A roadmap for the computation of persistent homology
 Nina Otter^{1, 3},
 Mason A Porter^{1, 2, 4}Email author,
 Ulrike Tillmann^{1, 3},
 Peter Grindrod^{1} and
 Heather A Harrington^{1}
Received: 30 January 2017
Accepted: 7 June 2017
Published: 9 August 2017
Abstract
Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of stateoftheart implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and realworld data sets to evaluate currently available opensource implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.
Keywords
persistent homology topological data analysis pointcloud data networks1 Introduction
The amount of available data has increased dramatically in recent years, and this situation — which will only become more extreme — necessitates the development of innovative and efficient dataprocessing methods. Making sense of the vast amount of data is difficult: on one hand, the sheer size of the data poses challenges; on the other hand, the complexity of the data, which includes situations in which data is noisy, highdimensional, and/or incomplete, is perhaps an even more significant challenge. The use of clustering techniques and other ideas from areas such as computer science, machine learning, and uncertainty quantification — along with mathematical and statistical models — are often very useful for data analysis (see, e.g., [1–4] and many other references). However, recent mathematical developments are shedding new light on such ‘traditional’ ideas, forging new approaches of their own, and helping people to better decipher increasingly complicated structure in data.
Techniques from the relatively new subject of ‘topological data analysis’ (TDA) have provided a wealth of new insights in the study of data in an increasingly diverse set of applications — including sensornetwork coverage [5], proteins [6–9], 3dimensional structure of DNA [10], development of cells [11], stability of fullerene molecules [12], robotics [13–15], signals in images [16, 17], periodicity in time series [18], cancer [19–22], phylogenetics [23–25], natural images [26], the spread of contagions [27, 28], selfsimilarity in geometry [29], materials science [30–33], financial networks [34, 35], diverse applications in neuroscience [36–43], classification of weighted networks [44], collaboration networks [45, 46], analysis of mobile phone data [47], collective behavior in biology [48], timeseries output of dynamical systems [49], naturallanguage analysis [50], and more. There are numerous others, and new applications of TDA appear in journals and preprint servers increasingly frequently. There are also interesting computational efforts, such as [51].
TDA is a field that lies at the intersection of data analysis, algebraic topology, computational geometry, computer science, statistics, and other related areas. The main goal of TDA is to use ideas and results from geometry and topology to develop tools for studying qualitative features of data. To achieve this goal, one needs precise definitions of qualitative features, tools to compute them in practice, and some guarantee about the robustness of those features. One way to address all three points is a method in TDA called persistent homology (PH). This method is appealing for applications because it is based on algebraic topology, which gives a wellunderstood theoretical framework to study qualitative features of data with complex structure, is computable via linear algebra, and is robust with respect to small perturbations in input data.
Types of data sets that can be studied with PH include finite metric spaces, digital images, level sets of realvalued functions, and networks (see Section 5.1). In the next two paragraphs, we give some motivation for the main ideas of persistent homology by discussing two examples of such data sets.
Finite metric spaces are also called pointcloud data sets in the TDA literature. From a topological point of view, finite metric spaces do not contain any interesting information. One thus considers a thickening of a point cloud at different scales of resolution and then analyzes the evolution of the resulting shape across the different resolution scales. The qualitative features are given by topological invariants, and one can represent the variation of such invariants across the different resolution scales in a compact way to summarize the ‘shape’ of the data.
By using homology, a tool in algebraic topology, one can measure several features of the spaces \(S_{\epsilon}\) — including the numbers of components, holes, and voids (higherdimensional versions of holes). One can then represent the lifetime of such features using a finite collection of intervals known as a ‘barcode.’ Roughly, the left endpoint of an interval represents the birth of a feature, and its right endpoint represents the death of the same feature. In Figure 1(b), we reproduce such intervals for the number of components (blue solid lines) and the number of holes (violet dashed lines). In Figure 1(b), we observe a dashed line that is significantly longer than the other dashed lines. This indicates that the data set has a longlived hole. By contrast, in this example one can potentially construe the shorter dashed lines as noise. (However, note that while widespread, such an intepretation is not correct in general; for applications in which one considers some short and mediumsized intervals as features rather than noise, see [52, 53].) When a feature is still ‘alive’ at the largest value of ϵ that we consider, the lifetime interval is an infinite interval, which we indicate by putting an arrowhead at the right endpoint of the interval. In Figure 1(b), we see that there is exactly one solid line that lives up to \(\epsilon=2.1\). One can use information about shorter solid lines to extract information about how data is clustered in a similar way as with linkageclustering methods [3].
One of the most challenging parts of using PH is statistical interpretation of results. From a statistical point of view, a barcode like the one in Figure 1(b) is an unknown quantity that one is trying to estimate; one therefore needs methods for quantitatively assessing the quality of the barcodes that one obtains with computations. The challenge is twofold. On one hand, there is a cultural obstacle: practitioners of TDA often have backgrounds in pure topology and are not wellversed in statistical approaches to data analysis [54]. On the other hand, the space of barcodes lacks geometric properties that would make it easy to define basic concepts such as mean, median, and so on. Current research is focused both on studying geometric properties of this space and on studying methods that map this space to spaces that have better geometric properties for statistics. In Section 5.4, we give a brief overview of the challenges and current approaches for statistical interpretation of barcodes. This is an active area of research and an important endeavor, as few statistical tools are currently available for interpreting results in applications of PH.
In the present article, we focus on persistent homology, but there are also other methods in TDA — including the Mapper algorithm [56], Euler calculus (see [57] for an introduction with an eye towards applications), cellular sheaves [57, 58], and many more. We refer readers who wish to learn more about the foundations of TDA to the article [59], which discusses why topology and functoriality are essential for data analysis. We point to several introductory papers, books, and two videos on PH at the end of Section 4.
The first algorithm for the computation of PH was introduced for computation over \(\mathbb{F}_{2}\) (the field with two elements) in [60] and over general fields in [61]. Since then, several algorithms and optimization techniques have been presented, and there are now various powerful implementations of PH [62–68]. Those wishing to try PH for computations may find it difficult to discern which implementations and algorithms are best suited for a given task. The field of PH is evolving continually, and new software implementations and updates are released at a rapid pace. Not all of them are welldocumented, and (as is wellknown in the TDA community), the computation of PH for large data sets is computationally very expensive.
To our knowledge, there exists neither an overview of the various computational methods for PH nor a comprehensive benchmarking of the stateoftheart implementations for the computation of persistent homology. In the present article, we close this gap: we introduce computation of PH to a general audience of applied mathematicians and computational scientists, offer guidelines for the computation of PH, and test the existing opensource published libraries for the computation of PH.
The rest of our paper is organized as follows. In Section 2, we discuss related work. We then introduce homology in Section 3 and introduce PH in Section 4. We discuss the various steps of the pipeline for the computation of PH in Section 5, and we briefly examine algorithms for generalized persistence in Section 6. In Section 7, we give an overview of software libraries, discuss our benchmarking of a collection of them, and provide guidelines for which software or algorithm is better suited to which data set. (We provide specific guidelines for the computation of PH with the different libraries in the Tutorial in Additional file 2 of the Supplementary Information (SI).) In Section 8, we discuss future directions for the computation of PH.
2 Related work
In our work, we introduce PH to nonexperts with an eye towards applications, and we benchmark stateoftheart libraries for the computation of PH. In this section, we discuss related work for both of these points.
There are several excellent introductions to the theory of PH (see the references at the end of Section 4.1), but none of them emphasizes the actual computation of PH by providing specific guidelines for people who want to do computations. In the present paper, we navigate the theory of PH with an eye towards applications, and we provide guidelines for the computation of PH using the opensource libraries javaPlex, Perseus, Dionysus, DIPHA, Gudhi, and Ripser. We include a tutorial (see Additional file 2 of the SI) that gives specific instructions for how to use the different functionalities that are implemented in these libraries. Much of this information is scattered throughout numerous different papers, websites, and even source code of implementations, and we believe that it is beneficial to the applied mathematics community (especially people who seek an entry point into PH) to find all of this information in one place. The functionalities that we cover include plots of barcodes and persistence diagrams and the computation of PH with Vietoris–Rips complexes, alpha complexes, Čech complexes, witness complexes, cubical complexes for image data. We also discuss the computation of the bottleneck and Wasserstein distances. We thus believe that our paper closes a gap in introducing PH to people interested in applications, while our tutorial complements existing tutorials (see, e.g. [69–71]).
We believe that there is a need for a thorough benchmarking of the stateoftheart libraries. In our work, we use twelve different data sets to test and compare the libraries javaPlex, Perseus, Dionysus, DIPHA, Gudhi, and Ripser. There are several benchmarkings in the PH literature; we are aware of the following ones: the benchmarking in [72] compares the implementations of standard and dual algorithms in Dionysus; the one in [73] compares the Morsetheoretic reduction algorithm with the standard algorithm; the one in [62] compares all of the data structures and algorithms implemented in PHAT; the benchmarking in [74] compares PHAT and its spinoff DIPHA; and the benchmarking in C. Maria’s doctoral thesis [75] is to our knowledge the only existing benchmarking that compares packages from different authors. However, Maria compares only up to three different implementations at one time, and he used the package jPlex (which is no longer maintained) instead of the javaPlex library (its successor). Additionally, the widely used library Perseus (e.g., it was used in [22, 27, 30, 31]) does not appear in Maria’s benchmarking.
3 Homology
Assume that one is given data that lies in a metric space, such as a subset of Euclidean space with an inherited distance function. In many situations, one is not interested in the precise geometry of these spaces, but instead seeks to understand some basic characteristics, such as the number of components or the existence of holes and voids. Algebraic topology captures these basic characteristics either by counting them or by associating vector spaces or more sophisticated algebraic structures to them. Here we are interested in homology, which associates one vector space \(H_{i}(X)\) to a space X for each natural number \(i \in\{0, 1,2, \dots \}\). The dimension of \(H_{0}(X)\) counts the number of path components in X, the dimension of \(H_{1}(X)\) is a count of the number of holes, and the dimension of \(H_{2}(X)\) is a count of the number of voids. An important property of these algebraic structures is that they are robust, as they do not change when the underlying space is transformed by bending, stretching, or other deformations. In technical terms, they are homotopy invariant.^{1}
It can be very difficult to compute the homology of arbitrary topological spaces. We thus approximate our spaces by combinatorial structures called ‘simplicial complexes,’ for which homology can be easily computed algorithmically. Indeed, often one is not even given the space X, but instead possesses only a discrete sample set S from which to build a simplicial complex following one of the recipes described in Sections 3.2 and 5.2.
3.1 Simplicial complexes and their homology
Definition 1
A simplicial complex ^{2} is a collection K of nonempty subsets of a set \(K_{0}\) such that \(\{v\}\in K\) for all \(v\in K_{0}\), and \(\tau\subset\sigma \) and \(\sigma\in K\) guarantees that \(\tau\in K\). The elements of \(K_{0}\) are called vertices of K, and the elements of K are called simplices. Additionally, we say that a simplex has dimension p or is a psimplex if it has a cardinality of \(p+1\). We use \(K_{p}\) to denote the collection of psimplices. The kskeleton of K is the union of the sets \(K_{p}\) for all \(p\in\{0,1,\dots, k\}\). If τ and σ are simplices such that \(\tau\subset\sigma\), then we call τ a face of σ, and we say that τ is a face of σ of codimension \(k'\) if the dimensions of τ and σ differ by \(k'\). The dimension of K is defined as the maximum of the dimensions of its simplices. A map of simplicial complexes, \(f : K \to L\), is a map \(f : K_{0}\to L_{0}\) such that \(f(\sigma)\in L\) for all \(\sigma\in K\).
We give an example of a simplicial complex in Figure 3(a) and an example of a map of simplicial complexes in Figure 3(b). Definition 1 is rather abstract, but one can always interpret a finite simplicial complex K geometrically as a subset of \(\mathbb {R}^{N}\) for sufficiently large N; such a subset is called a ‘geometric realization,’ and it is unique up to a canonical piecewiselinear homeomorphism. For example, the simplicial complex in Figure 3(a) has a geometric realization given by the subset of \(\mathbb{R}^{2}\) in Figure 3(c).
Definition 2
When working with simplicial complexes, one can modify a simplicial complex by removing or adding a pair of simplices \((\sigma, \tau)\), where τ is a face of σ of codimension 1 and σ is the only simplex that has τ as a face. The resulting simplicial complex has the same homology as the one with which we started. In Figure 3(a), we can remove the pair \((\{a, b, c\}, \{ b,c\})\) and then the pair \((\{ a,b\}, \{b \})\) without changing the Betti numbers. Such a move is called an elementary simplicial collapse [77]. In Section 5.2.6, we will see an application of this for the computation of PH.
In this section, we have defined simplicial homology over the field \(\mathbb{F}_{2}\) — i.e., ‘with coefficients in \(\mathbb{F}_{2}\).’ One can be more general and instead define simplicial homology with coefficients in any field (or even in the integers). However, when \(1 \neq1\), one needs to take more care when defining the boundary maps \(d_{p}\) to ensure that \(d_{p} \circ d_{p+1}\) remains the zero map. Consequently, the definition is more involved. For the purposes of the present paper, it suffices to consider homology with coefficients in the field \(\mathbb{F}_{2}\). Indeed, we will see in Section 4 that to obtain topological summaries in the form of barcodes, we need to compute homology with coefficients in a field. Furthermore, as we summarize in Table 2 (in Section 7), most of the implementations for the computation of PH work with \(\mathbb{F}_{2}\).
We conclude this section with a warning: changing the coefficient field can affect the Betti numbers. For example, if one computes the homology of the Klein bottle (see Section 7.1.1) with coefficients in the field \(\mathbb{F}_{p}\) with p elements, where p is a prime, then \(\beta_{0}(K)=1\) for all primes p. However, \(\beta_{1}(K)=2\) and \(\beta_{2}(K)=1\) if \(p=2\), but \(\beta_{1}(K)=1\) and \(\beta_{2}(K)=0\) for all other primes p. The fact that \(\beta_{2}(K)=0\) for \(p\ne2\) arises from the nonorientability of the Klein bottle. The treatment of different coefficient fields is beyond the scope of our article, but interested readers can peruse [78] for an introduction to homology and [76] for an overview of computational homology.
3.2 Building simplicial complexes
As we discussed in Section 3.1, computing the homology of finite simplicial complexes boils down to linear algebra. The same is not true for the homology of an arbitrary space X, and one therefore tries to find simplicial complexes whose homology approximates the homology of the space in an appropriate sense.
An important tool is the Čech (Č) complex. Let \(\mathcal {U}\) be a cover of X — i.e., a collection of subsets of X such that the union of the subsets is X. The ksimplices of the Čech complex are the nonempty intersections of \(k+1\) sets in the cover \(\mathcal {U}\). More precisely, we define the nerve of a collection of sets as follows.
Definition 3
Let \(\mathcal {U}=\{U_{i}\}_{i\in I}\) be a nonempty collection of sets. The nerve of \(\mathcal {U}\) is the simplicial complex with set of vertices given by I and ksimplices given by \(\{i_{0},\dots, i_{k}\} \) if and only if \(U_{i_{0}}\cap\cdots\cap U_{i_{k}}\ne\emptyset\).
If the cover of the sets is sufficiently ‘nice,’ then the Nerve Theorem implies that the nerve of the cover and the space X have the same homology [79, 80]. For example, suppose that we have a finite set of points S in a metric space X. We then can define, for every \(\epsilon>0\), the space \(S_{\epsilon}\) as the union \(\bigcup _{x\in S}B(x,\epsilon)\), where \(B(x,\epsilon)\) denotes the closed ball with radius ϵ centered at x. It follows that \(\{B(x,\epsilon )\mid x\in S\}\) is a cover of \(S_{\epsilon}\), and the nerve of this cover is the Čech complex on S at scale ϵ. We denote this complex by \(\check {C}_{\epsilon}(S)\). If the space X is Euclidean space, then the Nerve Theorem guarantees that the simplicial complex \(\check {C}_{\epsilon}(S)\) recovers the homology of \(S_{\epsilon}\).
From a computational point of view, the Čech complex is expensive because one has to check for large numbers of intersections. Additionally, in the worst case, the Čech complex can have dimension \(\mathcal {U}1\), and it therefore can have many simplices in dimensions higher than the dimension of the underlying space. Ideally, it is desirable to construct simplicial complexes that approximate the homology of a space but are easy to compute and have ‘few’ simplices, especially in high dimensions. This is a subject of ongoing research: In Section 5.2, we give an overview of stateoftheart methods to associate complexes to pointcloud data in a way that addresses one or both of these desiderata. See [80, 81] for more details on the Čech complex, and see [79, 80] for a precise statement of the Nerve Theorem.
4 Persistent homology
4.1 Filtered complexes and homology
Definition 4
The pth persistent homology of a filtered simplicial complex gives more refined information than just the homology of the single subcomplexes. We can visualize the information given by the vector spaces \(H_{p}(K_{i})\) together with the linear maps \(f_{i,j}\) by drawing the following diagram: at filtration step i, we draw as many bullets as the dimension of the vector space \(H_{p}(K_{i})\). We then connect the bullets as follows: we draw an interval between bullet u at filtration step i and bullet v at filtration step \(i+1\) if the generator of \(H_{p}(K_{i})\) that corresponds to u is sent to the generator of \(H_{p}(K_{i+1})\) that corresponds to v. If the generator corresponding to a bullet u at filtration step i is sent to 0 by \(f_{i,i+1}\), we draw an interval starting at u and ending at \(i+1\). (See Figure 5(b) for an example.) Such a diagram clearly depends on a choice of basis for the vector spaces \(H_{p}(K_{i})\), and a poor choice can lead to complicated and unreadable clutter. Fortunately, by the Fundamental Theorem of Persistent Homology [61], there is a choice of basis vectors of \(H_{p}(K_{i})\) for each \(i\in \{1,\dots, l\}\) such that one can construct the diagram as a welldefined and unique collection of disjoint halfopen intervals, collectively called a barcode.^{4} We give an example of a barcode in Figure 5(c). Note that the Fundamental Theorem of PH, and hence the existence of a barcode, relies on the fact that we are using homology with field coefficients. (See [61] for more details.)
There is a useful interpretation of barcodes in terms of births and deaths of generators. Considering the maps \(f_{i,j}\) written in the basis given by the Fundamental Theorem of Persistent Homology, we say that \(x\in H_{p}(K_{i})\) (with \(x \neq0\)) is born in \(H_{p}(K_{i})\) if it is not in the image of \(f_{i1,i}\) (i.e., \(f^{1}_{i1,i}(x)=\emptyset\)). For \(x \in H_{p}(K_{i})\) (with \(x \neq0\)), we say that x dies in \(H_{p}(K_{j})\) if \(j>i\) is the smallest index for which \(f_{i,j}(x)=0\). The lifetime of x is represented by the halfopen interval \([i,j)\). If \(f_{i,j}(x)\ne0\) for all j such that \(i< j\leq l\), we say that x lives forever, and its lifetime is represented by the interval \([i,\infty)\).
Remark 5
Note that some references (e.g., [80]) introduce persistent homology by defining the birth and death of generators without using the existence of a choice of compatible bases, as given by the Fundamental Theorem of Persistent Homology. The definition of birth coincides with the definition that we have given, but the definition of death is different. One says that \(x \in H_{p}(K_{i})\) (with \(x \neq0\)) dies in \(H_{p}(K_{j})\) if \(j>i\) is the smallest index for which either \(f_{i,j}(x)=0\) or there exists \(y\in H_{p}(K_{i'})\) with \(i'< i\) such that \(f_{i',j}(y)=f_{i,j}(x)\). In words, this means that x and y merge at filtration step j, and the class that was born earlier is the one that survives. In the literature, this is called the elder rule. We do not adopt this definition, because the elder rule is not welldefined when two classes are born at the same time, as there is no way to choose which class will survive. For example, in Figure 5, there are two classes in \(H_{0}\) that are born at the same stage in \(K_{1}\). These two classes merge in \(K_{2}\), but neither dies. The class that dies is \([a]+[c]\).
There are numerous excellent introductions to PH, such as the books [57, 80, 82, 83] and the papers [59, 84–88]. For a brief and friendly introduction to PH and some of its applications, see the video https://www.youtube.com/watch?v=h0bnG1Wavag. For a brief introduction to some of the ideas in TDA, see the video https://www.youtube.com/watch?v=XfWibrh6stw.
5 Computation of PH for data
5.1 Data
As we mentioned in Section 1, types of data sets that one can study with PH include finite metric spaces, digital images, and networks. We now give a brief overview of how one can study these types of data sets using PH.
5.1.1 Networks
One can construe an undirected network as a 1dimensional simplicial complex. If the network is weighted, then filtering by increasing or decreasing weight yields a filtered 1dimensional simplicial complex. To obtain more refined information about the network, it is desirable to construct higherdimensional simplices. There are various methods to do this. The simplest method, called a weight rank clique filtration (WRCF), consists of building a clique complex on each subnetwork. (See Section 5.2.1 for the definition of ‘clique complex.’) See [89] for an application of this method. Another method to study networks with PH consists of mapping the nodes of the network to points of a finite metric space. There are several ways to compute distances between nodes of a network; the method that we use in our benchmarking in Section 7 consists of computing a shortest path between nodes. For such a distance to be welldefined, note that one needs the network to be connected (although conventionally one takes the distance between nodes in different components to be infinity). There are many methods to associate an unfiltered simplicial complex to both undirected and directed networks. See the book [90] for an overview of such methods, and see the paper [91] for an overview of PH for networks.
5.1.2 Digital images
As we mentioned in Section 1, digital images have a natural cubical structure: 2dimensional digital images are made of pixels, and 3dimensional images are made of voxels. Therefore, to study digital images, cubical complexes are more appropriate than simplicial complexes. Roughly, cubical complexes are spaces built from a union of vertices, edges, squares, cubes, and so on. One can compute PH for cubical complexes in a similar way as for simplicial complexes, and we will therefore not discuss this further in this paper. See [76] for a treatment of computational homology with cubical complexes rather than simplicial complexes and for a discussion of the relationship between simplicial and cubical homology. See [55] for an efficient algorithm and data structure for the computation of PH for cubical data, and [92] for an algorithm that computes PH for cubical data in an approximate way. For an application of PH and cubical complexes to movies, see [73].
Other approaches for studying digital images are also useful. In general, given a digital image that consists of N pixels or voxels, one can consider this image as a point in a \(c\times N\)dimensional space, with each coordinate storing a vector of length c representing the color of a pixel or voxel. Defining an appropriate distance function on such a space allows one to consider a collection of images (each of which has N pixels or voxels) as a finite metric space. A version of this approach was used in [26], in which the local structure of natural images was studied by selecting \(3\times3\) patches of pixels of the images.
5.1.3 Finite metric spaces
As we mentioned in the previous two subsections, both undirected networks and image data can be construed as finite metric spaces. Therefore, methods to study finite metric spaces with PH apply to the study of networks and image data sets.
In some applications, points of a metric space have associated ‘weights.’ For instance, in the study of molecules, one can represent a molecule as a union of balls in Euclidean space [93, 94]. For such data sets, one would therefore also consider a minimum filtration value (see Section 5.2 for the description of such filtration values) at which the point enters the filtration. In Table 2(g), we indicate which software libraries implement this feature.
5.2 Filtered simplicial complexes
In Section 3.2, we introduced the Čech complex, a classical simplicial complex from algebraic topology. However, there are many other simplicial complexes that are better suited for studying data from applications. We discuss them in this section.
We summarize several types of complexes that are used for PH
Complex K  Size of K  Theoretical guarantee 

Čech  \(2^{\mathcal {O}(N)}\)  Nerve theorem 
Vietoris–Rips (VR)  \(2^{\mathcal {O}(N)}\)  Approximates Čech complex 
Alpha  \(N^{\mathcal {O}(\lceil d/2 \rceil)}\) (N points in \(\mathbb {R}^{d}\))  Nerve theorem 
Witness  \(2^{\mathcal {O}(L)}\)  For curves and surfaces in Euclidean space 
Graphinduced complex  \(2^{\mathcal {O}(Q)}\)  Approximates VR complex 
Sparsified Čech  \(\mathcal {O}(N)\)  Approximates Čech complex 
Sparsified VR  \(\mathcal {O}(N)\)  Approximates VR complex 
For the rest of this subsection \((X,\mathrm {d})\) denotes a metric space, and S is a subset of X, which becomes a metric space with the induced metric. In applications, S is the collection of measurements together with a notion of distance, and we assume that S lies in the (unknown) metric space X. Our goal is then to compute persistent homology for a sequence of nested spaces \(S_{\epsilon_{1}},S_{\epsilon _{2}},\dots, S_{\epsilon_{l}}\), where each space gives a ‘thickening’ of S in X.
5.2.1 Vietoris–Rips complex
In applications, one therefore usually only computes the VR complex up to some dimension \(k \llS1\). In our benchmarking, we often choose \(k=2\) and \(k=3\).
The paper [95] overviews different algorithms to perform both of the steps for the construction of the VR complex, and it introduces fast algorithms to construct the clique complex. For more details on the VR complex, see [80, 96]. For a proof of the approximation of the Čech complex by the VR complex, see [80]; see [97] for a generalization of this result.
5.2.2 The Delaunay complex
To avoid the computational problems of the Čech and VR complexes, we need a way to limit the number of simplices in high dimensions. The Delaunay complex gives a geometric tool to accomplish this task, and most of the new simplicial complexes that have been introduced for the study of data are based on variations of the Delaunay complex. The Delaunay complex and its dual, the Voronoi diagram, are central objects of study in computational geometry because they have many useful properties.
The complexity of the Delaunay complex depends on the dimension d of the space. For \(d\leq2\), the best algorithms have complexity \(\mathcal {O}(N \log N)\), where N is the cardinality of S. For \(d\geq3\), they have complexity \(\mathcal {O}(N^{\lceil d/2 \rceil})\). The construction of the Delaunay complex is therefore costly in high dimensions, although there are efficient algorithms for the computation of the Delaunay complex for \(d = 2\) and \(d = 3\). Developing efficient algorithms for the construction of the Delaunay complex in higher dimensions is a subject of ongoing research. See [98] for a discussion of progress in this direction, and see [99] for more details on the Delaunay complex and the Voronoi diagram.
5.2.3 Alpha complex
We continue to assume that S is a finite set of points in \(\mathbb {R}^{d}\). Using the Voronoi decomposition, one can define a simplicial complex that is similar to the Čech complex, but which has the desired property that (if the points S are in general position) its dimension is at most that of the space. Let \(\epsilon>0\), and let \(S_{\epsilon}\) denote the union \(\bigcup_{s\in S}B(s,\epsilon)\). For every \(s\in S\), consider the intersection \(V_{s}\cap B(s,\epsilon)\). The collection of these sets forms a cover of \(S_{\epsilon}\), and the nerve complex of this cover is called the alpha (α) complex of S at scale ϵ and is denoted by \(A_{\epsilon}(S)\). The Nerve Theorem applies, and it therefore follows that \(A_{\epsilon}(S)\) has the same homology as \(S_{\epsilon}\).
Furthermore, \(A_{\infty}(S)\) is the Delaunay complex; and for \(\epsilon <\infty\), the alpha complex is a subcomplex of the Delaunay complex. The alpha complex was introduced for points in the plane in [100], in 3dimensional Euclidean space in [101], and for Euclidean spaces of arbitrary dimension in [102]. For points in the plane, there is a wellknown speedup for the alpha complex that uses a duality between 0dimensional and 1dimensional persistence for alpha complexes [86]. (See [103] for the algorithm, and see [104] for an implementation.)
5.2.4 Witness complexes
Witness complexes are very useful for analyzing large data sets, because they make it possible to construct a simplicial complex on a significantly smaller subset \(L\subseteq S\) of points that are called ‘landmark’ points. Meanwhile, because one uses information about all points in S to construct the simplicial complex, the points in S are called ‘witnesses.’ Witness complexes can be construed as a ‘weak version’ of Delaunay complexes. (See the characterization of the Delaunay complex in [105].)
Definition 6
Let \((S,\mathrm {d})\) be a metric space, and let \(L\subseteq S\) be a finite subset. Suppose that σ is a nonempty subset of L. We then say that \(s\in S\) is a weak witness for σ with respect to L if and only if \(\mathrm {d}(s,a)\leq \mathrm {d}(s,b)\) for all \(a\in\sigma\) and for all \(b\in L\setminus\sigma\). The weak Delaunay complex \(\operatorname {Del}^{w}(L;S)\) of S with respect to L has vertex set given by the points in L, and a subset σ of L is in \(\operatorname {Del}^{w}(L;S)\) if and only if it has a weak witness in S.
To obtain nested complexes, one can extend the definition of witnesses to ϵwitnesses.
Definition 7
A point \(s\in S\) is a weak ϵwitness for σ with respect to L if and only if \(\mathrm {d}(s,a)\leq \mathrm {d}(s,b)+\epsilon\) for all \(a\in\sigma\) and for all \(b\in L\setminus \sigma\).
Now we can define the weak Delaunay complex \(\operatorname {Del}^{w}(L;S,\epsilon )\) at scale ϵ to be the simplicial complex with vertex set L, and such that a subset \(\sigma\subseteq L\) is in \(\operatorname {Del}^{w}(L;S,\epsilon)\) if and only if it has a weak ϵwitness in S. By considering different values for the parameter ϵ, we thereby obtain nested simplicial complexes. The weak Delaunay complex is also called the ‘weak witness complex’ or just the ‘witness complex’ in the literature.
The weak Delaunay complex was introduced in [105], and parametrized witness complexes were introduced in [106]. Witness complexes can be rather useful for applications. Because their complexity depends on the number of landmark points, one can reduce the complexity by computing simplicial complexes using a smaller number of vertices. However, there are theoretical guarantees for the witness complex only when S is the metric space associated to a lowdimensional Euclidean submanifold. It has been shown that witness complexes can be used to recover the topology of curves and surfaces in Euclidean space [107, 108], but they can fail to recover topology for submanifolds of Euclidean space of three or more dimensions [109]. Consequently, there have been studies of simplicial complexes that are similar to the witness complexes but with better theoretical guarantees (see Section 5.2.5).
5.2.5 Additional complexes
Overview of existing software for the computation of PH that have an accompanying peerreviewed publication (and also [ 68 ], because of its performance)
Software  javaPlex  Perseus  jHoles  Dionysus  PHAT  DIPHA  Gudhi  SimpPers  Ripser 

(a) Language  Java  C++  Java  C++  C++  C++  C++  C++  C++ 
(b) Algorithms for PH  standard, dual, zigzag  Morse reductions, standard  standard (uses javaPlex)  standard, dual, zigzag  standard, dual, twist, chunk, spectral sequence  twist, dual, distributed  dual, multifield  simplicial map  twist, dual 
(c) Coefficient field  \(\mathbb{Q}\), \(\mathbb{F}_{p}\)  \(\mathbb{F}_{2}\)  \(\mathbb{F}_{2}\)  \(\mathbb{F}_{2}\) (standard, zigzag), \(\mathbb{F}_{p}\) (dual)  \(\mathbb{F}_{2}\)  \(\mathbb{F}_{2}\)  \(\mathbb{F}_{p}\)  \(\mathbb{F}_{2}\)  \(\mathbb{F}_{p}\) 
(d) Homology  simplicial, cellular  simplicial, cubical  simplicial  simplicial  simplicial, cubical  simplicial, cubical  simplicial, cubical  simplicial  simplicial 
(e) Filtrations computed  VR, W, \(\mathrm{W}_{\nu}\)  VR, lower star of cubical complex  WRCF  VR, α, C̆    VR, lower star of cubical complex  VR, α, W, lower star of cubical complex    VR 
(f) Filtrations as input  simplicial complex, zigzag, CW  simplicial complex, cubical complex    simplicial complex, zigzag  boundary matrix of simplicial complex  boundary matrix of simplicial complex    map of simplicial complexes   
(g) Additional features  Computes some homological algebra constructions, homology generators  weighted points for VR    vineyards, circlevalued functions, homology generators           
(h) Visualization  barcodes  persistence diagram        persistence diagram       
5.2.6 Reduction techniques
Thus far, we have discussed techniques to build simplicial complexes with possibly ‘few’ simplices. One can also take an alternative approach to speed up the computation of PH. For example, one can use a heuristic (i.e., a method without theoretical guarantees on the speedup) to reduce the size of a filtered complex while leaving the PH unchanged.
For simplicial complexes, one such method is based on discrete Morse theory [114], which was adapted to filtrations of simplicial complexes in [115]. The basic idea of the algorithm developed in [115] is that one can compute a partial matching of the simplices in a filtered simplicial complex so that (i) pairs occur only between simplices that enter the filtration at the same step, (ii) unpaired simplices determine the homology, and (iii) one can remove paired simplices from the filtered complex without altering the PH. Such deletions are examples of the elementary simplicial collapses that we mentioned in Section 3.1. Unfortunately, the problem of finding an optimal partial matching was shown to be NP complete [116], and one thus relies on heuristics to find partial matchings to reduce the size of the complex.
One particular family of elementary collapses, called strong collapses, was introduced in [117]. Strong collapses preserve cycles of shortest length in the representative class of a generator of a hole [118]; this feature makes strong collapses useful for finding holes in networks [118]. A distributed version of the algorithm proposed in [118] was presented in [119] and adapted for the computation of PH in [120].
A method for the reduction of the size of a complex for clique complexes, such as the VR complex, was proposed in [121] and is called the tidyset method. Using maximal cliques, this method extracts a minimal representation of the graph that determines the clique complex. Although the tidyset method cannot be extended to filtered complexes, it can be used for the computation of zigzag PH (see Section 6) [122]. The tidyset method is a heuristic, because it does not give a guarantee to minimize the size of the output complex.
5.3 From a filtered simplicial complex to barcodes

a face of a simplex precedes the simplex;

a simplex in the ith complex \(K_{i}\) precedes simplices in \(K_{j}\) for \(j>i\), which are not in \(K_{i}\).
Once one has constructed the boundary matrix, one has to reduce it using Gaussian elimination.^{6} In the following subsections, we discuss several algorithms for reducing the boundary matrix.
5.3.1 Standard algorithm
5.3.2 Reading off the intervals

If \(\operatorname {low}(j)=i\), then the simplex \(\sigma_{j}\) is paired with \(\sigma_{i}\), and the entrance of \(\sigma_{i}\) in the filtration causes the birth of a feature that dies with the entrance of \(\sigma_{j}\).

If \(\operatorname {low}(j)\) is undefined, then the entrance of the simplex \(\sigma_{j}\) in the filtration causes the birth of a feature. It there exists k such that \(\operatorname {low}(k)=j\), then \(\sigma_{j}\) is paired with the simplex \(\sigma_{k}\), whose entrance in the filtration causes the death of the feature. If no such k exists, then \(\sigma_{j}\) is unpaired.
5.3.3 Other algorithms
After the introduction of the standard algorithm, several new algorithms were developed. Each of these algorithms gives the same output for the computation of PH, so we only give a brief overview and references to these algorithms, as one does not need to know them to compute PH with one of the publiclyavailable software packages. In Section 7.2, we indicate which implementation of these libraries is best suited to which data set.
As we mentioned in Section 5.3.1, in the worst case, the standard algorithm has cubic complexity in the number of simplices. This bound is sharp, as Morozov gave an example of a complex with cubic complexity in [123]. Note that in cases such as when matrices are sparse, complexity is less than cubic. Milosavljević, Morozov, and Skraba [124] introduced an algorithm for the reduction of the boundary matrix in \(\mathcal {O}(n^{\omega})\), where ω is the matrixmultiplication coefficient (i.e., \(\mathcal {O}(n^{\omega})\) is the complexity of the multiplication of two square matrices of size n). At present, the best bound for ω is 2.376 [125]. Many other algorithms have been proposed for the reduction of the boundary matrix. These algorithms give a heuristic speedup for many data sets and complexes (see the benchmarkings in the forthcoming references), but they still have cubic complexity in the number of simplices. Sequential algorithms include the twist algorithm [126] and the dual algorithm [72, 127]. (Note that the dual algorithm is known to give a speedup when one computes PH with the VR complex, but not necessarily for other types of complexes (see also the results of our benchmarking for the vertebra data set in Additional file 1 of the SI).) Parallel algorithms in a shared setting include the spectralsequence algorithm (see Section VII.4 of [80]) and the chunk algorithm [128]; parallel algorithms in a distributed setting include the distributed algorithm [74]. The multifield algorithm is a sequential algorithm that allows the simultaneous computation of PH over several fields [129].
5.4 Statistical interpretation of topological summaries
Once one has obtained barcodes, one needs to interpret the results of computations. In applications, one often wants to compare the output of a computation for a certain data set with the output for a null model. Alternatively, one may be studying data sets from the output of a generative model (e.g., many realizations from a model of random networks), and it is then necessary to average results over multiple realizations. In the first instance, one needs both a way to compare the two different outputs and a way to evaluate the significance of the result for the original data set. In the second case, one needs a way to calculate appropriate averages (e.g., summary statistics) of the result of the computations.
From a statistical perspective, one can interpret a barcode as an unknown quantity that one tries to estimate by computing PH. If one wants to use PH in applications, one thus needs a reliable way to apply statistical methods to the output of the computation of PH. To our knowledge, statistical methods for PH were addressed for the first time in the paper [130]. Roughly speaking, there are three current approaches to the problem of statistical analysis of barcodes. In the first approach, researchers study topological properties of random simplicial complexes (see, e.g., [131, 132]) and the review papers [133, 134]. One can view random simplicial complexes as null models to compare with empirical data when studying PH. In the second approach, one studies properties of a metric space whose points are persistence diagrams. In the third approach, one studies ‘features’ of persistence diagrams. We will provide a bit more detail about the second and third approaches.
In the second approach, one considers an appropriately defined ‘space of persistence diagrams,’ defines a distance function on it, studies geometric properties of this space, and does standard statistical calculations (means, medians, statistical tests, and so on). Recall that a persistence diagram (see Figure 5 for an example) is a multiset of points in \(\overline{\mathbb{R}}^{2}\) and that it gives the same information as a barcode. We now give the following precise definition of a persistence diagram.
Definition 8
A persistence diagram is a multiset that is the union of a finite multiset of points in \(\overline{\mathbb{R}}^{2}\) with the multiset of points on the diagonal \(\Delta=\{(x,y)\in\mathbb {R}^{2}\mid x=y\}\), where each point on the diagonal has infinite multiplicity.
In this definition, we include all of the points on the diagonal in \(\mathbb{R}^{2}\) with infinite multiplicity for technical reasons. Roughly, it is desirable to be able to compare persistence diagrams by studying bijections between their elements, and persistence diagrams must thus be sets with the same cardinality.
Given two persistence diagrams X and Y, we consider the following general definition of distance between X and Y.
Definition 9
Usually, one takes \(\mathrm {d}=L_{q}\) for \(q \in[1,\infty]\). One of the most commonly employed distance functions is the bottleneck distance \(W_{\infty}[L_{\infty}]\).
The development of statistical analysis on the space of persistence diagrams is an area of ongoing research, and presently there are few tools that can be used in applications. See [135–137] for research in this direction. Until recently, the library Dionysus [64] was the only library to implement computation of the bottleneck and Wasserstein distances (for \(\mathrm{d}=L_{\infty}\)); the library hera [138] implements a new algorithm [139] for the computation of the bottleneck and Wasserstein distances that significantly outperforms the implementation in Dionysus. The library TDA Package [140] (see [69] for the accompanying tutorial) implements the computation of confidence sets for persistence diagrams that was developed in [141], distance functions that are robust to noise and outliers [142], and many more tools for interpreting barcodes.
The third approach for the development of statistical tools for PH consists of mapping the space of persistence diagrams to spaces (e.g., Banach spaces) that are amenable to statistical analysis and machinelearning techniques. Such methods include persistence landscapes [143], using the space of algebraic functions [144], persistence images [145], and kernelization techniques [146–149]. See the papers [6, 52] for applications of persistence landscapes. The package Persistence Landscape Toolbox [150] (see [70] for the accompanying tutorial) implements the computation of persistence landscapes, as well as many statistical tools that one can apply to persistence landscapes, such as mean, ANOVA, hypothesis tests, and many more.
5.5 Stability
As we mentioned in Section 1, PH is useful for applications because it is stable with respect to small perturbations in the input data.
The first stability theorem for PH, proven in [151], asserts that, under favorable conditions, step (2) in the pipeline in Figure 6 is 1Lipschitz with respect to suitable distance functions on filtered complexes and the bottleneck distance for barcodes (see Section 5.4). This result was generalized in the papers [152–154]. Stability for PH is an active area of research; for an overview of stability results, their history and recent developments, see [82], Chapter 3.
6 Excursus: generalized persistence
7 Software
There are several publiclyavailable implementations for the computation of PH. We give an overview of the libraries with accompanying peerreviewed publication and summarize their properties in Table 2.
The software package javaPlex [66], which was developed by the computational topology group at Stanford University, is based on the Plex library [161], which to our knowledge is the first piece of software to implement the computation of PH. Perseus [65] was developed to implement Morsetheoretic reductions [115] (see Section 5.2.6). jHoles [162] is a Java library for computing the weight rank clique filtration for weighted undirected networks [89]. Dionysus [64] is the first software package to implement the dual algorithm [72, 127]. PHAT [62] is a library that implements several algorithms and data structures for the fast computation of barcodes, takes a boundary matrix as input, and is the first software to implement a matrixreduction algorithm that can be executed in parallel. DIPHA [63], a spinoff of PHAT, implements a distributed computation of the matrixreduction algorithm. Gudhi [67] implements new data structures for simplicial complexes and the boundary matrix. It also implements the multifield algorithm, which allows simultaneous computation of PH over several fields [129]. This library is currently under intense development, and a Python interface was just released in the most recent version of the library (namely, Version 2.0.0, whereas the version that we study in our tests is Version 1.3.1). The library ripser [68], the most recently developed software of the set that we examine, uses several optimizations and shortcuts to speed up the computation of PH with the VR complex. This library does not have an accompanying peerreviewed publication. However, because it is currently the bestperforming (both in terms of memory usage and in terms of walltime seconds^{8}) library for the computation of PH with the VR complex, we include it in our study. The library SimpPers [163] implements the simplicial map algorithm. Libraries that implement techniques for the statistical interpretation of barcodes include the TDA Package [140] and the Persistence Landscape Toolbox [150]. (See Section 5.4 for additional libraries for the interpretation of barcodes.) RIVET, a package for visualizing 2parameter persistent homology, is slated to be released soon [158]. We summarize the properties of the libraries for the computation of PH that we mentioned in this paragraph in Table 2, and we discuss the performance for a selection of them in Section 7.1.3 and in Additional file 1 of the SI. For a list of programs, see https://github.com/notter/PHroadmap.
7.1 Benchmarking
We benchmark a subset of the currently available opensource libraries with peerreviewed publication for the computation of PH. To our knowledge, the published opensource libraries are jHoles, javaPlex, Perseus, Dionysus, PHAT, DIPHA, SimpPers, and Gudhi. To these, we add the library ripser, which is currently the bestperforming library to compute PH with the VR complex. To study the performance of the packages, we restrict our attention to the algorithms that are implemented by the largest number of libraries. These are the VR complex and the standard and dual algorithms for the reduction of the boundary matrix. PHAT only takes a boundary matrix as input, so it is not possible to conduct a direct comparison of it with the other implementations. However, the fast data structures and algorithms implemented in PHAT are also implemented in its spinoff software DIPHA, which we include in the benchmarking. The software jHoles computes PH using the WRCF for weighted undirected networks, and SimpPers takes a map of simplicial complexes as input, so these two libraries cannot be compared directly to the other libraries. In Additional file 1 of the SI, we report benchmarking of some additional features that are implemented by some of the six libraries (i.e., javaPlex, Perseus, Dionysus, DIPHA, Gudhi, and Ripser) that we test. Specifically, we report results for the computation of PH with cubical complexes for image data sets and the computation of PH with witness, alpha, and Čech complexes.
 1.
Performance measured in CPU seconds and walltime (i.e., elapsed time) seconds.
 2.
Memory required by the process.
 3.
Maximum size of simplicial complex allowed by the software.
7.1.1 Data sets
In this subsection, we describe the data sets that we use for our benchmarking. We use data sets from a variety of different mathematical and scientific areas and applications. In each case, when possible, we use data sets that have already been studied using PH. Our list of data sets is far from complete; we view this list as an initial step towards building a comprehensive collection of benchmarking data sets for PH.
Data sets (1)–(4) are synthetic; they arise from topology (1), stochastic topology (2), dynamical systems (3), and from an area at the intersection of network theory and fractal geometry (4). (As we discuss below, data set (4) was used originally to study connection patterns in the cerebral cortex.) Data sets (5)–(12) are from empirical experiments and measurements: they arise from phylogenetics (5)–(6), image analysis (7), genomics (9), neuroscience (8), medical imaging (10), political science (11), and scientometrics (12).
In each case, these data sets are of one of the following three types: point clouds, weighted undirected networks, and grayscale digital images. To obtain a point cloud from a realworld weighted undirected network, we compute shortest paths using the inverse of the nonzero weights of edges as distances between nodes (except for the US Congress networks and the human genome network; see below). For the synthetic networks, the values assigned to edges are interpreted as distances between nodes, and we therefore use these values to compute shortest paths. We make all processed versions of the data sets that we use in the benchmarking available at https://github.com/notter/PHroadmap/tree/master/data_sets. We provide the scripts that we used to produce the synthetic data sets at https://github.com/notter/PHroadmap/tree/master/matlab/synthetic_data_sets_scripts.
 (1)Klein bottle. The Klein bottle is a onesided nonorientable surface (see Figure 8). We linearly sample points from the Klein bottle using its ‘figure8’ immersion in \(\mathbb{R}^{3}\) and size sample of 400 points. We denote this data set by Klein. Note that the image of the immersion of the Klein bottle does not have the same homotopy type as the original Klein bottle, but it does have the same singular homology^{9} with coefficients in \(\mathbb{F}_{2}\). We have \(H_{0}(B)=\mathbb{F}_{2}\), \(H_{1}(B)=\mathbb{F}_{2}\oplus\mathbb{F}_{2}\), and \(H_{2}(B)=\mathbb{F}_{2}\), where B denotes the Klein bottle and \(H_{i}(B)\) is the ith singular homology group with coefficients in \(\mathbb{F}_{2}\).
 (2)
Random VR complexes (uniform distribution) [165]. The parameters for this model are positive integers N and d; the random VR complex for parameters N and d is the VR complex \(\operatorname {VR}_{\epsilon}(X)\), where X is a set of N points sampled from \(\mathbb{R}^{d}\). (Equivalently, the random VR complex is the clique complex on the random geometric graph \(G(N,\epsilon)\) [166].) We sample N points uniformly at random from \([0,1]^{d}\). We choose \((N,d)=(50,16)\) and we denote this data set by random. The homology of random VR complexes was studied in [165].
 (3)
Vicsek biological aggregation model. This model was first introduced in [167] and was studied using PH in [48]. We implement the model in the form in which it appears in [48]. The model describes the motion of a collection of particles that interact in a square with periodic boundary conditions. The parameters for the model are the length l of the side of the square, the initial angle \(\theta_{0}\), the (constant) absolute value \(v_{0}\) for the velocity, the number N of particles, a noise parameter η, and the number T of time steps. The output of the model is a point cloud in 3dimensional Euclidean space in which each point is specified by its position in the 2dimensional box and its velocity angle (‘heading’). We run three simulations of the model using the parameter values from [48]. For each simulation, we choose two point clouds that correspond to two different time frames. See [48] for further details. We denote this data set by Vicsek.
 (4)
Fractal networks. These are selfsimilar networks introduced in [168] to investigate whether connection patterns of the cerebral cortex are arranged in selfsimilar patterns. The parameters for this model are natural numbers b, k, and n. To generate a fractal network, one starts with a fullyconnected network with \(2^{b}\) nodes. Two copies of this network are connected to each other so that the ‘connection density’ between them is \(k^{1}\), where the connection density is the number of edges between the two copies divided by the number of total possible edges between them. Two copies of the resulting network are connected with connection density \(k^{2}\). One repeats this type of connection process until the network has size \(2^{n}\), but with a decrease in the connection density by a factor of \(1/k\) at each step.
We define distances between nodes in two different ways: (1) uniformly at random, and (2) with linear weightdegree correlations. In the latter, the distance between nodes i and j is distributed as \(k_{i}k_{j}X\), where \(k_{i}\) is the degree of node i and X is a random variable uniformly distributed on the unit interval. We use the parameters \(b=5\), \(n=9\), and \(k=2\); and we compute PH for the weighted network and for the network in which all adjacent nodes have distance 1. We denote this data set by fract and distinguish between the two ways of defining distances between weights using the abbreviations ‘r’ for random, and ‘l’ for linear.
 (5)
Genomic sequences of the HIV virus. We construct a finite metric space using the independent and concatenated sequences of the three largest genes — gag, pol, and env — of the HIV genome. We take \(1\mbox{,}088\) different genomic sequences and compute distances between them by using the Hamming distance. We use the aligned sequences studied using PH in [23]. (The authors of that paper retrieved the sequences from [169].) We denote this data set by HIV.
 (6)
Genomic sequences of H3N2. This data set consists of \(1\mbox{,}000\) different genomic sequences of H3N2 influenza. We compute the Hamming distance between sequences. We use the aligned sequences studied using PH in [23]. We denote this data set by H3N2.
 (7)
Stanford Dragon graphic. We sample points uniformly at random from 3dimensional scans of the dragon [164], whose reconstruction we show in Figure 8. The sample sizes include \(1\mbox{,}000\) and \(2\mbox{,}000\) points. We denote these data sets by drag 1 and drag 2, respectively.
 (8)
C. elegans neuronal network. This is a weighted, undirected network in which each node is a neuron and edges represent synapses or gap junctions. We use the network studied using PH in [89]. (The authors of the paper used the data set studied in [170], which first appeared in [171].) Recall that for this example, and also for the other realworld weighted networks (except for the human genome network and the US Congress networks), we convert each nonzero edge weight to a distance by taking its inverse. We denote this data set by eleg.
 (9)
Human genome. This is a weighted, undirected network representing a sample of the human genome. We use the network studied using PH in [89]. (The authors of that paper created the sample using data retrieved from [172].) Each node represents a gene, and two nodes are adjacent if there is a nonzero correlation between the expression levels of the corresponding genes. We define the weight of an edge as the inverse of the correlation.^{10} We denote this data set by genome.
 (10)
Grayscale image: 3dimensional rotational angiography scan of a head with an aneurysm. This data set was used in the benchmarking in [74]. This data set is given by a 3dimensional array of size \(512\times512\times512\), where each entry stores an integer that represents the grey value for the corresponding voxel. We retrieved the data set from the repository [173]. We denote this data set by vertebra.
 (11)
US Congress rollcall voting networks. These two networks (the Senate and House of Representatives from the 104th United States Congress) are constructed using the procedure in [174] from data compiled by [175]. In each network, a node is a legislator (Senators in one data set and Representatives in the other), and there is a weighted edge between legislators i and j, where the weight \(w_{i,j}\) is a number in \([0,1]\) (it is equal to 0 if and only if legislators i and j never voted the same way on any bill) given by the number of times the two legislators voted in the same way divided by the total number of bills on which they both voted. See [174] for further details. We denote the networks from the Senate and House by senate and house, respectively. The network senate has 103 nodes, and the network house has 445 nodes. To compute shortest paths, we define the distance between two nodes i and j to be \(1w_{i,j}\). In the 104th Congress, no two politicians voted in the same way on every bill, so we do not have distinct nodes with 0 distance between them. (This is important, for example, if one wants to apply multidimensional scaling.)
 (12)
Network of network scientists. This is a weighted, undirected network representing the largest connected component of a collaboration network of network scientists [176]. Nodes represent authors and edges represent collaborations, and weights indicate the number of joint papers. The largest connected component consists of 379 nodes. We denote this data set by netwsc.
7.1.2 Machines and compilers
Performance of the software packages measured in walltime (i.e., elapsed time), and CPU seconds (for the computations running on the cluster)
Data set  (a) Computations on cluster: walltime seconds  

eleg  Klein  HIV  drag 2  random  genome  
Size of complex  4.4 × 10^{6}  1.1 × 10^{7}  2.1 × 10^{8}  1.3 × 10^{9}  3.1 × 10^{9}  4.5 × 10^{8} 
Max. dim.  2  2  2  2  8  2 
javaPlex (st)  84  747         
Dionysus (st)  474  1,830         
DIPHA (st)  6  90  1,631  142,559    9,110 
Perseus  543  1,978         
Dionysus (d)  513  145         
DIPHA (d)  4  6  81  2,358  5,096  232 
Gudhi  36  89  1,798  14,368    4,753 
Ripser  1  1  2  6  349  3 
Data set  (b) Computations on cluster: CPU seconds  

eleg  Klein  HIV  drag 2  random  genome  
Size of complex  4.4 × 10^{6}  1.1 × 10^{7}  2.1 × 10^{8}  1.3 × 10^{9}  3.1 × 10^{9}  4.5 × 10^{8} 
Max. dim.  2  2  2  2  8  2 
javaPlex (st)  284  1,031         
Dionysus (st)  473  1,824         
DIPHA (st)  68  1,360  25,950  1,489,615    130,972 
Perseus  542  1,974         
Dionysus (d)  513  145         
DIPHA (d)  39  73  1,276  37,572  79,691  3,622 
Gudhi  36  88  1,794  14,351    4,764 
Ripser  1  1  2  5  348  2 
Data set  (c) Computations on sharedmemory system: walltime seconds  

eleg  Klein  HIV  drag 2  genome  fract r  
Size of complex  3.2 × 10^{8}  1.1 × 10^{7}  2.1 × 10^{8}  1.3 × 10^{9}  4.5 × 10^{8}  2.8 × 10^{9} 
Max. dim.  3  2  2  2  2  3 
javaPlex (st)  13,607  1,358  43,861    28,064   
Perseus    1,271         
Dionysus (d)    100  142,055  35,366    572,764 
DIPHA (d)  926  13  773  4,482  1,775  3,923 
Gudhi  381  6  177  1,518  442  4,590 
Ripser  2  1  2  5  3  1,517 
Memory usage in GB for the computations summarized in Table 3
Data set  (a) Computations on cluster  

eleg  Klein  HIV  drag 2  random  genome  
Size of complex  4.4 × 10^{6}  1.1 × 10^{7}  2.1 × 10^{8}  1.3 × 10^{9}  3.1 × 10^{9}  4.5 × 10^{8} 
Max. dim.  2  2  2  2  8  2 
javaPlex (st)  <5  <15  >64  >64  >64  >64 
Dionysus (st)  1.3  11.6         
DIPHA (st)  0.1  0.2  2.7  4.9    4.8 
Perseus  5.1  12.7         
Dionysus (d)  0.5  1.1         
DIPHA (d)  0.1  0.2  1.8  13.8  9.6  6.3 
Gudhi  0.2  0.5  8.5  62.8    21.5 
Ripser  0.007  0.02  0.06  0.2  24.7  0.07 
Data set  (b) Computations on sharedmemory system  

eleg  Klein  HIV  drag 2  genome  fract r  
Size of complex  3.2 × 10^{8}  1.1 × 10^{7}  2.1 × 10^{8}  1.3 × 10^{9}  4.5 × 10^{8}  2.8 × 10^{9} 
Max. dim.  3  2  2  2  2  3 
javaPlex (st)  <600  <15  <700  >700  <700  >700 
Perseus    11.7         
Dionysus (d)    1.1  16.8  134.2    268.5 
DIPHA (d)  31.2  0.9  17.7  109.5  37.3  276.1 
Gudhi  15.4  0.5  10.2  62.8  21.4  134.8 
Ripser  0.2  0.03  0.07  0.2  0.07  155 
7.1.3 Tests and results
We now report the details and results of the tests that we performed. We have made the data sets, header file to measure memory, and other information related to the tests available at https://github.com/notter/PHroadmap. Of the six software packages that we study, four implement the computation of the dual algorithm, and four implement the standard algorithm. It is reported in [66] that javaPlex implements the dual algorithm, but the implementation of the algorithm has a bug and gives a wrong output. To our knowledge, this bug has not yet been fixed (at the time of writing), and we therefore test only the standard algorithm.
For the computations on the cluster, we compare the libraries running both the dual algorithm and the standard algorithm. The package DIPHA is the only one to implement a distributed computation. As a default, we run the software on one node and 16 cores; we only increase the number of nodes and cores employed when the machine runs out of memory. However, augmenting the number of nodes can make the computations faster (in terms of CPU seconds) for complexes of all sizes.^{12} We see this in our experiments, and it is also discussed in [74]. For the other packages, we run the computations on a single node with one core.
For computations on the sharedmemory system, we compare the libraries using only the dual algorithm if they implement it, and we otherwise use the standard algorithm. For the sharedmemory system, we run all packages (including DIPHA) on a single core.
For each software package in (a), we indicate in (b) the maximal size of the simplicial complex supported by it thus far in our tests
(a)  javaPlex  Dionysus  DIPHA  Perseus  Gudhi  Ripser  

st  st  d  st  d  st  d  d  
(b)  4.5⋅10^{8}  1.1⋅10^{7}  2.8 × 10^{9}  1.3⋅10^{9}  3.4⋅10^{9}  1⋅10^{7}  3.4⋅10^{9}  3.4⋅10^{9} 
7.2 Conclusions from our benchmarking
Our tests suggest that Ripser is the bestperforming library currently available for the computation of PH with the Vietoris–Rips complex, and in order of decreasing performance, that Gudhi and DIPHA are the nextbest implementations. For the computation of PH with cubical complexes, Gudhi outperforms DIPHA by a factor of 3 to 4 in terms of memory usage, and DIPHA outperforms Gudhi in terms of walltime seconds by a factor of 1 to 2 (when running on one core on a sharedmemory system). Both DIPHA and Gudhi significantly outperform the implementation in Perseus. For the computation of PH with the alpha complex, we did not observe any significant differences in performance between the libraries Gudhi and Dionysus. Because the alpha complex has fewer simplices than the other complexes that we used in our tests, further tests with larger data sets may be appropriate in future benchmarkings.
There is a huge disparity between implementations of the dual and standard algorithms when computing PH with the VR complex. In our benchmarking, the dual implementations outperformed standard ones both in terms of computation time (with respect to both CPU and walltime seconds) and in terms of the amount of memory used. This significant difference in performance and memory usage was also revealed for the software package Dionysus in [72]. However, note that when computing PH for other types of complexes, the standard algorithm may be better suited than the dual algorithm. (See, e.g., the result of our test for the vertebra data set in Additional file 1 of the SI.)
To conclude, in our benchmarking, the fastest software packages were Ripser, Gudhi, and DIPHA. For small complexes, the software packages Perseus and javaPlex are good choices, because they are the easiest ones to use. (They are the only libraries that need only to be downloaded and are then ‘plugandplay,’ and they have userfriendly interfaces.) Because the library javaPlex implements the computation of a variety of complexes and algorithms, we feel that it is the best software for an initial foray into PH computation.
Guidelines for which implementation is bestsuited for which data set, based on our benchmarking
Data type  Complex  Suggested software 

networks  WRCF  jHoles 
image data  cubical  Gudhi or DIPHA (st) 
distance matrix  VR  Ripser 
distance matrix  W  javaPlex 
points in Euclidean space  VR  Gudhi 
points in Euclidean space  Č  Dionysus 
points in Euclidean space  α (only in dim 2 and 3)  Dionysus ((st) in dim 2, (d) in dim 3) or Gudhi 
8 Future directions
We conclude by discussing some future directions for the computation of PH. As we saw in Section 5, much work has been done on step 2 (i.e., going from filtered complexes to barcodes) of the PH pipeline of Figure 6, and there exist implementations of many fast algorithms for the reduction of the boundary matrix. Step 1 (i.e., going from data to a filtered complex) of the PH pipeline is an active area of research, but many sparsification techniques (see, e.g., [97, 112]) for complexes have yet to be implemented, and more research needs to be done on steps 1 and 3 (i.e., interpreting barcodes; see, e.g., [130, 136, 143]) of the PH pipeline. In particular, it is important to develop approaches for statistical analysis of persistent homology.
We believe that there needs to be a communitywide effort to build a library that implements the algorithms and data structures for the computation of PH, and that it should be done in a way that new algorithms and methods can be implemented easily in this framework. This would parallel similar communitywide efforts in fields such as computational algebra and computational geometry, and libraries such as Macaulay2 [177], Sage [178], and CGAL [179].
We also believe that there is a need to create guidelines and benchmark data sets for the test of new algorithms and data structures. The methods and collection of data sets that we used in our benchmarking provide an initial step towards establishing such guidelines and a list of test problems.
9 List of abbreviations
 1.
α: alpha complex
 2.
d (following the name of a library): implementation of the dual algorithm
 3.
Č: Čech complex
 4.
PH: persistent homology
 5.
SI: Supplementary Information
 6.
st (following the name of a library): implementation of the standard algorithm
 7.
TDA: topological data analysis
 8.
VR: Vietoris–Rips complex
 9.
W: weak witness complex
 10.
\(\mathrm {W}_{\nu}\): parametrized witness complexes
 11.
WRCF: weight rank clique filtration
10 Availability of data and materials
The processed version of the data sets used in the benchmarking and the scripts written for the tutorial are available at https://github.com/notter/PHroadmap. The opensource libraries for the computation of PH studied in this paper are available at the references indicated in the associated citations.
Conversely, under favorable conditions (see [78], Corollary 4.33), these algebraic invariants determine the topology of a space up to homotopy — an equivalence relation that is much coarser (and easier to work with) than the more familiar notion of homeomorphy.
A pair \(( \{ M_{i}\}_{i\in I}, \{\phi_{i,j}: M_{i}\to M_{j}\}_{i\leq j} )\), where \((I,\leq)\) is a totally ordered set, such that for each i, we have that \(M_{i}\) is a vector space and the maps \(\phi_{i,j}\) are linear maps satisfying the functoriality property (1), is usually called a persistence module. With this terminology, the homology of a filtered simplicial complex is an example of persistence module.
Although the collection of intervals is unique, note that one has to choose a vertical order when drawing the intervals in the diagram, and there is therefore an ambiguity in the representation of the intervals as a barcode. However, there is no ambiguity when representing the intervals as points in a persistence diagram (see Figure 5(d)).
A set S of points in \(\mathbb{R}^{d}\) is in general position if no \(d+2\) points of S lie on a ddimensional sphere, and for any \(d'< d\), no \(d'+2\) points of S lie on a \(d'\)dimensional subspace that is isometric to \(\mathbb{R}^{d'}\). In particular, a set of points S in \(\mathbb{R}^{2}\) is in general position if no four points lie on a 2dimensional sphere and no three points lie on a line.
As we mentioned in Section 4, for the reduction of the boundary matrix and thus the computation of PH, it is crucial that one uses simplicial homology with coefficients in a field; see [61] for details.
This map is called ‘low’ in the literature, because one can think of it as indicating the index of the ‘lowest’ row — the one that is nearest to the bottom of the page on which one writes the boundary matrix — that contains a 1 in column j.
Singular homology is a method that assigns to every topological space homology groups encoding invariants of the space, in an analogous way as simplicial homology assigns homology groups to simplicial complexes. See [78] for an account of singular homology.
We note that the weight should be the correlation; this issue came to our attention when the paper was in press.
Note that we performed the computations for Gudhi and Ripser at a different point in time, during which the sharedmemory system was running the OS Ubuntu 16.04.01.
Based on the results of our tests, we think of small, medium, and large complexes, respectively, as complexes with a size of order of magnitude of up to 10 million simplices, between 10 million and 100 million simplices, and between 100 million and a billion simplices.
Declarations
Acknowledgements
We thank the Rabadan Lab at Columbia University for providing the HIV and H3N2 sequences used in [23] and Giovanni Petri for sharing the data sets used in [89]. We thank Krishna Bhogaonker, Adrian Clough, Patrizio Frosini, Florian Klimm, Yacoub Kureh, Vitaliy Kurlin, Robert MacKay, James Meiss, Dane Taylor, Leo Speidel, Parker Edwards, and Bernadette Stolz for helpful comments on a draft of this paper. We also thank the anonymous referees for their many helpful comments. The first author thanks Ulrich Bauer, Michael Lesnick, Hubert Wagner, and Matthew Wright for helpful discussions, and thanks Florian Klimm, Vidit Nanda, and Bernadette Stolz for precious advice. The authors would like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility (http://dx.doi.org/10.5281/zenodo.22558) in carrying out some of the computations performed in this work. The first author thanks the support team at the ARC for their assistance. NO and PG are grateful for support from the EPSRC grant EP/G065802/1 (The Digital Economy HORIZON Hub). HAH gratefully acknowledges EPSRC Fellowship EP/K041096/1. NO and UT were supported by The Alan Turing Institute through EPSRC grant EP/N510129/1. NO and HAH were supported by the EPSRC institutional grant D4D01270 BKA1.01.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Kaufman L, Rousseeuw PJ (1990) Finding groups in data: an introduction to cluster analysis. Wiley, New York MATHView ArticleGoogle Scholar
 Goldenberg A, Zheng AX, Fienberg SE, Airoldi EM (2010) A survey of statistical network models. Found Trends Mach Learn 2:129233 MATHView ArticleGoogle Scholar
 Gan G, Ma C, Wu J (2007) Data clustering: theory, algorithms, and applications. SIAM, Philadelphia MATHView ArticleGoogle Scholar
 Schaeffer SE (2007) Graph clustering. Comput Sci Rev 1:2764 MATHView ArticleGoogle Scholar
 de Silva V, Ghrist R (2007) Coverage in sensor networks via persistent homology. Algebraic Geom Topol 7:339358 MathSciNetMATHView ArticleGoogle Scholar
 KovacevNikolic V, Bubenik P, Nikolić D, Heo G (2014) Using persistent homology and dynamical distances to analyze protein binding. arXiv:1412.1394
 Gameiro M, Hiraoka Y, Izumi S, Kramár M, Mischaikow K, Nanda V (2015) A topological measurement of protein compressibility. Jpn J Ind Appl Math 32:117 MathSciNetMATHView ArticleGoogle Scholar
 Xia K, Wei GW (2014) Persistent homology analysis of protein structure, flexibility, and folding. Int J Numer Methods Biomed Eng 30:814844 MathSciNetView ArticleGoogle Scholar
 Xia K, Li Z, Mu L (2016) Multiscale persistent functions for biomolecular structure characterization. arXiv:1612.08311
 Emmett K, Schweinhart B, Rabadán R (2016) Multiscale topology of chromatin folding. In: Proceedings of the 9th EAI international conference on bioinspired information and communications technologies (formerly BIONETICS), BICT’15. ICST (Institute for Computer Sciences, SocialInformatics and Telecommunications Engineering), Brussels, pp 177180 Google Scholar
 Rizvi A, Camara P, Kandror E, Roberts T, Schieren I, Maniatis T, Rabadan R (2017) Singlecell topological RNAseq analysis reveals insights into cellular differentiation and development. Nat Biotechnol 35:551560. doi:10.1038/nbt.3854 View ArticleGoogle Scholar
 Xia K, Feng X, Tong Y, Wei GW (2015) Persistent homology for the quantitative prediction of fullerene stability. J Comput Chem 36:408422 View ArticleGoogle Scholar
 Bhattacharya S, Ghrist R, Kumar V (2015) Persistent homology for path planning in uncertain environments. IEEE Trans Robot 31:578590 View ArticleGoogle Scholar
 Pokorny FT, Hawasly M, Ramamoorthy S (2016) Topological trajectory classification with filtrations of simplicial complexes and persistent homology. Int J Robot Res 35:204223 View ArticleGoogle Scholar
 Vasudevan R, Ames A, Bajcsy R (2013) Persistent homology for automatic determination of humandata based cost of bipedal walking. Nonlinear Anal Hybrid Syst 7:101115 MathSciNetMATHView ArticleGoogle Scholar
 Chung MK, Bubenik P, Kim PT (2009) Persistence diagrams of cortical surface data. In: Prince JL, Pham DL, Myers KJ (eds) Information processing in medical imaging. Lecture notes in computer science, vol 5636. Springer, Berlin, pp 386397 View ArticleGoogle Scholar
 Guillemard M, Boche H, Kutyniok G, Philipp F (2013) Signal analysis with frame theory and persistent homology. In: 10th international conference on sampling theory and applications, pp 309312 Google Scholar
 Perea JA, Deckard A, Haase SB, Harer J (2015) Sw1pers: sliding windows and 1persistence scoring; discovering periodicity in gene expression time series data. BMC Bioinform 16:Article ID 257 View ArticleGoogle Scholar
 Nicolau M, Levine AJ, Carlsson G (2011) Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proc Natl Acad Sci USA 108:72657270 View ArticleGoogle Scholar
 DeWoskin D, Climent J, CruzWhite I, Vazquez M, Park C, Arsuaga J (2010) Applications of computational homology to the analysis of treatment response in breast cancer patients. Topol Appl 157:157164 MathSciNetMATHView ArticleGoogle Scholar
 Crawford L, Monod A, Chen AX, Mukherjee S, Rabadán R (2016) Topological summaries of tumor images improve prediction of disease free survival in glioblastoma multiforme. arXiv:1611.06818
 Singh N, Couture HD, Marron JS, Perou C, Niethammer M (2014) Topological descriptors of histology images. In: Wu G, Zhang D, Zhou L (eds) Machine learning in medical imaging. Lecture notes in computer science, vol 8679. Springer, Cham, pp 231239 Google Scholar
 Chan JM, Carlsson G, Rabadan R (2013) Topology of viral evolution. Proc Natl Acad Sci USA 110:1856618571 MathSciNetMATHView ArticleGoogle Scholar
 Cámara PG, Levine AJ, Rabadán R (2016) Inference of ancestral recombination graphs through topological data analysis. PLoS Comput Biol 12:Article ID e1005071 View ArticleGoogle Scholar
 Emmett K, Rosenbloom D, Camara P, Rabadan R (2014) Parametric inference using persistence diagrams: a case study in population genetics. arXiv:1406.4582
 Carlsson G, Ishkhanov T, de Silva V, Zomorodian A (2008) On the local behavior of spaces of natural images. Int J Comput Vis 76:112 View ArticleGoogle Scholar
 Taylor D, Klimm F, Harrington HA, Kramár M, Mischaikow K, Porter MA, Mucha PJ (2015) Topological data analysis of contagion maps for examining spreading processes on networks. Nat Commun 6:Article ID 7723 View ArticleGoogle Scholar
 Lo D, Park B (2016) Modeling the spread of the Zika virus using topological data analysis. arXiv:1612.03554
 MacPherson R, Schweinhart B (2012) Measuring shape with topology. J Math Phys 53:Article ID 073516 MathSciNetMATHView ArticleGoogle Scholar
 Kramár M, Goullet A, Kondic L, Mischaikow K (2013) Persistence of force networks in compressed granular media. Phys Rev E 87:Article ID 042207 View ArticleGoogle Scholar
 Kramár M, Goullet A, Kondic L, Mischaikow K (2014) Quantifying force networks in particulate systems. Physica D 283:3755 MathSciNetMATHView ArticleGoogle Scholar
 Hiraoka Y, Nakamura T, Hirata A, Escolar E, Matsue K, Nishiura Y (2016) Hierarchical structures of amorphous solids characterized by persistent homology. Proc Natl Acad Sci USA 113:70357040 View ArticleGoogle Scholar
 Lee Y, Barthel SD, Dłotko P, Mohamad Moosavi S, Hess K, Smit B (2017) Poregeometry recognition: on the importance of quantifying similarity in nanoporous materials. arXiv:1701.06953
 Leibon G, Pauls S, Rockmore D, Savell R (2008) Topological structures in the equities market network. Proc Natl Acad Sci USA 105:2058920594 MathSciNetMATHView ArticleGoogle Scholar
 Gidea M (2017) Topology data analysis of critical transitions in financial networks. arXiv:1701.06081
 Giusti C, Ghrist R, Bassett D (2016) Two’s company and three (or more) is a simplex. J Comput Neurosci 41:114 MathSciNetView ArticleGoogle Scholar
 Curto C (2017) What can topology tell us about the neural code? Bull, New Ser, Am Math Soc 54:6378 MathSciNetMATHView ArticleGoogle Scholar
 Dłotko P, Hess K, Levi R, Nolte M, Reimann M, Scolamiero M, Turner K, Muller E, Markram H (2016) Topological analysis of the connectome of digital reconstructions of neural microcircuits. arXiv:1601.01580
 Kanari L, Dłotko P, Scolamiero M, Levi R, Shillcock J, Hess K, Markram H (2016) Quantifying topological invariants of neuronal morphologies. arXiv:1603.08432
 Lord LD, Expert P, Fernandes HM, Petri G, Van Hartevelt TJ, Vaccarino F, Deco G, Turkheimer F, Kringelbach M (2016) Insights into brain architectures from the homological scaffolds of functional connectivity networks. Front Syst Neurosci 10:Article ID 85 View ArticleGoogle Scholar
 Bendich P, Marron JS, Miller E, Pieloch A, Skwerer S (2016) Persistent homology analysis of brain artery trees. Ann Appl Stat 10:198218 MathSciNetView ArticleGoogle Scholar
 Yoo J, Kim EY, Ahn YM, Ye JC (2016) Topological persistence vineyard for dynamic functional brain connectivity during resting and gaming stages. J Neurosci Methods 267:113 View ArticleGoogle Scholar
 Dabaghian Y, Brandt VL, Frank LM (2014) Reconceiving the hippocampal map as a topological template. eLife 3:Article ID e03476 View ArticleGoogle Scholar
 Sizemore A, Giusti C, Bassett D (2017) Classification of weighted networks through mesoscale homological features. J Complex Netw 5:245273 Google Scholar
 Pal S, Moore TJ, Ramanathan R, Swami A (2017) Comparative topological signatures of growing collaboration networks. In: Complex networks VIII. Springer, Cham, pp 201209 View ArticleGoogle Scholar
 Carstens CJ, Horadam KJ (2013) Persistent homology of collaboration networks. Math Probl Eng 2013:Article ID 815035 MathSciNetMATHView ArticleGoogle Scholar
 Bajardi P, Delfino M, Panisson A, Petri G, Tizzoni M (2015) Unveiling patterns of international communities in a global city using mobile phone data. EPJ Data Sci 4:Article ID 3 View ArticleGoogle Scholar
 Topaz CM, Ziegelmeier L, Halverson T (2015) Topological data analysis of biological aggregation models. PLoS ONE 10:Article ID e0126383 View ArticleGoogle Scholar
 Maletic S, Zhao Y, Rajkovic M (2015) Persistent topological features of dynamical systems. arXiv:1510.06933
 Zhu X (2013) Persistent homology: an introduction and a new text representation for natural language processing. In: Proceedings of the twentythird international joint conference on artificial intelligence, IJCAI ’13, Beijing, China AAAI Press, Menlo Park, pp 19531959 Google Scholar
 Wang B, Wei GW (2016) Objectoriented persistent homology. J Comput Phys 305:276299 MathSciNetMATHView ArticleGoogle Scholar
 Stolz BJ, Harrington HA, Porter MA (2017) Persistent homology of timedependent functional networks constructed from coupled time series. Chaos 27:Article ID 047410 MathSciNetView ArticleGoogle Scholar
 Bendich P, Marron JS, Miller E, Pieloch A, Skwerer S (2016) Persistent homology analysis of brain artery trees. Ann Appl Stat 10:198218 MathSciNetView ArticleGoogle Scholar
 Adler R (2014) TOPOS, and why you should care about it. IMS Bull 43:45 Google Scholar
 Wagner H, Chen C, Vuçini E (2012) Efficient computation of persistent homology for cubical data. In: Peikert R, Hauser H, Carr H, Fuchs R (eds) Topological methods in data analysis and visualization II. Mathematics and visualization. Springer, Berlin, pp 91106 View ArticleGoogle Scholar
 Singh G, Mémoli F, Carlsson G (2007) Topological methods for the analysis of high dimensional data sets and 3D object recognition. In: Eurographics symposium on pointbased graphics, pp 91100 Google Scholar
 Ghrist R (2014) Elementary applied topology, 1.0 edn Google Scholar
 Curry J (2013) Sheaves, cosheaves and applications. arXiv:1303.3255
 Carlsson G (2009) Topology and data. Bull Am Math Soc 46:255308 MathSciNetMATHView ArticleGoogle Scholar
 Edelsbrunner H, Letscher D, Zomorodian A (2002) Topological persistence and simplification. Discrete Comput Geom 28:511533 MathSciNetMATHView ArticleGoogle Scholar
 Zomorodian A, Carlsson G (2005) Computing persistent homology. Discrete Comput Geom 33:249274 MathSciNetMATHView ArticleGoogle Scholar
 Bauer U, Kerber M, Reininghaus J, Wagner H (2014) PHAT: persistent homology algorithms toolbox. In: Hong H, Yap C (eds) Mathematical software  ICMS 2014. Lecture notes in computer science, vol 8592. Springer, Berlin, pp 137143. Software available at https://code.google.com/p/phat/ Google Scholar
 Bauer U, Kerber M, Reininghaus J (2014) DIPHA (a distributed persistent homology algorithm). https://code.google.com/p/dipha/
 Morozov D Dionysus. http://www.mrzv.org/software/dionysus/
 Nanda V Perseus, the persistent homology software. http://www.sas.upenn.edu/~vnanda/perseus
 Tausz A, VejdemoJohansson M, Adams H (2014) JavaPlex: a research software package for persistent (co)homology. In: Hong H, Yap C (eds) Mathematical software  ICMS 2014. Lecture notes in computer science, vol 8592, pp 129136. Software available at http://appliedtopology.github.io/javaplex/ Google Scholar
 Maria C, Boissonnat JD, Glisse M, Yvinec M (2014) The Gudhi library: simplicial complexes and persistent homology. In: Hong H, Yap C (eds) Mathematical software  ICMS 2014. Lecture notes in computer science, vol 8592. Springer, Berlin, pp 167174. Software available at https://project.inria.fr/gudhi/software/ Google Scholar
 Bauer U (2016) Ripser. https://github.com/Ripser/ripser
 Fasy BT, Kim J, Lecci F, Maria C (2014) Introduction to the R package TDA. arXiv:1411.1830
 Bubenik P, Dłotko P (2017) A persistence landscapes toolbox for topological statistics. J Symb Comput 78:91114 MathSciNetMATHGoogle Scholar
 Adams H, Tausz A JavaPlex tutorial. https://github.com/appliedtopology/javaplex
 de Silva V, Morozov D, VejdemoJohansson M (2011) Dualities in persistent (co)homology. Inverse Probl 27:Article ID 124003 MathSciNetMATHView ArticleGoogle Scholar
 Nanda V (2012) Discrete Morse theory for filtrations. PhD thesis, Rutgers, The State University of New Jersey Google Scholar
 Bauer U, Kerber M, Reininghaus J (2014) Distributed computation of persistent homology. In: 2014 proceedings of the sixteenth workshop on algorithm engineering and experiments (ALENEX). SIAM, Philadelphia, pp 3138 View ArticleGoogle Scholar
 Maria C (2014) Algorithms and data structures in computational topology. PhD thesis, Université de NiceSophia Antipolis. http://wwwsop.inria.fr/members/Clement.Maria/docs/ClementMaria_PhDdissertation.pdf
 Kaczynski T, Mischaikow K, Mrozek M (2004) Computational homology. Applied mathematical sciences, vol 157. Springer, New York MATHGoogle Scholar
 Cohen MM (1970) A course in simple homotopy theory. Graduate texts in mathematics. Springer, New York Google Scholar
 Hatcher A (2002) Algebraic topology. Cambridge University Press, Cambridge MATHGoogle Scholar
 Björner A (1995) Topological methods. In: Graham R, Grötschel M, Lovász L (eds) Handbook of combinatorics. Elsevier, Amsterdam, pp 18191872 Google Scholar
 Edelsbrunner H, Harer J (2010) Computational topology: an introduction. Applied mathematics. Am. Math. Soc., Providence MATHGoogle Scholar
 Eilenberg S, Steenrod NE (1952) Foundations of algebraic topology. Princeton mathematical series. Princeton University Press, Princeton MATHView ArticleGoogle Scholar
 Oudot SY (2015) Persistence theory: from quiver representations to data analysis. AMS mathematical surveys and monographs, vol 209. Am. Math. Soc., Providence MATHView ArticleGoogle Scholar
 Zomorodian A (2009) Topology for computing. Cambridge monographs on applied and computational mathematics. Cambridge University Press, Cambridge MATHGoogle Scholar
 Weinberger S (2011) What is…persistent homology? Not Am Math Soc 58:3639 MathSciNetMATHGoogle Scholar
 Ghrist R (2008) Barcodes: the persistent topology of data. Bull Am Math Soc 45:6175 MathSciNetMATHView ArticleGoogle Scholar
 Edelsbrunner H, Harer J (2008) Persistent homology — a survey. In: Goodman JE, Pach J, Pollack R (eds) Surveys on discrete and computational geometry: twenty years later. Contemporary mathematics, vol 453. Am. Math. Soc., Providence, pp 257282 View ArticleGoogle Scholar
 Edelsbrunner H, Morozov D (2012) Persistent homology: theory and practice. In: Proceedings of the European congress of mathematics, pp 3150 Google Scholar
 Patania A, Vaccarino F, Petri G (2017) Topological analysis of data. EPJ Data Sci 6(1):7 View ArticleGoogle Scholar
 Petri G, Scolamiero M, Donato I, Vaccarino F (2013) Topological strata of weighted complex networks. PLoS ONE 8:Article ID e66506 View ArticleGoogle Scholar
 Jonsson J (2007) Simplicial complexes of graphs. Lecture notes in mathematics. Springer, Berlin MATHGoogle Scholar
 Horak D, Maletić S, Rajković M (2009) Persistent homology of complex networks. J Stat Mech Theory Exp 2009:Article ID P03034 MathSciNetGoogle Scholar
 Bendich P, Edelsbrunner H, Kerber M (2010) Computing robustness and persistence for images. IEEE Trans Vis Comput Graph 16:12511260 View ArticleGoogle Scholar
 Zhou W, Yan H (2014) Alpha shape and Delaunay triangulation in studies of proteinrelated interactions. Brief Bioinform 15:5464 View ArticleGoogle Scholar
 Xia K, Wei GW (2016) A review of geometric, topological and graph theory apparatuses for the modeling and analysis of biomolecular data. arXiv:1612.01735
 Zomorodian A (2010) Technical section: fast construction of the Vietoris–Rips complex. Comput Graph 34:263271 View ArticleGoogle Scholar
 Vietoris L (1927) Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen. Math Ann 97:454472 MathSciNetMATHView ArticleGoogle Scholar
 Kerber M, Sharathkumar R (2013) Approximate Čech complex in low and high dimensions. In: Cai L, Cheng SW, Lam TW (eds) 24th international symposium on algorithms and computation (ISAAC 2013). Lecture notes in computer science, vol 8283, pp 666676 Google Scholar
 Boissonnat JD, Devillers O, Hornus S (2009) Incremental construction of the Delaunay triangulation and the Delaunay graph in medium dimension. In: Proceedings of the twentyfifth annual symposium on computational geometry, SoCG ’09. ACM, New York, pp 208216 Google Scholar
 Goodman JE, O’Rourke J (eds) (2004) Handbook of discrete and computational geometry, 2nd edn. CRC Press, Boca Raton MATHGoogle Scholar
 Edelsbrunner H, Kirkpatrick D, Seidel R (1983) On the shape of a set of points in the plane. IEEE Trans Inf Theory 29:551559 MathSciNetMATHView ArticleGoogle Scholar
 Edelsbrunner H, Mücke EP (1994) Threedimensional alpha shapes. ACM Trans Graph 13:4372 MATHView ArticleGoogle Scholar
 Edelsbrunner H (1995) The union of balls and its dual shape. Discrete Comput Geom 13:415440 MathSciNetMATHView ArticleGoogle Scholar
 Kurlin V (2015) A onedimensional homologically persistent skeleton of an unstructured point cloud in any metric space. Comput Graph Forum 34:253262 View ArticleGoogle Scholar
 Kurlin V (2015) http://kurlin.org/projects/persistentskeletons.cpp
 de Silva V (2008) A weak characterisation of the Delaunay triangulation. Geom Dedic 135:3964 MathSciNetMATHView ArticleGoogle Scholar
 de Silva V, Carlsson G (2004) Topological estimation using witness complexes. In: Proceedings of the first Eurographics conference on pointbased graphics, pp 157166 Google Scholar
 Guibas LJ, Oudot SY (2008) Reconstruction using witness complexes. Discrete Comput Geom 40:325356 MathSciNetMATHView ArticleGoogle Scholar
 Attali D, Edelsbrunner H, Mileyko Y (2007) Weak witnesses for Delaunay triangulations of submanifolds. In: Proceedings of the 2007 ACM symposium on solid and physical modeling, SPM ’07. ACM, New York, pp 143150 View ArticleGoogle Scholar
 Boissonnat JD, Guibas LJ, Oudot SY (2009) Manifold reconstruction in arbitrary dimensions using witness complexes. Discrete Comput Geom 42:3770 MathSciNetMATHView ArticleGoogle Scholar
 Dey TK, Fan F, Wang Y (2013) Graph induced complex on point data. In: Proceedings of the twentyninth annual symposium on computational geometry, SoCG ’13. ACM, New York, pp 107116 Google Scholar
 Jyamiti research group (2013) GIComplex. http://web.cse.ohiostate.edu/~tamaldey/GIC/GICsoftware/
 Sheehy DR (2013) Linearsize approximations to the Vietoris–Rips filtration. Discrete Comput Geom 49:778796 MathSciNetMATHView ArticleGoogle Scholar
 Dey TK, Shi D, Wang Y (2016) SimBa: an efficient tool for approximating Ripsfiltration persistence via simplicial batchcollapse. In: 24th annual European symposium on algorithms (ESA 2016). LIPIcs  Leibniz international proceedings in informatics, vol 57. Schloss Dagstuhl  LeibnizZentrum für Informatik, Saarbrücken, pp 35:135:16 Google Scholar
 Robin F (1998) Morse theory for cell complexes. Adv Math 134:90145 MathSciNetMATHView ArticleGoogle Scholar
 Mischaikow K, Nanda V (2013) Morse theory for filtrations and efficient computation of persistent homology. Discrete Comput Geom 50:330353 MathSciNetMATHView ArticleGoogle Scholar
 Joswig M, Pfetsch ME (2006) Computing optimal Morse matchings. SIAM J Discrete Math 20:1125 MathSciNetMATHView ArticleGoogle Scholar
 Barmak JA, Minian EG (2012) Strong homotopy types, nerves and collapses. Discrete Comput Geom 47:301328 MathSciNetMATHView ArticleGoogle Scholar
 Wilkerson AC, Moore TJ, Swami A, Krim H (2013) Simplifying the homology of networks via strong collapses. In: 2013 IEEE international conference on acoustics, speech and signal processing (ICASSP), pp 52585262 View ArticleGoogle Scholar
 Wilkerson AC, Chintakunta H, Krim H, Moore TJ, Swami A (2013) A distributed collapse of a network’s dimensionality. In: 2013 IEEE global conference on signal and information processing, pp 595598 View ArticleGoogle Scholar
 Wilkerson AC, Chintakunta H, Krim H (2014) Computing persistent features in big data: a distributed dimension reduction approach. In: 2014 IEEE international conference on acoustics, speech and signal processing (ICASSP), pp 1115 View ArticleGoogle Scholar
 Zomorodian A (2010) The tidy set: a minimal simplicial set for computing homology of clique complexes. In: Proceedings of the twentysixth annual symposium on computational geometry, SoCG ’10. ACM, New York, pp 257266 Google Scholar
 Zomorodian A (2012) Topological data analysis. In: Zomorodian A (ed) Advances in applied and computational topology. Proceedings of symposia in applied mathematics, vol 70. Am. Math. Soc., Providence, pp 139 View ArticleGoogle Scholar
 Morozov D (2005) Persistence algorithm takes cubic time in worst case. BioGeometry News (Feb 2005), Department of Computer Science, Duke University Google Scholar
 Milosavljević N, Morozov D, Skraba P (2011) Zigzag persistent homology in matrix multiplication time. In: Proceedings of the twentyseventh annual symposium on computational geometry, SoCG ’11. ACM, New York, pp 216225 Google Scholar
 Coppersmith D, Winograd S (1990) Matrix multiplication via arithmetic progressions. J Symb Comput 9:251280 MathSciNetMATHGoogle Scholar
 Chen C, Kerber M (2011) Persistent homology computation with a twist. In: Proceedings of the 27th European workshop on computational geometry, pp 197200 Google Scholar
 de Silva V, Morozov D, VejdemoJohansson M (2011) Persistent cohomology and circular coordinates. Discrete Comput Geom 45:737759 MathSciNetMATHView ArticleGoogle Scholar
 Bauer U, Kerber M, Reininghaus J (2014) Clear and compress: computing persistent homology in chunks. In: Bremer PT, Hotz I, Pascucci V, Peikert R (eds) Topological methods in data analysis and visualization III. Mathematics and visualization. Springer, Cham, pp 103117 View ArticleGoogle Scholar
 Boissonnat JD, Maria C (2014) Computing persistent homology with various coefficient fields in a single pass. In: Schulz AS, Wagner D (eds) Algorithms  ESA 2014. Lecture notes in computer science, vol 8737. Springer, Berlin, pp 185196 Google Scholar
 Bubenik P, Kim PT (2007) A statistical approach to persistent homology. Homol Homotopy Appl 9:337362 MathSciNetMATHView ArticleGoogle Scholar
 Adler R, Bobrowski O, Weinberger S (2014) Crackle: the homology of noise. Discrete Comput Geom 52:680704 MathSciNetMATHView ArticleGoogle Scholar
 Young JG, Petri G, Vaccarino F, Patania A (2017) Construction of and efficient sampling from the simplicial configuration model. arXiv:1705.10298
 Adler RJ, Bobrowski O, Borman MS, Subag E, Weinberger S (2010) Persistent homology for random fields and complexes. In: Borrowing strength: theory powering applications  a festschrift for Lawrence D. Brown. IMS collections, vol 6. Institute of Mathematical Statistics, Beachwood, pp 124143 View ArticleGoogle Scholar
 Kahle M (2014) Topology of random simplicial complexes: a survey. In: Applied algebraic topology: new directions and applications. Contemporary mathematics, vol 620. Am. Math. Soc., Providence, pp 221241 Google Scholar
 Mileyko Y, Mukherjee S, Harer J (2011) Probability measures on the space of persistence diagrams. Inverse Probl 27:Article ID 124007 MathSciNetMATHView ArticleGoogle Scholar
 Turner K, Mileyko Y, Mukherjee S, Harer J (2014) Fréchet means for distributions of persistence diagrams. Discrete Comput Geom 52:4470 MathSciNetMATHView ArticleGoogle Scholar
 Munch E, Turner K, Bendich P, Mukherjee S, Mattingly J, Harer J (2015) Probabilistic Fréchet means for time varying persistence diagrams. Electron J Stat 9:11731204 MathSciNetMATHView ArticleGoogle Scholar
 Kerber M, Morozov D, Nigmetov A (2016). https://bitbucket.org/grey_narn/hera
 Kerber M, Morozov D, Nigmetov A (2016) Geometry helps to compare persistence diagrams. arXiv:1606.03357
 Fasy BT, Kim J, Lecci F, Maria C, Rouvreau V TDA: statistical tools for topological data analysis. https://cran.rproject.org/web/packages/TDA/index.html
 Fasy B, Lecci F, Rinaldo A, Wasserman L, Balakrishnan S, Singh A (2014) Confidence sets for persistence diagrams. Ann Stat 42:23012339 MathSciNetMATHView ArticleGoogle Scholar
 Chazal F, Fasy BT, Lecci F, Michel B, Rinaldo A, Wasserman L (2014) Robust topological inference: distance to a measure and kernel distance. arXiv:1412.7197
 Bubenik P (2015) Statistical topological data analysis using persistence landscapes. J Mach Learn Res 16:77102 MathSciNetMATHGoogle Scholar
 Adcock A, Carlsson E, Carlsson G (2013) The ring of algebraic functions on persistence bar codes. arXiv:1304.0530
 Chepushtanova S, Emerson T, Hanson E, Kirby M, Motta F, Neville R, Peterson C, Shipman P, Ziegelmeier L (2015) Persistence images: an alternative persistent homology representation. arXiv:1507.06217
 Kwitt R, Huber S, Niethammer M, Lin W, Bauer U (2015) Statistical topological data analysis  a kernel perspective. In: Cortes C, Lawrence ND, Lee DD, Sugiyama M, Garnett R (eds) Advances in neural information processing systems, vol 28. Curran Associates, Red Hook, pp 30523060 Google Scholar
 Reininghaus J, Huber S, Bauer U, Kwitt R (2015) A stable multiscale kernel for topological machine learning. In: 2015 IEEE conference on computer vision and pattern recognition (CVPR), pp 47414748 View ArticleGoogle Scholar
 Bobrowski O, Mukherjee S, Taylor J (2017) Topological consistency via kernel estimation. Bernoulli 23:288328 MathSciNetMATHView ArticleGoogle Scholar
 Zhu X, Vartanian A, Bansal M, Nguyen D, Brandl L (2016) Stochastic multiresolution persistent homology kernel. In: Proceedings of the twentyfifth international joint conference on artificial intelligence, IJCAI’16. AAAI Press, Palo Alto, pp 24492455 Google Scholar
 Dłotko P Persistence landscape toolbox. https://www.math.upenn.edu/~dlotko/persistenceLandscape.html
 CohenSteiner D, Edelsbrunner H, Harer J (2007) Stability of persistence diagrams. Discrete Comput Geom 37:103120 MathSciNetMATHView ArticleGoogle Scholar
 Chazal F, CohenSteiner D, Glisse M, Guibas LJ, Oudot SY (2009) Proximity of persistence modules and their diagrams. In: Proceedings of the twentyfifth annual symposium on computational geometry, SoCG ’09. ACM, New York, pp 237246 Google Scholar
 Bubenik P, Scott JA (2014) Categorification of persistent homology. Discrete Comput Geom 51:600627 MathSciNetMATHView ArticleGoogle Scholar
 Bubenik P, de Silva V, Scott J (2014) Metrics for generalized persistence modules. Found Comput Math 15:15011531 MathSciNetMATHView ArticleGoogle Scholar
 Carlsson G, de Silva V, Morozov D (2009) Zigzag persistent homology and realvalued functions. In: Proceedings of the twentyfifth annual symposium on computational geometry, SoCG ’09. ACM, New York, pp 247256 Google Scholar
 Dey TK, Fan F, Wang Y (2014) Computing topological persistence for simplicial maps. In: Proceedings of the thirtieth annual symposium on computational geometry, SoCG ’14. ACM, New York, pp 345354 View ArticleGoogle Scholar
 Carlsson G, Zomorodian A (2009) The theory of multidimensional persistence. Discrete Comput Geom 42:7193 MathSciNetMATHView ArticleGoogle Scholar
 Lesnick M, Wright M (2016) RIVET: the rank invariant visualization and exploration tool. http://rivet.online/
 Lesnick M, Wright M (2015) Interactive visualization of 2D persistence modules. arXiv:1512.00180
 Edelsbrunner H, Morozov D, Pascucci V (2006) Persistencesensitive simplification functions on 2manifolds. In: Proceedings of the twentysecond annual symposium on computational geometry, SoCG ’06. ACM, New York, pp 127134 View ArticleGoogle Scholar
 Perry P, de Silva V (2000– 2006) Plex. http://mii.stanford.edu/research/comptop/programs/
 Binchi J, Merelli E, Rucco M, Petri G, Vaccarino F (2014) jHoles: a tool for understanding biological complex networks via clique weight rank persistent homology. Electron Notes Theor Comput Sci 306:518 MathSciNetMATHView ArticleGoogle Scholar
 Jyamiti research group (2014) SimpPers. http://web.cse.ohiostate.edu/~tamaldey/SimpPers/SimpPerssoftware/
 Stanford University Computer Graphics Laboratory, The Stanford 3D scanning repository. https://graphics.stanford.edu/data/3Dscanrep
 Kahle M (2011) Random geometric complexes. Discrete Comput Geom 45:553573 MathSciNetMATHView ArticleGoogle Scholar
 Penrose M (2003) Random geometric graphs. Oxford University Press, Oxford MATHView ArticleGoogle Scholar
 Vicsek T, Czirók A, BenJacob E, Cohen I, Shochet O (1995) Novel type of phase transition in a system of selfdriven particles. Phys Rev Lett 75:12261229 MathSciNetView ArticleGoogle Scholar
 Sporns O (2006) Smallworld connectivity, motif composition, and complexity of fractal neuronal connections. Biosystems 85:5564 View ArticleGoogle Scholar
 Los Alamos National Laboratory, HIV database. http://www.hiv.lanl.gov/content/index
 Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small world’ networks. Nature 393(6684):440442 View ArticleGoogle Scholar
 White JG, Southgate E, Thomson JN, Brenner S (1986) The structure of the nervous system of the nematode Caenorhabditis elegans. Philos Trans R Soc Lond B, Biol Sci 314(1165):1340 View ArticleGoogle Scholar
 Davis TA, Hu Y (2011) The University of Florida sparse matrix collection. ACM Trans Math Softw 38:125. http://www.cise.ufl.edu/research/sparse/matrices MathSciNetMATHGoogle Scholar
 Volvis repository. http://volvis.org
 Waugh AS, Pei L, Fowler JH, Mucha PJ, Porter MA (2012) Party polarization in congress: a network science approach. arXiv:0907.3509. Data available at http://figshare.com/articles/Roll_Call_Votes_United_States_House_and_Senate/1590036
 Poole KT (2016) Voteview. http://voteview.com
 Newman MEJ (2006) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74:Article ID 036104 MathSciNetView ArticleGoogle Scholar
 Grayson DR, Stillman ME Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
 T. S. Developers, Sage mathematics software. http://www.sagemath.org
 The CGAL Project (2015) CGAL user and reference manual, 4.7 edn. CGAL Editorial Board Google Scholar