.
Revised: That is, (cid:98) G contains a ( k − 1) -simplex if and only if its k -core G k is not empty.

Original: For any i, k , we have the k -core of G i contained in G i by construction, i.e. G ki
Revised: For any i, k , we have the k -core of G i contained in G i by construction, i.e. G ki

Original: ⊂ G i .
Revised: ⊂ G i .

Original: This implies that the same holds for their clique complexes, i.e. (cid:98) G ki ⊂ (cid:98) G i .
Revised: This implies that the same holds for their clique complexes, i.e. (cid:98) G ki ⊂ (cid:98) G i .

Original: On the other hand, if one restricts the original ﬁltering function f : V → R to the vertices V k of the k -core of G , we have f : V k
Revised: On the other hand, if one restricts the original filtering function f : V → R to the vertices V k of the k -core of G , we have f : V k

Original: → R .
Revised: → R .

Original: By using the same thresholds for f : V k
Revised: By using the same thresholds for f : V k

Original: → R , we obtain the ﬁltration (cid:98) G k 0{+
+} ⊂ (cid:98) G k 1 ⊂ (cid:98) G k 2 ⊂ ...
Revised: → R , we obtain the filtration (cid:98) G k {+
+} ⊂ (cid:98) G k 1 ⊂ (cid:98) G k 2 ⊂ ...

Original: ⊂ (cid:98) G km .
Revised: ⊂ (cid:98) G km .

Original: This will induce the persistence diagram PD r ( G k ) for any dimension r .
Revised: This will induce the persistence diagram PD r ( G k ) for any dimension r .

Original: Since for any i, k , (cid:98) G ki ⊂ (cid:98) G i , we have the following diagram.
Revised: Since for any i, k , (cid:98) G ki ⊂ (cid:98) G i , we have the following diagram.

Original: [Equation]
Revised: [Equation]

Original: Notice that for any j ≥ k − 1 , if there is a j -cycle σ living in C j ( (cid:98) G ki ) , then we have σ ⊂ C j
Revised: Notice that for any j ≥ k − 1 , if there is a j -cycle σ living in C j ( (cid:98) G ki ) , then we have σ ⊂ C j

Original: ( (cid:98) G i ) as (cid:98) G ki ⊂ (cid:98) G i .
Revised: ( (cid:98) G i ) as (cid:98) G ki ⊂ (cid:98) G i .

Original: In the following, we show that for these cycles, the converse is also true, and we show the equivalence in the homology level.
Revised: In the following, we show that for these cycles, the converse is also true, and we show the equivalence in the homology level.

Original: Remark 1.
Revised: Remark 1.

Original: [Restriction of f to V k ] Notice that the ﬁltering function f : V k
Revised: [Restriction of f to V k ] Notice that the filtering function f : V k

Original: → R on V k , the vertices of G k , is deﬁned directly by restricting values of f : V → R to the subset V k ⊂ V .
Revised: → R on V k , the vertices of G k , is defined directly by restricting values of f : V → R to the subset V k ⊂ V .

Original: In particular, if f : V → R is a function coming from the graph attributes (such as vertex degree), then f : V k
Revised: In particular, if f : V → R is a function coming from the graph attributes (such as vertex degree), then f : V k

Original: → R may not be the same function coming from the graph attributes induced by the graph G k .
Revised: → R may not be the same function coming from the graph attributes induced by the graph G k .

Original: For example, let f be the degree function on V , the vertices of G .
Revised: For example, let f be the degree function on V , the vertices of G .

Original: Then, for any w ∈ V k , f ( w ) is the degree of w in G , not its degree in G k .
Revised: Then, for any w ∈ V k , f ( w ) is the degree of w in G , not its degree in G k .

Original: While the k -core graph G k changes, we do not update the values of f on
Revised: While the k -core graph G k changes, we do not update the values of f on

Original: V k according to its attribute deﬁnition in G k , but we keep the same values in the original function f : V → R for the remaining vertices in V k ⊂ V .
Revised: V k according to its attribute definition in G k , but we keep the same values in the original function f : V → R for the remaining vertices in V k ⊂ V .

Original: In graph terms, this corresponds to computing vertex ﬁltering (activation) values on the original graph but using the edges of the reduced graph to extract simplices.
Revised: In graph terms, this corresponds to computing vertex filtering (activation) values on the original graph but using the edges of the reduced graph to extract simplices.

Original: 4.2 CoralTDA Reduction
Revised: 4.2 CoralTDA Reduction

Original: Our CoralTDA technique shows that lower degree vertices do not affect higher persistence diagrams, i.e., CoralTDA yields exact results.
Revised: Our CoralTDA technique shows that lower degree vertices do not affect higher persistence diagrams, i.e., CoralTDA yields exact results.

Original: Note that in the following result, even though the graph size changes, we keep the same ﬁltering function f : V → R with the original values.
Revised: Note that in the following result, even though the graph size changes, we keep the same filtering function f : V → R with the original values.

Original: See Remark 1 for further details.
Revised: See Remark 1 for further details.

Original: We give the proof of the following theorem in Appendix.
Revised: We give the proof of the following theorem in Appendix.

Original: Theorem 2.
Revised: Theorem 2.

Original: Let G be an unweighted connected graph.
Revised: Let G be an unweighted connected graph.

Original: Let f : V → R be a ﬁltering function on
Revised: Let f : V → R be a filtering function on

Original: G .
Revised: G .

Original: Let PD k
Revised: Let PD k

Original: ( G , f ) represent the k th persistence diagram for the sublevel ﬁltration of the clique complexes.
Revised: ( G , f ) represent the k th persistence diagram for the sublevel filtration of the clique complexes.

Original: Let (cid:98) G k be the k -core of G .
Revised: Let (cid:98) G k be the k -core of G .

Original: Then, for any j ≥ k
Revised: Then, for any j ≥ k

Original: [Equation]
Revised: [Equation]

Original: Outline of the proof: We show that for any nontrivial k -homology class σ in the original clique complex (cid:98) G , a generating k -cycle S in this homology class also lives in a much smaller subcomplex: the clique complex of the ( k + 1) -core
Revised: Outline of the proof: We show that for any nontrivial k -homology class σ in the original clique complex (cid:98) G , a generating k -cycle S in this homology class also lives in a much smaller subcomplex: the clique complex of the ( k + 1) -core

Original: ( (cid:98) G k +1 ).
Revised: ( (cid:98) G k +1 ).

Original: That is, we prove that any vertex in the k -cycle S must have a degree at least k + 1 where this degree count comes only from the k -simplices of S , and removing the lower degree vertices from G has no effect on the existence of such S .
Revised: That is, we prove that any vertex in the k -cycle S must have a degree at least k + 1 where this degree count comes only from the k -simplices of S , and removing the lower degree vertices from G has no effect on the existence of such S .

Original: We give the proof of the theorem in Appendix.
Revised: We give the proof of the theorem in Appendix.

Original: The above result indicates that k th persistence diagram information can be obtained by only considering the ( k + 1) -core of a graph.
Revised: The above result indicates that k th persistence diagram information can be obtained by only considering the ( k + 1) -core of a graph.

Original: CoralTDA is an effective tool for reducing computational costs to compute higher persistence diagrams.
Revised: CoralTDA is an effective tool for reducing computational costs to compute higher persistence diagrams.

Original: See Figure 4 for reduction results for various datasets.
Revised: See Figure 4 for reduction results for various datasets.

Original: Remark 3.
Revised: Remark 3.

Original: [Higher PDs in Random Networks vs. Real-Life Networks] Note that by Kahle’s seminal result [35], to observe nontrivial Betti numbers for higher dimensionsin Erdós-Rényi graphs G ( n, p ) , the average degree must be very high.
Revised: [Higher PDs in Random Networks vs. Real-Life Networks] Note that by Kahle’s seminal result [35], to observe nontrivial Betti numbers for higher dimensions the average degree must be very high.

Original: In particular, for a graph G ( n, p ) , in order to have nontrivial k th -homology in its clique complex, Kahle proved that for p = n α , α should be between − 1 /k and − 1 / ( k + 1) .
Revised: In particular, for a graph G ( n, p ) , in order to have nontrivial k th -homology in its clique complex, Kahle proved that for p = n α , α should be between − 1 /k and − 1 / ( k + 1) .

Original: In terms of average degree n × p , this means the average degree should be between n ( k − 1) /k and n k/ ( k +1) .
Revised: In terms of average degree n × p , this means the average degree should be between n ( k − 1) /k and n k/ ( k +1) .

Original: For instance, for dimension k = 2 , the average degree should be between √ n and 3 √ n 2 .
Revised: For instance, for dimension k = 2 , the average degree should be between √ n and 3 √ n 2 .

Original: For a graph order of n = 1000 , this implies that the average degree should be between 31 and 100 to have a nontrivial second homology in random networks.
Revised: For a graph order of n = 1000 , this implies that the average degree should be between 31 and 100 to have a nontrivial second homology in random networks.

Original: However, in real-life networks, our results show that higher Betti numbers are prevalent in much sparser graphs (see ﬁg. 2 and appendix ﬁg. 10).
Revised: However, in real-life networks, our results show that higher Betti numbers are prevalent in much sparser graphs (see fig. 2 and appendix fig. 10).

Original: These ﬁndings can be further used to derive error bounds and the associated loss of topological information when G ( n, p ) is employed to approximate real-world network phenomena, for instance, in the case of synthetic power grid networks and other cyber-physical systems.
Revised: These findings can be further used to derive error bounds and the associated loss of topological information when G ( n, p ) is employed to approximate real-world network phenomena, for instance, in the case of synthetic power grid networks and other cyber-physical systems.

Original: In the following, we give another effective method to reduce the size of a graph G without affecting the persistence diagrams PD r ( G ) for any dimension r ≥ 0 .
Revised: In the following, we give another effective method to reduce the size of a graph G without affecting the persistence diagrams PD r ( G ) for any dimension r ≥ 0 .

Original:  PrunIT Algorithm
Revised:  PrunIT Algorithm

Original: This section introduces another effective reduction technique for computing persistence diagrams of graphs induced by a ﬁltering function.
Revised: This section introduces another effective reduction technique for computing persistence diagrams of graphs induced by a filtering function.

Original: In particular, we show that for a graph G , and ﬁltering function f : V → R , removing (pruning) speciﬁc vertices from the graph does not change the persistent homology at any level.
Revised: In particular, we show that for a graph G , and filtering function f : V → R , removing (pruning) specific vertices from the graph does not change the persistent homology at any level.

Original: The result is valuable because the algorithm may reduce the vertex set considerably (Table 1).
Revised: The result is valuable because the algorithm may reduce the vertex set considerably (Table 1).

Original: Furthermore, as our experiments show, the reduced vertex set can signiﬁcantly lower the simplex count, leading to much shorter computational times for persistent homology (see
Revised: Furthermore, as our experiments show, the reduced vertex set can significantly lower the simplex count, leading to much shorter computational times for persistent homology (see

Original: Figure 4 and appendix Figure 7).
Revised: Figure 4 and appendix Figure 7).

Original: In algebraic topology, homotopy is a very effective tool to compute topological invariants like homology, and fundamental group [28].
Revised: In algebraic topology, homotopy is a very effective tool to compute topological invariants like homology, and fundamental group [28].

Original: These topological invariants are homotopy invariant, meaning that if two spaces are homotopy equivalent, then their corresponding topological invariants are the same
Revised: These topological invariants are homotopy invariant, meaning that if two spaces are homotopy equivalent, then their corresponding topological invariants are the same

Original: , e.g., X ∼ Y ⇒ H
Revised: , e.g., X ∼ Y ⇒ H

Original: i ( X ) =
Revised: i ( X ) =

Original: H i
Revised: H i

Original: ( Y ) .
Revised: ( Y ) .

Original: We give a very natural homotopy construction to simplify a graph in the following.
Revised: We give a very natural homotopy construction to simplify a graph in the following.

Original: For a given ﬁltering function f : V → R , let (cid:98) G
Revised: For a given filtering function f : V → R , let (cid:98) G

Original: i be the clique complex of G i which induces the sublevel ﬁltration (cid:98) G 

Revised: i be the clique complex of G i which induces the sublevel filtration (cid:98) G

Original: ⊂ (cid:98) G 1 ⊂ (cid:98) G 2 ⊂ ...
Revised: ⊂ (cid:98) G 1 ⊂ (cid:98) G 2 ⊂ ...

Original: ⊂ (cid:98) G m .
Revised: ⊂ (cid:98) G m .

Original: Let PD k
Revised: Let PD k

Original: ( G , f ) represent the k th persistence diagram for the sublevel ﬁltration { (cid:98) G i } as described above.
Revised: ( G , f ) represent the k th persistence diagram for the sublevel filtration { (cid:98) G i } as described above.

Original: Now, we deﬁne dominated vertices in G .
Revised: Now, we define dominated vertices in G .

Original: Deﬁne the neighborhood of u 0 as N ( u 0 ) =
Revised: Define the neighborhood of u 0 as N ( u 0 ) =

Original: { u 0 } ∪ { v ∈
Revised: { u 0 } ∪ { v ∈

Original: V | e u 0 v ∈ E} .
Revised: V | e u 0 v ∈ E} .

Original: In particular, N ( u 0 ) ⊂ V is the set of all vertices adjacent to u 0 , and u 0 itself.
Revised: In particular, N ( u 0 ) ⊂ V is the set of all vertices adjacent to u 0 , and u 0 itself.

Original: Deﬁnition 4.
Revised: Definition 4.

Original: A vertex u is dominated by the vertex v in G if N ( u ) ⊂ N ( v ) .
Revised: A vertex u is dominated by the vertex v in G if N ( u ) ⊂ N ( v ) .

Original: If there is such a vertex v , we call u a dominated vertex of G (see Figure 3).
Revised: If there is such a vertex v , we call u a dominated vertex of G (see Figure 3).

Original: Removing a vertex u from a graph G creates the natural subgraph of G obtained by removing the vertex u and all adjacent edges from G , i.e.
Revised: Removing a vertex u from a graph G creates the natural subgraph of G obtained by removing the vertex u and all adjacent edges from G , i.e.

Original: G − { u } = G (cid:48)
Revised: G − { u } = G ′

Original: =
Revised: =

Original: ( V (cid:48) , E (cid:48) ) where V (cid:48)
Revised: ( V ′ , E ′ ) where V ′ =

Original: = V − { u } , and E (cid:48) =
Revised: V − { u } , and E ′ =

Original: E − { e uw ∈
Revised: E − { e uw ∈

Original: E} for any w .
Revised: E} for any w .

Original: We can alternatively express these via the star notion.
Revised: We can alternatively express these via the star notion.

Original: The star St ( u ) of a vertex u is the union of all simplices which contains u .
Revised: The star St ( u ) of a vertex u is the union of all simplices which contains u .

Original: Then, u is dominated by v if St ( u ) ⊂
Revised: Then, u is dominated by v if St ( u ) ⊂

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

Original: Similarly, removing a vertex u from G corresponds to removing St ( u ) from the clique complex (cid:98) G , i.e. (cid:98) G − St ( u )
Revised: Similarly, removing a vertex u from G corresponds to removing St ( u ) from the clique complex (cid:98) G , i.e. (cid:98) G − St ( u )

Original: =
Revised: =

Original: (cid:98) G (cid:48) .
Revised: (cid:98) G ′ .

Original: A useful result is that removing a dominated vertex does not affect the homotopy type of the corresponding clique complexes.
Revised: A useful result is that removing a dominated vertex does not affect the homotopy type of the corresponding clique complexes.

Original: Lemma 5.
Revised: Lemma 5.

Original: Let u be a dominated vertex in G .
Revised: Let u be a dominated vertex in G .

Original: Let G (cid:48) =
Revised: Let G ′ =

Original: G − { u } .
Revised: G − { u } .

Original: Then the clique complexes (cid:98) G and (cid:98) G (cid:48) are homotopy equivalent, i.e. (cid:98) G ∼ (cid:98) G (cid:48) .
Revised: Then the clique complexes (cid:98) G and (cid:98) G ′ are homotopy equivalent, i.e. (cid:98) G ∼ (cid:98) G ′ .

Original: Proof:
Revised: Proof:

Original: Notice that G (cid:48) is a subgraph of G , and hence (cid:98) G (cid:48) is a subcomplex in (cid:98) G .
Revised: Notice that G ′ is a subgraph of G , and hence (cid:98) G ′ is a subcomplex in (cid:98) G .

Original: Let u be dominated by v in G .
Revised: Let u be dominated by v in G .

Original: Then, we can write a deformation retract from (cid:98) G to (cid:98) G (cid:48) by pushing the edge e uv starting from u toward v .
Revised: Then, we can write a deformation retract from (cid:98) G to (cid:98) G ′ by pushing the edge e uv starting from u toward v .

Original: In other words, by using the simplicial coordinates, one can deﬁne a homotopy
Revised: In other words, by using the simplicial coordinates, one can define a homotopy

Original: F : (cid:98) G × I → (cid:98) G which is identity on (cid:98) G (cid:48) and pushing all the faces in (cid:98) G − (cid:98) G (cid:48) to the corresponding faces in (cid:98) G (cid:48) .
Revised: F : (cid:98) G × I → (cid:98) G which is identity on (cid:98) G ′ and pushing all the faces in (cid:98) G − (cid:98) G ′ to the corresponding faces in (cid:98) G ′ .

Original: This gives a homotopy equivalence (cid:98) G ∼ (cid:98) G (cid:48) .
Revised: This gives a homotopy equivalence (cid:98) G ∼ (cid:98) G ′ .

Original: To visualize, in Figure 3, one can push vertex 1 in the clique complex (cid:98) G towards vertex 3 along the edge between them.
Revised: To visualize, in Figure 3, one can push vertex 1 in the clique complex (cid:98) G towards vertex 3 along the edge between them.

Original: After the push, the 2 -simplices [1,2,3] and [1,3,4] are pushed to the edges [2,3] and [3,4] respectively.
Revised: After the push, the 2 -simplices [1,2,3] and [1,3,4] are pushed to the edges [2,3] and [3,4] respectively.

Original: See [1, 7] and [11, Lemma 2.2] for details.
Revised: See [1, 7] and [11, Lemma 2.2] for details.

Original: (cid:3)
Revised: □

Original: Remark 6.
Revised: Remark 6.

Original: [Collapsing] Note that this collapsing operation is adaptation of a well-known notion called deformation retract in algebraic topology in a simplicial complex setting [28].
Revised: [Collapsing] Note that this collapsing operation is adaptation of a well-known notion called deformation retract in algebraic topology in a simplicial complex setting [28].

Original: This operation keeps the homotopy type the same, and hence the homology does not change with this reduction.
Revised: This operation keeps the homotopy type the same, and hence the homology does not change with this reduction.

Original: In [1, 7, 11], this is called folding ( G folds onto G − { u } ) or a strong collapse .
Revised: In [1, 7, 11], this is called folding ( G folds onto G − { u } ) or a strong collapse .

Original: In these papers, the algorithm reduces simplicial complexes in the ﬁltration one by one so that its associated clique complex keeps the same homotopy type.
Revised: In these papers, the algorithm reduces simplicial complexes in the filtration one by one so that its associated clique complex keeps the same homotopy type.

Original: Our contribution here is to adapt this operation to the graph ﬁltrations and deﬁne a smaller subgraph before the ﬁltration step so that the induced simplicial complexes are homotopy equivalent.
Revised: Our contribution here is to adapt this operation to the graph filtrations and define a smaller subgraph before the filtration step so that the induced simplicial complexes are homotopy equivalent.

Original: Since we prune the graph at the beginning of the process, our algorithm signiﬁcantly reduces the computational costs for the induced persistence diagrams.
Revised: Since we prune the graph at the beginning of the process, our algorithm significantly reduces the computational costs for the induced persistence diagrams.

Original: In the following, we introduce the PrunIT Algorithm by showing that removing a dominated vertex does not change the persistence diagrams of the graph.
Revised: In the following, we introduce the PrunIT Algorithm by showing that removing a dominated vertex does not change the persistence diagrams of the graph.

Original: We give the proof in Appendix C.
Revised: We give the proof in Appendix C.

Original: Theorem 7.
Revised: Theorem 7.

Original: Let G =
Revised: Let G =

Original: ( V , E ) be an unweighted graph, and f : V → R be a ﬁltering function.
Revised: ( V , E ) be an unweighted graph, and f : V → R be a filtering function.

Original: Let u ∈ V be dominated by v ∈ V and f ( u ) ≥ f ( v ) .
Revised: Let u ∈ V be dominated by v ∈ V and f ( u ) ≥ f ( v ) .

Original: Then, removing u from G does not change the persistence diagrams for sublevel ﬁltration, i.e. for any k ≥
Revised: Then, removing u from G does not change the persistence diagrams for sublevel filtration, i.e. for any k ≥

Original: [Equation]
Revised: [Equation]

Original: Outline of the proof: The main idea is to employ the collapsing idea in the simplicial complexes of the ﬁltration (cid:98) G 0 ⊂ (cid:98) G 1 ⊂ (cid:98) G 2 ⊂ · · · ⊂ (cid:98) G m in a suitable way.
Revised: Outline of the proof: The main idea is to employ the collapsing idea in the simplicial complexes of the filtration (cid:98) G 0 ⊂ (cid:98) G 1 ⊂ (cid:98) G 2 ⊂ · · · ⊂ (cid:98) G m in a suitable way.

Original: In particular, the lemma above shows that if a vertex u is dominated by a vertex v in (cid:98) G i , then removing St ( u ) from (cid:98) G i does not change the homotopy type.
Revised: In particular, the lemma above shows that if a vertex u is dominated by a vertex v in (cid:98) G i , then removing St ( u ) from (cid:98) G i does not change the homotopy type.

Original: Hence, if we ensure that when u ﬁrst appears in the ﬁltration { (cid:98) G i } , the dominated vertex v is already there, then u can be removed from all the simplicial complexes in the ﬁltration; removing u from the original graph before building the simplicial complexes does not affect the homotopy type of complexes in the ﬁltration.
Revised: Hence, if we ensure that when u first appears in the filtration { (cid:98) G i } , the dominated vertex v is already there, then u can be removed from all the simplicial complexes in the filtration; removing u from the original graph before building the simplicial complexes does not affect the homotopy type of complexes in the filtration.

Original: The condition f ( u ) ≥ f ( v ) makes sure that whenever u exists in { (cid:98) G i } , the dominant vertex is already there, and u can be removed from all simplicial complexes, and hence from the graph G .
Revised: The condition f ( u ) ≥ f ( v ) makes sure that whenever u exists in { (cid:98) G i } , the dominant vertex is already there, and u can be removed from all simplicial complexes, and hence from the graph G .

Original: We give the proof of the theorem in Appendix C.
Revised: We give the proof of the theorem in Appendix C.

Original: Notice that the primary condition to remove dominated vertices from the graph ensures that the dominated vertex enters the ﬁltration after its dominating counterpart.
Revised: Notice that the primary condition to remove dominated vertices from the graph ensures that the dominated vertex enters the filtration after its dominating counterpart.

Original: With the PrunIT Algorithm, we show that removing the dominated vertex does not change the homotopy type of the simplicial complexes in the ﬁltration.
Revised: With the PrunIT Algorithm, we show that removing the dominated vertex does not change the homotopy type of the simplicial complexes in the filtration.

Original: As homotopy equivalence implies the equivalences of homology groups at all levels, the reduction with this algorithm works in all dimensions.
Revised: As homotopy equivalence implies the equivalences of homology groups at all levels, the reduction with this algorithm works in all dimensions.

Original: Furthermore, while coral reduction works above the corresponding dimension ( j > k ) , the PrunIT algorithm works in any dimension.
Revised: Furthermore, while coral reduction works above the corresponding dimension ( j > k ) , the PrunIT algorithm works in any dimension.

Original: Remark 8.
Revised: Remark 8.

Original: [Superlevel Filtration]
Revised: [Superlevel Filtration]

Original: The same proof applies to the superlevel ﬁltration by changing the condition f ( u ) ≥ f ( v ) to f ( u ) ≤ f ( v ) in the theorem.
Revised: The same proof applies to the superlevel filtration by changing the condition f ( u ) ≥ f ( v ) to f ( u ) ≤ f ( v ) in the theorem.

Original: In particular, if PD v k ( G , f ) represents the k th PD for superlevel ﬁltration, then with the condition f ( u ) ≤ f ( v ) , we would have PD v k ( G , f ) =
Revised: In particular, if PD v k ( G , f ) represents the k th PD for superlevel filtration, then with the condition f ( u ) ≤ f ( v ) , we would have PD v k ( G , f ) =

Original: PD v k ( G − { u } , f ) for any k ≥ 0 .
Revised: PD v k ( G − { u } , f ) for any k ≥ 0 .

Original: Notice that if one takes f to be the degree function and uses the superlevel ﬁltration, then the theorem automatically holds for any dominated vertex as deg ( u ) ≤ deg ( v ) when u is dominated by v .
Revised: Notice that if one takes f to be the degree function and uses the superlevel filtration, then the theorem automatically holds for any dominated vertex as deg ( u ) ≤ deg ( v ) when u is dominated by v .

Original: Remark 9.
Revised: Remark 9.

Original: [Detecting Dominating Vertices]
Revised: [Detecting Dominating Vertices]

Original: The dominating vertices can be computed by using the following approach (Algorithm is given in appendix Section B).
Revised: The dominating vertices can be computed by using the following approach (Algorithm is given in appendix Section B).

Original: Let A =
Revised: Let A =

Original: ( a ij ) be the adjacency matrix for a graph G .
Revised: ( a ij ) be the adjacency matrix for a graph G .

Original: Given v i 0 ∈ V , consider all j ’s with a i 

Revised: Given v i 0 ∈ V , consider all j ’s with a i

Original: j = 1 .
Revised: j = 1 .

Original: Check if v i 0 is dominated by v j by comparing the rows R i 0 and R j , i.e. for any k (cid:54) =
Revised: Check if v i 0 is dominated by v j by comparing the rows R i 0 and R j , i.e. for any k ̸ =

Original: j with a i 0 k
Revised: j with a i 0 k

Original: = 1 , check whether a j k = 1 .
Revised: = 1 , check whether a j k = 1 .

Original: If this holds, v j dominates v i 0 .
Revised: If this holds, v j dominates v i 0 .

Original: Removing i th 0 row R i 

Revised: Removing i th 0 row R i

Original: and i th 0 column C i 0 from A corresponds to removing v i 0 from G .
Revised: and i th 0 column C i 0 from A corresponds to removing v i 0 from G .

Original: Essentially, vertex v i 0 is compared to each neighbor v j by checking whether v i 0 is already a neighbor of each of v j ’s neighbors.
Revised: Essentially, vertex v i 0 is compared to each neighbor v j by checking whether v i 0 is already a neighbor of each of v j ’s neighbors.

Original: These checks require iterating over each vertex, searching vertex neighbors in the graph and getting the neighbors of each neighbor.
Revised: These checks require iterating over each vertex, searching vertex neighbors in the graph and getting the neighbors of each neighbor.

Original: The computational complexity is therefore O ( |V| × d 2 ) where d is the average degree in the graph.
Revised: The computational complexity is therefore O ( |V| × d 2 ) where d is the average degree in the graph.

Original: While our main focus is the most common method, sublevel/superlevel ﬁltration, in the application of
Revised: While our main focus is the most common method, sublevel/superlevel filtration, in the application of

Original: PH in graph setting, our PrunIt algorithm works perfectly well with another common method, power ﬁltration [3], as well, i.e. removing a dominated vertex does not change persistence diagrams.
Revised: PH in graph setting, our PrunIt algorithm works perfectly well with another common method, power filtration [3], as well, i.e. removing a dominated vertex does not change persistence diagrams.

Original: Theorem 10.
Revised: Theorem 10.

Original: [PrunIt for Power Filtration] Let G =
Revised: [PrunIt for Power Filtration] Let G =

Original: ( V , E ) be an unweighted connected graph.
Revised: ( V , E ) be an unweighted connected graph.

Original: Let (cid:100) PD k
Revised: Let (cid:100) PD k

Original: ( G ) represent k th persistence diagram of G with power ﬁltration.
Revised: ( G ) represent k th persistence diagram of G with power filtration.

Original: Let u ∈ V be dominated by any other vertex in V .
Revised: Let u ∈ V be dominated by any other vertex in V .

Original: Then, for any k ≥ 1 , 
Revised: Then, for any k ≥ 1 , (cid:100) PD k ( G ) = (cid:100) PD k

Original: [Equation]
Revised: ( G − { u } ) .

Original: The proof of this result is given in appendix Section C.2.
Revised: The proof of this result is given in appendix Section C.2.

Original: Combining the CoralTDA and PrunIT Algorithms
Revised: Combining the CoralTDA and PrunIT Algorithms

Original: Even though both algorithms are quite effective by themselves, we signiﬁcantly reduce the computational costs (Figure 6) by combining them as follows.
Revised: Even though both algorithms are quite effective by themselves, we significantly reduce the computational costs (Figure 6) by combining them as follows.

Original: For a given graph G =
Revised: For a given graph G =

Original: ( V , E ) and ﬁltering function f : V → R , one can start by trimming all dominated vertices with respect to f , and get a smaller graph G (cid:48) .
Revised: ( V , E ) and filtering function f : V → R , one can start by trimming all dominated vertices with respect to f , and get a smaller graph G ′ .

Original: We have already proven that PD k ( G ) =
Revised: We have already proven that PD k ( G ) =

Original: PD k ( G (cid:48) ) .
Revised: PD k ( G ′ ) .

Original: Then, one can take the k -core of this smaller graph G (cid:48) to compute higher persistence diagrams of the original graph G as before.
Revised: Then, one can take the k -core of this smaller graph G ′ to compute higher persistence diagrams of the original graph G as before.

Original: In particular, by applying both reduction algorithms, for any k ≥ 0 , we obtain
Revised: In particular, by applying both reduction algorithms, for any k ≥ 0 , we obtain

Original: [Equation]
Revised: [Equation]

Original:  Experiments and Discussion
Revised:  Experiments and Discussion

Original: We apply our new approaches to three types of datasets.
Revised: We apply our new approaches to three types of datasets.

Original: The details of datasets are provided in
Revised: The details of datasets are provided in

Original: Table 1 and appendix Table 2.
Revised: Table 1 and appendix Table 2.

Original: Graph classiﬁcation datasets consists of biological kernel [37] and ego networks from TWITTER and FACEBOOK [42].
Revised: Graph classification datasets consists of biological kernel [37] and ego networks from TWITTER and FACEBOOK [42].

Original: Node classiﬁcation datasets includes CITESEER and CORA [50] and Open
Revised: Node classification datasets includes CITESEER and CORA [50] and Open

Original: Graph Benchmark citation (paper cites paper)
Revised: Graph Benchmark citation (paper cites paper)

Original: OGB-ARXIV and OGB-MAG
Revised: OGB-ARXIV and OGB-MAG

Original: [32] networks.
Revised: [32] networks.

Original: Large networks dataset contains 11 large networks of 100K-1M vertices from the Stanford Repository [40].
Revised: Large networks dataset contains 11 large networks of 100K-1M vertices from the Stanford Repository [40].

Original: We used an AMD Ryzen 5 2100 MHZ 4 core computer in our R, Python and Java experiments.
Revised: We used an AMD Ryzen 5 2100 MHZ 4 core computer in our R, Python and Java experiments.

Original: We evaluate both algorithms by comparing vertex and edge sets and the total run time for the reduced graph with respect to the original graph.
Revised: We evaluate both algorithms by comparing vertex and edge sets and the total run time for the reduced graph with respect to the original graph.

Original: In the rest of this manuscript, we compute the vertex set reduction as 100 × ( |V| − |V (cid:48) | ) / |V| where V (cid:48) is the vertex count in the reduced graph.
Revised: In the rest of this manuscript, we compute the vertex set reduction as 100 × ( |V| − |V ′ | ) / |V| where V ′ is the vertex count in the reduced graph.

Original: Edge and time reductions are computed similarly.
Revised: Edge and time reductions are computed similarly.

Original: 6.1 Reduction on Graph Classiﬁcation Datasets
Revised: 6.1 Reduction on Graph Classification Datasets

Original: In this task, our goal is to evaluate the reduction of computational costs when we use the CoralTDA and PrunIT algorithms on datasets chosen from different graph classiﬁcation tasks.
Revised: In this task, our goal is to evaluate the reduction of computational costs when we use the CoralTDA and PrunIT algorithms on datasets chosen from different graph classification tasks.

Original: We used one of the most commonly used functions in these experiments, the degree function with sublevel ﬁltration.
Revised: We used one of the most commonly used functions in these experiments, the degree function with sublevel filtration.

Original: In Figure 4, we show the vertex reduction when using CoralTDA for computations of PD k ( G ) for dimensions (Betti) k
Revised: In Figure 4, we show the vertex reduction when using CoralTDA for computations of PD k ( G ) for dimensions (Betti) k

Original: = 1 to k = 5 .
Revised: = 1 to k = 5 .

Original: At dimension k = 4 and k = 5 , CoralTDA reduces 10 datasets by 100%, i.e., these datasets have trivial PD k ( G ) for k ≥ 4 .
Revised: At dimension k = 4 and k = 5 , CoralTDA reduces 10 datasets by 100%, i.e., these datasets have trivial PD k ( G ) for k ≥ 4 .

Original: Even at smaller dimensions, CoralTDA can reduce the vertex set by 25%-75%.
Revised: Even at smaller dimensions, CoralTDA can reduce the vertex set by 25%-75%.

Original: Figure 5a shows reduction percentages by the PrunIT algorithm.
Revised: Figure 5a shows reduction percentages by the PrunIT algorithm.

Original: FIRSTMM and SYNNEW datasets are reduced by less than 10%; however the other 11 datasets are reduced by at least 35%.
Revised: FIRSTMM and SYNNEW datasets are reduced by less than 10%; however the other 11 datasets are reduced by at least 35%.

Original: The lower reduction on FIRSTMM and SYNNEW are due to stronger cores on the networks.
Revised: The lower reduction on FIRSTMM and SYNNEW are due to stronger cores on the networks.

Original: SYNNEW is
Revised: SYNNEW is

Original: [Figure]
Revised: [Figure]

Original: [Figure]

Original: [Figure]
Revised: [Figure]

Original: synthetically created, but FIRSTMM is created from 3d point cloud data and categories of various household objects.
Revised: synthetically created, but FIRSTMM is created from 3d point cloud data and categories of various household objects.

Original: We believe that the physical proximity of similar objects (e.g., chairs are close to each other) in a household creates a denser community structure in the FIRSTMM dataset, which in turn results in strong cores.
Revised: We believe that the physical proximity of similar objects (e.g., chairs are close to each other) in a household creates a denser community structure in the FIRSTMM dataset, which in turn results in strong cores.

Original: We further report reductions in computational time (Figure 8), edge set (Figure 9), and simplex count (Figure 7) in the Appendix.
Revised: We further report reductions in computational time (Figure 8), edge set (Figure 9), and simplex count (Figure 7) in the Appendix.

Original: 6.2 Reduction on Node Classiﬁcation Datasets
Revised: 6.2 Reduction on Node Classification Datasets

Original: In this task, our goal is to compute the reduction of computational costs by using CoralTDA and
Revised: In this task, our goal is to compute the reduction of computational costs by using CoralTDA and

Original: PrunIT algorithms on datasets chosen from node classiﬁcation tasks.
Revised: PrunIT algorithms on datasets chosen from node classification tasks.

Original: The CoralTDA results are computed over CITESEER and CORA networks and shown in Figure 4 with more than 20% reduction for the ﬁrst and higher dimensional persistence.
Revised: The CoralTDA results are computed over CITESEER and CORA networks and shown in Figure 4 with more than 20% reduction for the first and higher dimensional persistence.

Original: In node classiﬁcation, we can also analyze the k-hop ( k ≥ 1 ) neighborhood of a vertex with topological features (such as Betti-0) and use the computed persistence diagram to classify the vertex.
Revised: In node classification, we can also analyze the k-hop ( k ≥ 1 ) neighborhood of a vertex with topological features (such as Betti-0) and use the computed persistence diagram to classify the vertex.

Original: Such an approach has yielded SOTA results with signiﬁcant improvement in accuracy by using 0-dimensional persistence [19].
Revised: Such an approach has yielded SOTA results with significant improvement in accuracy by using 0-dimensional persistence [18].

Original: However, the computational costs of persistent homology are non-negligible in large graphs, even for 0-dimensional features.
Revised: However, the computational costs of persistent homology are non-negligible in large graphs, even for 0-dimensional features.

Original: For example, in Open Graph
Revised: For example, in Open Graph

Original: Benchmark datasets [32], one must compute persistence diagrams for each vertex in 100k to 111M vertex graphs.
Revised: Benchmark datasets [32], one must compute persistence diagrams for each vertex in 100k to 111M vertex graphs.

Original: We apply the PrunIT algorithm to two graphs, Arxiv and MAG, from the Open Graph Benchmark to compute time reduction in persistence diagram computations.
Revised: We apply the PrunIT algorithm to two graphs, Arxiv and MAG, from the Open Graph Benchmark to compute time reduction in persistence diagram computations.

Original: We follow the approach in [19] and extract the 1-hop neighborhood of each ego vertex.
Revised: We follow the approach in [18] and extract the 1-hop neighborhood of each ego vertex.

Original: We use the degree function as ﬁltering function as before.
Revised: We use the degree function as filtering function as before.

Original: In Figure 5b, we show the reduction in computational time for 0-dimensional persistence.
Revised: In Figure 5b, we show the reduction in computational time for 0-dimensional persistence.

Original: We compute the time costs of PrunIT by considering all the algorithm steps: ﬁnding and removing the dominated vertices, creating an induced graph with the vertices, and running 0-dimensional persistent homology on the graph by using vertex degrees as the ﬁltering function.
Revised: We compute the time costs of PrunIT by considering all the algorithm steps: finding and removing the dominated vertices, creating an induced graph with the vertices, and running 0-dimensional persistent homology on the graph by using vertex degrees as the filtering function.

Original: As Figure 5b shows, we see more than 25% reduction in computation time in most graphs.
Revised: As Figure 5b shows, we see more than 25% reduction in computation time in most graphs.

Original: Speciﬁcally, on average, computation times of 0-dimensional persistence on OGB-ARXIV networks are reduced by 37%, and those of
Revised: Specifically, on average, computation times of 0-dimensional persistence on OGB-ARXIV networks are reduced by 37%, and those of

Original: OGB-MAG networks are reduced by 23%.
Revised: OGB-MAG networks are reduced by 23%.

Original: The results show that we can mitigate the computational costs of persistence homology by using the PrunIT algorithms.
Revised: The results show that we can mitigate the computational costs of persistence homology by using the PrunIT algorithms.

Original: 6.3 Reduction on Large Networks
Revised: 6.3 Reduction on Large Networks

Original: Our goal is to combine PrunIt and CoralTDA algorithms to achieve the maximum vertex and edge reduction in large networks in these experiments.
Revised: Our goal is to combine PrunIt and CoralTDA algorithms to achieve the maximum vertex and edge reduction in large networks in these experiments.

Original: Table 1 shows that on the biggest network of com-youtube , we eliminate 59% of the vertices when we only apply the PrunIt (on average 62% in all datasets).
Revised: Table 1 shows that on the biggest network of com-youtube , we eliminate 59% of the vertices when we only apply the PrunIt (on average 62% in all datasets).

Original: The reduction is as high as 95% (in emailEuAll ).
Revised: The reduction is as high as 95% (in emailEuAll ).

Original: Similarly, PrunIt creates signiﬁcant edge reduction; 40% of all edges are removed on
Revised: Similarly, PrunIt creates significant edge reduction; 40% of all edges are removed on

Original: [Table]
Revised: [Table]

Original: average.
Revised: average.

Original: Figure 6 shows the reduction when we apply both CoralTDA and PrunIt on large networks.
Revised: Figure 6 shows the reduction when we apply both CoralTDA and PrunIt on large networks.

Original: Even for low cores of 2 and [Table] the combined algorithms reach a vertex reduction rate of 78%.
Revised: Even for low cores of 2 and 3, the combined algorithms reach a vertex reduction rate of 78%.

Original: These results show that our algorithms can effectively reduce large networks to more manageable sizes
Revised: These results show that our algorithms can effectively reduce large networks to more manageable sizes.

Original: [Table]

Original:  Conclusion
Revised:  Conclusion

Original: We have proposed two new highly effective algorithms to signiﬁcantly reduce the computational costs of TDA methods on graphs.
Revised: We have proposed two new highly effective algorithms to significantly reduce the computational costs of TDA methods on graphs.

Original: While coral reduction is very effective for higher persistence diagrams,
Revised: While coral reduction is very effective for higher persistence diagrams,

Original: PrunIt is highly efﬁcient, in general.
Revised: PrunIt is highly efficient, in general.

Original: Our experiments have showed that even for lower dimensional topological features, such as k = 1 , for some datasets our methods can reduce graph order by up to 95%, which alleviates computational costs substantially.
Revised: Our experiments have showed that even for lower dimensional topological features, such as k = 1 , for some datasets our methods can reduce graph order by up to 95%, which alleviates computational costs substantially.

Original: Furthermore, in most graph datasets we reduce graph sizes by 100% for 3rd or higher dimensions.
Revised: Furthermore, in most graph datasets we reduce graph sizes by 100% for 3rd or higher dimensions.

Original: Our methods provides a novel solution for efﬁcient application of the powerful TDA methods on large networks and build a bridge between the graph theory and TDA, opening a pathway for broader applicability of topological graph learning in practice.
Revised: Our methods provides a novel solution for efficient application of the powerful TDA methods on large networks and build a bridge between the graph theory and TDA, opening a pathway for broader applicability of topological graph learning in practice.

Original:  Acknowledgments
Revised:  Acknowledgments

Original: This material is based upon work sponsored by the Canadian NSERC Discovery Grant RGPIN2020-05665, NSF of USA under award number ECCS 2039701, OAC-1828467, DMS-1925346,
Revised: This material is based upon work sponsored by the Canadian NSERC Discovery Grant RGPIN2020-05665, NSF of USA under award number ECCS 2039701, OAC-1828467, DMS-1925346,

Original: CNS-2029661, OAC-2115094, ARO award W911NF-17-1-0356, and Simons Collaboration Grant # 579977.
Revised: CNS-2029661, OAC-2115094, ARO award W911NF-17-1-0356, and Simons Collaboration Grant # 579977.

Original: Part of this material is also based upon work supported by (while serving at) the National
Revised: Part of this material is also based upon work supported by (while serving at) the National

Original: Science Foundation.
Revised: Science Foundation.

Original: Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reﬂect the views of the National Science
Revised: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science

Original: Foundation.
Revised: Foundation.