Randomly Weighted d-complexes: Minimal Spanning Acycles and Persistence Diagrams
RRANDOMLY WEIGHTED d − COMPLEXES: MINIMAL SPANNINGACYCLES AND PERSISTENCE DIAGRAMS
PRIMOZ SKRABA, GUGAN THOPPE, AND D. YOGESHWARAN
Abstract.
This paper has three key parts. The first part concerns spanning acycles onsimplicial complexes which are higher dimensional analogues of spanning trees. We give acompendium of their basic properties lending further credence to this analogy and acting asa stepping stone for study of minimal spanning acycles on infinite complexes. We also givea simplicial version of the Kruskal’s and Jarn`ık-Prim-Dijkstra’s algorithms for generatingminimal spanning acycles. In the second part, using simplicial Kruskal’s algorithm, weprove that the set of face-weights of a minimal spanning acycle is same as the set of ‘deathtimes’ in the associated persistence diagram provided the weights on faces induce a filtrationof the complex. Further, we prove a stability result for the face-weights in a minimalspanning acycle, and hence for death and birth times under the L p matching distance forany p ∈ { , . . . , ∞} . In the third part, we consider randomly weighted d − complexes on n vertices. In the generic case, all faces up to dimension d − d − faceweights are perturbation of some i.i.d. distribution. Using the above stability result, weshow that if the maximum perturbation converges in probability to 0 sufficiently fast then,the suitably scaled extremal weights in the minimal d − spanning acycle and extremal deathtimes in the persistence diagram of the ( d − − th homology converge to the Poisson pointprocess on R with intensity e − x d x . We lastly show that the ζ (3) limit of Frieze [29] ontotal edge-weight of a random minimal spanning tree and asymptotics of lifetime sums ofpersistence diagrams of randomly weighted complexes by Hiraoka and Shirai [33] also holdfor suitable noisy versions. Introduction
This paper investigates extremal properties of persistence diagrams and minimal spanningacycles of a ‘mean-field’ model of weighted random complexes. As we shall see, intimatelyrelated to the above two terms is what we refer to as the “nearest face” distance. In themuch more popular scenario of weighted graphs (which are a special case of weighted com-plexes), persistence diagrams correspond to the vanishing thresholds of number of connectedcomponents, minimal spanning acycles correspond to minimal spanning trees and nearestface distance is nothing but the nearest neighbour distance. That connectivity and nearestneighbour distance are intertwined was conspicuous even in the earliest work on connectivity
Mathematics Subject Classification.
Primary : 60C05, 05E45 Secondary : 60G70, 60B99, 05C80 .
Key words and phrases.
Random complexes, Persistent diagrams, Minimal spanning acycles, Point pro-cesses, Weak convergence, Stability.Research supported by EU project Proasense FP7-ICT-2013-10(612329).Research supported by URSAT, ERC Grant 320422.Research supported by DST-INSPIRE faculty award. a r X i v : . [ m a t h . P R ] J a n hresholds for random graphs by Erd˝os and R´enyi [27]. Coincidentally, three years earlierthe relation between minimal spanning trees and vanishing of connected components in agraph was used in Kruskal’s algorithm to construct a minimal spanning tree [44] and therelation was also used later in the seminal work of Frieze [29]. On the other hand, connec-tions between largest nearest neighbour distances and longest edges of a minimal spanningtree on randomly weighted graphs were at the heart of studies in [32, 61, 3, 54, 35]. Morecomplete accounts of the theory of random graphs and in particular, the relation betweenthe above three quantities can be found in [7, 64, 37, 55, 30]. Random complexes:
In the recent years, motivated by applications in topological dataanalysis, the study of random graphs has been extended to the study of random complexes.Though we now narrow our focus to a specific random complex, we alert the reader ofexistence of a richer theory of random complexes and topological data analysis [10, 39, 6, 40,20]. Before describing random complexes, we define (simplicial) complexes, which are higherdimensional versions of a graph.
Definition 1.1. An (abstract) simplical complex K on a finite ground set V is a collectionof subsets of V such that if σ ∈ K and ∅ (cid:54) = σ ⊂ σ , then σ ∈ K as well. The elementsof K are called simplices or faces and the dimension of a simplex σ is | σ | − , with | · | heredenoting the cardinality. A d − face of K is a face of K with dimension d. Given a complex K , we denote the d -faces of K by F d ( K ) and its d -skeleton by K d (i.e.,the sub-complex of K consisting of all faces of dimension at most d ) for any d ≥
0. We use σ, τ to denote faces and the dimension of the face shall not be explicitly mentioned unlessrequired. If a (simplicial) complex consisted only of 0-faces and 1-faces then it is a graph orin other words, the 1-skeleton of a complex is a graph. Associated to each simplicial complexis a collection of non-negative integers denoted β ( K ) , β ( K ) , . . . , called the Betti numbers (see Section 2.1 for detailed definitions) which are a measure of connectivity of the simpli-cial complex. Informally, the d − th Betti number counts the number of ( d + 1)-dimensionalholes in the complex or equivalently the number of independent non-trivial cycles formed by d -faces. Two points to note at the moment are: (i) β ( K ) is one less than the number ofconnected components in the graph formed by 0-faces and 1-faces and (ii) if the dimensionof K (maximum of dimension of faces) is d , then β j ( K ) = 0 for all j ≥ d + 1.The probabilistic model of interest to us is the one introduced by Linial and Meshulam[45] and then extended by Meshulam and Wallach [51]. This model, called as the random d -complex and denoted by Y n,d ( p ), consists of all faces on n vertices (i.e., ground set V = [ n ] := { , . . . , n } ) with dimension at most ( d −
1) and each d -face is included with probability p independently. Y n, ( p ) is the classical Erd˝os-R`enyi graph on n vertices with edge-connectionprobability p . Like Erd˝os-R`enyi graph is a mean-field model of pairwise interactions, therandom d -complex can be considered as a model of higher-order interactions. Though a We work with reduced Betti numbers defined using field coefficients throughout the article. imple model, it has spanned a rich literature in the recent years [45, 51, 14, 18, 19, 47]. Thefocus of many studies on random d -complexes have been the two non-trivial Betti numbersof the complex: β d − ( · ) and β d ( · ). The starting part of our study is the following fine phasetransition result for β d − ( Y n,d ( p )). Lemma 1.2. [62] , [41, Theorem 1.10] Fix d ≥ . Consider Y n,d ( p n ) with p n = d log n + c − log( d !) n for some fixed c ∈ R . Then, as n → ∞ , β d − ( Y n,d ( p n )) ⇒ Poi ( e − c ) , where Poi ( λ ) stands forthe Poisson random variable with mean λ and ⇒ denotes convergence in distribution. Additionally, one has that β d − ( Y n,d ( p n )) → np n − d log n → ∞ and β d − ( Y n,d ( p n )) →∞ if np n − d log n → −∞ . These were proven by Erd˝os and R´enyi [27] in 1959 for d = 1,much later by Linial and Meshulam [45] in 2006 for d = 2 and shortly thereafter in 2009 for d ≥ n vertices having all faces up to dimension d. A standard couplingof random graphs that can be extended to random complexes, is to endow all the d -faceswith i.i.d. uniform [0 ,
1] weights and all other faces having dimension d − U n,d and call it the uniformlyweighted d − complex. Let U n,d ( t ) denote the subcomplex of U n,d with only those faces withweight less than or equal to t. Then, clearly, for t ∈ [0 , , U n,d ( t ) has the same distribution as Y n,d ( t ). Topological considerations imply that β d − ( U n,d ( t )) is a non-increasing, step functionin t, i.e., a.s. a non-increasing jump process. Let D = { D i } (called as death times of the( d − D characterisesthe persistence diagram of U n,d (see Subsection 2.1.2). Defining the scaled point process P Dn,d := { nD i − d log n + log( d !) } , we can rephrase Lemma 1.2 as P Dn,d ( c, ∞ ) ⇒ Poi( e − c ) , where, for R ⊆ R , P Dn,d ( R ) := |{ i : nD i − d log n + log( d !) ∈ R }| . See Figure 1(a) for asimulation of death times of the ( d − U ,d ( t ) for d = 1 , . . . ,
4. Oneof our main results is that the above distributional convergence can be extended to weakconvergence of the corresponding point processes in vague topology.
Theorem 1.3. As n → ∞ , P Dn,d converges in distribution to P poi , where P poi is the Poissonpoint process with intensity e − x d x on R . Theorem 1.3 gives convergence of the extremal points of the persistence diagram of auniformly weighted d -complex. We say extremal points because only the points near themaximum contribute to the scaled point process and the scaling maps many points in thepersistence diagram to −∞ . A weak convergence result as above along with the continuousmapping theorem yields asymptotic distribution of various statistics of P Dn,d . For example,one can infer the asymptotic joint distribution of the largest m points in the persistence iagram whereas Lemma 1.2 will give only result about the one-dimensional marginal dis-tributions. Further, our results might be useful in deriving asymptotics for other summarystatistics of persistence diagrams such as persistence landscapes [8], homological scaffolds [56]or accumulative persistence function [5]. An important paradigm in topological data analysisis that extremal points of a persistence diagram encode meaningful topological informationabout the underlying structure. Viewed in this light, our result completely characterizes theextremal behaviour of the persistence diagram of the uniformly weighted d -complex.We would like to emphasize that to the best of our knowledge this is the first such completecharacterization of extremal persistence diagram of a model of random complex in higherdimensions and to nobody’s surprise, it is for the simplest model of a random complex. Forthe case d = 1, which corresponds to β , Theorem 1.3 for a model of random geometricgraphs can be deduced from the results of [54, 35]. As with many other scaling limit results,one would expect our result to hold for many other random complex models such as randomclique complexes [39], random geometric complexes [6], etc. But this is beyond the scope ofthe current paper. As for our proof, it mainly involves an application of the factorial momentmethod to show convergence of the point process of largest nearest face distances and usingthis to approximate the extremal persistence diagram of the uniformly weighted d -complex.Though such an approach should not surprise someone familiar with the proof technionquesof [45, 51, 41], the theorem significantly extends the connection between nearest neighbourdistances and connectivity of random graphs to random complexes. We discuss importantextensions of these results a little later. Having discussed so far persistence diagrams andnearest face distances, we now discuss the more novel component of the paper. Minimal Spanning Acycle:
A spanning tree of a graph on a vertex set V is easily de-scribed in topological terms as a set of edges S such that β ( V ∪ S ) = β ( V ∪ S ) = 0 , i.e., V ∪ S is connected and has no cycles. As an higher-dimensional generalization, the followingdefinition due to Kalai [42] is natural. Definition 1.4 (Spanning acycle) . Consider a complex K of dimension at least d , d ≥ .A subset S of d -faces is called a spanning acycle if β d − ( K d − ∪ S ) = 0 (spanning) and β d ( K d − ∪ S ) = 0 (acyclicity). A subset S of d -faces is called a maximal acycle if it is anacycle and maximal w.r.t. inclusion of d -faces. Though the definition of a spanning acycle appears natural and seems to be merely replac-ing the appropriate indices in the definition of a spanning tree, what is not obvious is thatthis is a good higher-dimensional generalization of a spanning tree and enjoys the many niceproperties spanning trees do. An algebraic description of a spanning tree is that it is theset of columns that form a basis for the column space of the incidence matrix or boundarymatrix, i.e., the matrix ∂ whose rows are indexed by vertices and columns by edges suchthat the i, j − th entry is 1 if the vertex i belongs to the edge j and 0 otherwise . Such a For simplicity, we are assuming our underlying field F = Z here, i.e., all vector spaces involved are Z -vector spaces. D i m e n s i o n Linial-Meshulam Death Times for 30 points 0 0.2 0.4 0.6 0.8 100.20.40.60.81 Dimension=0Dimension=1Dimension=2Dimension=3Dimension=4Erd˝os-R´enyi Persistence Diagram for 50 points
Figure 1. (Left) The death times corresponding to the uniformly weightedrandom d -complex built on 30 points in different dimensions. (Right) Thepersistence diagram for an Erd˝os-Renyi clique complex for 50 points.description (described in an earlier version of the article - [59, Appendix B]) also holds forspanning acycles and underpins many of our proof ideas even though it is not mentionedexplicitly. Further, once we assign weights to faces, that is we consider a weighted complex K , one can naturally define a minimal spanning acycle. Since we deal with only finite com-plexes, existence of a minimal spanning acycle is guaranteed once a spanning acycle exists.Though Kalai’s definition of a spanning acycle and enumeration of number of spanning cycles(a generalization of Cayley’s formula for spanning trees) is more than three decades old, itis receiving increased attention in the last few years [4, 24, 36, 43, 33, 34, 48, 46, 50].Our second significant contribution is to add to this burgeoning literature on spanningacycles and the nascent literature on random spanning acycles. Firstly, we list a numberof basic properties of (minimal) spanning acycles analogous to those known for (minimal)spanning trees. We expect this enumeration of properties (see Section 3.1) - existence,uniqueness, cut property, cycle property, exchange property - to be useful in further researchon (minimal) spanning acycles. We also present the simplicial Kruskal’s and Jarn´ık-Prim-Dijkstra’s algorithm (see Section 3.2) to generate minimal spanning acycles. We believe thatmany of the properties we have enumerated - especially the Prim’s algorithm and Lemma3.25 - are important steps towards study of minimal spanning acycles on infinite complexesand in particular Euclidean minimal spanning acycles. For analogous results about minimalspanning trees, see [49, Chapter 11], and for the Euclidean case, refer to [1, 2].Though many of the above properties can be proved independently using matroid theory[65, 53]) and such a viewpoint has been beneficial to the study of spanning acycles (see[4, 34] and in particular [43, Section 4]), we provide self-contained proofs here using tools rom combinatorial topology. It is possible that some of these results have been implicitlyused in the literature but we have not been able to find explicit mention of these propertieselsewhere. We now highlight an inclusion-exclusion identity which relies heavily on Mayer-Vietoris exact sequence from algebraic topology. If γ d ( K ) were to denote the cardinality ofa spanning acycle and K , K are two finite complexes, then γ d ( K ) + γ d ( K ) = γ d ( K ∪ K ) + γ d ( K ∩ K ) + β (ker ν d − ) , where the homomorphism ν d − is precisely defined in Theorem 3.7, ker stands for kerneland β ( · ) denotes the rank. The matroidal property of spanning acycles shall only yield aninequality. Higher-dimensional connectivity can be studied via hypergraphs as well and webriefly discuss this relation with hypergraph connectivity in Section 3.2, in particular show-ing that if the ( d − K are hypergraph connected, so is the spanningacycle. We also wish to point out that properties of spanning acycles are preserved undersimplicial isomorphisms but not necessarily under homotopy equivalence. Persistent Diagrams and Minimal Spanning Acycles.
Having introduced minimalspanning acycles and discussed their basic properties, we now preview their connection topersistence diagrams. Let K be a weighted complex with real valued injective weight func-tion w such that K ( t ) := w − ( ∞ , t ] is a simplicial complex for all t ∈ R . Let d ≥ β d ( K ) = 0. Let β d ( t ) = β d ( K ( t )) . We remark that β d ( t ) is a jumpfunction. The times of positive jumps are birth times B = { B i } of the persistence diagramand the times of negative jumps are death times D = { D i } of the persistence diagram asdescribed earlier. Persistent diagram (or homology) is more than merely keeping track ofdeath and birth times and in particular considers the (complex) pairings of birth times withtheir “corresponding” death times (For example, see Figure 1(b) for a persistence diagramof Erd˝os-R`enyi clique complexes). In this article we shall focus only on their two projections- birth and death times.An easy consequence of the above informal description that can be justified via Fubini’stheorem is the following identity for lifetime sum of persistence diagrams :(1.1) L d := (cid:88) i ( D i − B i ) = (cid:90) ∞ β d ( t )d t. Assume that β d − ( K ) = β d − ( K ) = 0. This assures us that M d and M d − , respectively the d − th and ( d − − th minimal spanning acycle, exist and are unique (see Lemmas 3.6 and2.6). Under the above conditions, we have one of our important results (Theorem 3.23) :(1.2) D d = { w ( σ ) : σ ∈ M d } , B d − = { w ( σ ) : σ ∈ F d − \ M d − } , where B d − , D d are respectively the birth and death times of the H d − ( · ) persistence diagramof K . The simplicial version of Kruskal’s algorithm and the incremental algorithm for per-sistence (which is also a greedy algorithm) are the crucial tools in the above proof. As a Our results hold for non-injective monotonic weight functions as well but birth and death times need tobe defined appropriately. orollary of the above, we derive the following relation(1.3) L d − = (cid:90) ∞ β d − ( t )d t = (cid:88) σ ∈ M d w ( σ ) − (cid:88) σ ∈F d − \ M d − w ( σ ) = w ( M d )+ w ( M d − ) − w ( F d − ) , where w ( S ) = (cid:80) σ ∈ S w ( σ ) for a subset S of simplices. For d = 1 (assuming K ⊂ w − (0)),the above relation can be derived from Kruskal’s algorithm and, for d ≥
2, this relation wasderived recently in [33, Theorem 1.1] using different techniques. This latter paper and, inparticular, their derivation of (1.3) served as our stimulus to investigate minimal spanningacycles. Apart from the striking simplicity of the result connecting minimal spanning acyclesto persistence diagram, we believe the relation can be useful in studying either of them usingthe other. Since much of the complication in understanding persistent homology arises fromthe complex pairing of birth and death times, the above result is useful in understandingdeath or birth times individually and in certain cases, this shall yield useful information (e.g.lifetime sum) even without knowledge of the pairings.As a trivial corollary to Theorem 1.3, we can deduce point process convergence of extremalweights of a minimal spanning acycle on uniformly weighted d -complexes. To the best of ourknowledge, such a result (though not surprising) is not known even for complete graphs withi.i.d. uniform [0 , k ( · )persistence diagrams) to those for minimal spanning acycles. Stability Results.
Stability results (e.g. [25, Section VIII.2], [11, 12, 15, 16]) are animportant cog in the wheel of topological data analysis and provide a theoretical justificationfor the robustness of persistent homology. While L ∞ stability (or bottleneck stability ) is themost standard form of stability proven for persistence diagrams, L p stability for p ≥ Theorem 1.5.
Let K be a finite complex with two weight functions f, g ; both of which inducea filtration on K . Let B f = { B i } ; D f = { D i } ; B g = { B (cid:48) i } ; D g = { D (cid:48) i } be the respective birth and death times in the H d ( · ) and H d − ( · ) persistence diagrams of f, g respectively. Let Π D be the set of bijections from D f to D g and similarly, Π B , the set ofbijections from B f to B g . Then for any p ∈ { , . . . , ∞} , max { inf π ∈ Π D (cid:88) i | D i − π ( D (cid:48) i ) | p , inf π ∈ Π B (cid:88) i | B i − π ( B (cid:48) i ) | p } ≤ (cid:88) σ ∈F d | f ( σ ) − g ( σ ) | p , For p = ∞ and a sequence { x i } i ≥ , in the usual manner (cid:80) i | x i | p should be read as sup i | x i | . s part of the proof of the above stability result, we show that changing weights of m ( m ≥
1) faces can change at most m death times and m birth times by the difference betweenthe weights on the faces. One might suspect that the L ∞ stability in the above theorem canbe deduced from bottleneck stability of persistence diagrams by a projection argument. Wewould like to point out that this is not the case. This is mainly due to the fact that thediagonal plays a special role in the definition of bottleneck stability of persistence diagramswhereas there is no such equivalent for bottleneck distance between point processes on R .We explore consequences of the above stability result in the context of weighted randomcomplexes. In particular, we consider the generically weighted d -complex L (cid:48) n,d whose d -faces σ have weights φ ( σ ) = w ( σ ) + (cid:15) n ( σ ) where w ( σ ) are i.i.d. with a strictly increasing andLipschitz continuous distribution and (cid:15) n ( σ )’s are a arbitrary collection of random variables.Regardless of any dependence properties of (cid:15) n ( σ )’s, we show that if n sup σ | (cid:15) n ( σ ) | → e − x d x on R .In other words, Theorem 1.3 holds for both death times and face-weights of the minimalspanning acycles under the perturbed set-up provided the perturbations decay sufficientlyfast. We also mention special cases where the above condition can be verified easily.Reconsider the uniformly weighted d − complex U n,d . In this case, we have that the lifetimesum L n, (see (1.1)) is equal to the weight of the minimal spanning tree (see (1.3)). Acelebrated result in the theory of minimal spanning trees by Frieze ([29]) shows that(1.4) L n, a.s. → ζ (3) = ∞ (cid:88) k =1 k − = 1 . , where ζ ( · ) is Riemann’s zeta function. In higher dimensions, Hiraoka-Shirai [33, Theorem1.2] showed recently that E [ L n,d ] = Θ( n d − ). The existence of a limit and its value stillremains open for d ≥
2. As a consequence of our stability result (Theorem 1.5), we triv-ially get that the above results of Frieze and Hiroaka-Shirai hold in expectation for thenoisy random complex L (cid:48) n,d provided sup σ E [ (cid:15) n ( σ )] = o ( n − ). While it is believed that in-troducing weak dependencies between the random variables will not affect the asymptotics,it is not often easy to rigorously prove such a statement. In general, our results shall helpone to easily establish limit theorems for a “noisy” version of any random complex modelonce it has been established for the random complex model without noise. For example,in [33, Theorem 6.10], an upper and lower bound for the expected lifetime was shown. Itis possible to extend these bounds to a suitable noisy version of the random clique complexes. Organisation of Paper:
The next section - Section 2 - gives in detail the necessary topo-logical (Section 2.1) and probabilistic preliminaries (Section 2.2) . Section 3 is exclusivelydevoted to studying various properties of minimal spanning acycles (Section 3.1), algorithmsto generate them (Section 3.2), their connection to persistence diagrams (Section 3.3) and In our effort to make this article reasonably self-contained as well as accessible to the trio of probabilists,combinatorialists and topologists, we might have erred on the side of details rather than terseness. crucial stability result (Section 3.4). In the next two sections, we study weighted sim-plicial complexes wherein the weights have been generated randomly and prove our pointprocess convergence results. In Section 4, the weights are independent and identically dis-tributed (i.i.d.) uniform [0 ,
1] random variables and in Section 5, the weights are either i.i.d.with a more general distribution F or a perturbation of the same. The appendix (Section5.1)collects results regarding point process convergence in vague topology.2. Preliminaries
We describe here the basic notions of simplicial homology and persistent homology. Inan earlier version of the article (see [59, Appendix B]), we have rephrased our topologicalnotions in the language of matrices for ease of understanding. After topological preliminariesare covered here, we review notions regarding point processes and their weak convergence.2.1.
Topological Notions.
We restrict ourselves to studying simplicial complexes and shallalways choose our coefficients from a field F . In this regard, 0 stands for additive identity, 1stands for multiplicative identity and − F = Z in which case 1 = − Simplicial Homology.
This provides an algebraic tool to study the topology of a sim-plicial complex. For a good introduction to algebraic topology, see [31], and for simplicialcomplexes and homology, see [25, 52].Let K be a simplicial complex (see Definition 1.1). We assume throughout that all oursimplicial complexes are defined over a finite set V . Further, we denote the cardinality of F d ( K ) by f d ( K ). The 0-faces of K are also called as vertices . When obvious, we shall omitthe reference to the underlying complex K in the notation. A d -simplex σ is often representedas [ v , . . . , v d ] to explicitly indicate the subset of V generating the simplex σ .An orientation of a d -simplex is given by an ordering of the vertices and denoted by[ v , . . . , v d ] . Two orderings induce the same orientation if and only if they differ by an evenpermutation of the vertices. In other words, for a permutation π on [ d ],[ v , . . . , v d ] = ( − sgn ( π ) [ v π (0) , . . . , v π ( d ) ] . We assume that each simplex in our complex is assigned a specific orientation (i.e., ordering).Let F be a field. A simplicial d -chain is a formal sum of oriented d -simplices (cid:80) i c i σ i , c i ∈ F . The free abelian group generated by the d -chains is denoted by the C d ( K ), the d − th chaingroup, i.e., C d ( K ) := (cid:40)(cid:88) i c i σ i : c i ∈ F , σ i ∈ F d ( K ) (cid:41) . Clearly C d ( K ) is a F -vector space. We shall set C − = F and C d = 0 for d = − , − , . . . . Fora vector space, we shall denote the rank by β ( · ). Thus, β ( C d ) = f d for d ≥
0. For d ≥ , wedefine the boundary operator ∂ d : C d −→ C d − first on each d − simplex as below and then xtend it linearly on C d : ∂ d ([ v , . . . , v d ]) = d (cid:88) i =0 ( − i [ v , . . . , ˆ v i , . . . , v d ] , where ˆ v i denotes that v i is to be omitted. ∂ is defined by setting ∂ ([ v ]) = 1 for all v ∈ F . Itcan be verified that ∂ d is a linear map of vector spaces and more importantly that ∂ d − ◦ ∂ d = 0for all d ≥ , i.e., boundary of a boundary is zero. When the context is clear, we will dropthe dimension d from the subscript of ∂ d . Note that the free abelian group of d -chains is defined only using F d ( K ). When we use asubset S ⊂ F d ( K ) of d -faces rather than the entire collection of d -faces to generate the freeabelian group, we shall call the corresponding free abelian (sub)group of d -chains as C d ( S ).In other words, C d ( S ) = C d ( K d − ∪ S ). Also, for a d -chain (cid:80) i a i σ i , the support denoted as supp ( (cid:80) i a i σ i ) is { σ i : a i (cid:54) = 0 } ⊂ F d . For F = Z , the support characterizes the d -chain.The d − th boundary space denoted by B d is im ∂ d +1 and the d − th cycle space Z d is ker ∂ d .Elements of Z d are called cycles or d -cycles to be more specific. The d -dimensional (reduced) homology group is then defined as the quotient group(2.1) H d = Z d B d . Again, since we are working with field coefficients, B d , Z d and H d are all F − vector spaces.The d − th Betti number of the complex β d ( K ) is defined to be the rank of the vector space H d .Respectively, let b d ( K ) := β ( B d ) and z d ( K ) := β ( Z d ) denote the ranks of the d − th boundaryand d − th cycle spaces respectively. Thus we have that β d = z d − b d . Note that we drop theadjective reduced henceforth, but all our homology groups and Betti numbers are indeedreduced ones. Some authors prefer to use ˜ H d and ˜ β d to denote reduced homology groupsand Betti numbers respectively, but we refrain from doing so for notational convenience.However, under such a notation, we note that β d − ˜ β d = 1[ d = 0]. This gives an easy way totranslate results for reduced Betti numbers to Betti numbers and vice-versa. We denote theEuler-Poincar´e characteristic by χ and the Euler-Poincar´e formula holds as follows:(2.2) χ ( K ) = ∞ (cid:88) j =0 ( − j f j ( K ) = 1 + ∞ (cid:88) j =0 ( − j β j ( K ) . An important property of homology groups that is often of use is the following: If K , K are two complexes such that the function h : K → K is a simplicial map (i.e., σ =[ v , . . . , v d ] ∈ K implies that h ( σ ) = [ h ( v ) , . . . , h ( v d )] ∈ K for all d ≥ h ∗ : H d ( K ) → H d ( K ) called the induced homomorphism between thehomology groups. One of the natural simplicial maps is the inclusion map from a complex K to K such that K ⊂ K . One of the tools in algebraic topology we use is the Mayer-Vietorissequence . Below we state it explicitly and its implication for Betti numbers.A sequence of vector spaces V , . . . , V l and linear maps ν i : V i → V i +1 , i = 1 , . . . , l − exact if im ν i = ker ν i +1 for all i = 1 , . . . , l − Reduced is used to refer to the convention that C − = F instead of C − = 0. emma 2.1. (Mayer-Vietoris Sequence [52, Theorem 25.1] , [22, Corollary 2.2] .)Let K and K be two finite simplicial complexes and K = K ∩ K (i.e., K is the complexformed from all the simplices that are in both K and K ). Then the following are true: (1) The following is an exact sequence, and, furthermore, the homomorphisms ν d areinduced by the respective inclusions: · · · → H d ( K ) ν d → H d ( K ) ⊕ H d ( K ) → H d ( K ∪ K ) → H d − ( K ) ν d − → H d − ( K ) ⊕ H d − ( K ) → · · · (2) Furthermore, (2.3) β d ( K ∪ K ) = β d ( K ) + β d ( K ) + β (ker ν d ) + β (ker ν d − ) − β d ( K ) . Persistent Homology. A filtration of a simplicial complex K is a sequence of subcom-plexes {K ( t ) : t ∈ R } satisfying ∅ = K ( −∞ ) ⊆ K ( t ) ⊆ K ( t ) ⊆ K ( ∞ ) = K for all −∞ < t ≤ t < ∞ . Put differently, the filtration {K ( t ) : t ∈ R } describes howto build K by adding collections of simplices at a time. Persistent homology provides aformal tool to understand how the topology of a filtration evolves as the filtration parameterchanges. For more complete introduction and survey of persistent homology, see [25, 9, 10].We now describe the natural filtration associated with weighted simplicial complexes . Con-sider a simplicial complex K weighted by w : K → R satisfying w ( σ ) ≤ w ( τ ) , whenever σ, τ ∈ K and σ ⊂ τ. Functions having this property are called monotonic functions in [25,Chapter VIII]. As w is monotone, {K ( t ) : t ∈ R } with K ( t ) := w − ( −∞ , t ] forms a sublevelset filtration of K . Further note that w induces a partial order on the faces of K . Assumingaxiom of choice, this partial order can always be extended to a total order [63]. Let < l denoteone such total order. We make the standing assumption that for a given weight function w ,the same total order < l is chosen and used throughout the article.One can now view the above sublevel set filtration associated with ( K , w ) in a dynamicfashion: as the parameter t evolves over R , K gets built one face at a time respecting thetotal order < l . In this way, with the addition of faces, the topology of K evolves. Clearly(2.4) K ( σ − ) := { σ ∈ K : w ( σ ) < l w ( σ ) } denotes the complex right before the face σ is to be added. Thus given a monotonic weightfunction w , we can construct a filtration with respect to the chosen total order < l . Weshall call this filtration the canonical filtration associated with the total order < l or a linearfiltration of the weight function w .To track the changes in topology, akin to the definition for homology given in (2.1), wedefine the ( t , t )- persistent homology group as the quotient groupH t ,t d = Z t d Z t d ∩ B t d , t ≤ t . he information for all pairs ( t , t ) can be encoded in a unique interval representationcalled a persistence barcode [67] or equivalently a persistence diagram [15]. Before giving thedefinition, we first note that for a finite simplicial complex endowed with a total ordering < l , we can reindex the filtration by assigning a natural number to each simplex. We referto this as a discrete filtration w N corresponding to the monotonic weight function w i.e., w N ( σ ) < w N ( τ ) iff w ( σ ) < l w ( τ ). Note that there is a bijection between total orders < l andweight function w N . Thus, the discrete filtration has a natural, well-defined projection π back to the original function values, π ( i ) (cid:55)→ w ( σ ) | w N ( σ ) = i Definition 2.2.
Given a simplicial complex K with a monotonic function w and the cor-responding discrete filtration w N : K → N , the d − th persistence diagram Dgm( K , w N ) isthe multiset of points in the extended grid N such that the each point ( i, j ) in the diagramrepresents a distinct class (i.e., a topological feature) in H d ( w − N ( −∞ , t ]) for all t ∈ [ i, j ) and is not a topological feature in t / ∈ [ i, j ) . The persistence diagram Dgm( K , w ) is thendefined as the projection of the multiset of points under π , i.e., ( i, j ) ∈ Dgm( K , w N ) iff ( π ( i ) , π ( j )) ∈ Dgm( K , w ) . This differs from the typical definition of a persistence diagram, where the existence anduniqueness of the persistence diagram is defined in terms of an algebraic decomposition intointerval modules see [13, 21]. For technical reasons, this approach generally discards thepoints on the diagonal, i.e., topological features which are both born and die at time t . Inthe above definition, the total order guarantees that there are no points on the diagonalof the discrete filtration. However, since we deal with the restricted setting of piece-wiseconstant functions on finite simplicial complexes, we do not lose any information, indeed wekeep more of the chain level information. We then transform the persistence diagram backto the original monotone function. After the transformation, points may lie on the diagonaland as we see, we will require these points.Our definition is used implicitly in [67], which first identified the algebraic decompositionas a consequence of the structure theorem of finitely generated modules over a principleideal domain. This applies in this setting since the homology groups of finite simplicialcomplexes are always finitely generated. Therefore, we could have equivalently defined thediagram using the decomposition directly as done in Corollary 3.1 in [67], as the modifiedSmith Normal Form of the boundary operator [60]. We believe that our definition is moreaccessible to a non-algebraic audience and is included for completeness. But more importantfor us are birth and death times defined below. Definition 2.3.
The death times (respectively birth times ) of the filtration associated with ( K , w ) are equal to the multiset of y -coordinates ( x -coordinates) of points in Dgm( K , w ) . We shall now discuss the notion of negative and positive faces which are vital tools for ourproofs. We begin with an application of the Mayer-Vietoris exact sequences for complexes. emma 2.4. ( [22, Section 3] ) Let K be a simplicial complex on vertex set V and σ ⊂ V be aset of cardinality d + 1 in V for some d ≥ . Further, assume that σ / ∈ K but ∂σ ∈ C d − ( K ) .Then β j ( K ∪ σ ) = β j ( K ) for all j / ∈ { d − , d } . Further, one and only one of the followingtwo statements hold: (1) β d − ( K ∪ σ ) = β d − ( K ) − . (2) β d ( K ∪ σ ) = β d ( K ) + 1 . From the definition of the cycle and boundary spaces, the above two numbered statementscan be interpreted equivalently in the following manner which shall be useful for us: β d − ( K ∪ σ ) = β d − ( K ) − ⇔ b d − ( K ∪ σ ) = b d − ( K ) + 1 ⇔ ∂σ / ∈ ∂ ( C d ( K )) , (2.5) β d ( K ∪ σ ) = β d ( K ) + 1 ⇔ z d ( K ∪ σ ) = z d ( K ) + 1 ⇔ ∂σ ∈ ∂ ( C d ( K )) . (2.6) Definition 2.5 (Positive and Negative faces) . Let K be a complex with vertex set V and σ ⊂ V be a set of cardinality d + 1 for some d ≥ . Further assume that σ / ∈ K but ∂σ ∈ C d − ( K ) (cid:54) = 0 . Such a σ is called a negative face w.r.t. K if β d − ( K ∪ σ ) = β d − ( K ) − . σ is called a positive face if it is not negative, i.e., β d ( K ∪ σ ) = β d ( K ) + 1 . This is useful for understanding how the topology evolves in the linear filtration of w (recall(2.4)). If σ is a d − face, then Lemma 2.4 shows that the relationship between the topologyof the setup before and after addition of σ is as follows: (i) β j ( K ( σ − ) ∪ σ ) = β j ( K ( σ − )) forall j / ∈ { d, d − } , (ii) one and exactly one of the following is true:(2.7) β d − ( K ( σ − ) ∪ σ ) = β d − ( K ( σ − )) − β d ( K ( σ − ) ∪ σ ) = β d ( K ( σ − )) + 1 . As in Definition 2.5, when (2.7) holds (respectively (2.8) holds) σ will be called a negativeface (positive face) w.r.t. the natural filtration of ( K , w ). We emphasize that the total order < l uniquely determines the label of faces as either positive or negative. The above discussioncan be neatly converted to an algorithm to generate birth and death times of the persistencediagram with respect to a given linear filtration of the weight function w . Algorithm 1
Incremental Persistence Algorithm
Input: K , w Main Procedure: F = F d while F (cid:54) = ∅• remove a face σ with minimum weight (w.r.t < l ) from F . Set d = dim ( σ ). • if σ is negative w.r.t. K ( σ − ) , i.e., if β d − ( K ( σ − ) ∪ σ ) = β d − ( K ( σ − )) − w ( σ ) to D d − else add w ( σ ) to B d . Output B d , D d , for all d ≥ he above algorithm is a simplification of the persistence algorithm in [26, Fig. 5] whichalso used negative and positive simplices. The simplification in our algorithm essentially liesin turning a blind eye to the information about the pairing between the birth and deathtimes. The equivalence of negative faces with death times (and hence positive faces withbirth times) was established in [67, Fig. 9]. These algorithms extended the incrementalalgorithm for computing Betti numbers in [22]. We summarize the algorithm, especially forease for future reference, as follows : Let σ be a d -face in K .(2.9) w ( σ ) ∈ D d − ⇔ (2.7) holds or equivalently w ( σ ) ∈ B d ⇔ (2.8) holdsWe end this subsection reiterating a remark with respect to our proofs. Remark 2.6.
As already explained, if K is a weighted simplicial complex, there is a uniquetotal ordering of the faces if the weight function is injective otherwise, it is only a partialordering. However, this partial ordering can be extended to a total order. This correspondencebetween monotonic weights and total orders shall be used to simplify many of our proofs. Weshall often prove many statements for weighted simplicial complexes with unique weights andappeal to this correspondence in extending the proof to general monotonic weight functions.Equivalently, one can prove results for w N and then use the natural projection π to obtainthe corresponding result for monotonic weight function w . Spanning acycles.
As made clear in the title, the other key object of our study isthe spanning acycle , which has been already introduced in Definition 1.4. We now discussthe definition in more detail. Apart from being more restrictive than that in [33, 42], ourdefinition differs from that of [33] in its use of field coefficients over integer coefficients.Clearly, in the case of d = 1, S is a minimal spanning tree on the graph K . Strictlyspeaking, the above definition is that of a d -spanning acycle but since in most cases thedimension d will be clear from the context, we shall not explicitly refer to the dimension d .Also, we shall say that a subset S is spanning or an acycle to refer to the respective equalitiesin the definition 1.4. We postpone further discussion of spanning acycles to Section 3 wherewe answer many natural questions arising about spanning acycles.Recall that for any S ⊆ K , w ( S ) = (cid:80) σ ∈ S w ( σ ) as the weight of S . Suppose the simplicialcomplex K was a weighted complex with weight function w , we can define the weight ofany spanning acycle S ⊂ F d as w ( S ). A minimal spanning acycle of K is a spanning acycle S ⊂ F d with the minimum weight. In other words, if we denote the set of d -spanning acyclesof K by S d ( K ), then S ∈ S d ( K ) is a minimal spanning acycle if(2.10) w ( S ) = min S ∈S d ( K ) w ( S ) . Spanning trees and more generally connectivity in the case of graphs can be extended ina multitude of ways to higher-dimensions. Betti numbers and acycles represent one possible(and indeed a very satisfying) generalization to higher dimensions. Another common gen-eralization is the notion of a hypergraph . In this context, one can define a hypergraph ona simplicial complex by considering all the faces as hyper-edges. We shall discuss brieflyhypergraph-connectivity of spanning acycles (see Section 3.2). .2. Probabilistic notions.
We give here a brief introduction to point processes on R witha special focus on Poisson point processes. We discuss further about point processes andstate some results required for a self-contained treatment in Section A. For a more detailedreading on weak convergence of point processes, we refer the reader to [58, Chapter 3]. Let B ( R ) be the Borel σ − algebra of subsets in R . A point measure on R is a map from B ( R ) to the set of natural numbers, i.e., it is a Radon(locally-finite) counting measure. A point measure m is represented as m ( · ) = (cid:80) ∞ i =1 δ x i ( · ) , for some countable but locally-finite collection of points { x i } in R and where δ x ( · ) denotesthe delta measure at x . Alternatively, we define the support of the point measure m , denotedby supp ( m ) as the multi-set { x i } . A point measure is simple if m ( { x } ) ≤ x ∈ R . Let M p ( R ) denote the set of all point measures on R . Also let C + c ( R ) denote the set of allcontinuous, non-negative functions f : R → R with compact support. For f ∈ C + c ( R ) and m = (cid:80) ∞ i =1 δ x i ∈ M p ( R ) , define(2.11) m ( f ) := (cid:90) R f dm = ∞ (cid:88) i =1 f ( x i ) . Let m n , m ∈ M p ( R ) . We will say that m n converges vaguely to m, denoted m n v → m, if forevery f ∈ C + c ( R ) , m n ( f ) → m ( f ). Using this notion of vague convergence, one defines thevague topology on M p ( R ) . That is, a subset of M p ( R ) is vaguely closed if it includes all itslimit points w.r.t. vague convergence. The sub-base for this topology consists of open setsof the form { m ∈ M p ( R ) : m ( f ) ∈ ( s, t ) } , f ∈ C + c ( R ) , s, t ∈ R , s < t. A point process P on R is a random variable taking values in the space ( M p ( R ) , M p ( R ))where M p ( R ) denotes the Borel σ -algebra generated by the vague topology. A point processis called simple if P ( { x } ) ≤ Definition 2.7.
Let µ : R → [0 , ∞ ) be locally integrable ( (cid:82) A µ ( x ) dx < ∞ for all bounded A ⊂ R ). A point process P on R is said to be a Poisson point process with intensity function µ if the following two properties hold. (1) If A , . . . , A m ⊆ R are disjoint, then P ( A ) , . . . , P ( A m ) are independent randomvariables. (2) For any A ⊆ R , P ( A ) is a Poisson random variable with mean (cid:82) A µ ( x ) dx. Definition 2.8.
Let P n , P be point processes on R . These processes need not be defined onthe same probability space. We will say that P n converges weakly to P , denoted P n ⇒ P , if E [ f ( P n )] → E [ f ( P )] for all continuous and bounded f : ( M p ( R ) , M p ( R )) → R . This is equivalent to saying lim n →∞ P { P n ∈ A } = P { P ∈ A } for all A ∈ M p ( R ) such that P { P ∈ ∂A } = 0 . Here ∂A denotes the boundary of A. n alternative topology on M p ( R ) that arises naturally in computational topology is theso-called bottleneck distance d B . Note that we require a modified definition for point measuresin R rather than the more standard definition for persistence diagrams (e.g. [11, 25]). Definition 2.9.
For m , m ∈ M p ( R ) d B ( m , m ) := inf γ sup x : supp ( m ) | x − γ ( x ) | , where the infimum is over all possible bijections γ : supp ( m ) → supp ( m ) between themulti-sets. If no bijection exists, set d B ( m , m ) = ∞ . Though this is not a metric in the classical sense, taking min { d B , } we get a metric on M p ( R ). More importantly, the topology induced by d B and min { d B , } are the same. Weshall prove in Lemma A.2 that this topology is stronger than that of vague topology.3. Minimal spanning acycles
Recall the notion of spanning acycles from Definition 1.4. This section explores manyinteresting combinatorial properties of (minimal) spanning acycles which are of independentinterest. To avoid tedium, we do not always single out the results for the case of minimalspanning tree, i.e., d = 1. Since these are classical results in combinatorial optimization andgraph theory, one can refer to [17, 66] for graph-theoretic (and expectedly simpler) proofsof these results for the minimal spanning tree. However, we mention that our proofs do notneed any modification for the case d = 1. We start with elementary simplicial homologyresults that will be used often in the paper. The following lemma is a corollary of Lemma 2.4. Corollary 3.1.
Let K be a simplicial complex with S ⊂ S ⊂ F d . Then β d − ( K d − ) ≥ β d − ( K d − ∪ S ) ≥ β d − ( K d − ∪ S ) ≥ β d − ( K d ) = β d − ( K ) ≥ . Lemma 3.2.
Let K be a simplicial complex, σ ∈ K be a d -face and let S ⊂ S ⊂ F d be suchthat σ / ∈ S . If σ is a positive face for K d − ∪ S , then σ is a positive face for K d − ∪ S aswell. Conversely, if σ is a negative face for K d − ∪ S , then σ is a negative face for K d − ∪ S as well.Proof. If σ is a positive face for K d − ∪ S , then it follows from (2.6) that ∂σ ∈ ∂ ( C d ( K d − ∪ S )) . Since ∂ ( C d ( K d − ∪ S )) ⊂ ∂ ( C d ( K d − ∪ S )), from (2.6), we have that σ is positive for K d − ∪ S . The second statement follows from negation of the implication since a simplexmust be either positive or negative. (cid:3) Lemma 3.3.
Let K be a simplicial complex. For all S ⊂ F d , if β d − ( K d − ∪ S ) = m >β d − ( K ) then there exists a σ ∈ F d \ S such that β d − ( K d − ∪ S ∪ σ ) = m − .Proof. Let S ⊂ F d such that β d − ( K d − ∪ S ) = m > β d − ( K ) and suppose that all σ ∈F d \ S are positive faces w.r.t. K d − ∪ S . Let S ⊂ F d be such that S ⊃ S. Then fromLemma 3.2, we have that any σ ∈ F d \ S is positive w.r.t. K d − ∪ S . From this, we have β d − ( K d − ∪ S ) = β d − ( K d − ∪ S ) for any S ⊃ S . Taking S = F d , we obtain the necessarycontradiction that β d − ( K ) = β d − ( K d − ∪ S ) . Hence there is a negative face σ ∈ F d \ S w.r.t. K d − ∪ S . (cid:3) .1. Basic properties.
From Corollary 3.1, we get the following necessary condition forexistence of a spanning acycle: If a spanning acycle exists for a complex K , then β d − ( K ) =0. We shortly prove that it is sufficient as well. From Definition 1.4, Corollary 3.1 andLemma 2.4, we get the converse to Lemma 3.3 which applies to spanning acycles as well. Lemma 3.4.
Let S ⊂ F d be such that β d − ( K d − ∪ S ) = β d − ( K ) . Then any σ ∈ F d \ S isa positive face w.r.t. K d − ∪ S . In particular, this holds when S is a spanning acycle. It is known that the bases of a matroid have the same cardinality called the rank . Thusthe cardinality of any maximal acycle is the same and we now show that it can be expressedexplicitly. For a complex K , let γ d ( K ) := f d ( K ) − β d ( K d ). By evaluating χ ( K d ) − χ ( K d − )using the Euler-Poincar´e formula (2.2), we derive that(3.1) γ d ( K ) = β d − ( K d − ) − β d − ( K d ) = β d − ( K d − ) − β d − ( K ) , where the later equality follows from Corollary 3.1. Another use of Euler-Poincar´e formulagives us the following neat result that is a slight generalization of [33, Lemma 3.4] thoughthe proof is very much the same. Lemma 3.5.
For a simplicial complex K and a subset S ⊂ F d of d -faces, any two of thefollowing three statements imply the third. (1) β d − ( K d − ∪ S ) = β d − ( K ) . (2) β d ( K d − ∪ S ) = 0 . (3) | S | = γ d ( K ) .Proof. By applying the Euler-Poincar´e formula to ( − d [ χ ( K d − ∪ S ) − χ ( K d )] and re-arranging the terms, we derive the below identity which proves the lemma.(3.2) β d ( K d − ∪ S ) + γ d ( K ) − | S | − β d − ( K d − ∪ S ) + β d − ( K ) = 0 . (cid:3) In particular, if S ⊂ F d is a spanning acycle of a simplicial complex K with β d − ( K ) = 0 , then we get using (3.1) that | S | = f d ( K ) − β d ( K d ) = β d − ( K d − ).As mentioned earlier, the next result shows that β d − ( K ) = 0 is a sufficient conditionfor the existence of a spanning acycle. Enroute, we also prove that any two of the threeconditions in Lemma 3.5 characterize maximal acycles as well as show that the rank of thematroid of acycles, i.e., the cardinality of a maximal acycle, is nothing but γ d ( K ). In case of d = 1, a maximal acycle is nothing but a maximal spanning forest of the graph K . Lemma 3.6.
Let K be a simplicial complex. Then every maximal acycle of d − faces hascardinality γ d ( K ) . Consequently, if β d − ( K ) = 0 , we have that there exists a spanning acycle.Proof. By Corollary 3.1, β d − ( K d − ∪ S ) − β d − ( K ) ≥ S ⊂ F d . Using this in (3.2), | S | ≤ γ d ( K ) for any acycle S ⊂ F d . Hence, to show the desired result, it now suffices toshow that there exists an acycle with cardinality γ d ( K ) . From (3.1), β d − ( K d − ) = γ d ( K ) + β d − ( K ) . Since, by definition, γ d ( K ) ≥ , it follows fromLemma 3.3 that, starting with an empty set, we can inductively construct a set S ⊂ F d uch that | S | = γ d ( K ) and β d − ( K d − ∪ S ) = β d − ( K ) . By Lemma 3.5, it follows that β d ( K d − ∪ S ) = 0 implying that S is an acycle with | S | = γ d ( K ) as desired. If β d − ( K ) = 0 , then this S is also a spanning acycle. (cid:3) The above two lemmas characterize d − maximal acycles as well as their cardinality. Inthe proof of Lemma 3.5, it was essential to work with the restriction of K to its d − skeleton,i.e., K d . We now give alternate expressions in which there is no need for this restriction.As before, let K be a simplicial complex and let d ≥ . Since f d +1 ( K d ) = 0, we have that b d ( K d ) = 0 and so β d ( K d ) = z d ( K d ) = z d ( K ) = β d ( K ) + b d ( K ). Combining this with the factthat f d ( K ) = z d ( K ) + b d − ( K ) and γ d ( K ) = f d ( K ) − β d ( K d ) , we get that(3.3) γ d ( K ) = b d − ( K ) . Separately, using (3.1) for d + 1 and γ d ( K ) = f d ( K ) − β d ( K d ) , we get that(3.4) γ d ( K ) = f d ( K ) − β d ( K ) − γ d +1 ( K ) . As b d ( K ) is monotone, it follows from (3.3) that γ d ( K ) is also monotone, i.e.,(3.5) K ⊂ K ⇒ γ d ( K ) ≤ γ d ( K ) . By using the submodular inequality for ranks of matroids (see [65, Sec 1.6]), we get thatfor any two complexes K , K such that K d − = K d − (i.e., they differ only in faces withdimension d and above) γ d ( K ) + γ d ( K ) ≥ γ d ( K ∪ K ) + γ d ( K ∩ K ) . We now strengthen this to a neat equality which is applicable in greater generality.
Theorem 3.7.
Let K , K be two complexes and let d ≥ . Then (3.6) γ d ( K ) + γ d ( K ) = γ d ( K ∪ K ) + γ d ( K ∩ K ) + β (ker ν d − ) , where ν d − : H d − ( K ∩ K ) → H d − ( K ) ⊕ H d − ( K ) is the induced linear map.Proof. The identity (3.6) trivially holds for d > dim( K ∪ K ) since all expressions on theright and left are zero. Let d ≤ dim( K ∪ K ) . The key tool now is the identity (2.3) forBetti numbers. Substituting (3.4) in (2.3), we get that f d ( K ) − γ d ( K ) − γ d +1 ( K ) + f d ( K ) − γ d ( K ) − γ d +1 ( K ) = f d ( K ∩ K ) − γ d ( K ∩ K ) − γ d +1 ( K ∩ K ) − β (ker ν d )+ f d ( K ∪ K ) − γ d ( K ∪ K ) − γ d +1 ( K ∪ K ) − β (ker ν d − ) . The f d terms cancel out due to inclusion-exclusion and hence the expression simplifies to γ d ( K ) + γ d +1 ( K ) + γ d ( K ) + γ d +1 ( K ) = γ d ( K ∩ K ) + γ d +1 ( K ∩ K ) + β (ker ν d )+(3.7) γ d ( K ∪ K ) + γ d +1 ( K ∪ K ) + β (ker ν d − )Firstly, let d = dim ( K ∪K ) and so γ d +1 ( · ) = 0 for all K , K , K ∪K , K ∩K . Furthermore,by exactness β (ker ν d ) = 0. Substituting into (3.7), we obtain (3.6) for d = dim ( K ∪ K ).From (3.7), it is easy to see that if (3.6) holds for d + 1, then it holds for d as well. Usingthis recursively and that (3.6) holds for d = dim ( K ∪ K ), the proof follows. (cid:3) emma 3.8 (Exchange property) . Let S ⊂ F d be a spanning acycle of a simplicial complex K and let σ ∈ F d \ S. Then, for any d -face σ ∈ S such that σ is part of a d -cycle containing σ , S ∪ σ \ σ is also a spanning acycle.Proof. By Lemma 3.4, σ ∈ F d \ S is a positive face w.r.t. K d − ∪ S . So β d ( K ∪ S ∪ σ ) = 1 . Let C be the d -cycle in S ∪ σ. Clearly, σ ∈ C . So (cid:80) τ ∈ C ∩ S a τ ∂τ = − ∂σ for some collectionof non-zero F − valued coefficients { a τ } . Suppose that for σ ∈ S, S ∪ σ \ σ is not a spanning acycle. Then by Lemma 2.4, weobtain that β d ( K d − ∪ S ∪ σ \ σ ) = β d − ( K d − ∪ S ∪ σ \ σ ) = 1. Let C be the d -cyclein K d − ∪ S ∪ σ \ σ . Clearly, C (cid:54)⊂ S as S is a spanning acycle. Hence σ ∈ C and wederive that for some collection of non-zero a (cid:48) τ ∈ F , (cid:80) τ ∈ ( C ∩ S ) a (cid:48) τ ∂τ = − ∂σ . Setting a τ = 0for τ ∈ C \ C and similarly for a (cid:48) τ , we derive that (cid:88) τ ∈ ( C ∪ C ) ∩ S ( a (cid:48) τ − a τ ) ∂τ = (cid:88) τ ∈ ( C ∩ S ) a τ ∂τ − (cid:88) τ ∈ ( C ∩ S ) a τ ∂τ = ∂σ − ∂σ = 0 . But since S is a spanning acycle, the above implies that ∀ τ ∈ ( C ∪ C ) ∩ S , a τ = a (cid:48) τ and hence C = C . So, we have that σ / ∈ C if S ∪ σ \ σ is not a spanning acycle. By contraposition,we have that if σ ∈ C , then S ∪ σ \ σ is a spanning acycle. (cid:3) Lemma 3.9.
Let S , S ⊂ F d be two distinct spanning acycles of a simplicial complex K and let σ ∈ S \ S , a d -face. Then, there exists a d -face σ ∈ S such that S ∪ σ \ σ isalso a spanning acycle.Proof. Clearly, β d − ( K d − ∪ S \ σ ) = 1 and β d ( K d − ∪ S \ σ ) = 0 . As S is a spanning acycle, β d − ( K d − ∪ S ∪ S \ σ ) = 0. From the β d − relations, it follows that there exists σ ∈ S and S (cid:48) ⊂ S such that σ is negative w.r.t. K d − ∪ S (cid:48) ∪ S \ σ . So by Lemma 3.2, we have that σ is also a negative face for K d − ∪ S \ σ . Hence by Lemma 2.4, β d ( K d − ∪ S ∪ σ \ σ ) = 0 . It thus follows that S ∪ σ \ σ is also a spanning acycle. (cid:3) Lemma 3.10 (Cycle property) . Let K be a weighted simplicial complex having a cycle C ⊂F d . Let σ ∈ C be such that its weight is strictly larger than other that of the other d -facesin C. Then σ / ∈ M for any minimal spanning acycle M .Proof. Let M be a minimal spanning acycle on K such that σ ∈ M. Clearly C (cid:54)⊂ M. Hencethere exists σ ∈ C \ M. From Lemma 3.8, it follows that M ∪ σ \ σ is a spanning acycle.But w ( M ∪ σ \ σ ) < w ( M ) , a contradiction. This proves the desired result. (cid:3) Lemma 3.11 (Uniqueness) . Let K be a simplicial complex weighted by w : K → R which isinjective on F d . If a minimal spanning acycle exists, then it must be unique.Proof. Suppose that S and M are two distinct minimal spanning acycles. Let σ be the d -facewith least weight such that σ ∈ S (cid:77) M and without loss of generality, assume σ ∈ S . Thenthere is a d − cycle C ⊂ M ∪ σ such that σ ∈ C . Since C (cid:54)⊂ S, there exists a d -face σ ∈ M \ S that is part of a d -cycle containing σ . By the choice of σ , w ( σ ) > w ( σ ) . From Lemma 3.8, M ∪ σ \ σ is a spanning cycle. But w ( M ∪ σ \ σ ) < w ( M ), a contradiction. (cid:3) emark 3.12. Suppose the weight function w is not injective on F d but nevertheless mono-tonic on K . Then as discussed in Remark 2.6, this weight function shall yield a total orderon K and so on F d as well. In such a case, the above theorem guarantees that the minimalspanning acycle is unique with respect to the chosen total order. Algorithms.
Having investigated some basic properties of minimal spanning acycles,we now turn to the question of algorithms to generate a minimal spanning acycle. Again,we take our inspiration from the spanning tree and matroid literature. One of the propertiesenjoyed by matroids is the fact that greedy algorithms can be designed to output a minimalbasis [65, Chapter 19]. The greedy algorithm for matroids is an extension of the well-known Kruskal’s algorithm [44] to generate minimal spanning trees. We shall describe herethis algorithm for simplicial complexes. We later also extend the Jarn´ık-Prim-Dijkstra’salgorithm [38, 57, 23]. As with minimal spanning trees, we emphasize that these algorithmsare important in theoretical analysis of minimal spanning acycles but for actual generationof minimal spanning acycles, it might be possible to design better algorithms.Let K be a simplicial complex weighted by w : K → R . By Lemma 2.4, every σ ∈ F d iseither positive or negative, but not both, with respect to a subcomplex K such that σ / ∈ K .Using this, we give the simplicial Kruskal’s algorithm below. Algorithm 2
Simplicial Kruskal’s Algorithm
Input: d ≥ , K , w Main Procedure: set S = ∅ , F = F d while F (cid:54) = ∅ and β d − ( K d − ∪ S ) (cid:54) = 0 • remove a face σ with minimum weight from F (w.r.t. < l ). • if σ is negative w.r.t. K d − ∪ S, then add σ to S . Output M = S . Theorem 3.13.
Let K be a weighted simplicial complex with β d − ( K ) = 0 and let M be theoutput of the simplicial Kruskal’s algorithm. Then, M is a minimal spanning acycle.Proof. We shall assume that the weight function w is injective. For the general case, similararguments can be carried out by using Remarks 2.6 and 3.12. From Lemmas 3.6 and 3.11,it follows that there is a unique minimal spanning acycle which we denote by M . We now show that M is a spanning acycle. Clearly, β d ( K d − ) = 0 and by our algorithmand Lemma 2.4, it remains the same at every stage of the algorithm and so β d ( K d − ∪ M ) = 0,proving that M is an acycle. Clearly, each face in F d \ M is positive with respect to K d − ∪ M .Hence, M is spanning as using Lemma 3.2 we have that β d − ( K d − ∪ M ) = β d − ( K d − ∪ M ∪ F d \ M ) = β d − ( K ) = 0 , For the proof of minimality, we argue as in the Kruskal’s algorithm for minimal spanningtree. We prove that at any stage of the algorithm, S ⊂ M . Assuming that the above claimis true, M ⊂ M . Since M and M are both spanning acycles, M = M . e shall prove the claim inductively. Trivially, this is true for S = ∅ . Suppose that theclaim holds for S at some stage of the algorithm i.e., S ⊂ M but S (cid:40) M. This impliesthat there does exist a d − face in F d \ S which is negative w.r.t. K d − ∪ S and hence, fromLemma 3.4, S (cid:54) = M . Let σ be the next face that is added to S and suppose that σ / ∈ M . Clearly β d ( K d − ∪ M ∪ σ ) = 1 . Hence there exists a d − cycle C in K d − ∪ M ∪ σ containing σ. Since β d ( K d − ∪ S ∪ σ ) = 0 , C (cid:54)⊂ S ∪ σ and so there exists σ ∈ C ∩ M \ S. Clearly,either w ( σ ) < w ( σ ) or w ( σ ) > w ( σ ) as w is injective. Suppose that w ( σ ) < w ( σ ) . FromLemma 3.8, it follows that M ∪ σ \ σ is spanning acycle with w ( M ∪ σ \ σ ) < w ( M ) , acontradiction. Suppose that w ( σ ) > w ( σ ) . Since S (cid:40) M , σ ∈ M \ S, and M is a spanningacycle, σ is negative w.r.t. K d − ∪ S by Lemma 3.2. Thus, it follows that the algorithmwould have chosen σ before σ, a contradiction and hence proves the claim. (cid:3) As with minimal spanning trees, the Kruskals’ algorithm has a number of useful conse-quences. We shall list some consequences now and defer the more important consequences tothe next two sections. We first define a (topological) notion of a cut for simplicial complexes.
Definition 3.14 (Cut) . Let d ≥ . Given a simplicial complex K with β d − ( K ) = 0 , a subset C ⊂ F d is a cut if β d − ( K − C ) > and for any C (cid:40) C , β d − ( K − C ) < β d − ( K − C ) . Lemma 3.15 (Cut Property) . Let K be a weighted simplicial complex with β d − ( K ) = 0 . Let C ⊂ F d be a cut. Then C ∩ S (cid:54) = ∅ for any spanning acycle S and every minimum weightface in C belongs to some minimal spanning acycle.Proof. The first conclusion follows from the definition of cut and Lemma 3.3. Now for thesecond part. Let σ be a minimum weight face in the cut C and let < l be a total order inwhich this is the unique minimum weight face in the cut C . Consider the simplicial Kruskal’salgorithm under this < l and let S be the acycle constructed when σ is the minimum weightface in F . Clearly K d − ∪ S ⊂ K − C . Setting C = C \ σ , the cut property implies that β d − ( K − C ) < β d − ( K − C ) . Thus σ is negative w.r.t. K − C and, by Lemma 3.2, is also negative w.r.t. K d − ∪ S . Hence σ will be added to the minimal spanning acycle by the simplicial Kruskal’s algorithm . (cid:3) Let K be a simplicial complex and let τ ∈ F d − . Then σ ∈ F d is said to be a coface of τ, if τ ⊂ σ. Since the set of all cofaces of a ( d − Corollary 3.16.
Let K be a weighted simplicial complex with β d − ( K ) = 0 . Then for anyspanning acycle S , supp ( ∂S ) is the set of all ( d − -faces with a d -coface. Further, let τ ∈ F d − ( K ) and σ = arg min { w ( σ ) : σ ∈ F d ( K ) , τ ⊂ σ. } . Then σ ∈ M for some minimalspanning acycle M . Lemma 3.17.
Let K be a weighted simplicial complex and let K ⊂ K be a subcomplex.Further, let β d − ( K ) = 0 and M be the minimal spanning acycle on K with respect tosome total order < l . Then there exists a minimal spanning acycle M on K such that M ∩ K ⊂ M . roof. We first fix a total order < l on K and hence on K as well. Let M be the minimalspanning acycle chosen with respect to this total order < l (see Remark 3.12). Let σ ∈ M ∩K . Consider K ( σ − ) , K ( σ − ) with respect to < l as in Remark 2.6. Since the simplicial Kruskal’salgorithm generates the minimal spanning acycle M (Theorem 3.13), and since σ ∈ M, itfollows that σ is negative w.r.t. K ( σ − ) . Combining this with the fact that K ( σ − ) ⊂ K ( σ − ) , itfollows from Lemma 3.2 that σ is also negative face w.r.t K ( σ − ) . So the simplicial Kruskal’salgorithm for K would necessarily choose σ. The desired result now follows. (cid:3)
Algorithm 3
Simplicial Prims’s Algorithm
Input: d ≥ , K , w Main Procedure:
Choose an arbitrary ( d − σ Set S = ∅ , M arked = ∅ , V = σ while M arked (cid:54) = K d and β d − ( K d − ∪ S ) (cid:54) = 0 • G ← cofaces(V) • G = G − M arked • τ ← arg min G f ( G ) (w.r.t. < l ) • if τ is negative w.r.t. K d − ∪ S then S = S ∪ τ . • M arked = M arked ∪ τ • V = V ∪ supp( ∂τ ) Output M = S .We now discuss the Jarn`ık-Prim-Dijkstra’s algorithm to generate minimal spanning acycleson weighted simplicial complexes. The details are given in Algorithm 3. The algorithmbegins at an arbitrary ( d − d − faces that do not create a cycle. The set V serves as the higher dimensional proxy for the connected component in the 1-D case. Beforewe prove the correctness of the algorithm, we mention that this algorithm is a local greedyalgorithm in contrast to Kruskal’s algorithm which is a global greedy algorithm and hencethis is more suitable for generating minimal spanning acycles on infinite complexes. Indeed,this procedure is used to define minimal spanning trees on infinite graphs in [1, Lemma 1].Prim’s algorithm is related to hypergraph connectivity which we define now. Definition 3.18 (Hypergraph connectivity) . Let K be a simplicial complex and let d ≥ . Let G Hd ( K ) be the graph with vertex set V = F d − and edge set E = { ( σ , σ ) ∈ F d − × F d − : σ ∪ σ ∈ F d } . A set S ⊂ F d − is said to be d -hypergraph connected in K if S is connected in G Hd ( K ) . Thecomplex K is said to be d -hypergraph connected if G Hd ( K ) is connected. When the dimension associated is clear, we shall speak of hypergraph connectivity omittingthe dimension. We start with a crucial lemma for our hypergraph connectivity results. emma 3.19. Let K be a d -complex. Suppose C is a d -acycle in K such that C ∪ σ is a d -cycle (i.e., there is a d -chain in Z d ( K ) whose support is C ∪ σ and no d -chain in Z d ( K ) has support C ) then supp ( ∂ C ) is d -hypergraph connected in K d − ∪ C . As a consequence supp ( ∂σ ) ⊂ supp ( ∂ C ) is also d -hypergraph connected in K d − ∪ C .Proof. Let σ and C be as assumed. This means that there exists field coefficients a τ , τ ∈ C (all non-trivial) such that ∂σ + (cid:80) τ ∈ C a τ ∂τ = 0 but for all field coefficients b τ , τ ∈ C (not alltrivial) (cid:80) τ ∈ C b τ ∂τ (cid:54) = 0 . If supp ( ∂ C ) be not hypergraph connected in K d − ∪ C then so is F d − . So, let F d − = A ∪ B such that there is no τ ∈ C such that supp ( ∂τ ) ∩ A (cid:54) = ∅ and supp ( ∂τ ) ∩ B (cid:54) = ∅ .Thus we can partition C into subsets A and B depending upon whether supp ( ∂τ ) ⊂ A or supp ( ∂τ ) ⊂ B respectively. Thus we get that C = A ∪ B with A ∩ B = ∅ . By definitionof C , we have that A ∪ σ and B ∪ σ are acyclic. We shall now derive a contradiction byshowing that C ∪ σ is acyclic.Define the complexes K A := K d − ∪ A , K B = K d − ∪ B and K σ := { τ ∈ K : τ ⊂ σ } i.e., σ viewed as a complete d -complex. Note that K A ∪ K B = K d − and K A ∩ K B = K d − . Nowto show K d − ∪ C ∪ σ is acyclic, we use the Mayer-Vietoris long exact sequence (see Lemma2.1). We set K = K A ∪ A ∪ K σ and K = K B ∪ B ∪ K σ . By definition of A , B , A, B , wehave that K and K are both complexes. Further, we have that K ∪ K = K d − ∪ C ∪ σ and K ∩ K = K d − ∪ K σ . Thus applying the Mayer-Vietoris long exact sequence, we have · · · −→ H d ( K d − ∪ K σ ) −→ H d ( K A ∪ A ∪ K σ ) ⊕ H d ( K B ∪ B ∪ K σ ) −→ H d ( K d − ∪ C ∪ K σ ) −→ H d − ( K d − ∪ K σ ) −→ · · · Since K , K are acyclic, acyclicity of K d − ∪ C ∪ σ follows from exactness of the abovesequence and the fact that H d ( K d − ∪ K σ ) and H d − ( K d − ∪ K σ ) are both trivial. Thetriviality of the latter two homology groups follows from noting that F d ( K d − ∪ K σ ) = { σ } , F d − ( K d − ∪ K σ ) = supp ( ∂σ ) , F d − ( K d − ∪ K σ ) = F d − ( K ) i.e., K d − ∪ K σ is nothingbut ( d − d -simplex attached. (cid:3) Proposition 3.20.
Let K be a d − hypergraph connected simplicial complex with β d − ( K ) = 0 .Then the subcomplex K d − ∪ S is d -hypergraph connected for any spanning acycle S .Proof. Excluding the trivial case of S = F d , let S (cid:40) F d . Then there exists σ ∈ F d \ S .Then S is a d -acycle in K ∗ = K d − ∪ S ∪ σ such that S ∪ σ is a d -cycle in K ∗ by Lemma 3.2.Since K is hypergraph connected, by corollary 3.16, we have that F d − = supp ( ∂S ) and byLemma 3.19, supp ( ∂S ) is hypergraph connected in K ∗ \ σ = K d − ∪ S as needed. (cid:3) As a special case, consider a d -dimensional complex K on n vertices with a complete( d − |F d − | = (cid:0) nd (cid:1) . Suppose that β d − ( K ) = 0. By Lemma 3.6, there exists aspanning acycle S ⊂ F d ( K ). Clearly S is also a spanning acycle for the complete d -complex K ∗ and K ∗ is clearly d -hypergraph connected. Hence K d − ∪ S is d -hypergraph connectedand since d -hypergraph connectivity is preserved under addition of d -faces, we get that K isalso d -hypergraph connected. Thus, we have shown that a d -dimensional complex K on n ertices with a complete ( d − β d − ( K ) = 0 is d -hypergraph connected. Thisis nothing but [41, Theorem 1.7]. In fact the above arguments can be easily repeated toextend both Proposition 3.20 and [41, Theorem 1.7] to the following corollary. Corollary 3.21.
Let K be a simplicial complex such that β d − ( K ) = 0 and there exists a d -hypergraph connected complex K such that K ⊂ K with K d − = K d − . Then K is also d -hypergraph connected. Theorem 3.22.
Prim’s algorithm returns the minimal spanning acycle if β d − ( K ) = 0 and K is d -hypergraph connected.Proof. We proceed as in the proof of simplicial Kruskal’s algorithm. So, we shall againonly prove under the assumption that the d -faces have unique weights and then appeal toRemarks 2.6 and 3.12 to extend the proof to the general case. Again, the assumption ofunique weights on F d , guarantees existence of a unique minimal spanning tree (Lemmas 3.6and 3.11) which we denote by M .We first note that all d -simplices are considered during the execution of Prim’s algorithm.This follows from the assumption of hypergraph connectivity. Regardless of the startingplace, all ( d − V by the assumption of hypergraphconnectivity. Therefore, all cofaces, i.e., d -simplices, will be considered. This also impliesthat the algorithm terminates. Further, same argument as that in the proof of Kruskal’salgorithm will give that M is a spanning acycle.It remains to argue minimality of M . This also we shall do inductively as in the proofof Kruskal’s algorithm. We shall show that at every stage of the algorithm S ⊂ M . Thiscertainly holds for the initial step S = ∅ . Let S be the spanning acycle constructed at someintermediate step (called current step in rest of the proof) of the algorithm, i.e., S (cid:40) M and S ⊂ M . Suppose that σ be the next d -face added to S and suppose σ / ∈ M . This impliesthat all d -faces considered after the current step but before σ were positive faces with respectto K d − ∪ S . Also, K d − ∪ M ∪ σ contains a d -cycle C and for all σ ∈ C \ σ , M ∪ σ \ σ is a spanning acycle by the exchange property (Lemma 3.8). Note that C (cid:40) S ∪ σ since S ∪ ( C \ σ ) ⊂ M . Because of unique weights, w ( σ ) < w ( σ ) or w ( σ ) > w ( σ ). The formerpossibility leads to an easy contradiction as w ( M ∪ σ \ σ ) < w ( M ). Thus we have that w ( σ ) > w ( σ ) for all σ ∈ C \ σ from which also we shall derive a contradiction.Define S ∗ consists of those d -simplices that are d -hypergraph connected to supp ( ∂S ) via d -faces of strictly lower weight than σ . More formally, S ∗ = { τ ∈ F d \ S : w ( τ ) < w ( σ ) , supp ( ∂τ ) is d -hypergraph connected to supp ( ∂S ) in K ( σ − ) } . The Prim’s algorithm shall consider all d -faces in S ∗ before considering σ .Since C is a cycle and C \ σ is an acycle, supp ( ∂ ( C \ σ )) is d -hypergraph connected in K ( σ − ) by Lemma 3.19. Since supp ( ∂σ ) ⊂ supp ( ∂ ( C \ σ )) and supp ( ∂σ ) ∩ supp ( ∂S ) (cid:54) = ∅ ,we have that supp ( ∂ ( C \ σ )) ∩ supp ( ∂S ) (cid:54) = ∅ . Hence, C \ ( S ∪ σ ) ⊂ S ∗ . Thus, the Prim’salgorithm shall consider all d -faces in C \ ( S ∪ σ ) before considering σ .Now we divide into two cases. Suppose all τ ∈ C \ ( S ∪ σ ) were considered at a stage ofthe Prim’s algorithm before the current step, then since S ∪ ( C \ σ ) is acyclic, by Lemma 3.2 ach τ should have been added to the spanning acycle at that stage of the algorithm leadingto the contradiction that τ ∈ S .Consider the next case that there exists a τ ∈ C \ ( S ∪ σ ) which is considered at a stepafter the current step. As τ ∈ S ∗ , τ is considered before σ by the Prim’s algorithm. Since σ is the next d -simplex added after S , the acycle at the step when τ is considered is still S .But since S ∪ ( C \ σ ) is acyclic, τ is a negative face with respect to K d − ∪ S and thus τ isadded to S contradicting the fact σ is the next d -face added to S .Thus we get that σ ∈ M and hence S ∪ { σ } ⊂ M completing the proof. (cid:3) Persistence Diagrams and Minimal Spanning Acycles.
Here we highlight theconnection between persistence diagrams and minimal spanning acycles. The minimal span-ning acycle represents the persistence boundary basis w.r.t. the sublevel set filtration inducedby weights on the simplices. To illustrate this explicitly, we first recount the incrementalalgorithm for Betti numbers from [22]. From the decomposition of a filtration into persis-tence diagram, it follows that a positive simplex generates a new homology class and henceforms a new cycle, while a negative simplex bounds an existing homology class and hence isa boundary. The incremental algorithm distinguishes between the two cases by constructinga set of basis vectors for the (graded) boundary operator in the order of the filtration. Inthis setting, the boundary of each simplex is reduced w.r.t. the existing basis and if it isfound to be linearly independent, it is a negative simplex and the reduced vector is added tothe basis. Since such a simplex does not generate a cycle, it is part of the minimal spanningacycle. By maintaining the basis in this manner, we can simulate Kruskal’s algorithm. Thefact that we do not add any simplices that generate cycles follows from the fact that bydefinition the boundary chain of such a simplex reduces to 0. The converse fact, that theexistence of a pivot implies that it reduces the ( k − Theorem 3.23.
Let K be a weighted d -complex with β d − ( K ) = 0 . Let D be the point-set ofdeath times in the persistence diagram of the H d − ( K ) with the canonical filtration induced bythe weights and let B be the point-set of birth times in the persistence diagram of the H d ( K ) with the canonical filtration induced by the weights. Then we have that D = { w ( σ ) : σ ∈ M } , B = { w ( σ ) : σ ∈ F d \ M } , where M is a d -minimal spanning acycle of K and F d are the d -simplices of K .Proof. We shall only need to prove the result for death times D . Because every d -simplexis either positive or negative with respect to K ( σ − ) (see (2.7) and (2.8)). Further since bythe incremental algorithm (Algorithm 1), negative simplices correspond to death times andpositive simplices correspond to birth time ((2.9)), the result for D implies that for B .We again only consider the case when the filtration values are unique and appeal to Remark2.6 to complete the proof in the general case. Note that in the general case, we use the same otal ordering for the incremental algorithm (Algorithm 1) generating death and birth timesas well as the simplicial Kruskal’s algorithm (Algorithm 2).By uniqueness of weights on F d , we know that the Kruskal’s algorithm gives us the minimalspanning acycle M . Firstly, note that by the relation (2.5), the condition to add σ to S inKruskal’s algorithm is equivalent to ∂ ( C d ( S )) (cid:40) ∂ ( C d ( S ∪ σ )). And similarly, we can observethat the incremental algorithm adds c = w ( σ ) to D if ∂ ( C d ( K ( c − ))) (cid:40) ∂ ( C d ( K ( c − ) ∪ σ )).Let c be a value in the filtration, i.e., there exists σ ∈ K such that w ( σ ) = c . Let M ( c ) de-note the acycle generated by Kruskal’s algorithm on K ( c ) , i.e., M ( c ) = M ∩K ( c ) and similarly,we shall use the notation M ( c − ) as well. By the above discussion on Kruskal’s algorithm andincremental algorithm, our proof is complete if we show that ∂ ( C d ( M ( c ))) = ∂ ( C d ( K ( c ))).Trivially, ∂ ( C d ( M ( c ))) ⊂ ∂ ( C d ( K ( c ))) and we shall now show the other inclusion.Suppose the other inclusion does not hold, then there exists a τ ∈ F d ( K ( c )) \ M ( c ) suchthat ∂τ / ∈ ∂ ( C d ( M ( c ))). Let w ( τ ) = b < c . Then clearly ∂τ / ∈ ∂ ( C d ( M ( b − ))) and so by(2.5), τ will be a negative face with respect to K d − ∪ M ( b − ). So, Kruskal’s algorithm wouldhave added σ to the acycle M ( b − ) contradicting the assumption that τ / ∈ M ( c ). Thus, wehave ∂ ( C d ( M ( c ))) = ∂ ( C d ( K ( c ))) and the proof is complete. (cid:3) The above result has strong applications for random complexes as will be seen in the nextsection. Firstly, we obtain [33, Theorem 1.1] (see (1.3)) as an easy corollary of our previoustheorem and the representation (1.1). Further, we can easily prove a fundamental uniquenessresult for minimal spanning acycles relying upon this correspondance and the uniqueness ofpersistence diagrams [67, Theorem 2.1],[13, Theorem 1.3], [21, Theorem 1.1] . Theorem 3.24.
Let K be a weighted d -complex such that β d − ( K d ) = 0 and M , M be two d -minimal spanning acycles in K . Let c ∈ [0 , ∞ ) . Then we have that |{ σ ∈ M : w ( σ ) = c }| = |{ σ ∈ M : w ( σ ) = c }| . In the case of unique weights, the minimal spanning acycle is unique making the abovetheorem trivially true. In the case of non-unique weights, the minimal spanning acycle weobtain will depend on our choice of extension to a total order. However, the above theoremstates that the weights of the minimal spanning acycle will be independent of this choice.We now give an alternative characterization of a minimal spanning acycle that followsfrom the proof of Theorem 3.23. Such a characterization of a minimal spanning tree hasbeen very useful in the study of minimal spanning trees on infinite graphs ([49, Chapter11], [2, Proposition 2.1]). A similar characterization for minimal spanning tree is calledas the creek-crossing criterion in [2]. We have mentioned earlier that simplicial Jarn`ık-Prim-Dijkstra’s algorithm can also be used to define minimal spanning acycles on infinitecomplexes analogous to the graph case. However, we wish to point out now that thesedifferent characterizations do not coincide even in the infinite graph case ([2, Proposition2.1]) and so the analogous question for complexes is moot. Uniqueness follows from certain assumptions on finiteness and the KrullRemakSchmidt theorem of iso-morphisms of indecomposible subgroups, which always hold in the setting of finite simplicial complexes. emma 3.25. Let K be a weighted simplicial complex with β d − ( K ) = 0 . Let σ ∈ F d and M be the minimal spanning acycle with respect to a total order < l extending the partial orderinduced by w . Then σ ∈ M iff ∂σ / ∈ ∂ ( C d ( K ( σ − ))) . Proof.
From the proof of Theorem 3.23, we know that ∂ ( C d ( M ∩ K ( σ − )) = ∂ ( C d ( K ( σ − )) . Thus by Kruskal’s algorithm and (2.5), we have that σ ∈ M iff ∂σ / ∈ ∂ ( C d ( K ( σ − ))). (cid:3) Combining Lemma 3.25 with the fact that if ∂σ ∈ ∂ ( C d ( K ( σ − ))) , then by Lemma 3.19,supp( ∂σ ) is d − hypergraph connected, we have the following corollary. Corollary 3.26.
Let K be a weighted simplicial complex with β d − ( K ) = 0 . Let M be theminimal spanning acycle with respect to a total order < l extending the partial order inducedby w and σ ∈ F d . If supp( ∂σ ) is not d − hypergraph connected in K ( σ − ) , then σ ∈ M. Stability Results. [Proof of Theorem 1.5] Again, it suffices to prove the theorem fordeath times and the proof for birth times is quite identical. Secondly, due to Theorem 3.23,we shall prove the stability result for weights of a minimal spanning acycle. We shall alsoassume 0 ≤ p < ∞ and the extension to p = ∞ follows by a standard limiting argument.Let M, M (cid:48) be the two minimal spanning acycles corresponding to f, f (cid:48) . We begin with thefollowing case: where f, f (cid:48) differ precisely on one simplex σ and f ( σ ) = a, f (cid:48) ( σ ) = a (cid:48) , | a − a (cid:48) | = c . In this case, we shall show that | M (cid:52) M (cid:48) | ≤
2, where (cid:52) denotes the symmetric differencebetween the two sets. Since
M, M (cid:48) have equal cardinalities, | M (cid:52) M (cid:48) | ∈ { , } . If M (cid:52) M (cid:48) = ∅ ,we are done since the identity map between the simplices in M, M (cid:48) gives thatinf π (cid:88) σ ∈ M | f ( σ ) − f (cid:48) ( π ( σ )) | p ≤ c p = (cid:88) σ ∈F d | f ( σ ) − f (cid:48) ( σ ) | p . In the other case, M (cid:52) M (cid:48) = { σ , σ } with σ ∈ M, σ ∈ M (cid:48) and one of the σ i ’s is σ . Further,we shall also show that | f ( σ ) − f (cid:48) ( σ ) | ≤ c . This again shows thatinf π (cid:88) σ ∈ M | w ( σ ) − w ( π ( σ )) | p ≤ c p = (cid:88) σ ∈F d | f ( σ ) − f (cid:48) ( σ ) | p . By a recursive application of the above case, we can prove the theorem for the general caseof f, f (cid:48) differing in many simplices.Thus we shall now on, focus only on the case of f, f (cid:48) differing only at σ ∈ F d . Withoutloss of generality, assume that f, f (cid:48) assign distinct weights to distinct faces and the case ofnon-distinct weights can be proved by appealing again to Remarks 2.6 and 3.12. Given aset I ⊂ R , we shall use M ( I ) = { σ ∈ M : w ( σ ) ∈ I } and similarly for M (cid:48) . Also as before, M ( a ) = M (( −∞ , a ]) , M ( a − ) = M (( −∞ , a )). We shall break the proof into four cases wherethe first two take care of the trivial cases when M (cid:52) M (cid:48) = ∅ . We shall assume that both M, M (cid:48) are generated by simplicial Kruskal’s algorithm (Algorithm 2).
Case 1:
Suppose σ ∈ M and a > a (cid:48) , i.e., f ( σ ) > f (cid:48) ( σ ). In this case, since M ( a ) , M (cid:48) ( a )are both maximal acycles in K ( a ) we have that | M ( a ) | = | M (cid:48) ( a ) | and further by Kruskal’salgorithm, M ( a (cid:48) − ) = M (cid:48) ( a (cid:48) − ). By Lemma 3.2, K d − ∪ M (cid:48) ( a (cid:48) − ) ∪ σ is acyclic and hence ∈ M (cid:48) . Further, since K d − ∪ M ( a − ) ∪ σ is acyclic, M (( a (cid:48) , a )) ⊂ M (cid:48) and hence M ( a ) = M (cid:48) ( a )since M ( a ) , M (cid:48) ( a ) have equal cardinalities. So, by Kruskal’s algorithm M = M (cid:48) . Case 2:
Suppose σ / ∈ M and a (cid:48) > a . In this case, M ( a (cid:48) − ) = M (cid:48) ( a (cid:48) − ) and further K d − ∪ M (cid:48) ( a (cid:48) − ) ∪ σ is cyclic by Lemma 3.4 and so again by Kruskal’s algorithm M = M (cid:48) . Case 3:
Suppose σ ∈ M and a < a (cid:48) . If σ ∈ M (cid:48) , then arguing as in Case 1, we have that M = M (cid:48) . So let σ / ∈ M (cid:48) . We shall show that M (cid:48) (( a, a (cid:48) )) \ M (( a, a (cid:48) )) = { τ } and further that M (cid:52) M (cid:48) = { σ, τ } . Also, we have that f (cid:48) ( τ ) − f ( σ ) ≤ a (cid:48) − a = c as needed.To show the same, note that by Kruskal’s algorithm M ( a − ) = M (cid:48) ( a − ) and | M ( a (cid:48) ) | = | M (cid:48) ( a (cid:48) ) | . So, if we show that M ( a, a (cid:48) ) ⊂ M (cid:48) ( a, a (cid:48) ) then because of the equality of cardinalities,there exists a τ with w ( τ ) ∈ ( a, a (cid:48) ) such that M (cid:48) ( a (cid:48) ) = M ( a ) ∪ M (( a, a (cid:48) )) ∪ τ .We shall prove M (( a, a (cid:48) )) ⊂ M (cid:48) (( a, a (cid:48) )) by contradiction. Let M (( a, a (cid:48) )) = { τ , . . . , τ k } and τ i be the first simplex (in increasing order of weights) such that τ i / ∈ M (cid:48) . This meansthat some other simplex τ / ∈ M ( a, a (cid:48) ) that should have been added in M (cid:48) before τ i thatcreates a cycle along with τ i and also a cycle with σ . More formally, there exists a τ / ∈ M with a < f ( τ ) < f ( τ i ) such that K d − ∪ M ( a − ) ∪ M (( a, f ( τ ))) ∪ τ is acyclic but K d − ∪ M ( a − ) ∪ M (( a, f ( τ ))) ∪ τ ∪ τ i and K d − ∪ M ( a − ) ∪ M (( a, f ( τ ))) ∪ σ ∪ τ are cyclic. Thesethree statements together imply that there exist b, b (cid:48) non-zero such that ∂τ − b∂σ ∈ ∂ ( C d ( M ( f ( τ ) − )) − σ ) ; ∂τ i − b (cid:48) ∂τ ∈ ∂ ( C d ( M ( f ( τ ) − )) − σ ) . The above two statements imply that ∂τ i − bb (cid:48) ∂σ ∈ ∂ ( C d ( M ( f ( τ ) − )) − σ ), a contradictionto the acyclicity of M . Hence τ i ∈ M (cid:48) , ∀ i = 1 , . . . , k and so M (( a, a (cid:48) )) ⊂ M (cid:48) (( a, a (cid:48) )) asrequired. Thus, M ( a (cid:48) ) (cid:52) M (cid:48) ( a (cid:48) ) = { σ, τ } .Now we need to show that M ( a (cid:48) , ∞ ) = M (cid:48) ( a (cid:48) , ∞ ) to conclude that M (cid:52) M (cid:48) = { σ, τ } . Since M ( a (cid:48) ) \ M (cid:48) ( a (cid:48) ) = { σ } and by Kruskal’s algorithm σ is positive with respect to M (cid:48) ( a (cid:48) ) = M ( a − ) ∪ M ( a, a (cid:48) ) ∪ τ , we have that ∂σ ∈ ∂ ( C d ( M (cid:48) ( a (cid:48) ))). So, ∂ ( C d ( M ( a (cid:48) ))) = ∂ ( C d ( M (cid:48) ( a (cid:48) )))and hence by (2.5) and (2.6), negativity and positivity of simplices in Kruskal’s algorithmremain unchanged after a (cid:48) whether we are considering M or M (cid:48) . Case 4:
Suppose σ / ∈ M and a (cid:48) < a . Then either σ / ∈ M (cid:48) or σ ∈ M (cid:48) . If σ / ∈ M (cid:48) , then M = M (cid:48) as in Case 2. If σ ∈ M (cid:48) , arguing as in Case 3, we have that M (( a (cid:48) , a )) \ M (cid:48) (( a (cid:48) , a )) = { τ } , M ( a, ∞ ) = M (cid:48) ( a, ∞ ) and hence M (cid:52) M (cid:48) = { σ, τ } with f ( τ ) − f (cid:48) ( σ ) ≤ a − a (cid:48) = c . (cid:3) Random d − complex : I.I.D. uniform weights We shall now consider randomly weighted complexes on n vertices. We consider first thesimplest model where the weights are i.i.d. uniform on all possible d -faces and 0 elsewhere.We prove one of our main theorems - Theorem 1.3 - and the analogous result for weights ofa minimal spanning acycle. Definition 4.1.
Let d ≥ be some integer. Consider n vertices and let K dn be the complete d − skeleton on them. Let w : K dn → [0 , be the weight function with the following properties: (1) w ( σ ) = 0 for σ ∈ (cid:83) d − i =0 F i ( K dn ) , and (2) { w ( σ ) : σ ∈ F d ( K dn ) } are i.i.d. uniform random variables on [0 , . he uniformly weighted d − complex U n,d is the simplicial complex K dn weighted by w. Thecanonical filtration associated with U n,d is denoted using {U n,d ( t ) : t ∈ [0 , } . That is, U n,d ( t ) := { σ ∈ K dn : w ( σ ) ≤ t } . Note that U n,d (0) is almost surely (a.s.) the complete ( d − − dimensional skeleton on n vertices. Further, note that the weights on the d -faces are a.s. distinct. The well-knownrandom d -complex Y n,d ( t ), introduced in [45, 51] and defined before Lemma 1.2 is the sameas U n,d ( t ) in distribution. For ease of use, we shall write σ ∈ U n,d to mean σ ∈ K dn . Similarly, F i ( U n,d ) shall mean F i ( K dn ) and so on.Fix d ≥ . Viewing U n,d both as a randomly weighted simplicial complex and a filtrationof random complexes, we are interested in the distributions of the following three point sets.(1) Nearest neighbour distances of the ( d − − faces, i.e., { C ( σ ) : σ ∈ F d − ( U n,d ) } . Here,for σ ∈ F d − ( U n,d ) , (4.1) C ( σ ) = min τ ∈F d ( U n,d ) ,τ ⊃ σ w ( τ ) . (2) Death times { D i } in the persistence diagram of H d − ( U n,d ) (see Definition 2.3 ).(3) Weights in the d − minimal spanning acycle M of U n,d − { w ( σ ) : σ ∈ M } (see (2.10)).Our key result here is that all the above three point sets, under appropriate scaling convergeto a Poisson point process as the number of vertices go to infinity. We use factorial momentmethod to show convergence of the first point process and then show that this is a goodenough approximation for the second point process. This yields convergence of the secondpoint process and Theorem 3.23 easily gives the convergence of the third point process.4.1. Extremal nearest neighbour distances.
Fix σ ∈ F d − ( U n,d ) . Then C ( σ ) as definedin (4.1) denotes the nearest neighbour distance of σ. By considering the filtration {U n,d ( t ) : t ∈ [0 , } , note that σ is isolated (not part of any d − face) exactly between times 0 and C ( σ )in {U n,d ( t ) : t ∈ [0 , } . The first coface of σ appears at t = C ( σ ) . For each σ ∈ F d − ( U n,d ) , let ¯ C ( σ ) := nC ( σ ) − d log n + log( d !) and the scaled point set is(4.2) P Cn,d := { ¯ C ( σ ) : σ ∈ F d − ( U n,d ) } Viewing the latter as a point process for any R ⊆ R , we set(4.3) P Cn,d ( R ) := |{ σ ∈ F d − ( U n,d ) : ¯ C ( σ ) ∈ R }| . For any c ∈ R , let P Cn,d ( c, ∞ ) ≡ P Cn,d (( c, ∞ )) . Let N n,d − ( p ) denote the number of isolated( d − − faces in Y n,d ( p ) . Since U n,d ( p ) has the same distribution as Y n,d ( p ) , it follows that P Cn,d ( np − d log n + log( d !) , ∞ ) has the same distribution as N n,d − ( p ) . From Lemma 1.2 weknow that, as n → ∞ , N n,d − ( p n ) converges to Poi( e − c ) (for some fixed c ∈ R ), the Poissonrandom variable with mean e − c , whenever(4.4) p n = d log n + c − log( d !) n . From this, we have P Cn,d ( c, ∞ ) ⇒ Poi( e − c ) as n → ∞ . We now extend this to a multivariateconvergence, thereby proving convergence of point processes P Cn,d . heorem 4.2. As n → ∞ with p n as in (4.4) , P Cn,d converges in distribution to the Poissonpoint process P poi , where P poi is as in Theorem 1.3.Proof. Let I := ∪ mj =1 ( a j − , a j ] ⊆ R be an arbitrary but fixed finite disjoint union ofintervals. Since P poi is simple and does not contain atoms, as per Lemma A.1, it suffices toestablish the following two statements to prove weak convergence of the point process P Cn,d : ( i ) lim n →∞ E [ P Cn,d ( I )] = E [ P poi ( I )] and ( ii ) P Cn,d ( I ) d ⇒ P poi ( I ) as n → ∞ . Further, by the method of factorial moments, both the above statements hold if for all l ≥ E [( P Cn,d ( I )) ( l ) ] → (cid:18)(cid:90) I e − x dx (cid:19) (cid:96) = E [( P poi ( I )) ( l ) ] , where for m ∈ N , m ( l ) = m ( m − . . . ( m − l + 1) denotes its l − th factorial moment. Restof the proof concerns proving (4.5).Let (cid:96) ≥ (cid:96) − th factorial moment of P Cn,d ( I ) by M ( (cid:96) ) n,d . For σ ∈ F d − ( U n,d )and R ⊆ R , define the indicator 1( σ ; R ) = [ ¯ C ( σ ) ∈ R ]. Then clearly P Cn,d ( I ) = (cid:88) σ ∈F d − ( U n,d ) σ ; I ) . From this, it is not difficult to see that M ( (cid:96) ) n,d = (cid:88) σσσ ∈ I ( (cid:96) ) n,d E (cid:34) (cid:96) (cid:89) i =1 σ i ; I ) (cid:35) , where I ( (cid:96) ) n,d := { σσσ ≡ ( σ , . . . , σ (cid:96) ) : σ i ∈ F d − ( U n,d ) and no two of σ , . . . , σ (cid:96) are same } . To simplify the computation of M ( (cid:96) ) n,d , we group the faces σσσ ∈ I ( (cid:96) ) n,d which give the same valuefor E (cid:104)(cid:81) (cid:96)i =1 σ i ; I ) (cid:105) . We do this as follows. For σσσ ∈ I ( (cid:96) ) n,d , let γ ( σσσ ) ≡ ( | ∩ i ∈ S σ i | : S ⊆ { , . . . , (cid:96) } , | S | ≥ σσσ, σ (cid:48) σ (cid:48) σ (cid:48) ∈ I ( (cid:96) ) n,d , we will say that both have similar intersectiontype, denoted by σσσ ∼ σ (cid:48) σ (cid:48) σ (cid:48) , if there exists a permutation π of the faces in σ (cid:48) σ (cid:48) σ (cid:48) such that γ ( σσσ ) = γ ( π ( σ (cid:48) σ (cid:48) σ (cid:48) )) . It is easy to see that ∼ is an equivalence relation. Let Γ := { [ σσσ ] } denote thequotient of I ( (cid:96) ) n,d under ∼ with [ σσσ ] denoting the equivalence class of σσσ. Since the number ofways in which (cid:96) distinct ( d − − faces can intersect each other is finite, we have that thenumber of equivalence classes in Γ , i.e., | Γ | , is upper bounded by some constant (w.r.t. n ).Indeed | Γ | depends on d and (cid:96), but these are fixed a priori in our setup. Lastly, note thatfor σσσ ∈ I ( (cid:96) ) n,d , the cardinality of its equivalence class | [ σσσ ] | indeed depends on n. Fix σσσ ≡ ( σ , . . . , σ (cid:96) ) and σ (cid:48) σ (cid:48) σ (cid:48) ≡ ( σ (cid:48) , . . . , σ (cid:48) (cid:96) ) in I ( (cid:96) ) n,d such that σσσ ∼ σ (cid:48) σ (cid:48) σ (cid:48) . Then E (cid:34) (cid:96) (cid:89) i =1 σ i ; I ) (cid:35) = E (cid:34) (cid:96) (cid:89) i =1 σ (cid:48) i ; I ) (cid:35) . ence M ( (cid:96) ) n,d can be rewritten as M ( (cid:96) ) n,d = (cid:88) [ σσσ ] ∈ Γ (cid:88) σ (cid:48) σ (cid:48) σ (cid:48) ∈ I ( (cid:96) ) n,d : σ (cid:48) σ (cid:48) σ (cid:48) ∼ σσσ E (cid:34) (cid:96) (cid:89) i =1 σ (cid:48) i ; I ) (cid:35) = (cid:88) [ σσσ ] ∈ Γ | [ σσσ ] | E (cid:34) (cid:96) (cid:89) i =1 σ i ; I ) (cid:35) . Counting the number of ways of choosing (cid:96) distinct ( d − − faces from a total of (cid:0) nd (cid:1) , wehave | I ( (cid:96) ) n,d | = (cid:96) ! (cid:0) ( nd ) (cid:96) (cid:1) . Since, for each [ σσσ ] ∈ Γ , [ σσσ ] ⊂ I ( (cid:96) ) n,d , we write | [ σ ][ σ ][ σ ] | = c n ([ σσσ ]) | I ( (cid:96) ) n,d | , forsome c n ([ σσσ ]) ∈ [0 , . Clearly(4.6) (cid:88) [ σσσ ] ∈ Γ c n ([ σσσ ]) = 1 . Hence it follows that M ( (cid:96) ) n,d = (cid:88) [ σσσ ] ∈ Γ c n ([ σσσ ]) (cid:96) ! (cid:18)(cid:0) nd (cid:1) (cid:96) (cid:19) E (cid:34) (cid:96) (cid:89) i =1 σ i ; I ) (cid:35) . Since for every σ ∈ F d − ( U n,d ) , σ ; I ) = m (cid:88) j =1 σ ; ( a j − , a j ]) = m (cid:88) j =1 (1( σ ; ( a j − , ∞ )) − σ ; ( a j , ∞ ))) , it follows that for any σσσ ≡ ( σ , . . . , σ (cid:96) ) ∈ I ( (cid:96) ) n,d , (cid:96) (cid:89) i =1 σ i ; I ) = (cid:88) ( α ,...,α (cid:96) ) ∈{ ,..., m } (cid:96) (cid:96) (cid:89) i =1 ( − α i +1 σ i ; ( a α i , ∞ )) , where, for any R ⊆ R , R (cid:96) denotes the cartesian product of R taken (cid:96) times. Hence,(4.7) M ( (cid:96) ) n,d = (cid:88) [ σσσ ] ∈ Γ c n ([ σσσ ]) (cid:88) ( α ,...,α (cid:96) ) ∈{ ,..., m } (cid:96) (cid:96) ! (cid:18)(cid:0) nd (cid:1) (cid:96) (cid:19) ( − (cid:80) i α i + l E (cid:34) (cid:96) (cid:89) i =1 σ i ; ( a α i , ∞ )) (cid:35) . By the scaling of C ( σ ), note that, for any σ ∈ F d − ( U n,d ) and any a ∈ R , σ, ( a, ∞ )) = [ C ( σ ) > a + d log n − log( d !) n ]Combining this with (4.1) and (2) from Definition 4.1, observe that (cid:96) ! (cid:18)(cid:0) nd (cid:1) (cid:96) (cid:19) E (cid:34) (cid:96) (cid:89) i =1 σ i ; ( a α i , ∞ )) (cid:35) ∼ n d(cid:96) ( d !) (cid:96) (cid:96) (cid:89) i =1 (cid:18) − a α i + d log n − log( d !) n (cid:19) n − κ i . Here κ , . . . , κ (cid:96) ≥ σ , . . . , σ (cid:96) . From this, irrespective of κ , . . . , κ (cid:96) , we havelim n →∞ (cid:96) ! (cid:18)(cid:0) nd (cid:1) (cid:96) (cid:19) E (cid:34) (cid:96) (cid:89) i =1 σ i ; ( a α i , ∞ )) (cid:35) = e − (cid:80) (cid:96)i =1 a αi . ubstituting this in (4.7) and using (4.6), we derive (4.5) as follows :lim n →∞ M ( (cid:96) ) n,d = (cid:32) m (cid:88) j =1 [ e − a j − e − b j ] (cid:33) (cid:96) = (cid:18)(cid:90) I e − x dx (cid:19) (cid:96) . (cid:3) Extremal death times.
We now discuss death times in the persistence diagram. First,we need the following lemma, which explains why nearest neighbour distances approximatedeath times in a persistence diagram.
Lemma 4.3.
Fix d ≥ . Let N d − ( Y n,d ( p n )) be the number of isolated ( d − − faces in Y n,d ( p n ) with p n as in (4.4) . Then lim n →∞ E | β d − ( Y n,d ( p n )) − N d − ( Y n,d ( p n )) | = 0 . This lemma essentially follows from ideas in the proofs in [41, Theorem 1.10]. But, to thebest of our knowledge, it has not been explicitly mentioned anywhere. The proof for the case d ≥ { D i } denotes the set of all death times. Scaleeach death time D i → nD i − d log n + log( d !) =: ¯ D i and let(4.8) P Dn,d = { ¯ D i } denote the set of scaled death times related to H d − ( U n,d ) . For p n as defined in (4.4), observethat for a c fixed and n large enough we have P Dn,d ( c, ∞ ) = β d − ( U n,d ( p n )) , P Cn,d ( c, ∞ ) = N d − ( U n,d ( p n )) . The above observation along with Lemma 4.3 yields the following easy corollary.
Corollary 4.4.
Let c ∈ R be arbitrary but fixed. Then lim n →∞ E | P Dn,d ( c, ∞ ) − P Cn,d ( c, ∞ ) | = 0 . Now we are ready to prove the main theorem of the section.
Proof of Theorem 1.3.
Let I := ∪ mj =1 ( a j − , a j ] ⊆ R be some finite union of disjoint intervals.Since P poi is simple and does not contain atoms, again as per Lemma A.1, to prove thedesired result, it suffices to show that:( i ) lim n →∞ E [ P Dn,d ( I )] = E [ P poi ( I )] and ( ii ) P Dn,d ( I ) d ⇒ P poi ( I ) as n → ∞ . From triangle inequality,(4.9) | P Dn,d ( I ) − P Cn,d ( I ) | ≤ m (cid:88) j =1 | P Dn,d ( a j , ∞ ) − P Cn,d ( a j , ∞ ) | . Combining this with Corollary 4.4 and the factorial moment convergence in (4.5), we get (i).Arguing as above, we also have | P Dn,d ( I ) − P Cn,d ( I ) | → n → ∞ . Combining this with Slutsky’s theorem [28, Chapter 3, Corollary 3.3] and the distributionalconvergence of P Cn,d ( I ) proven in Theorem 4.2, we obtain (ii) as desired. (cid:3) .3. Extremal weights in the d − minimal spanning acycle. Again fix d ≥ . Viewing U n,d as a weighted simplicial complex, let M denote its d − minimal spanning acycle. And let(4.10) P Mn,d := { nw ( σ ) − d log n + log( d !) : σ ∈ M } denote the set of scaled weights of the faces in the d − minimal spanning acycle of U n,d . UsingTheorems 3.23 and 1.3, we get the following theorem immediately.
Theorem 4.5. As n → ∞ , P Mn,d converges in distribution to the Poisson point process P poi ,where P poi is as in Theorem 1.3. Recall from Corollary 3.16 and Theorem 3.23, we have that P Cn,d ⊂ P Dn,d = P Mn,d . But aswe have shown, the extremal points of the three point sets coincide asymptotically.5.
Random d − complexes : I.I.D. Generic Weights with Perturbation Here, in contrast to the previous section, we deal with simplicial complexes whose d − faceweights are perturbations of some generic i.i.d. distribution. Our key result here is that ifthe perturbations decay sufficiently fast, then the point process convergence results from theprevious section continue to hold. The proof is a transparent consequence of our stabilityresult (Theorem 1.5). We first define our model. Definition 5.1.
Let d ≥ be some integer. Consider n vertices and let K dn be the complete d − skeleton on them. Let φ (cid:48) : K dn → [0 , be the weight function with the following properties: (1) φ (cid:48) ( σ ) = 0 for σ ∈ (cid:83) d − i =0 F i ( K dn ) , and (2) φ (cid:48) ( σ ) = φ ( σ ) + (cid:15) n ( σ ) for σ ∈ F d ( K dn ) , where { φ ( σ ) : σ ∈ F d ( K dn ) } are real valuedi.i.d. random variables with some generic distribution F : R ⊆ R → [0 , perturbedrespectively by { (cid:15) n ( σ ) : σ ∈ F d ( K dn ) } , another set of real valued random variables (notnecessarily independent or identically distributed).The generically weighted d − complex with perturbation L (cid:48) n,d is the simplicial complex K dn weighted by φ (cid:48) . Associated with L (cid:48) n,d is the canonical simplicial process given by the filtration {L (cid:48) n,d ( t ) : t ∈ R } , where L (cid:48) n,d ( t ) = { σ ∈ K dn : φ (cid:48) ( σ ) ≤ t } . For ease of use, we shall write σ ∈ L (cid:48) n,d to mean σ ∈ K dn . Similarly, F i ( L (cid:48) n,d ) shall mean F i ( K dn ) and so on. Now consider the following three scaled point processes on R . (1) P C (cid:48) n,d := { n F ( C (cid:48) ( σ )) − d log n + log( d !) : σ ∈ F d − ( L (cid:48) n,d ) } , where, for σ ∈ L (cid:48) n,d ,C (cid:48) ( σ ) := min τ ∈F d ( L (cid:48) n,d ) ,τ ⊃ σ φ (cid:48) ( τ ) . (2) P D (cid:48) n,d := { n F ( D (cid:48) i ) − d log n + log( d !) } , where { D (cid:48) i } denotes the set of death times inthe persistence diagram of H d − ( L (cid:48) n,d ) (see Definition 2.3).(3) P M (cid:48) n,d := { n F ( φ (cid:48) ( σ )) − d log n + log( d !) : σ ∈ M (cid:48) } , where M (cid:48) is a d − minimal spanningacycle in L (cid:48) n,d (see (2.10)).One can look at the simplicial complex K dn weighted by φ alone, which we shall refer to as L n,d . With respect to this L n,d , define C ( σ ) , D i , and M, and P Cn,d , P Dn,d , and P Mn,d , exactly as bove. Also, similarly define F i ( L n,d ) etc. In relation to the perturbation random variables,let (cid:107) (cid:15) n (cid:107) ∞ := max σ ∈F d ( L (cid:48) n,d ) | (cid:15) n ( σ ) | ; (cid:107) (cid:15) n (cid:107) := (cid:88) σ ∈F d ( L (cid:48) n,d ) | (cid:15) n ( σ ) | . Theorem 5.2.
Suppose that F is continuous and strictly increasing. Then the point pro-cesses P Cn,d , P Dn,d , and P Mn,d , converge in distribution to P poi as n → ∞ . Proof.
Clearly, { F ( φ ( σ )) } σ ∈F d ( L n,d ) are i.i.d. uniform [0 ,
1] random variables. The desiredresult is now immediate from Theorems 4.2, 1.3 and 4.5 (cid:3)
We now state our main perturbation result (Theorem 5.3) and then describe Corollaries 5.5and 5.6, which give simpler bounds to verify assumptions of this result. Note that weadditionally need Lipschitz continuity of F . Theorem 5.3.
Suppose that F is Lipschitz continuous and strictly increasing. If n (cid:107) (cid:15) n (cid:107) ∞ → in probability, then each of P C (cid:48) n,d , P D (cid:48) n,d , and P M (cid:48) n,d converges in distribution to P poi . We need a lemma before we prove the above result. The first inequality is straightforwardand the next two follow from Theorem 1.5 for p = ∞ , and Theorem 3.23. Lemma 5.4.
For fixed n, d ≥ , we have the following inequalities: max σ ∈F d − ( L (cid:48) n,d ) | C (cid:48) ( σ ) − C ( σ ) | ≤ (cid:107) (cid:15) n (cid:107) ∞ , inf γ max i | D (cid:48) i − γ ( D i ) | ≤ || φ (cid:48) − φ || ∞ ≤ (cid:107) (cid:15) n (cid:107) ∞ , where the infimum is over all possible bijections γ : { D (cid:48) i } → { D i } , and inf γ max i | φ (cid:48) ( σ (cid:48) i ) − γ ( φ ( σ i )) | ≤ || φ (cid:48) − φ || ∞ ≤ (cid:107) (cid:15) n (cid:107) ∞ , where the infimum is over all possible bijections γ : { φ (cid:48) ( σ (cid:48) ) : σ (cid:48) ∈ M (cid:48) } → { φ ( σ ) : σ ∈ M } . Proof of Theorem 5.3.
We only show that P D (cid:48) n,d ⇒ P poi as n → ∞ using Lemma 5.4, asthe other results follow similarly. Let d v be the vague metric given in (A.1). Suppose weshow that d v ( P D (cid:48) n,d , P Dn,d ) → n → ∞ . Then using Slutsky’s theorem ([28,Chapter 3, Corollary 3.3]), Theorem 5.2, and Lemma A.1, it follows that P D (cid:48) n,d ⇒ P poi asdesired. It thus suffices to prove that d v ( P D (cid:48) n,d , P Dn,d ) → n → ∞ . Let d B be as in Definition 2.9. Then by Lemma 5.4, we have that d B ( P D (cid:48) n,d , P Dn,d ) ≤ ζn (cid:107) (cid:15) n (cid:107) ∞ . here we have assumed that the Lipschitz constant is ζ . Now by assumption, d B ( P D (cid:48) n,d , P Dn,d ) → (cid:15) ∈ (0 ,
1) and choose δ = (cid:15)k ∈ (0 ,
1) for some k ≥ . Then we have P { d v ( P D (cid:48) n,d , P Dn,d ) > (cid:15) } ≤ P { d v ( P D (cid:48) n,d , P Dn,d ) > (cid:15), d B ( P D (cid:48) n,d , P Dn,d ) ≤ δ } + P { d v ( P D (cid:48) n,d , P Dn,d ) > (cid:15), d B ( P D (cid:48) n,d , P Dn,d ) > δ }≤ P { λ P Dn,d ( K (cid:15) ) δ > (cid:15)/ } + P { d B ( P D (cid:48) n,d , P Dn,d ) > δ } , ≤ P (cid:26) P Dn,d ( K (cid:15) ) > k λ (cid:27) + P { d B ( P D (cid:48) n,d , P Dn,d ) > δ } . (5.1)For the second inequality, we have used (A.2) with λ and the compact set K (cid:15) as given there.Since the second term in (5.1) converges to 0 as n → ∞ , using Theorem 5.2, we have thatlim sup n →∞ P { d V ( P D (cid:48) n,d , P Dn,d ) > (cid:15) } ≤ lim n →∞ P (cid:26) P Dn,d ( K (cid:15) ) > k λ (cid:27) = P (cid:26) P poi ( K (cid:15) ) > k λ (cid:27) . Now letting k → ∞ , we have that d V ( P D (cid:48) n,d , P Dn,d ) → (cid:3) Except for a few trivial cases, for dependent random variables { (cid:15) n ( σ ) } (even if they areidentically distributed), determining the distribution of the maximum (cid:107) (cid:15) n (cid:107) ∞ is not easy andhence might restrict application of Theorem 5.1. The next two corollaries require simpleexpectation bounds to verify the assumptions of Theorem 5.3 for the case of dependent (cid:15) n ( σ )’s and for ease of stating the results, we consider only identically distributed weights. Corollary 5.5.
For each n, let { ψ ( σ ) : σ ∈ F d ( L (cid:48) n,d ) } have the same distribution as thereal valued random variable ψ which, for some s > , satisfies E [ e s | ψ | ] < ∞ . Define (cid:15) n ( σ ) = a − n ψ ( σ ) where a n is a sequence such that a n = ω ( n log n ) . If F is Lipschitz continuous andstrictly increasing, then, each of P C (cid:48) n,d , P D (cid:48) n,d , and P M (cid:48) n,d converges in distribution to P poi .Proof. Using Jensen’s inequality for the second inequality below and since |F d ( L (cid:48) n,d ) | ≤ n d +1 ,s E [ (cid:107) (cid:15) n (cid:107) ∞ ] ≤ a n log e s E (cid:20) max σ ∈F d ( L(cid:48) n,d ) | ψ ( σ ) | (cid:21) ≤ a n log E (cid:104) e s max σ ∈F d ( L(cid:48) n,d ) | ψ ( σ ) | ] (cid:105) = 1 a n log E (cid:34) max σ ∈F d ( L (cid:48) n,d ) e s | ψ ( σ ) | (cid:35) ≤ a n log( n d +1 E [ e s | ψ | ]) . Hence n || (cid:15) n || ∞ → n → ∞ . The result now follows from Theorem 5.3. (cid:3)
Corollary 5.6.
For each n, let { (cid:15) n ( σ ) : σ ∈ F d ( L (cid:48) n,d ) } be identically distributed randomvariables with E [ (cid:15) n ( σ )] = o ( n − d − ) for each σ. If F is Lipschitz continuous and strictlyincreasing, then, each of P C (cid:48) n,d , P D (cid:48) n,d , and P M (cid:48) n,d converges in distribution to P poi .Proof. Using Markov’s inequality and (cid:107) (cid:15) n (cid:107) ∞ ≤ (cid:107) (cid:15) n (cid:107) , the result follows from Theorem 5.3. (cid:3) Here w is the small omega notation. .1. Limits for Expected Lifetime Sums :
Reconsider the perturbed random complex L (cid:48) n,d as in Definition 5.1. For simplicity, assume that the generic distribution F is uniform[0 ,
1] distribution, i.e., the weights φ ( σ ) are i.i.d. uniform [0 ,
1] random variables. Recallthat {L (cid:48) n,d ( t ) : t ∈ [0 , } yields a canonical filtration of the simplicial complex. Further, let L n,d , L (cid:48) n,d denote the lifetime sums (see (1.1)) respectively in the two models L n,d and L (cid:48) n,d .Then by our stability result (Theorem 1.5), we have | L n,d − L (cid:48) n,d | ≤ (cid:88) σ ∈F d | (cid:15) n ( σ ) | = 2 (cid:107) (cid:15) n (cid:107) . Now combining this observation with [29] and [33, Theorem 1.2] (see (1.4)) and furtherassuming that sup σ ∈F d E [ | (cid:15) n ( σ ) | ] = o ( n − ), we obtain E [ L (cid:48) n, ] → ζ (3) , E [ L (cid:48) n,d ] = Θ( n d − ) , for d ≥ , where ζ is Riemann’s zeta function. We have thus extended the lifetime sum results foruniformly weighted d − complexes to noisy/perturbed versions of the same. Acknowledgements
This research was supported through the program “Research in Pairs” by the Mathema-tisches Forschungsinstitut Oberwolfach in 2015.
Appendix A. Convergence of Point processes
We discuss here convergence of point processes under vague topology. For notations anddefinitions, see Subsection 2.1. Firstly, it is known that ( M p ( R ) , M p ( R )) is metrizable as acomplete, separable metric space ([58, Chapter 3, Proposition 3.17]). In the proof of thisproposition, the vague metric d v has been defined which we describe next.Let { G i } be the collection of open intervals in R with rational end points and let { h j } be suitable piece-wise linear approximations to the indicator function of these sets. Thefunctions h j are chosen so that they lie in C + c ( R ) and are Lipschitz continuous (while it isnot explicitly highlighted, from the definition of h j in the proof of [58, Chapter 3, Proposition3.17], one can check that it is Lipschitz). Then for m , m ∈ M p ( R ) , (A.1) d v ( m , m ) := ∞ (cid:88) j =1 − exp {−| m ( h j ) − m ( h j ) |} j ≤ ∞ (cid:88) j =1 min {| m ( h j ) − m ( h j ) | , } j . Since probability measures on Polish spaces (i.e., complete, separable metric spaces) aremore tractable, it is of importance to us that M p ( R ) is a Polish space. The following is anoft-used result to prove weak convergence to a simple point process on R . Lemma A.1. [58, Proposition 3.22]
Let { P n } , P be point processes on R with P beingsimple. Let I ( R ) be the collection of all finite union of intervals in R . Suppose that for each I ∈ I ( R ) , with P { P ( ∂I ) = 0 } = 1 , we have lim n →∞ P { P n ( I ) = 0 } = P { P ( I ) = 0 } and lim n →∞ E [ P n ( I )] = E [ P ( I )] < ∞ . Then P n ⇒ P in M p ( R ) . e now prove a lemma that will be useful when combining results from computationaltopology (which uses bottleneck distance) and point process theory (vague topology). Lemma A.2.
The topology of bottleneck distance on M p ( R ) is stronger than that of vaguetopology on M p ( R ) .Proof. The desired result follows if we show that for all m ∈ M p ( R ) and (cid:15) >
0, there exists δ > m, (cid:15) ) such that d B ( m , m ) ≤ δ implies that d v ( m , m ) < (cid:15) .Set δ = d B ( m , m ) and, without loss of generality, assume that δ < . Further let γ : supp ( m ) → supp ( m ) be the bijection such that max x ∈ supp( m ) | x − γ ( x ) | ≤ δ . Now, thefollowing is enough to prove the above claim: For a given (cid:15) >
0, there exists constant λ anda compact set K (cid:15) (depending on (cid:15) alone) such that the following bound holds:(A.2) d v ( m , m ) ≤ λm ( K (cid:15) ) δ + (cid:15) . To prove the above bound, first note that for a given (cid:15) >
0, from (A.1) we can choose k such that the following holds: d v ( m , m ) ≤ k (cid:88) j =1 | m ( h j ) − m ( h j ) | + (cid:15) . Let us set K j to be the compact support of h j and λ j to be the Lipschitz constant. By thedefinition of Bottleneck distance, we have that for any compact set K (A.3) m ( K ) ≤ m ( K δ ) ≤ m ( K δ ) , where K δ := { x ∈ R : ∃ y ∈ K s.t. | x − y | ≤ δ } . Fix a j ∈ { , . . . , k } and set M = supp ( m ) , M = supp ( m ). Now by definition of m ( h j ) and that h j is a λ j -Lipschitz contin-uous function supported on K j , we have that | m ( h j ) − m ( h j ) | = | (cid:88) x ∈ M h j ( x ) − (cid:88) x ∈ M h j ( x ) | = | (cid:88) x ∈ M [ h j ( x ) − h j ( γ ( x ))] |≤ λ j [ m ( K j ) + m ( K j )] δ ≤ λ j m ( K j ) δ, where in the last inequality we have used (A.3) and the fact that δ <
1. Now setting λ = (cid:80) kj =1 λ j and K (cid:15) = ∪ kj =1 K j , we get the desired bound (A.2). (cid:3) References
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E-mail address : [email protected] (GT) Faculty of Electrical Engineering, Technion - Israel Institute of Technology,Haifa, Israel
E-mail address : [email protected] (DY) Statistics and Mathematics Unit, Indian Statistical Institute, Bengaluru, India
E-mail address : [email protected]@isibang.ac.in