Shiquan REN Chengyuan WU Jie WU

Abstract In this paper,the authors study further properties and applications of weighted homology and persistent homology.The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced.For applications,the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network.They also prove a theorem to calculate the mod p2 weighted persistent homology provided with some information on the mod p weighted persistent homology.

Keywords Algebraic topology,Persistent homology,Weighted persistent homology,Bockstein spectral sequence

1 Introduction

Persistent homology is a recent branch of applied algebraic topology that has applications in data analysis(see[9]),image processing and recognition(see[1,13]),and more(see[5,16]).It is also a subject of active research,from both the computational(see[17])and theoretical(see[10-11])points of view.

The purpose of this paper is to reformulate some classical computational tools in homology theory in the context of weighted persistent homology.An important computational tool in homology theory is the Mayer-Vietoris sequence which can largely shorten the computations by handling well the subcomplexes.The Bockstein spectral sequence is a classical tool for recovering the integral homology from modphomology.In the situation of weighted persistent homology,it is convenient to consider the chains with coefficients in Z/2Z(see[7,19,35]).It is important to explore the Bockstein spectral sequence on weighted persistent homology as a tool to recover the integral weighted persistent homology so that one can obtain more topological information on weighted data.

In this paper,we study further properties and applications of weighted homology and persistent homology.Weighted simplicial homology(see[15,30])is a generalization of simplicial homology,that reduces to the usual simplicial homology when all the simplices have the same nonzero weight.For weighted simplicial complexes,we allow weights in a commutative ringRwith unity.When considering weighted homology,we require the ringRto be an integral domain.In[30],it is shown that weighted persistent homology can tell apart filtrations that ordinary persistent homology does not distinguish.For example,if there is a point considered as special,weighted persistent homology can tell when a cycle containing the point is formed or has disappeared.Hence,weighted persistent homology is a richer invariant than persistent homology.

Our approach to weighted persistent homology is to weight the boundary map.There are also various other approaches to adding weight to persistent homology(see[4,12,19,29]).

The Mayer-Vietoris sequence is an important tool in algebraic topology to study the homology of a space.In Section 3,we state and verify the Mayer-Vietoris sequence for weighted homology.

In Section 4,we show that a descending chain of ideals gives rise to a filtration of weighted simplicial complexes.An application is an algorithm(see Subsection 4.1)to construct a filtration of weighted simplicial complexes from a weighted network(weighted graph).This is related to the concept of Weight Rank Clique filtration(see[29])which is used in the study of complex networks(see[2,6,29,33]).A key feature is that in the process,we only construct the clique complex once,as opposed to the Weight Rank Clique filtration where multiple constructions of the clique complex is needed.In Subsection 4.2,we also illustrate an application in relation to Stanley-Reisner theory where a filtration could be set up so as to gather new information about the weighted simplicial complexes.

Next,we prove that over a field F,the weighted homology groupsHn(K,w;F)are isomorphic to the usual unweighted homology groupsHn(K;F).This is described in greater detail in Section 5.

In Section 6,we develop the Bockstein spectral sequence for weighted homology.The motivation behind using the Bockstein spectral sequence is that in persistent homology algorithms(see[7,35]),the homology is usually computed with field coefficients.However,the integral homology groups contain more information than the homology groups with field coefficients.The Bockstein spectral sequence allows us to“unravel”the integral homology from the modphomology.In the process,we prove a theorem(see Theorem 6.3)that allows us to calculate the modp2weighted persistent homology provided some conditions on the modppersistent homology are satisfied.

The final part of this paper(see Section 7)is about the generalized Bockstein spectral sequence,where we consider coefficients in an integral domainR.A potential application is in algebraic geometry where recently there has been some interest in the usage of weighted simplicial complexes(see[18,20])with weights in a ringR.

2 Background

In this section,we review the background necessary for the subsequent sections.We begin by reviewing weighted simplicial homology(see[15,30]),and then weighted persistent homology(see[30]).

2.1 Weighted simplicial homology

Weighted simplicial homology(see[15,30])is a generalization of simplicial homology.Every simplex has a weight in a ringR,and the boundary map is weighted accordingly.When all the simplices have the same weighta∈R{0},the resulting weighted homology is the same as the usual simplicial homology.We list some of the key definitions and results below.

2.2 Weighted persistent homology

3 The Mayer-Vietoris Sequence and Weighted Homology

The Mayer-Vietoris sequence for weighted simplicial homology was first studied briefly in[15,p.235].We prove that the Mayer-Vietoris sequence is exact for weighted simplicial homology,using an approach based on[27,p.142],which is different from the approach given in[15].In this section,we letRbe an integral domain.

We will need the following Lemma 3.1,which is also known as the Zig-zag lemma(see[27,p.136]).

4 Further Properties of Weighted Simplicial Complexes

(2)At each step t of the increasing weight ranking we consider the thresholded simplicial complex L(∊t),i.e.,the subcomplex1Let σ ∈L(∊t).For any nonempty τ ⊆σ,we have w(τ)|w(σ)and hence w(τ)≤w(σ)< ∊t.Thus τ ∈L(∊t),and hence L(∊t)is indeed a subcomplex.of(K,w)consisting of all simplices of weights smaller than

We have the following equivalent statements:

4.1 Application

The Weighted Rank Simplicial filtration in Definition 4.2 provides an alternative way to construct a filtration of(weighted)simplicial complexes from a weighted network(weighted graph)Ω.First we construct the clique complexKfrom Ω,and assign postive integer weights to makeKinto a weighted simplicial complex(K,w).This can be done as follows:

(1)Set all 0-simplices(vertices)inKto have weight 1.

(2)Rank the weight of links(edges)of Ω in increasing/decreasing order(depending on which edges the user wishes to appear first in the resulting filtration).

(3)Set the weight of each 1-simplex(edge)inKto be 2k,wherekis its rank in the weight ranking of Ω(edges can have the same rank if they have the same weight in Ω).

(4)For higher dimensional simplices,its weight is set to be the product of all the weights of the 1-simplices contained in it.

Then we carry out Weight Rank Simplicial filtration to obtain a filtrationof(K,w).The filtrationcan be described in terms of complements of preimage of ideals,as shown in Theorem 4.2.

Though the weights of the form 2kcan be very large integers,in practice we only need to store and compute the exponentk.Divisibility can be checked easily since 2k1|2k2if and only ifk1≤k2.

Note that in the entire process,we only construct the clique complex once.In general,it is desirable to reduce the number of times we construct the clique complex(see[34]).

4.2 Filtration related to the Stanley-Reisner Ideal of a Weighted Simplicial Complex

Similarly,Proposition 4.3 can be stated in the language of weighted persistent homology.

Propositions 4.1 and 4.3 show that the Stanley-Reisner filtration can distingush between WSCs(with suitably chosen weights).In turn,weighted persistent homology is a possible tool to study the Stanley-Reisner filtration.In our brief discussion,we show that there is some promise in applying weighted persistent homology to study algebraic geometry/ combinatorial commutative algebra through the connection with Stanley-Reisner theory.

5 Weighted Homology over a Field with Weight Function w : K→

The mapψis clearly linear.SinceandhenceIfthen sinceσiare distinct basis elements ofCn(K),thusfor alli.Henceai=0 for alli,andHenceψis injective.For surjectivity,we observe that anycan be written in the formby settingai=biw(σi),where we can similarly check2The proof thatindeed lies in ker ∂is similar to the part where we verify that.thatindeed lies in ker∂.

Therefore,we have shown thatψis a vector space isomorphism.

Theorem 5.1Let(K,w)be a finite(or finite-type3A WSC(K,w)is said to be of finite-type if for each n,the number of n-simplices in K is finite.)WSC with all weights of simplices.

Figure 1 Simplicial complex with 3 vertices x, y, z.

6 Bockstein Spectral Sequence and Weighted Persistent Homology

Both spectral sequences and persistent homology are related to filtrations,hence it is natural to explore the relationship between them.In[3],Basu and Parida derived formulas which expresses the relationship between spectral sequences and persistent homology.In[3],all homology groups are taken with coefficients in a field.In[31],Romero et al.studied persistent Z-homology using spectral sequences.We refer the reader to[14,24-25]for an overview of spectral sequences.

In this section we consider the Bockstein spectral sequence applied to weighted homology and weighted persistent homology.We will give a brief introduction to the Bockstein spectral sequence and refer the reader to[24,Chapter 24],[25,Chapter 10]and[28,Chapter 7]for more details.The motivation behind using the Bockstein spectral sequence is that in persistent homology algorithms(see[7 ,35]),most of the time the homology is computed with field coefficients,for instanceHowever,the integral homology groups contain more information than the homology groups with field coefficients.The Bockstein spectral sequence allows us to“unravel”the integral homology from the modphomology.Since the standard unweighted homology is a special case of weighted homology,the below results also hold for unweighted homology.

6.1 Bockstein homomorphism for weighted homology

Recall the following results from[25,p.455],which we adapt to the context of weighted homology.Consider the short exact sequence of coefficient rings where redris reduction modr,

The Bockstein spectral sequence is obtained from the long exact sequence in Lemma 6.1 when we view it as an exact couple.

6.2 The Bockstein spectral sequence for weighted homology

Letpbe a prime number.In a way similar to the previous Subsection 6.1,we can construct a long exact sequence associated to the short exact sequence of coefficients,

Notice that in the long exact sequence(see Lemma 6.1),two out of every three terms is the same.Hence,we can interpret the long exact sequence as an exact couple(see[8,25]):

We define theE1-term to beThe first differential is defined to bed1=redp∗◦∂=β,the Bockstein homomorphism.The resulting Bockstein spectral sequence is singly-graded.

we get ther-th order Bockstein operator as the connecting homomorphism.

By an argument similar to[25,Proposition 10.4],we obtain the following theorem.

6.3 Applications

For a finite(or finite-type)WSC(K,w),a complete knowledge of the Bockstein spectral sequences for all primespallows us to recover completely the integral weighted homologyH∗(K,w).From Theorem 6.1,theE∞term tells us the torsion-free part ofH∗(K,w).Moreover,by Proposition 6.1,the rank of the differentialdrtells us the number of summands ofin the integral weighted homology.

Hence,in the event that the Bockstein spectral sequence is known or has already been computed,we can skip the calculation of the integral weighted homology,and instead derive it from the Bockstein spectral sequence.We illustrate the above idea with an example.

Example 6.1Consider the WSC(K,w)shown in Figure 2.

Figure 2 WSC(K,w)with the following weights: w(v0)= w(v1)= w(v2)=1, w([v0,v1])=w([v1,v2])=w([v0,v2])=4.

6.4 Application to weighted persistent homology

In particular,the second statement of Lemma 6.2 has applications to calculate the modp2weighted persistent homology provided with some information about the modppersistent homology.We describe it in the following theorem.

Figure 3 The filtration of WSCs with the following weights: w([v0,v1])=2, w(σ)=1 for all other simplices σ /=[v0,v1].

7 Generalized Bockstein Spectral Sequence for Weighted Homology

In[22,pp.465-490],a generalized Bockstein spectral sequence of the cochain complexC∗with respect to a fixed elementtin the center of a ringAwas studied.In this section,we study and develop a generalized Bockstein spectral sequence in the context of weighted homology.

7.1 The generalized Bockstein spectral sequence for weighted homology

Letpbe a prime element in the integral domainR.To set up the generalized Bockstein spectral sequence,we view the long exact sequence as an exact couple:

Figure 4 WSC(K,w)with the following weights: w(v0)= w(v1)= w(v2)=1, w([v0,v1])=w([v1,v2])=w([v0,v2])=x2.

We first compute the generalized Bockstein spectral sequence forp=x.We obtain the following results: