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Journal of the Physical Society of Japan 91, 091013 (2022)
https://doi.org/10.7566/JPSJ.91.091013

Special Topics
Hyper-Ordered Structures: Recent Progress and Future Perspectives

Persistent Homology Analysis for Materials Researchand Persistent Homology Software: HomCloudIppei Obayashi1,4+
1, Takenobu Nakamura2, and Yasuaki Hiraoka3,4Cyber-Physical Engineering Informatics Research Core (Cypher), Okayama University, Okayama 700-8530, Japan
2
National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568, Japan
3
Kyoto University Institute for Advanced Study, WPI-ASHBi, Kyoto University, Sakyo, Kyoto 606-8501, Japan
4
Center for Advanced Intelligence Project, RIKEN, Chuo, Tokyo 103-0027, Japan(Received November 30, 2021; accepted March 1, 2022; published online May 12, 2022)

This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using themathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using someexamples. We also review some applications of persistent homology to materials researches and software for persistenthomology data analysis. HomCloud, one of persistent homology software, is especially featured in this paper.

1.

Introduction

This paper introduces the concept of persistent homology(PH), a data analysis technique based on the mathematicalconcept of topology. PH utilizes topological structures suchas connected components, holes, rings, and voids tocharacterize the shape of data at multiple scales.
We often have the opportunity to analyze the geometricstructure of materials science data. From atomic-scale dataobtained by molecular dynamics simulations to larger-scaledata obtained by various types of microscopy, materials
science provides many spatial structures. It is a typicalproblem to investigate how such structures are correlatedwith physical parameters and the physical properties of thematerials.
We need to quantify the geometric structures to investigate
the problems. PH enables us to summarize the shape of data
quantitatively using mathematics. From the viewpoint ofmaterials informatics, PH provides descriptors of the shape ofdata.
The output of PH is a persistence diagram or a persistence
barcode. The diagram and barcode have the same information, and only how to visualize is diﬀerent. Therefore we will
always use the persistence diagram in this paper.
This paper is organized as follows: Section 2 introduces
the history and recent research trends in PH. Section 3
describes the theoretical foundations of PH, which will help
the readers to understand how PH extracts the geometricfeatures of data. Section 4 introduces the applications of PH
to materials researches, which will help readers to understandwhat kind of data is suitable for PH. Section 5 introduces
the PH software, especially HomCloud, which is mainly
developed by Obayashi, one of the authors. Section 6summarizes this review paper and gives some concluding
remarks.
2.

A Brief History of PH

The concept of PH ﬁrst appeared in the paper1) by
Edelsbrunner et al. in 2002. They studied topologicalpersistence in a growing sequence of simplicial complexesin R3 . Then, Zomorodian and Carlsson2) introduced amathematical framework of PH based on ﬁnitely-generated
graded modules over polynomial rings k½t with one variablet, where k is a ﬁeld. This framework generalized the original

concept of PH and enlarged the applicability of PH into widerclasses of geometric problems. Furthermore, they clariﬁed
the so-called structure theorem of PH, i.e., under a certain
ﬁniteness condition, any PH can be uniquely decomposed
into a direct sum of intervals (see Sect. 3.2). This property
provides a natural visualization of PH, called persistence
diagrams, by plotting points with birth and death endpoints
of intervals on the plane. The persistence diagrams show
topological summaries of our input data in a multi-scale way.
In order for PH to be applied to practical data analysis, it
should be clariﬁed whether the persistence diagram is stablewith respect to small perturbations. This property will be
important for applying PH to practical problems since thosedata often contain noises or errors, and the output of PH
should be stable to those small perturbations for obtaining
essential data structure. The question of this type was ﬁrst
solved by Cohen-Steiner et al.,3) and they positively proved
this property of persistence diagrams. Several generalizationsof this result have also been reported later.4,5)
The structure and stability theorem explained here are
the most important properties in applying PH to practicalproblems. The former provides a compact descriptor showing
a topological multi-scale structure of data, and the latter
guarantees their stability with respect to small perturbations.
In fact, PH has been recently applied in various ﬁelds ofscience, including sensor networks,6) materials science,7,8)
biological evolution,9) biomolecular structural analysis,10)
brains,11) cosmology,12) etc. In those applications, a signiﬁcant property of PH characterizing topological structures in a
multi-scale way plays an important role in clarifying new
insights. In this paper, we focus on the applications tomaterials science and show some of those successful
examples.
We also remark that these various applications give strong
motivations for further mathematical studies of PH. For
example, some methods of inverse problems and machinelearnings of PH are actually developed by considering
practical applications into materials science.13,14) For detailsof the recent development of PH-based machine learning and
its applications, we refer to the survey paper.15)
3.

Foundation of PH

This section describes how to characterize the shape ofdata quantitatively in the form of persistence diagrams. PH is091013-1

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Special Topics

I. Obayashi et al.

Fig. 1. Filtration and PD (a) Input data (b) Pointcloud and discs with r ¼ b1 (c) r ¼ b2 (d) r ¼ d2 (e) r ¼ d1 (f ) PD.

Fig. 2. PDs for regular tetrahedral and octahedral points. (a) PD1 fortetrahedral points (b) PD2 for tetrahedral points (c) a regular tetrahedron(d) PD1 for octahedral points (e) PD2 for octahedral points (f ) a regularoctahedron.

available for various kinds of data such as pointclouds andbitmaps, but we mainly consider pointclouds as input data inthis paper.
A pointcloud is a ﬁnite set of points. A typical pointcloudis an atomic conﬁguration data. We can analyze such datausing PH. The outline of PH is explained by the examples in
Figs. 1 and 2.
The pointcloud Fig. 1(a) has no ring or hole since all
points are separated, but the pointcloud looks like it has tworings due to the proximity of the points. To constructtopological structures on the data, we put discs on all points
whose radii are the same, and holes appear as inFigs. 1(b)–1(e). We can count the number of holes in theseﬁgures.
The problem here is to determine the radius of discs. The
number of holes changes when the radius changes. The
fundamental idea of PH is to consider the changing processinstead of ﬁxing the radius. When the radius r gradually
becomes larger from zero, holes appear and disappear as inFigs. 1(b)–1(e). The increasing process of the shapes is called
a ﬁltration, and the theory of PH makes pairs of appearance
and disappearance of holes. In Fig. 1, ðb1 ; d1 Þ and ðb2 ; d2 Þ arepaired. The radius of appearance is called a birth time, theradius of disappearance is called a death time, and the pair ofbirth and death times is called a birth-death pair. The set ofbirth-death pairs with multiplicity is called a persistence
diagram (PD). The PD is often visualized by a scatter plot or
2D histogram [Fig. 1(f )]. We remark that the birth times anddeath times are sometimes squared according to conventions
in computational geometry and topological data analysis.
PH is applicable to any dimensional data. For 3D data,
we use spheres instead of discs. In homology theory, 3D
geometric objects have three types of homology information,and each type is characterized by dimension. 0D homology
has information about connectivity, 1D homology has
information about holes or rings, and 2D homology has
information about cavities or bubbles. Corresponding 0D,
1D, and 2D PDs are available. Since we consider rings in
Fig. 1, in Fig. 1(f ) is a 1D PD.
One advantage of PH is its mathematical background. The
structure theorem of PH gives the algorithm to compute thediagram, and the theorem also ensures the uniqueness of the
diagram. It means that the same input gives the same
diagram, unlike the Monte-Carlo method or stochastic
gradient descent. The stability theorem of PH ensures thatthe small change of the input data causes only small changesof output PD. These theorems play an important role inreliable data analysis.
To understand PDs, we examine some PDs for typicalpointclouds. Figures 2(a) and 2(b) show the PDs of regulartetrahedral points [Fig. 2(c)] where a is the distance between
two points. PD1 for regular tetrahedral points
pﬃﬃﬃ has three birthdeath pairs, and all ofpthem
are
ða=2;
a=
3Þ. PD2 has one
ﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ
birth-death pair ða= 3; a 3=8Þ. Three pairs in PD1
correspond to triangles of the tetrahedron, and one pair in
PD2 corresponds to the void at p
theﬃﬃﬃ center of the points. a=2 ishalf the lengthp
ofﬃﬃﬃﬃﬃﬃﬃ
the
edge,
a=
3 is the circumradius of the
ﬃ
triangle, and a 3=8 is the circumradius of the tetrahedron.
Probably readers will wonder why PD1 has three pairs, notfour. The reason comes from the mathematical theory ofhomology. We will discuss the background in Sect. 3.2.
We can apply the same idea to the regular octahedral points.
Figures 2(d) and 2(e) show the PDs for the pointcloud.
PD1 forpthe
ﬃﬃﬃ regular octahedral points has seven pairs at
ða=2; a= 3Þ. These pairs correspond to the triangles of theoctahedron. The PD1 does not have pairs corresponding to theeighth triangle and the square for the same reason
ﬃﬃﬃ
pﬃﬃﬃ aspthe
tetrahedron. PD2 has one birth-death pair at ða= 3; a= 2Þ
corresponding to the void at the center of the octahedron.
The PDs of hcp crystalline structure are in some sense the
combination of PDs for tetrahedral points and octahedral
points. p
PD1
ﬃﬃﬃ for the hcp has only birth-death pairs at
ða=2; a= 3Þ corresponding
theﬃﬃﬃﬃﬃﬃﬃ
triangles,
andpﬃﬃﬃPD2 phas
ﬃﬃﬃ
pﬃﬃﬃto pﬃbirth-death pairs at ða= 3; a 3=8Þ and ða= 3; a= 2Þ
corresponding to the tetrahedral sites and octahedral sites in
the hcp structure. Indeed fcc crystalline structure also has the091013-2

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Special Topics

I. Obayashi et al.

tetrahedral sites and octahedral sites, and the PDs for the fcc
points are completely the same as hcp.
3.1 Geometric representations
The concept of PH is applicable to various data. We now
describe some geometric representations to understandavailable data for PH.
One important geometric representation is called asimplicial complex. A simplicial complex is composed ofpoints, line segments, triangles, tetrahedrons, and higher
dimensional counterparts.
Formally saying, an n-dimensional simplicial complex
is a ﬁnite set of k-simplices for k ¼ 0; . . . ; n, which are
represented by k þ 1 vertices. A 0-simplex is a point,
1-simplex is a line segment, 2-simplex is a triangle, and3-simplex is a tetrahedron.
For an n-simplex σ and k < n, a k-simplex included in σ iscalled a face of σ if the all vertices of the k-simplex is also thevertices of σ.
A ﬁnite set of simplices X is called a simplicial complex if
X satisﬁes the following two conditions:(1) If  2 X, any face of σ is contained in X
(2) The intersection of two simplices in X is their common
face
Condition (2) means that two simplices are glued together bya lower-dimensional simplex.
A purely combinatorial description of a simplicial complex
is called an abstract simplicial complex. For a ﬁnite set V,a family of subsets of V, Σ, is called abstract simplicial
complex if the following two conditions hold:• For any v 2 V, fvg 2 • If  2  and   ,  2 
Of course, We can regard a simplicial complex as an abstract
simplicial complex. An abstract simplicial complex is helpful
to represent a geometric object on a computer.
One important simplicial complex is an alpha complex.16)
We can construct an alpha complex with radius parameter r
from pointcloud as a subset of Delaunay triangulation using
the Voronoi diagram. An alpha complex with parameter r
must have the same topological information as the union
discs model shown in Fig. 1. Therefore we usually use analpha complex to represent the union discs model such as
Figs. 1(b)–1(e).
We explain the construction of an alpha complex using theexample shown in Fig. 3. Figure 3(a) shows a pointcloud and
its Voronoi diagram, (b) shows the overlay image of (a) anddiscs, and (c) shows the alpha complex of the pointcloud. The
triple intersection P in (b) corresponds to the triangle P0 in(c), and the double intersections in (b) (shown in dotted lines)
correspond to edges in (c). Since there exists no tripleintersection at Q in (b), the alpha complex has no triangle at
Q0 in (c). As shown in Figs. 3(b) and 3(c) have the same
topological information. For example, both (b) and (c) have
one hole (Q and Q0 ) and two connected components. Nerve
theorem from abstract topology ensures that the union ofdiscs and the alpha complex gives the same topologicalinformation. A ﬁltration by alpha complexes is called analpha ﬁltration.
We can extend the idea of the alpha complex to 3D or
higher dimensional spaces by considering quadruple intersections and more.

Fig. 3. Construction of an alpha complex (a) Pointcloud and Voronoi
diagram (b) Overlay image of discs and Voronoi diagram (c) Alpha complex.

Another important simplicial complex for data analysis isa clique complex and Vietoris–Rips complex. We can deﬁne
a clique complex from an undirected graph. For a graph
G ¼ ðV; EÞ, a set of vertices C ¼ fv0 ; . . . ; vk g is called aclique if any pair ðvi ; vj Þ is adjacent, that is, the graph G has
an edge between vi and vj . The set of all cliques is called aclique complex. We easily show that the clique complex
satisﬁes the conditions of an abstract simplicial complex, andwe write it as XðGÞ. By using clique complexes, we canconstruct ﬁltration from a weighted graph. For a weightw : E ! R and a 2 R, Ga ¼ ðVa ; Ea Þ is a subgraph of G,
where:
Va ¼ V;Ea ¼ fe 2 E j wðeÞ  ag:

ð1ÞXðGa Þ gradually increases when a increases, so fXðGa Þga2R
gives a ﬁltration of clique complexes.
By considering a ﬁnite number of points V on a metricspace, we can construct a ﬁltration called a Vietoris–Rips
ﬁltration using the complete graph on V and the weightfunction wðfvi ; vJ gÞ ¼ dðvi ; vj Þ, where dðvi ; vj Þ is the distance between two points.
Vietoris–Rips ﬁltrations are sometimes used for PH data
analysis. In the study of molecular phylogenetics by Chenet al.,9) they use Vietoris–Rips ﬁltrations using a genomic
distance between genomic sequences.
One advantage of Vietoris–Rips ﬁltrations is that they canbe used as long as pairwise distances are available. At thesame time, the disadvantages are high computation cost andlack of geometric correspondence as in alpha ﬁltrations.
We also use cubical complexes to analyze pixel and voxel
data. A cubical ﬁltration consists of points, line segments,
squares, cubes, and higher-dimensional counterparts. We canconstruct a ﬁltration of cubical complexes by considering
a level function on pixels. The concept can be directly
applicable to gray-scale bitmap data. This type of ﬁltration
is called “level-set ﬁltration”. Some studies17–20) apply a
cubical ﬁltration to 2D or 3D binary bitmap data using
distance transform.

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Finally, we explain the idea of PH using the example inFig. 5. In this ﬁgure, the numbers of holes are 0; 1; 2; 1; 0from left to right. The sets of linearly independent rings
without ﬁlling with triangles are fg; fAg; fB0 ; C0 g; fB00 g; fg.
To understand the relationship between these sets, weconsider the change of basis. By changing the basis from
fB0 ; C0 g to fB0 þ C0 ; C0 g, we can consider the following
relationship called interval decomposition, where  means
the vanishment of the hole:
fg ! fAg ! fB0 þ C0 ; C0 g ! fB00 g! fg
A! A0 ¼ B0 þ C0
C

Fig. 4. (a) Tetrahedron (b) Tetrahedron looking from above (c) Four ringsin the tetrahedron (d) A ring C in (c) is ﬁlled with a triangle.

3.2 Mathematical foundation
PH is based on the theory of homology, a part of topology
theory. Since this paper is intended for readers who areinterested in the application of PH, we do not attempt to fully
formulate the mathematics of PH. Instead, we will illustrate
the mathematical ideas using some examples.
First, we consider the problem of counting the number ofrings in a tetrahedral skeleton. The tetrahedron [Fig. 4(a)]seems to have four rings, but it looks like three rings when
we see the tetrahedron from above [Fig. 4(b)]. This isbecause the outer ring D in Fig. 4(c) looks like the
combination of the three rings A, B, and C in some sense.
The theory of homology justiﬁes intuition using linear
algebra. In homology theory, each ring is regarded as a vector,
and we can justify the equality A þ B þ C ¼ D by assuming
the rule that adding the same edge twice equals zero. An easy
way to justify the rule is to consider Z=2Z-vector space. The
equality means that the four rings A; B; C; D are not linearly
independent. We can consider the linear space of all rings, andwe can count the number of linearly independent rings bycomputing the dimension of the linear space. In Figs. 4(a) and
4(b), the tetrahedron skeleton has three linearly independentrings. This is because PD1 for regularpﬃﬃﬃ tetrahedral points has
three birth-death pairs at ða=2; a= 3Þ, not four. The same istrue for the regular octahedral points.
Next, we consider the counting of holes in Fig. 4(d). The
ﬁgure seems to have two holes since the hole C in Fig. 4(c) is
ﬁlled with a triangle and the rest are A and B. In other words,
we can count the number of holes by z  b, where z is thenumber of linearly independent rings, and b is the number oflinearly independent rings ﬁlled by triangles. We can realize
the idea by canceling out the ring C in an algebraic way
called the quotient. By quotient, we can compute the linear
space of all unremoved rings. In fact, z  b is the dimensionof the quotient linear space. Of course, the same idea isavailable to count the number of cavities.

0! A00 ¼ B00
!

!





ð2Þ
In the above relationship, A ¼ B is justiﬁed due to cancelingout rule. We can say that the hole A appears at Fig. 5(b)disappearance at Fig. 5(e), and another hole C0 appears at
Fig. 5(c) and disappears at Fig. 5(d). We can make pairs ofappearance and disappearance of the holes mathematically inthat way. The theory of PH ensures that we can always ﬁnd
such proper bases. This is the fundamental idea of PH.
00

4.

00

Applications to Materials ResearchPH, especially for point clouds, is currently being used as atool to extract structures in various materials. In Ref. 21, twoof the authors already give some case studies to show how
PH works in representative structures found in material
systems. Since then, many interesting applications have been
investigated, including granular materials, network former,
and polymer. There are several properties of these systems
that explain the eﬀectiveness of PH feature extraction. They
are not found in the introduction of the previous case studies.
Now, we will focus on applications of the point clouds andpresent both the validity of using PDs as a descriptor forthese materials and the validity of geometric interpretations
for them. We then will present some notable recent results foreach of these materials.
4.1 PD as a descriptor of disordered system
PH has the potential to become a universal tool for
representing complex structures in material science. Oneof the central issues in materials science is to express thediﬀerence in physical properties by the diﬀerence instructure. This approach has been very eﬀective for regularly
arranged structures, such as crystals, and extremely disordered structures, such as gases. Apart from these two
extremes, however, it is diﬃcult to take such an approach
because of the diﬃculty in quantitatively evaluating complex
structures. For example, there can be structures partiallyordered in some regions while disordered in other regions, or
structures in an intermediate stage of ordering. To quantify
them, the method must be able to express the diversity of thestructure. Since PD can be regarded as a kind of distributionfunction, it can represent those features.

Fig. 5. Increase sequence of simplicial complexes.

091013-4

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Special Topics

When extracting a characteristic shape from a disordered
conﬁguration of particles, we may face the problem of thearbitrariness of the threshold, as mentioned in Sect. 3 as the
problem of determining the radius of the disk. For example,
in order to introduce hydrogen bonds, metal complexes,
polyhedra of metal clusters, etc., it is necessary to determine
an appropriate threshold. Usually, thresholds are set based on
physical and chemical properties, but PH provides analternative solution to this problem. PH evaluates the degreeof robustness of the thresholds by sweeping them. Holes witha small d  b, that is, holes that disappear soon after they
appear as the radius parameter increases, are not detected
unless the threshold is ﬁnely tuned, and vice versa. In otherwords, holes far from the diagonal b ¼ d in the PD can beextracted even if some numerical noise is added. Therefore,
we can assume that such holes have important information
for the disordered conﬁguration.
4.2 Comparison with conventional methods
To gain a better understanding of the uniqueness of PH,
it will be helpful to compare it with conventional methods
used in material science such as Voronoi analysis,22) ringstatistics,23) bond orientational order parameter,24) and radial
distribution function.25) The Voronoi analysis and the ringstatistics provide discrete variable indicators. Due to thediscreteness, two conﬁgurations close to each other may be
identiﬁed as diﬀerent indices. In PH, the stability theorem
allows us to avoid this problem.
The bond orientational order parameter is a continuous
variable indicator. Together with the two mentioned above,
this indicator expresses the degree of disorder in terms ofsimilarity to the crystalline structure. When the structure ofinterest is relatively close to the crystal, these quantities work
for interpreting the structure. However, they are not suitable
for representing disordered shapes far from the crystalline
structure. Whereas, PH provides interpretations that are notbased on similarity to crystals.
The radial distribution function is a quantity that does notrely on the similarity of crystal and can be applicable to adisordered structure. In fact, it can completely express thedisordered structure of the simple liquid. However, it is notsuitable for expressing complex shapes composed of many
particles, which can be expressed by PH.
As can be seen from these comparisons, PH is suitable fordescribing shapes consisting of a large number of particles
embedded in a structure that is far from the crystal structure.
Then, the degree of the disorder can be expressed as a
distribution function. A material system that has the beneﬁtof applying PH has these characteristics.
4.3 An object with disordered structure
We can get an overview of the systems in which PH iscurrently successfully applied by analyzing the reasons forthe realization of disordered structure from the viewpoint ofmaterial science. There are three possible reasons for thedisordered structure: 1. because of the property of the
microscopic constituents of the material, 2. because of theproperty of the process of preparing the material, 3.
because the material is composed of macroscopic components with heterogeneity and friction. Here we will describe
them.

I. Obayashi et al.

Soft matter, such as polymers, often has disorderedstructures on the atomic and molecular scales. In molecularsystems that constitute soft materials, there are intermolecularpotential and intramolecular potential such as a torsional
potential. The magnitude of these energies is of the sameorder as the thermal energy at room temperature. As a result,both single molecules and molecular assemblies are allowed
to have various structures in soft matter and take quite
disordered structures.
The solid-state with a disordered structure, called glass oramorphous, is obtained by quenching the material while
maintaining the disorder that appeared in the liquid state.
Many multi-component systems reach the glassy state by thisprocedure. In addition to the soft matter, alloys and ceramics
can also achieve a glassy disordered structure.
Granular consisting of macroscopic particles also often
have disordered structures. Unlike atoms or molecules, thegranular particles have diversity in size and interactions.
Therefore, when they are densely packed, it is not necessarily
arranged regularly like crystals. Even if particles have auniform property in the simulation, they may not always form
crystals and solidify in a disordered structure due to friction.
Therefore, in both experiments and simulations, granularsystems often show a disordered structure.
For these reasons, these material systems often havestructures that are diﬃcult to quantify, and PH is required tobe quantiﬁed there.
4.4 Material systems with geometric interpretation
We can interpret the shape extracted by PH if the physicalprocess of creating the point cloud is governed by geometry.
In other words, the validity of interpreting the point cloud as
a complex consisting of an expanding sphere accompanying
each point is justiﬁed on the basis of materials science. For amono-disperse system, the construction is always justiﬁed,
but for a multi-component system, it is not always. It is nontrivial that each particle has an intrinsic quantity corresponding to its radius. Only if this property is present can we deﬁnea physically interpretable contact between a particle andanother adjacent particle. In a system with N kinds ofparticles, there are NðN þ 1Þ=2 kinds of inter-particle
distances. Since there must be N radii of particles, the degreeof indeﬁniteness is NðN  1Þ=2. The radius is well deﬁned
only when this indeﬁniteness is not present.
Since the conﬁguration of the system is determined by theinter-particle potential between the particles, let us consider
the conditions under which the potential has the distancecorresponding to the radius. For example, consider a binary
system (N ¼ 2) with a Lennard-Jones potential UAB ðrÞ ¼
AB ½ðAB =rÞ12  ðAB =rÞ6 , where A and B represent the typeof particle. The length of the interaction between component A and component B is determined by AB . As
mentioned above, there are three length parameters,AA ; AB ; BB , corresponding to the type of distance between
the particles. If AB is expressed using the arithmetic meanof AA ; BB , we can introduce the radius as A ¼ AA orB ¼ BB . Here, the condition of the arithmetic-mean has
eliminated the indeﬁniteness for the radius. The same applies
to the system with N kinds of particles. Especially in the caseof LJ, the equation given by this arithmetic mean is called theLorentz–Bertelot combining rule.26) Thus, the combining rule091013-5

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guarantees the validity of considering the radius of theparticle and makes the virtual contact in alpha shapephysically meaningful.
Many polymer systems currently analyzed might beconsidered to satisfy the combining rule. The interaction
between granular particles always satisﬁes it whether in anexperimental or simulated system because the particle ismacroscopic. The details will be discussed later respectively.
In the case of network former, radius and contact can bereasonably introduced from another justiﬁcation. This will
also be discussed later.
4.5 Achievements in material systems
In the following, we will introduce some notable recent
results of PH in the material system. Most of the analyses arepoint clouds generated by particle simulations, and some data
are generated by processing data from the Reverse Monte
Carlo method based on the experimental measurement.
4.5.1 Granular
Granular systems have been studied extensively since thevery early days of PH’s application to material systems.
Unlike the atomic and molecular scales, granular particles
are macroscopic objects, and the radius and shape are welldeﬁned without ambiguity. In fact, the interactions between
granular particles are usually represented by the contact force
such as repulsive Hertzian or harmonic. These potentials
satisfy the combining rule, which guarantees that PH works
eﬀectively. Concerning extraction of structure, local structures close to crystalline structure and deformation withgeometric constraints have been studied.8)
In addition to the extraction of structure, PH also helps us
to quantify the force-chain network. Due to the contact force,the mechanical properties at the particle scale are characterized by a force chain network. This is a unique property togranular systems. The bulk mechanical properties are thenfound to be related to the Betti number of the force chain
network. The work by Kodic et al.27) is one of the very earlystudies of PH in material systems, which found that friction
and polydispersity are expressed as diﬀerences in the force
chain network and that the force chain network changes
qualitatively at the Jamming transition point.
PH is also useful for expressing the process dependency of
the force chain network. The static mechanical properties of
granular depend on the process. This subject is one of thecentral issues of research of granular systems, it is often
addressed in TDA. In Ref. 28, it was found that PD1 canrepresent the history dependence of the tapping operation of
the granular system even for the system with the samedensity. This is in contrast to liquids, where density is a state
quantity that represents the physical properties of the system.
In addition to the static properties, the transient properties arealso characterized by PH. Changes in the force chain network
associated with the impact have been quantiﬁed for both the
granular system29) and the suspension system.30)
4.5.2 Network former
Network former, such as silica (SiO2), is another material
where PH is actively applied. In the case of network former,
the combining rule may not hold for the interaction
parameters.31) However, there is another reason that justiﬁes

I. Obayashi et al.

interpreting the extracted shapes. Since the closest particle
type to each particle is limited, a radius can be introduced foreach particle species. In the realized particle conﬁguration,indeﬁniteness to determine radius does not practically makea problem because the type of particle with the closest
coordination to each particle is restricted.
As an example, let us consider silica glass. For the silica at
low temperatures, silicon (Si) is always at the center of atetrahedral structure consisting of oxygen (O), and no twosilicons are directly next to each other. Therefore, the types ofinter-particle distances are limited to Si–O and O–O. The
radius of the oxygen can be determined from the nearest
neighbor distance of O–O, and the radius of Si can be
determined from the nearest neighbor distance of Si–O.
Consequently, we can interpret the shape realized by silica
through the union of balls that appeared in PH. If there are
additional elements, there is no guarantee that the interpretation will work. However, we may ensure the validity of theinterpretation by considering additives as secondary eﬀects
added to the backbone structure of silica.
PH was ﬁrst applied to network former by Hiraoka et al.7)
There, the origin of the medium-range order (MRO) of silica
glass was discussed based on two results. These results might
be considered a prototype of the subsequent research on theuse of PH in materials. We will describe them in detail.
The ﬁrst result is a relationship between PD and the ﬁrst
sharp diﬀraction peak (FSDP). Glassy systems are believed
to show MRO instead of the long-range order for the
crystalline system. As for the silica glass, MRO is detectedby FSDP of the structure factor SðqÞ. In fact, FSDP was
observed for the conﬁguration data obtained by the molecular
dynamics simulation of silica (Fig. 6 bottom right). PD1
obtained from the same conﬁguration data shows curves CP(red box), CO (green box), and BO (blue box) in Fig. 6 left
panel. Then, it was found that the histogram of wave-number
converted from d for these curves has the same support
around the FSDP (Fig. 6 upper right). From this, the lengthscale of the MRO was concluded to be the size of the ringassociated with the curve of PD1.
The second result is an expression of the geometricconstraints that appear as curves in the silica PD. A curve in
the PD sometimes represents the geometric constraint for the
point cloud data. CP and the lower curve in Fig. 6 correspondto the constraint of Si–O bond length in the large ring andtetrahedral structures of SiO4, respectively. In contrast, thegeometric constraint represented by CO is more complicated.
Figure 7 expresses the geometric constraint of CO . By
extracting the three oxygen atoms that generate CO andmaking a triangle by them, the shortest side L1 , the second
shortest side L2 of the triangle, and angle θ between them(Fig. 7 upper left) are drawn as a three-dimensional scatterplot for each triangle (Fig. 7 upper right). The scatter plots
from the directions of A, B, and C in the ﬁgure are also
depicted in the lower left, center, and right panels of Fig. 7,
respectively. The points of ðL1 ; L2 ; Þ are distributed around asurface. This feature is not seen when L1 , L2 , and θ areindependent, which means that the distribution represents ageometric constraint.
After the research, a detailed interpretation of the geometric constraints has been investigated using a simpliﬁed
model.32) Moreover, it was found that PD has the ability to

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A1=C

C B
A2=C
A3=C
A4=B

2
q Å

2.5

0.1
0.05
0
1.5
Sq

1
0.5
0

1

1.5

3

Fig. 6. (Color) PD for the silica glass obtained from the molecular
dynamics simulation (left panel). FSDP of structure factor (q) (rightbottom panel) and the histogram of death d for curve as a function ofwave-number (right top panel). Red, green, and blue colors correspond to
CP , CO , and BO , respectively.

[degree]
90

L1

80

L2

C
70

B
A
2.8
2.2
2.4
L1 [Å]

2.6
2.6 2.4

L2 [Å]

90[degree] 80

70
A

B

C

Fig. 7. (Color) The schematic of three oxygen atoms that generate CO anddeﬁnition of L1 , L2 , and θ (upper left panel). The bird’s-eye view of scatterplot ðL1 ; L2 ; Þ (upper right panel). The identical plots from the direction A(lower left panel), B (lower center panel), and C (lower right panel). The
color represents the distance from the center of curve CO .

estimate the glass transition temperature using the support
vector machine.33) PH has also been applied to experimentally obtained conﬁguration data for various network
formers.34,35) Amorphous ice is another example of a network
former, where the medium-range order of the hydrogenbonding network is quantiﬁed by PH.36)
4.5.3 Polymer
Polymers are another material where PH is currently being
actively applied. As mentioned earlier, soft matter systems
are very compatible with PH analysis because of the need to
extract characteristic shapes from the disordered conﬁguration. In many cases, coarse-grained models are used inmolecular simulations of polymers. In general, there is no
guarantee that the combining rule holds for the interaction
potential of coarse-grained models. For example, thecombining rule breaks down when charged particles orheterogeneous particles are treated as a single particle incoarse-graining. However, since the system currently underanalysis does not meet these conditions, such contribution is

I. Obayashi et al.

expected to be small. Therefore, the PH interpretation canbe applied to data obtained by coarse-grained molecular
dynamics.
In the case of polymer systems, the correlation withmaterial properties is actively discussed rather than theextraction of the structure itself, compared to the two
examples mentioned above. For example, correlations with
dielectric constant,37) shear response,38) and crazing process39) were discovered. As a study that focuses on the
structure itself, a method to quantify the threading of ring
polymers has been proposed.40)
4.5.4 Other systems
In addition to these representative systems, PH has also
been applied to other material systems. In Pd40Ni40P20 bulk
metallic glasses, the medium-range ordered structure was
extracted by applying PH to the conﬁguration data of both
total and each component.41) For the toy model of amorphoussolids by simulation, the eﬀectiveness of PH in describing thestructural changes during plastic deformation was evaluatedbased on machine learning,42) the yielding was associatedwith a decrease in the number of robust holes introduced by
PH.43) In the experiment, PH was used to express the local
structure of two-dimensional binary colloidal conﬁguration
conﬁned at the gas-liquid interface.44) In addition to point
clouds, PH has been applied to the visualization of energy
landscape45) and characterization of spatial patterns such asphase separation structures of magnetic materials46) andpolymers.47)
4.6 Perspectives for material science
We have presented some recent applications of PH, mainly
to granular materials, network formers, and polymers. It is
expected that PH will be applied to other material systems in
the future. For example, soft matter other than polymers, suchas liquid crystals, emulsions, vesicles, and colloidal gels, will
be promising applications of PH because of the need to
extract characteristic structures embedded in disordered
conﬁgurations. In addition to PH for point clouds, it is also
promising to use PH for the quantiﬁcation of systems withbubbles, ﬁllers, and porous shapes because their shapes are
dominated by holes.
5.

Software

For the applications of PH, software is important. The
development of algorithms and software has progressed inparallel with the development of theories. The paper by
Edelsbrunner et al.1) showed an algorithm to compute a PD,
and the algorithm has been reﬁned theoretically and practically by subsequent researches.2,48–52) Parallel and distributed
algorithms has been also studied.53,54)
Many researchers on PH have developed various data
analysis software using PH. Table I shows the list of
software.
The developers have their interests and analysis targets.
For example, Ripser focuses on the eﬃcient algorithm tocompute PDs for Vietoris–Rips ﬁltrations. A benchmark62)
showed that Ripser is one of the fastest software
compared to competitors. PHAT and Dipha are developedby the same people as Ripser, and they also have good
performance.

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Table I.
Javaplex
Perseus55)
PHAT 56)
Dipha54)
Ripser 57)
Gudhi
Dionysus
R-TDA
EIRENE51)
CubicalRipser
RIVET
Ripserer.jl58)
scikit-tda59)
giotto-tda60)
jHoles61)
HomCloudSpecial Topics

List of software.

I. Obayashi et al.

To improve the software quality, the developers of
HomCloud do several practices. One practice is dogfooding;that is, the developers use HomCloud for daily data
analysis. Close communication between developers andusers is eﬀective for software improvement, and dogfooding
is one extreme way. Another practice is continuous
integration. After a code is uploaded to the code repository,
HomCloud is automatically built and tested. Continuous
integration is essential to keep the quality and portability ofthe software.
HomCloud internally uses various third-party components
to reduce the development cost. For example, HomClouduses PHAT, Ripser, and Dipha to compute PDs. These are
known for their good performance; therefore we use them.
CGAL is also used to compute alpha ﬁltrations. Python’s
standard scientiﬁc computing libraries, such as NumPy,
SciPy, and Matplotlib are also used.

http://appliedtopology.github.io/javaplex/
http://people.maths.ox.ac.uk/nanda/perseus/
https://bitbucket.org/phat-code/phat/https://github.com/DIPHA/diphahttps://github.com/Ripser/ripser
https://gudhi.inria.fr/
https://mrzv.org/software/dionysus/
https://mrzv.org/software/dionysus2/
https://cran.r-project.org/package=TDA
http://gregoryhenselman.org/eirene/
https://github.com/CubicalRipser
https://github.com/rivetTDA/rivet
https://mtsch.github.io/Ripserer.jl/dev/
https://scikit-tda.org
https://github.com/giotto-ai/giotto-tda(No URL is available)
https://homcloud.dev

Gudhi is more interested in computational geometry.
Gudhi collaborates with CGAL (Computational Geometry
Algorithms Library, https://www.cgal.org/), a famous computer geometry library, and works with CGAL on thedevelopment. Gudhi has various representations of geometricobjects.
R-TDA provides an interface from the R language. In fact,
R-TDA uses Gudhi, PHAT, and Dyonisys as a backend. RTDA is a bridge between R and the TDA world.
Each software has its advantages and features.
5.1 HomCloudIn this subsection, we introduce our software, HomCloud.
Obayashi, one of the authors of this paper, mainly develops
HomCloud.
HomCloud is free software, and you can download
HomCloud from https://homcloud.dev. You can freely use,
copy, modify, and redistribute the software under GPL(https://www.gnu.org/licenses/gpl-3.0.html).
HomCloud focuses on applications, mainly to materials
science. We use HomCloud to analyze atomic conﬁguration
given by molecular dynamical simulations and reverse
Monte-Carlo and pixel data and voxel data given by anelectron microscope. Of course, other data than materials
science is also available. HomCloud has been already used invarious scientiﬁc researches, including materials science,17,34–38,40,63–69) geology,20,70) structural biology,71) andmedical image analysis.72,73)
HomCloud has useful functionalities such as visualization,
inverse analysis, machine learning. Especially, inverse
analysis of HomCloud, which detects a ring or a cavity
corresponding to each birth-death pair, is the most advanced
among other software.
HomCloud has two types of interface, command-line
interface and python interface. Since Python has a richscientiﬁc computing ecosystem, you can combine the outputof HomCloud with the ecosystem using the Python interface.
HomCloud is available on Windows, Linux, and macOS,
including Apple Silicon Mac. You can also use HomCloudon Google Colaboratory, which allows you to write andexecute Python code in the web browser. You can try
HomCloud on Colaboratory without installing HomCloud on
your machine.

5.1.1 Installing HomCloud
Since HomCloud is written in Python, you need to install
Python before installing HomCloud. After installing Python,
you can easily install HomCloud using pip command, whichis the de facto standard package management system for
Python. We recommend pip to install HomCloud on Linux,
Intel macOS, and Windows. Another installation option isconda (https://conda.io). Conda is the open-source packagemanagement and environment management system mainly
developed by Anaconda Inc. (https://www.anaconda.com).
Conda is a famous tool among data science people.
HomCloud prepares conda packages for Windows, Linux,
and Apple Silicon Mac. We recommend conda for Apple
Silicon Mac since conda-forge has the most extensive
package for Apple Silicon Mac. The installation manual isavailable at https://homcloud.dev/install-guide/index.en.html.
5.1.2 Basic input=output
HomCloud accepts the following data.
• 2D=3D pointcloud (alpha ﬁltrations are used)• Any dimensional bitmap data, both binary data, andgrayscale data (cubical ﬁltrations are used)• Distance matrix (Vietoris–Rips ﬁltrations are used)• Abstract simplicial complex with weights
We can assign an initial radius to each particle for a 3D
pointcloud to reﬂect the size of the particle using weightedalpha shape.74) This functionality is useful to reﬂect thephysical radii such as ionic radii. We can also use periodic
boundary conditions for 3D pointcloud.
The output of HomCloud is a PD. You can plot ahistogram by HomCloud. The backend of plotting is
Matplotlib, so you can create fancy ﬁgures by utilizing
Matplotlib. You can also output the list of birth and deathtimes.
5.1.3 Inverse analysis
HomCloud has advanced inverse analysis features.
Figure 8 shows the outline of the functionality. In a PD,
each birth-death pair corresponds to a ring or a cavity. It isvery helpful to identify the structure to analyze PDs, but
the identiﬁcation is not easy since many candidates exist.
Mathematical optimization is used to select the tightest
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I. Obayashi et al.

same dimension as input vectors, we can visualize the learned
results in histograms. The histograms show which birth-deathpairs are important. After identifying important birth-deathpairs, we can apply inverse analysis to map the learned
results onto the original data.14)
5.1.5 Other functionalities
HomCloud has some other utilities. For example,
HomCloud supports bottleneck and Wasserstein distances
to compare PDs using HERA.81) You can plot the slice of a
PD histogram using HomCloud.
5.1.6 HomCloud examples
As explained above, HomCloud has a Python interface andcommand-line interface. Comprehensive tutorials (both inEnglish and Japanese) are available on HomCloud’s website(https://homcloud.dev/basic-usage.en.html). You can learn
how to analyze pointclouds, 2D or 3D bitmap data anddistance matrices in those tutorials using Jupyter notebook(https://jupyter.org/). Tutorials for both the Python interface
and command-line interface are available. We also prepare atutorial about PH with machine learning.
Now we introduce the Python interface. You can try thecodes on jupyter notebook or other integrated development
environments such as VSCode.
First, HomCloud, Numpy, and Matplotlib are imported.
import homcloud.interface as hc
import numpy as np
import matplotlib.pyplot as plt

Fig. 8. (Color) Inverse analysis.

implements some methods.77,79) We can apply diﬀerent
methods depending on the purpose, input type, and performance. The inverse analysis is available for all types of input,
but which method to use depends on the data type.
We can easily visualize the result of inverse analysis using
HomCloud. We can also output the geometric information of
the result in several forms. We can compare the output withother information for further investigation.
5.1.4 Machine learning
HomCloud supports machine learning using PDs. Machinelearning can ﬁnd hidden patterns that are common to many
PDs. Since many machine learning methods require feature
vectors or Gram matrices as input data, we need to convert
PDs into vectors or matrices.15) HomCloud supports
persistence image80) for a vectorization method. Persistence
image vectors are computed from the histograms of diagrams.
Intuitively saying, the values of bins of the histogram areused as vector elements. The following techniques are used to
improve the performance of learning.
• 2D Gaussian ﬁlter to reﬂect the adjacency of bins
• Weight function to reﬂect the importance of birth-deathpairs depending on the distance from the diagonal
The advantage of persistence image is its simplicity. We
can intuitively understand the method. We can also convert
a vector into a histogram in reverse. The advantage is
especially useful if we use linear machine learning models
such as linear regression, logistic regression, and principal
component analysis. Since the models give vectors with the

The input pointcloud data should be a NumPy 2D array.
The following code reads the data from a text ﬁle.
pointcloud = np.loadtxt("pointcloud.txt")
The text ﬁle should be as follows. These numbers are thevalues of the X, Y, and Z coordinates of each point. The X,
Y, and Z values are separated by spaces or tabs.
5.043 -16.116 -4.787
7.184 -16.066 -3.850-8.529 -1.029 -0.326:::
The PDs are computed as follows:
hc.PDList.from_alpha_filtration(
pointcloud,save_to="pointcloud.pdgm",
save_boundary_map=True)
The PDs are saved into a ﬁle “pointcloud.pdmg”. You canload the information of the 1D PD by the following code:
pdlist = hc.PDList("pointcloud.pdgm")
pd1 = pdlist.dth_diagram(1)Since “pointcloud.pdgm” has the information of PDs of
all dimensions, hc.PDList("pointcloud.pdgm") returns
PDList object and we get PD object by pdlist.dth_diagram(1).

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The following code will give you the list of birth times as a
NumPy array.
pd1.births# => array([0.140, 0.139, …,
Of course, deaths is also available to get death times. You
can construct a histogram of birth-death pairs and plot it
using HomCloud.
pd1.histogram((-0.2, 3.0), 256).plot(colorbar={"type": "log"})
plt.show()
You can specify the range and the number of bins as above.
(-0.2, 3.0) is the range of birth and death times, and 256 is
the number of bins in both axes. We can also modify thecolor bar of the plot. Since HomCloud uses Matplotlib as abackend, you can create good appearance ﬁgures by calling
the Matplotlib functions.
You can compute an optimal volume, one importantfeature of HomCloud, by the following codes:
pair = pd1.nearest_pair_to(0.72, 2.1)
optimal_volume = pair.optimal_volume()
The ﬁrst line picks up a birth-death pair nearest to ð0:72; 2:1Þand the second line computes the optimal volume of the pair.
You can get access to the coordinates of points on thecorresponding ring or cavity as follows.

6.

I. Obayashi et al.

Concluding Remarks

PH gives a way to summarize the shape of data
quantitatively and it is helpful to analyze materials data frommicro-scale data to macro-scale data. PH is now rapidly
developed from theory, software, and to applications tovarious ﬁelds including materials science. The collaboration
between theory and applications of PH is quite active.
One example of collaboration is the study of multiparameter persistence. In PH, scale information is encoded ina ﬁltration, enabling us to extract multi-scale geometricstructures. If we want to integrate another information such as
noise reduction level or temporal information, it is natural tointroduce another ﬁltration axis. Carlsson and Zomorodian82)
ﬁrst proposed multi-parameter persistence and rank invariant
to characterize it. The paper also showed the theoreticaldiﬃculty of multi-parameter persistence. Now mathematicians tackle this mathematical problem to give a bettercharacterization,83–85) and in the future, they will provide
feedback for applications.
Software is essential for connecting theory to applications.
A lot of PH software has been developed according to thetheoretical interests and applications of developers. One of
the software, HomCloud, is introduced in this paper.
HomCloud has some advanced features such as inverse
analysis and machine learning support and has been applied
to data analysis of materials.
This paper introduces PH for readers interested in theapplication to materials science and gives an intuitive
explanation of PH. Some review papers and textbooks,62,86–91) including a textbook in Japanese,92) have been
written for readers who are interested in further mathematicaldetails and algorithms.
Acknowledgments This work was partially sponsored by JSPS KAKENHI
JP19H00834, JP20H05884, JP18H01188, JST Presto JPMJPR1923, JST CREST
Mathematics (JPMJCR15D3), JST MIRAI Program (JPMJMI18G3), and Council
for Science, Technology and Innovation (CSTI), Cross-ministerial Strategic
Innovation Promotion Program (SIP) and “Materials Integration” for revolutionary design system of structural materials (Funding agency: JST).

optimal_volume.boundary_points()# => [[0.24, 0.36, 0.39],# [0.02, 0.10, 0.68],#:
You can also get access to the edges of a ring or the faces of acavity using boundary method. HomCloud also can visualize the rings and cavities as in Fig. 8.
5.1.7 Performance remarks
Since HomCloud uses eﬃcient software such as PHAT or
Ripser as a backend, HomCloud has good performance. You
can analyze 1,000,000 points in 3D space or 300  300 
300 voxel data in ﬁve or ten minutes with a recent PC with32 GB memory. Larger data requires a larger computer. Somebenchmark results are available at https://homcloud.dev/
benchmarks.html.
5.1.8 Future plans
Our developer team has a plan to continuously improve
HomCloud. New features will be implemented in parallelwith theoretical research. Performance improvements and UI
enhancements will be made as needed. We will also
implement new methods proposed by previous researches.
Now we have the idea to implement vectorization andkernel methods for machine learning other than persistence
image.

+

i.obayashi@okayama-u.ac.jp
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Ippei Obayashi was born in Wakayama Prefercture,
Japan in 1980. He obtained his B.Sc. (2004), M.Sc.
(2006), and Ph.D. (2010) degree from Kyoto
University. He was a postdoctoral researcher at
Kyoto University for the CREST mathematics
program (2010–2015), an assistant professor (2015–2018) and an associate professor (2018) at AIMR,
Tohoku University, a research scientist at RIKENAIP (2018–2021). He has been a professor at
Okayama University since 2021. He has worked ontheory and applications of dynamical systems and topological data analysis.
He is now especially interested in persistent homology, one of the main toolsof topological data analysis. He has also developed HomCloud, persistenthomology-based data analysis software.

Takenobu Nakamura was born in Yamaguchi
Prefecture, Japan in 1978. He obtained his B.Sc.
(2002) degree from Keio University, and M.Sc.
(2004), Ph.D. (2007) degree from the University ofTokyo. He was a postdoctoral researcher at AIST for
the CREST program (2007–2012), a postdoctoral
researcher at FZ Jülich in Germany (2012), an
assistant professor (2012–2017) at AIMR Tohoku
University, and Senior Researcher at AIST (2017–).
He has worked on theory and simulation of nonequilibrium soft matter and the application of topological data analysis tochemical physics.
Yasuaki Hiraoka was born in Oita prefecture, Japan
in 1978. He obtained his B.Eng. degree (2000),
M.Sc. degree (2002), D.Sc. degree (2005) from
Osaka University, Japan. He belonged to Hiroshima
University (2006–2011), Kyushu University(2011–2015), Tohoku University (2015–2018) as
an assistant professor, associate professor andprofessor, and now works at Kyoto University(Director of Center for Advanced Study, Professor,
and Deputy Director of ASHBi). He has beenworking on applied mathematics, especially topological data analysis andapplied topology.

091013-12

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The Author(s)