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 with 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 thesuccessful application of TDA in real-world studies, particularly in conjunctionwith 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 socialnetworks often have hundreds of thousands of vertices, making the application ofexisting 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 buteffective algorithms to compute the exact persistence diagrams of large graphs toaddress 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 issufﬁcient 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 introducea pruning algorithm for graphs to compute their persistence diagrams by removingthe 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 indicate that the newapproach 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.

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, fromgraph-structured 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, it has become quite popular in various ML tasks, ranging from bioinformatics [KNBNH16, XW14, NPL + 15] to ﬁnance [LPRS08, GK18], to material science [IOH17, NHH + 15] and network analysis [SGB17, HMR09, CH13] (see the literatureoverviews [OPT + 17a, AQO + 20, CM21] and  TDA applications library [Giu22]).
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, 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]).

Original: Furthermore, recently there has emerged a highly active research area that combines the PH machinery with geometricdeep learning (GDL) methods [HKNU17, KFH17, ZLJ + 19, ZW19, HDBM + 22, CCG21b].
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 subtlerpatterns in the data shape dynamics at multiple resolution scales, which are not accessible with moreconventional, non-topological methods [Car09].
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 oftopological spaces (ﬁltration) induced from the data, and record the evolution of topological feeaturesin 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 howlong such features persist throughout the considered ﬁltration of a scale parameter, convey a criticalinsight 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 [RBB19, CW20, ZYCW20, HGR + 20], link prediction [BAS + 18, YMG + 21] and anomalydetection [BNP16, OBSDK + 21].
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,its wider usage is obstructed by prohibitive computational costs.
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 thecomplexity of the standard PH algorithm is cubic in the number of simplices [OPT + 17b], so oneneeds 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: In turn, computation of higherlevel 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 largenetworks by introducing two new efﬁcient algorithms which signiﬁcantly reduce the cost of computingpersistence diagrams (PD) for large real-world networks: CoralTDA and PrunIT Algorithms.
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 Reduction: Based on the observation that many vertices in large real-world networkshave low degrees andas such do not contribute to PDs in higher dimensions, we developed the CoralTDA algorithm (Theorem 2) where we prove that ( k + 1) -core
Revised: CoralTDA Algorithm: 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 tocompute 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 persistencediagram PD k ( G )without losing any 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 lowerdimensional topological features, such as k = 1 , we reduce the graph order by up to 73% forsome datasets (See Figure 3).
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 indicate that many real-life data sets exhibit nontrivialsecond 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 otherhand, our reduction reaches 100% for the third or higher dimensions in several networks, implyingthat 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 ourunderstanding of the existence of higher-order dimensional holes and their role in the organization ofcomplex 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 tofacilitate 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 agiven 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 thegraph does not change PDs at any level, provided that the dominated vertex enters the ﬁltration afterthe 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 broadcategory of large graphs from 100K to 1M vertices, and it can reach 95% vertex reduction (see Table1).
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 Table 1).

Original: Further, when  CoralTDA and PrunIT algorithms are combined, they signiﬁcantly reduce the graphsizes for the computation of PDs (Figure 5).
Revised: Further, when we combine CoralTDA and PrunIT algorithms, we can signiﬁcantly reduce the graph sizes for the computation of PDs (Figure 6).

Original: The key novelty of our contributionscan be summarized as follows:
Revised: We summarize the key novelty of our contributions as follows:

Original: • We show that the graphs’higher ( k + 1 ) persistence diagrams only depend on their k -cores.
Revised: • We show that the graphs’ ( k + 1 )st 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 graphsize 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 methods, highly successful TDA tools can be applied to very large graphs andlarge datasets where previously its use was obstructed 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 agiven 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 anambient 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 inducedby the pairwise distances of data points (Vietoris-Rips ﬁltration) and keeping track of the topologicalchanges in this sequence [ZC05, EH10].
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 PHconstruction 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 identiﬁerto deﬁne PH in the point cloud setting is the pairwise distances of points, the principal identiﬁer inthe 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 worksvery differently ina 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 thepersistence 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 [MW19].
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 [MN13].
Revised: Mischaikow and Nanda brought the discrete Morse Theory of geometric topology to the combinatorial setting [43].

Original: In [Oba18, ˇCV21, DHM19, EH16], the authors studied the same problem withdifferent 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 couldreduce 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 [KSRS19] by applyingthe techniques developed in [MN13] 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 ofk th persistence diagram (PD) is O ( n 3 ) where n is the number of k -simplices [OPT + 17b], [MN13]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 toﬁnd the critical k -simplices in each ﬁltration step, the computational complexity O ( m 2 × n log n ) isnot 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 (PH).
Revised: This part provides a background on the theory of persistent homology.

Original: Homology H k
Revised: Homology H k

Original: ( X ) isan essential invariant in algebraic topology, which captures the information of the k -dimensionalholes (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 connectedcomponent 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 controlledtopological 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 persistenthomology, see [EH10, DW22].
Revised: For basic background on persistent homology, see [23, 22].

Original: There are several ways to use PH in a network setting, such asthe power ﬁltration or using differentcomplexes (e.g., Vietoris-Rips, ˇ Cech complexes) to construct the ﬁltration for a given ﬁlteringfunction [AAEF19].
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/superlevelﬁltrations 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 ﬁltrationsare the most common methods because one can inject domain information into the PH process if thechosen ﬁltering function comes from the network domain (e.g., atomic number for 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 persistenthomology 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 canbe deﬁned similarly by considering the generating sets { f ( v r ) ≥ α i } instead of { f ( v r ) ≤ α i } above.
Revised: Superlevel ﬁltrations can be deﬁned similarly by considering the generating sets { f ( v r ) ≥ α i } instead of { f ( v r ) ≤ α i } above.

Original: Here, (cid:98) G i can be taken as the different simplicial complexes induced by G i which gives different typesof ﬁltrations [AAEF19].
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 thesimplicial 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 thehomology 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 (connected component, loop, 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 deathtime.
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 persistencediagram 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 ﬁlteringfunction 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 edgesconnected 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: [Sei83].

Original: In other words, G k is the largest subgraph of G whereall 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, vertex 1belongs to 0-core as it is disconnected from the graph.
Revised: Here, vertexbelongs to 0-core as it is disconnected from the graph.

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

Original: When we use vertex degree asthe ﬁltering function and allow graph cliques of size three atmost, the only one-dimensional hole (shown with the red circle)appears at degree 4 forthe 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 becausetheir 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 and1-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 manyareas such as graph similarity matching [NMLV18], graph clustering [GMTV14], network visualization [GTV11], anomalydetection [SBS + 13] and robustness analysis [BLMTM20].
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 verticeswhose degree falls below a k , until it deletes all vertices fromthe 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 arethe 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 thecomplexity to O ( m + n ) “by keeping an in-memory array of all possible degree values and keepingtrack of bin boundaries” [BZ03].
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: Recall that 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 degeneracyof 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 (cid:54) = ∅} .

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 impliesthat 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 theoriginal 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: Byusing 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 0 ⊂ (cid:98) G k 1 ⊂ (cid:98) G k 2 ⊂ ...

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

Original: Thiswill 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 theequivalence 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 verticesof 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, iff : 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: → Rmay 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 win 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 functionf : 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 computingvertex ﬁltering (activation) values on the original graph, but using the edges of the reduces graph toextract 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 sizechanges, 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 forfurther details.
Revised: See Remark 1 for further details.

Original: We give the proof of the following theorem in Appendix C.
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 cliquecomplex (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 Smust have a degree at least k + 1 where this degree count comes only from the k -simplices of S , andremoving 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 theproof of the theorem in Appendix C.
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 tocompute higher persistence diagrams.
Revised: CoralTDA is an effective tool for reducing computational costs to compute higher persistence diagrams.

Original: See Figure 3 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’sseminal result [Kah09], to observe nontrivial Betti numbers for higher dimensions in Erdós-Rényigraphs 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 in Erdós-Rényi graphs G ( n, p ) , the average degree must be very high.

Original: In particular, for a graph G ( n, p ) , in orderto have nontrivial k th -homology in its clique complex, Kahle proved that for p = n α , α should bebetween − 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 degreeshould 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 degreeshould 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 averagedegree should be between 31 and 100 to have  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 muchsparser graphs (see Appendix, ﬁgs. 9 and  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 errorbounds and the associated loss of topological information when G ( n, p ) is employed to approximatereal-world network phenomena, for instance, in the case of synthetic power grid networks and othercyber-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 affectingthe 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 diagramsof 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 agiven graph G , andﬁltering function f : V → R , removing (pruning) speciﬁc vertices from the graph does not changethe 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 thevertex 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 persistenthomology (see Figure 3 and  Figure 6).
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 Figure 4 and appendix Figure 7).

Original: In algebraic topology, homotopy is a very effective tool to compute topological invariants likehomology, and fundamental group [Hat02].
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 invariantsare 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 tosimplify agiven 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 thesublevel 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 diagramfor 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 vertexv , we call u a dominated vertex of G (see Figure 2).
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 Gobtained 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 (cid:48)

Original: =
Revised: =

Original: ( V (cid:48) , E (cid:48) )
Revised: ( V (cid:48) , E (cid:48) )

Original: where V (cid:48)
Revised: where V (cid:48)

Original: =
Revised: =

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

Original: E −{ e uw ∈ E}for any w .
Revised: E − { e uw ∈ 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, uis 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 ufrom G corresponds to removing St ( u ) from the clique complex (cid:98) G , i.e.(cid:98) G − St ( u ) = (cid:98) G (cid:48) .
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 ) = (cid:98) G (cid:48) .

Original: A useful result is that removing a dominated vertexdoes 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 (cid:48)

Original: =
Revised: =

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

Original: Thenthe 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 (cid:48) are homotopy equivalent, i.e. (cid:98) G ∼ (cid:98) G (cid:48) .

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 (cid:48) is a subgraph of G , and hence (cid:98) G (cid:48) is a subcomplex in (cid:98) G .

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

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

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 facesin (cid:98) G (cid:48) .
Revised: 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) .

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 (cid:48) .

Original: To visualize, in Figure 2, one can push vertex 1 inthe 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 [Ada13, BP18] and[BFJ10, Lemma 2.2] for details.
Revised: See [1, 7] and [11, Lemma 2.2] for details.

Original: [Equation]
Revised: (cid:3)

Original: Remark 6.
Revised: Remark 6.

Original: [Collapsing] Note that this collapsing operation is adaptation of a well-known notioncalled deformation retract in algebraic topology in a simplicial complex setting [Hat02].
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: Thisoperation keeps the homotopy type the same, and hence the homology does not change with thisreduction.
Revised: This operation keeps the homotopy type the same, and hence the homology does not change with this reduction.

Original: In [Ada13, BP18, BFJ10], 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 itsassociated 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 thisoperation to the graph ﬁltrations and deﬁne a smaller subgraph before the ﬁltration step so that theinduced 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 ofthe process, our algorithm signiﬁcantly reduces the computational costs for the induced persistencediagrams.
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 vertexdoes 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 ofthe 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 showsthat 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 thehomotopy 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 dominatedvertex 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 thehomotopy 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 wheneveru exists in { (cid:98) G i } , the dominant vertex is already there, and u can be removed from all simplicialcomplexes, 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 thedominated 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 simplicialcomplexes 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 groupsat 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 coralreduction works above the corresponding dimension ( j > k ) , the PrunIT algorithm works in anydimension.
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 thecondition 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 thek 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 thesuperlevel 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 beeasily computed byusing the following approach (Algorithm is given in  Section A.1).
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 adjacencymatrix 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 jby 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 (cid:54) =

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 thisholds, 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 removingv 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 alreadya 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 vertexneighbors 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 istherefore 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, powerﬁltration [AAEF19], too. We want to thank the reviewer for inquiring about this generalization.
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 v ∈ Vand f ( u ) ≥ f ( v ) .
Revised: Let u ∈ V be dominated by any other vertex in V .

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

Original: [Equation]
Revised: [Equation]

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: While our PrunIt algorithm generalizes to power ﬁltrations as above, unfortunately our CoralTDAreduction does not extend to power ﬁltrations (see Remark 11).

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 5) 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 (cid:48) .

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 (cid:48) ) .

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 (cid:48) 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: Tables 1 and  2.
Revised: Table 1 and appendix Table 2.

Original: Graph classiﬁcation datasets consists of biological kernel [KKM + 16] and ego networks from TWITTER and FACEBOOK 
Revised: Graph classiﬁcation datasets consists of biological kernel [37] and ego networks from TWITTER and FACEBOOK [42].

Original: [ML12]. Node classiﬁcation datasets includes CITESEER and 
Revised: Node classiﬁcation datasets includes CITESEER and CORA [50] and Open

Original: CORA [RA15] and Open Graph Benchmark citation (paper cites paper)
Revised: Graph Benchmark citation (paper cites paper)

Original: OGB-ARXIV and OGB-MAG [HFZ + 20] networks.
Revised: OGB-ARXIV and OGB-MAG [32] networks.

Original: Large networks dataset contains 11 large networks of 100K-1Mvertices from the Stanford Repository [LK14].
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 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 reducedgraph 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 setreduction 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 (cid:48) | ) / |V| where V (cid:48) is the vertex count in the reduced graph.

Original: Edge andtime 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 our CoralTDAand 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 ofthe 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: We further report reductions in computational time (Figure 7), edge set (Figure 8), and simplex count(Figure 6) in the Appendix. In Figure 3, we show the vertex reduction when using CoralTDA for computations of PD k ( G ) fordimensions (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 datasetsby 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, CoralTDAcan reduce the vertex set by 25%-75%.
Revised: Even at smaller dimensions, CoralTDA can reduce the vertex set by 25%-75%.

Original: Figure 4a shows reduction percentages by the PrunIT algorithm.
Revised: Figure 5a shows reduction percentages by the PrunIT algorithm.

Original: FIRSTMM and SYNNEW datasetsare 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: Thelower 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. SYNNEW is

Original: SYNNEW issynthetically created, but FIRSTMM is created from 3d point cloud data and categories of varioushousehold objects.
Revised: [Figure] [Figure] [Figure] 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 toeach other) in a household creates a denser community structure in the FIRSTMM dataset, which inturn 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.

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 arecomputed over CITESEER and CORA networks and shown in Figure 3 with more than 20% reductionfor 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 withtopological 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 using0-dimensional persistence [CCG21a].
Revised: Such an approach has yielded SOTA results with signiﬁcant improvement in accuracy by using 0-dimensional persistence [19].

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 [HFZ + 20], one must compute persistence diagrams for each vertex in 100k to111M 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 (See Table 2), 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 approachin [CCG21a] and extract the 1-hop neighborhood of each ego vertex.
Revised: We follow the approach in [19] and extract the 1-hop neighborhood of each ego vertex.

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

Original: In Figure 4b, we show the reduction in computational time for 0dimensional 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 algorithmsteps: ﬁnding and removing the dominated vertices, creating an induced graph with the vertices, andrunning 0-dimensional persistent homology on the graph by using vertex degrees as the ﬁlteringfunction.
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 4b 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 networksare reduced by 37%, and those of OGB-MAG networks are reduced by 23%.
Revised: Speciﬁcally, on average, computation times of 0-dimensional persistence on OGB-ARXIV networks are reduced by 37%, and those of OGB-MAG networks are reduced by 23%.

Original: The results show that wecan 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 edgereduction in large networks in these experiments.[Table]
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] Table 1 shows that on the biggest network of com-youtube , we eliminate 59% of the verticeswhen 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% (inemailEuAll ).
Revised: The reduction is as high as 95% (in emailEuAll ).

Original: Similarly, PrunIt creates signiﬁcant edge reduction; 40% of all edges are removed onaverage.
Revised: Similarly, PrunIt creates signiﬁcant edge reduction; 40% of all edges are removed on [Table]

Original:  Figure 5 shows the reduction when we apply both CoralTDA and PrunIt on large networks.
Revised: average. Figure 6 shows the reduction when we apply both CoralTDA and PrunIt on large networks.

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

Original: Theseresults 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: [Figure]
Revised: [Table]

Original:  Conclusion
Revised:  Conclusion

Original: We have proposed two new highly effective algorithms to signiﬁcantly reduce the computational costsof 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,pruning algorithm is highly efﬁcient, in general.
Revised: While coral reduction is very effective for higher persistence diagrams, PrunIt is highly efﬁcient, in general.

Original: Our experiments have showed that even for lowerdimensional topological features, such as k = 1 , for some datasets our methods can reduce graphorder 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 graphdatasets, 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 novelsolution for efﬁcient application of the powerful TDA methods on large networks and build a bridgebetween the graph theory and TDA, opening a pathway for broader applicability of topological graphlearning in practice.Checklist For all authors... (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s
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:  contributions and scope? [Yes] See abstract for details. (b) Did you describe the limitations of your work? [Yes] See Section section
Revised:  Acknowledgments

Original: (c) Did you discuss any potential negative societal impacts of your work?

Original: [Yes] See

Original: Appendix D.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Original: them?

Original: [Yes]

Original: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] Please see the

Original: proofs given in Appendix C.

Original: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] Please see the proofs

Original: given in Appendix.

Original: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original: mental results (either in the supplemental material or as a URL)?

Original: [Yes]

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters,

Original: how they

Original: were chosen)?

Original: [Yes] Please see Section 6.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experi-

Original: ments multiple times)?

Original: [Yes] Our experiments report averages with deviations (See

Original: Figure 5).

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., type

Original: of GPUs, internal cluster, or cloud provider)?

Original: [Yes] The experimental seeting is given in Section 6. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
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: [Yes]

Original: (b) Did you mention the license of the assets? [Yes] (c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
Revised: CNS-2029661, OAC-2115094, ARO award W911NF-17-1-0356, and Simons Collaboration Grant # 579977.

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] (e) Did you discuss whether the data you are using/curating contains personally identiﬁable
Revised: Part of this material is also based upon work supported by (while serving at) the National

Original: information or offensive content?

Original: [N/A] We use graph data only.

Original: Our datasets do not

Original: contain personally identiﬁable information or offensive content.

Original: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?

Original: [N/A]

Original: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?

Original: [N/A]

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?

Original: [N/A]

Original: Appendix

Original: A Datasets:

Original: The following table shows characteristics of graphs in our datasets.

Original: [Table]

Original: A.1 Algorithms

Original: This section gives the CoralTDA reduction (Algorithm 1) and PrunIT algorithm (Algorithm 2).

Original: Ourpseudocode is optimized for clarity.

Original: Our implementation is available at omitted for double blind.

Original: Algorithm 1 . CoralTDA G → G k + ﬂag =
Revised: Science Foundation.

Original: true

Original: G k +1 =

Original: G while ﬂag is true and G is not empty do ﬂag=

Original: false for u ∈ V do if | N ( u ) | <

Original: ( k + 1) then ﬂag=true

Original: G k +1 =

Original: G k +1 \ u end if end for end while return G k +

Original: In Algorithm 1, we repeatedly delete vertices that have less than k+1 edges and return the resultinggraph.

Original: In Algorithm 2, we search for dominated neighbors of a vertex (lines 7–14) and delete such aneighbor (if exists) from the graph.

Original: We continue until we cannot ﬁnd a dominated vertex.

Original: Lines 4–24has a complexity of O ( | V | 2 ) .

Original: The outer loop (line 3) may run | V | , however we can parallelize thealgorithm for each vertex.

Original: B Further Reduction Results

Original: B.1 Reduction for Graph and Node Classiﬁcation Datasets

Original: In the following Figures 6 to 8, we report further reduction percentages for each PD k ( G ) for thegraph and node classiﬁcation datasets given in Table 2.

Original: Here, horizontal direction represents the

Original: Algorithm

Original: G → G (cid:48) 1: ﬂag=true 2: G (cid:48)

Original: =

Original: G 3: while ﬂag is true do 4: ﬂag=false 5: for u ∈ V do 6: for v ∈ N ( u ) do 7: flag 1 → 2 = flag 2 → 1 = false 8: for w ∈ V \ { u, v } do 9: if e uw ∈ E ∧ e vw / ∈ E then 10: flag 1 → 2 = true 11: end if 12: if e uw / ∈ E ∧ e vw ∈ E then 13: flag 2 → 1 = true 14: end if 15: if flag 1 → 2 = true ∧ flag 1 → 2 = false then 16:

Original: G (cid:48)

Original: =

Original: G \ v and flag = true 17: else if flag 1 → 2 = false ∧ flag 1 → 2 = true then 18:

Original: G (cid:48)

Original: =

Original: G \ u and flag = true 19: else if flag 1 → 2 = false ∧ flag 1 → 2 = false then 20:

Original: G (cid:48)

Original: =

Original: G \ u and flag = true 21: end if 22: end for 23: end for 24: end for 25: end while 26: return G (cid:48) dimension k for PD k ( G ) .

Original: Edge and simplex reduction ﬁgures closely resemble the vertex reductionin Figure 3.

Original: In time reduction of Figure 7, we see three datasets (OHSU, FACEBOOK, TWITTER)with diverging behavior as follows.

Original: OHSU, despite losing many of vertices in core reduction, does not have a time reduction as muchas the other kernel datasets.

Original: We explain this behavior with small graph orders but high coreness in

Original: OHSU.

Original: Note that unlike any other kernel dataset, vertex reduction does not reach 100% in OHSUgraphs.

Original: Calling core decomposition on OHSU graphs adds a time cost that takes away the beneﬁt ofcoralTDA to some extent.

Original: As a result, time reduction is at most 25%.

Original: A similar dynamic is seen in TWITTER and FACEBOOK, where graphs have high cores.

Original: As aresult, most vertices cannot be peeled away in core reduction (Figure 3 shows at most 20% vertexreduction).

Original: B.2 Clustering Coefﬁcient and Higher Persistence Diagrams

Original: While the proposed reduction algorithms provide a computationally efﬁcient framework for higherorder topological summaries of graphs, in many real-world applications, we still need to addressthe important question of whether a given network exhibits a nontrivial higher ( k ≥ 2) persistencediagram PD k

Original: ( G ) .

Original: When the ( k + 1) -core G k +1 is sufﬁciently small, the answer immediately followsfrom our result.

Original: However, the problem is substantially more challenging in applications involving large complex networks such as blockchain transactions and protein interaction graphs.

Original: The goal isto assess whether for k = 2 , 3 , PD k ( G ) is trivial or not.

Original: Indeed, such higher homological featuresmight be associated, for example, with money laundering patterns or drug-target interactions.

Original: Our experiments show that many real-life data sets tend to exhibit nontrivial second and thirdpersistence diagrams (see Figures 9 and 10).

Original: We compare these datasets by considering theirclustering coefﬁcients CC ( G ) .

Original: Our observations indicate that when CC ( G ) is too low or too high,then the higher-order persistence diagrams associated with these data tend to be trivial.

Original: Intuitively,when CC ( G ) is too low, the graph G tends to be too sparse to form a k -cycle and, hence, is unlikelyto produce a nontrivial PD k ( G ) .

Original: When CC ( G ) is too high, then the graph G is too dense and everyk -cycle is ﬁlled immediately by a ( k + 1) -complex in (cid:98) G .

Original: We formulate this phenomenon in the formof the following conjecture.

Original: Conjecture: For k ≥ 2 , there are 0 < α k < β k < 1 such that when CC ( G ) >

Original: β k and CC ( G ) <

Original: α k ,then PD k ( G ) = ∅ with high probability.

Original: However, clustering coefﬁcient has the computational complexity of O ( |E| 1 . 48 )

Original: [Boo16], hence theusing the coefﬁcient may not seem as an improvement over coral reduction, which has a complexity of

Original: O ( |E| + |V| ) .

Original: However, note that the clustering coefﬁcient can be iteratively computed as the averageof vertex clustering coefﬁcients (in a parallel setting).

Original: In this approach, a stopping condition can beapplied to terminate early when the coefﬁcient can be approximated.

Original: The gain in computational timemay help coral technique on extremely large graphs or their induced subgraphs.

Original: C Proofs of the Theorems:C.1 Proof of CoralTDA

Original: First, we give the proof of Theorem 2.

Original: We use the same notation introduced in Section 4.

Original: Theorem 2: Let G be an unweighted connected graph.

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

Original: G .

Original: Let PD k

Original: ( 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 .

Original: Then, for any j ≥ k

Original: [Equation]

Original: Proof:

Original: For simplicity, we prove the theorem for j = k .

Original: Then, we give the generalization to j > kcase.

Original: Fix k ≥ 1 .

Original: In order to show the theorem, we need to prove

Original: that ( b, d ) ∈ PD k

Original: ( (cid:98) G )

Original: if and only if( b, d ) ∈ PD k

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

Original: In other words, if a k th -homology class σ = [ S ] is born at (cid:98) G b and dies at (cid:98) G d ,then the same homology class [ S ] is born at (cid:98) G k +1 b and dies at (cid:98) G k +1 d .

Original: We ﬁrst prove that the inclusion map (cid:98) G

Original: k

Original: [-+
-]
Revised: [-+
-]

Original: i (cid:44) → (cid:98) G i in the Diagram 1 induces an isomorphism for thehomology groups

Original: H k

Original: ( (cid:98) G i )

Original: =

Original: H k

Original: ( (cid:98)

Original: G k +1 i ) for any 0 ≤ i ≤ m .

Original: Then, the proof of the theorem willfollow by the equivalence of the induced persistence modules.

Original: For simplicity, we omit the subscript i .

Original: Assume σ ∈ H k ( (cid:98) G ) .

Original: In particular, σ is a k -homology class of(cid:98) G .

Original: This means there exists a k -cycle S in the chain complex C k ( (cid:98) G ) with σ =

Original: [ S ] .

Original: We claim that anyvertex v in S must have degree at least k + 1 , and

Original: hence S ∈ C k ( (cid:98) G k +1 ) .

Original: As S is k -cycle ∂ k S = 0 ∈ C k − 1 ( (cid:98) G ) .

Original: Let S

Original: =

Original: (cid:80) Ni =1 c i ∆ i where { ∆ i } are k -simplices in thesimplicial complex (cid:98) G .

Original: Then, ∂S = ∂ (cid:80)

Original: i c i ∆ i = (cid:80) i c i ∂ ∆ i .

Original: Notice that if ∆ i ⊂ (cid:98) G , this implies anyvertex of ∆ i =

Original: [ w i 0 , w i 1 , . . .

Original: , w ik ] has at least degree k in G as ∆ i has k +1 vertices { w i 0 , w i 1 , . . .

Original: , w ik } ,and they are all pairwise connected by an edge in G .

Original: Now, ∂ k ∆ i = (cid:80) kr =0 ( − 1) r

Original: Ω ir where Ω ir =[ w i 0 , w i 1 , . . .

Original: , w ir − 1 , w ir

Original: +1 , . . .

Original: , w ik ] are k − 1 simplices in (cid:98) G for 0 ≤ r ≤ k ,

Original: i.e. Ω

Original: ir is a ( k − 1) -faceof

Original: ∆ i .

Original: As S being a k -cycle, and ∂ k S = 0 , the sum (cid:80) i c i ∂ ∆ i = 0 .

Original: This implies that in the sum any( k − 1) -chain Ω ir must cancel out with another Ω js ⊂ ∂ ∆ j where ∆ j contains Ω ir and a vertex w jswhich is not a vertex in ∆ i .

Original: Therefore, any vertex in ∆ i is connected to both all other vertices { w ir }in ∆ i and another vertex w js ∈ ∆

Original: j ⊂ S .

Original: Hence, any vertex in ∆ i has degree k + 1 .

Original: This can begeneralized to any vertex in S .

Original: Hence, any vertex in S has degree at least k + 1 .

Original: This proves byinduction that S ⊂ (cid:98) G k +1 as follows.

Original: All vertices in S has degree k + 1 in G .

Original: Notice that the verticesof S has degree ≥ k + 1 because of the other vertices in S , not with the help of outsider vertices.

Original: So,as long as all the vertices in S are still in i

Original: -core

Original: G i , then any vertex of S will still have degree ≥ k + 1in G i .

Original: Therefore, By going inductively on the core index i , none of the vertices of S are removedfrom G i for 1 ≤ i ≤ k + 1 .

Original: Therefore, S is a k -cycle in (cid:98) G k +1 ,

Original: i.e. S ∈ C k ( (cid:98) G k +1 ) .

Original: This proves for ∂ k : C k ( (cid:98) G )

Original: → C k − 1 ( (cid:98) G ) and ∂ k +1 k : C k ( (cid:98) G k +1 )

Original: → C k − 1 ( (cid:98) G k +1 ) , we have ker ∂ k =ker

Original: ∂ k +1 k

Original: (Recall (cid:98) G k +1 ⊂ (cid:98) G ).

Original: Notice that any vertex in any ( k + 1) -simplex in (cid:98) G has degree atleast k + 1 .

Original: Then, C k +1 ( (cid:98) G ) =

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

Original: This implies im ∂

Original: k

Original: [-+
-]
Revised: [-+
-]

Original: =

Original: im ∂

Original: k +1 k +1 where ∂

Original: k +1 k +1 :

Original: C k +1 ( (cid:98) G k +1 )

Original: → C k ( (cid:98) G k +1 ) .

Original: Then, H k ( (cid:98) G )

Original: =

Original: H k

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

Original: This proves any k -cycle S must beproduced by the k -simplices in (cid:98) G k +1 , and lower degree vertices cannot belong to S .

Original: Also, k -core of G i is equal to i th step of the ﬁltration induced by f : V k

Original: → R , i.e. ( G k ) i =

Original: ( G i ) k .

Original: Hence, if σ is born in H k ( (cid:98) G b ) , then it is also born in H k

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

Original: If it dies in H k

Original: ( (cid:98) G d ) , it also dies in

Original: H k ( (cid:98) G k +1 d ) .

Original: This proves PD k ( G ) =

Original: PD k ( G k +1 ) .

Original: For the generalization in j > k case, one only needs to consider the same process for higher ordercycles.

Original: For some j > k , let S be a j -cycle in (cid:98) G , i.e. C j ( (cid:98) G ) .

Original: Then, by above any vertex in S musthave degree at least j + 1 .

Original: This implies S must be j -cycle in (cid:98) G k +1 , too.

Original: Similarly, any vertex in a ( j + 1) -simplex must have degree ≥ k + 1 .

Original: [-This means C j +
-]
Revised: [-This means C j +
-]

Original: ( (cid:98) G ) =

Original: C j +1 ( (cid:98) G k +1 ) .

Original: This implies H j ( (cid:98) G ) =

Original: H j ( (cid:98) G k +1 ) for any j ≥ k .

Original: This proves that the inclusion(cid:98)

Original: G k

Original: +1 i (cid:44) → (cid:98) G i induces an isomorphism for the homology groups

Original: H j

Original: ( (cid:98) G i )

Original: =

Original: H j ( (cid:98) G

Original: k

Original: [-+
-]
Revised: [-+
-]

Original: i ) for any j ≥ k .

Original: This shows the equivalence of the induced persistence modules.

Original: The proof of the theorem follows.

Original: (cid:3)

Original: C.2 Proof of PrunIT

Original: Now, we give the proof of Theorem 7.

Original: We use the same notation given in Section 5.

Original: Theorem 7: Let G =

Original: ( V , E ) be an unweighted graph, and f : V → R be a ﬁltering function. Let u ∈ V be dominated by v ∈ V and f ( u ) ≥ f ( v ) . Then, removing u from G does not change the persistence diagrams for sublevel ﬁltration, i.e. for any k ≥ [Equation]
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: Proof:

Original: Let (cid:98) G 0 ⊂ (cid:98) G 1 ⊂ (cid:98) G 2 ⊂ · · · ⊂ (cid:98) G m be the sequence of clique complexes in the inducedsublevel ﬁltration.

Original: Let α

Original: i 0 − 1 < f ( v ) ≤ α i 0 .

Original: This means for any j ≥ i 0 , v ∈ G j , and hence v is avertex in (cid:98) G j .

Original: Since f ( u ) ≥ f ( v ) , α i 1 − 1 < f ( u ) ≤ α i 1 for some i 1 ≥ i 0 .

Original: Similarly, this implies uﬁrst appears in G i 1 , and u ∈ (cid:98) G j for j ≥ i 1 .

Original: In particular, v belongs to all G j containing u .

Original: Let G (cid:48)

Original: =

Original: G − { u } .

Original: [-Deﬁne the sublevel ﬁltration (cid:98) G (cid:48) 
-]
Revised: [-Deﬁne the sublevel ﬁltration (cid:98) G (cid:48) 
-]

Original: ⊂ (cid:98) G (cid:48) 1 ⊂ (cid:98) G (cid:48) 2 ⊂ ...

Original: ⊂ (cid:98) G (cid:48) m for the same ﬁlteringfunction (restricted to V (cid:48) )

Original: f : V (cid:48)

Original: → R with the same threshold set I .

Original: For any j < i 1 , (cid:98) G j = (cid:98) G (cid:48)

Original: j as u (cid:54)∈ G j .

Original: Fix j ≥ i 1 .

Original: Then, v ∈ (cid:98) G j as f ( u ) ≥ f

Original: ( v ) .

Original: As v dominates uin G , v dominates u in G j , too.

Original: Then, G j folds onto G (cid:48)

Original: j = G j − { u } .

Original: Then, by Lemma 5, we havehomotopy equivalence (cid:98) G j ∼ (cid:98) G (cid:48)

Original: j .

Original: Hence, for any j ≥ i 1 , this gives (cid:98) G j ∼ (cid:98) G (cid:48) j .

Original: Recall that for any j < i 1 , (cid:98) G j = (cid:98) G (cid:48)

Original: j .

Original: Therefore, for any 0 ≤ j ≤ m , the inclusion (cid:98) G (cid:48) j (cid:44) → (cid:98) G j inducesan isomorphism between the homology groups

Original: H k

Original: ( (cid:98) G (cid:48)

Original: j ) (cid:39) H k

Original: ( (cid:98) G j ) .

Original: This proves the equivalence ofthe induced persistence modules, and hence the corresponding persistence diagrams, i.e. PD k

Original: ( G ) =

Original: PD k ( G (cid:48) ) for any k ≥ 0 .

Original: The proof follows.

Original: (cid:3)

Original: Now, we prove Theorem 10.

Original: Recall that n th power of graph G =

Original: ( V , E ) is deﬁned as G n =

Original: ( V , E n )

Original: Where E n = { e uv | d ( u, v ) ≤ n } .

Original: In other words, E n is obtained by adding new edges to Econnecting all the vertices whose graph distance ≤ n .

Original: Then, the power ﬁltration of G is deﬁned as(cid:98)

Original: G 0 ⊂ (cid:98) G 1 ⊂ (cid:98) G 2 ⊂ · · · ⊂ (cid:98) G N where (cid:98) G n is the clique complex of G n , n th -power of G and (cid:98) G 0 representthe vertex set V [AAEF19].

Original: Theorem 10: [PrunIt for Power Filtration] Let G =

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

Original: Let (cid:100) PD k

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

Original: Let u ∈ V be dominated by v ∈ V .

Original: Then, for any k ≥ 1 ,

Original: [Equation] Proof:
Revised: Foundation.

Original: Notice that power ﬁltration for a connected graph is trivial in dimension 0 as all featuresbut one dies at threshold 1 .

Original: So we will assume k ≥ 1 .

Original: Let H =

Original: G − { u } .

Original: We will show that (cid:98) G n ishomotopy equivalent to (cid:98) H n for n ≥ 1 .

Original: Let { v = w 0 , w 1 , w 2 , . . .

Original: , w m } be all adjacent vertices to u in G .

Original: As u is dominated byv , { w 1 , w 2 , . . .

Original: , w m } will be adjacent to v , too.

Original: We claim that for any vertex z (cid:54)

Original: =

Original: u in G , d ( z, v ) ≤ d ( z, u ) where d ( z, z (cid:48) ) is the length of the shortest path from z to z (cid:48) and each edgehas length 1 .

Original: In particular, any shortest path γ from z to u must go through one of the adjacentvertices { v, w 1 , w 2 , . . .

Original: , w m } .

Original: Let τ ⊂ γ be the segment starting at z and ending with one ofthese adjacent vertices, say w i .

Original: Then, d ( z, u ) = (cid:107) γ (cid:107) =

Original: (cid:107) τ (cid:107) + 1 .

Original: Now, consider d ( z, v ) .

Original: Sinceγ (cid:48)

Original: =

Original: τ ∪ [ w i , v ] is a path from z to v , this implies d ( z, v ) ≤ (cid:107)

Original: τ (cid:107) + 1 =

Original: d

Original: ( z, u ) .

Original: Now, we claim that if u is dominated by v in G , then u is dominated by v in G n for any n .

Original: All weneed to show that any adjacent vertex to u in G n is also adjacent to v .

Original: If z is adjacent to u in G n , thend ( z, u ) ≤ n in G by the deﬁnition of G n .

Original: By above, we have d ( z, v ) ≤

Original: d ( z, u ) ≤ n .

Original: This implies zis adjacent to v , too.

Original: Hence, u is dominated by v .

Original: Furthermore, as v dominates u in G , no shortestpath goes through u unless the endpoint is u .

Original: Therefore, the distances in G and H would be same forany two vertices z, z

Original: (cid:48) ∈ V − { u } .

Original: Then, by Lemma 5, (cid:98) G n homotopy equivalent to (cid:98) H n for any n ≥ 1 .

Original: Then, consider (cid:100) PD k

Original: ( G ) for k ≥ 1 .

Original: For k ≥ 1 , any k -dimensional topological feature will be born in(cid:98) G n where n ≥ 1 .

Original: As (cid:98) G n homotopy equivalent to (cid:98) H n for any n ≥ 1 , the proof follows.

Original: (cid:3)

Original: Remark 11.

Original: [CoralTDA for Power Filtration] Of course, after the previous result, the naturalfollow-up question is "Does CoralTDA also extends to power ﬁltrations?".

Original: Unfortunately, the answerto this question is "No".

Original: The reason is by the nature of power ﬁltration, one keeps adding edgesto the existing vertices, and the degrees of the vertices increase in each step.

Original: This means k -coressigniﬁcantly changes during this process.

Original: A simple counterexample can be found in [Ada13, Corollary6.7], where the author completely classiﬁes the topological types of clique complexes of the powersof cyclic graphs { C n } , i.e., A graph which forms a circle.

Original: e.g., C 5 is a connected graph with 5 vertexand 5 edges.

Original: By deﬁnition, any 3 -core of a cyclic graph is empty.

Original: However, Adamazsek’s result showthat for any dimension k , there exists topological features (cid:100) PD k ( C n ) is nontrivial for n ≥ 2 k + 3 .

Original: IfcoralTDA could be extended to power ﬁltrations, then (cid:100) PD k ( C n ) would be trivial for any k ≥ 3 .

Original: Remark 12.

Original: [Combining Reduction Algorithms with Graph Filtration Learning]

Original: In [HGR + 20], theauthors studied a fundamental question, which is to ﬁnd the “best ﬁltering function” for the givenclassiﬁcation problem.

Original: Indeed, the “right” ﬁltering function is often the key behind the classiﬁcationperformance.

Original: Under the assumption that dominated vertex value is greater than its dominatingvertex value, we can invoke our PrunIt result and run the “Graph Filtration Learning” algorithm of[HGR + 20] on the pruned graph which will lead to computational and (possibly) performance gains.

Original: For node classiﬁcation tasks, it would be also wise to check the relation between the class differencesbetween dominated and their dominating vertices.

Original: With similar reasoning, our CoralTDA methodcan also be combined with the “Graph Filtration Learning” algorithm and other methods in.

Original: Remark 13.

Original: [PrunIT vs. Strong Collapse]

Original: In [BP18, BP20, BPP21], the authors introduced Strong

Original: Collapse method to effectively reduce the computational costs for the persistence diagrams.

Original: In thiscontext, Strong Collapse method is deﬁned for any simplicial complex sequence, whereas PrunITise speciﬁcally deﬁned for the graph setting.

Original: The main difference between Strong Collapse and

Original: PrunIt is that PrunIT captures the dominated vertices in the graph at once, before ﬁltration while

Original: Strong Collapse needs to be applied in each ﬁltration step.

Original: In particular, for a given graph G and theﬁltering function f , PrunIT detects the dominated vertices in the graph, and by removing them, givesa smaller graph G (cid:48) with the same persistence diagrams PD ( G , f ) =

Original: PD ( G (cid:48) , f ) .

Original: Since the detectionof dominated vertices is done in the graph itself, the PrunIt algorithm works very fast.

Original: On the otherhand, Strong Collapse method is designed to work with ﬂag complexes.

Original: Therefore, if one wants toapply Strong Collapse method to compute PD ( G , f ) , one needs to go to the ﬁltration sequence (ﬂagcomplexes) (cid:98) G 1 ⊂ (cid:98) G 2 ⊂ · · · ⊂ (cid:98) G N , and apply the Strong Collapse method for each ﬂag complex (cid:98) G nfor 1 ≤ n ≤ N in the the ﬁltration sequence.

Original: This means one needs to detect dominated vertices foreach (cid:98) G n and remove them.

Original: Therefore, especially if the ﬁltration sequence is long ( N is large), PrunITwould be a more efﬁcient method in graph setting.

Original: We run experiments on the on Email-Enron dataset with 36K nodes and 183K edges (see Table 1)for PrunIt and Strong Collapse methods with degree ﬁltering function.

Original: We provide the results in thesupplementary material.

Original: In particular, we tried two different threshold step sizes δ 1 = 4 and δ 2 = 12which give ﬁltration sequence of lengths N 1 = 346 and N 2 = 116 respectively.

Original: In Table 3, we givethe computation times in seconds for PrunIT and Strong Collapse algorithm to eliminate dominatedvertices before PH computations.

Original: For PH computation step, recall that PH computation is cubicin terms of count of simplices [OPT + 17b].

Original: In Table 3, we also give these simplex counts for eachmethod.

Original: The details can be found in the supplementary material.

Original: [Table]

Original: D Broader Impact

Original: The proposed methodology makes an important step toward addressing the major existing roadblockin bringing the powerful machinery of TDA to the analysis of large-scale networks, from online socialmedia to gene-to-gene interactions in bioinformatics.

Original: Undoubtedly, the application of TDA in learningsuch large-scale networks will have a substantial positive impact in a broad range of applicationssuch as classiﬁcation of anomalous subgraphs in blockchain transaction networks, discovering linksbetween side effects and drugs, and drug re-purposing.

Original: However, the critical negative impact ofthe proposed methodology is associated with our current inability to accurately quantify the loss oftopological information due to pruning and the inﬂuence of such loss on the ﬁnal learning task.

Original: Thisis a fundamental question that needs to be addressed in the future.