aa r X i v : . [ m a t h . P R ] D ec RANDOM GEOMETRIC COMPLEXES
MATTHEW KAHLE
Abstract.
We study the expected topological properties of ˇCechand Vietoris-Rips complexes built on random points in R d . We findhigher dimensional analogues of known results for connectivity andcomponent counts for random geometric graphs. However, higherhomology H k is not monotone when k > k > Introduction
The random geometric complexes studied here are simplicial com-plexes built on an i.i.d. random points in Euclidean space R d . Weidentify here the basic topological features of these complexes. In par-ticular, we identify intervals of vanishing and non-vanishing for eachhomology group H k , and give asymptotic formulas for the expectedrank of homology when it is non-vanishing.There are several motivations for studying this. The area of topo-logical data analysis has been very active lately [29, 12], and there isa need for a probabilistic null hypothesis to compare with topologicalstatistics of point cloud data [8].One approach to this problem was taken by Niyogi, Smale, andWeinberger [24], who studied the model where n points are sampleduniformly and independently from a compact manifold M embeddedin R d , and estimates were given for how large n must be in order to“learn” the topology of M with high probability. Their approach wasto take balls of radius r centered at the n points and approximate themanifold by the ˇCech complex; provided that r is chosen carefully, oncethere are enough balls to cover the manifold, one has a finite simplicial Date : October 23, 2018.Supported in part by Stanford’s NSF-RTG grant in geometry & topology. complex with the homotopy type of the manifold so in particular onecan compute homology groups and so on.The main technical innovation in [24] is a geometric method forbounding above the number of random balls needed to cover the man-ifold, given some information about the curvature of the manifold’sembedding. The assumption here is that one already knows how large r must be, or that one at least has enough information about the ge-ometry of the embedding of M in order to determine r . (In a secondarticle, they are able to recapture the topology of the manifold, even inthe more difficult setting when Gaussian noise is added to every sam-pled point [25]. Still, one needs some information about the embeddingof the manifold.)In this article we study both random Vietoris-Rips and ˇCech com-plexes for fairly general distributions on Euclidean space R d , and mostimportantly, allowing the radius of balls r to vary from 0 to ∞ . Weidentify thresholds for non-vanishing and vanishing of homology groups H k and also derive asymptotic formulas and bounds on expectationsof the Betti numbers β k in terms of n and r . It is well understood incomputational topology that persistent homology is more robust thanhomology alone (see for example the stability results of Cohen-Steiner,Edelsbrunner, and Harer [10]), and one might not know anything aboutthe underlying space, so in practice one computes persistent homologyover a wide regime of radius [29].There is also a close connection to geometric probability, and inparticular the theory of geometric random graphs. Some of our resultsare higher-dimensional analogues of thresholds for connectivity andcomponent counts in random geometric graphs due to Penrose [26],and we must also use Penrose’s results several times. However, animportant contrast is that the properties studied here are decidedlynon-monotone. In particular, for each k there is an interval of radius r for which the homology group H k = 0, and with the expected rankof homology E [ β k ] roughly unimodal in the radius r , but we also showthat for large enough or small enough radius, H k = 0.This paper can also be viewed in the context of several recent articleson the topology of random simplicial complexes [21, 23, 2, 18, 19, 27].This article discusses a fairly general framework for random complexes,since one has the freedom to choose the underlying density function,hence an infinite- dimensional parameter space.The probabilistic method has given non-constructive existence proofs,as well as many interesting and extremal examples in combinatorics ANDOM GEOMETRIC COMPLEXES 3 [1], geometric group theory [15], and discrete geometry [22]. Randomspaces will likely provide objects of interest to topologists as well.The problems discussed here were suggested, and the basic regimesdescribed, in Persi Diaconis’s MSRI talk in 2006 [11]. Some of theresults in this article may have been discovered concurrently and in-dependently by other researchers; it seems that Yuliy Barishnikov andShmuel Weinberger have also thought about similar things [3]. How-ever, we believe that this article fills a gap in the literature and hopethat it is useful as a reference.1.1.
Definitions.
We require a few preliminary definitions and con-ventions.
Definition 1.1.
For a set of points X ⊆ R d , and positive distance r ,define the geometric graph G ( X ; r ) as the graph with vertices V ( G ) = X and edges E ( G ) = {{ x, y } | d ( x, y ) ≤ r } . Definition 1.2.
Let f : R d → R be a probability density function,let x , x , . . . be a sequence of independent and identically distributed d -dimensional random variables with common density f , and let X n = { x , x , . . . , x n } . The geometric random graph G ( X n ; r ) is the geomet-ric graph with vertices X n , and edges between every pair of vertices u, v with d ( u, v ) ≤ r . Throughout the article we make mild assumptions about f , in par-ticular we assume that f is a bounded Lebesgue-measurable function,and that Z R d f ( x ) dx = 1(i.e. that f actually is a probability density function).In the study of geometric random graphs [26] r usually depends on n , and one studies the asymptotic behavior of the graphs as n → ∞ . Definition 1.3.
We say that G ( X n ; r n ) asymptotically almost surely(a.a.s.) has property P if Pr ( G ( X n ; r ) ∈ P ) → as n → ∞ . The main objects of study here are the ˇCech and Vietoris-Rips com-plexes on X n , which are simplicial complexes built on the geomet-ric random graph G ( X n ; r ). A historical comment: the Vietoris-Ripscomplex was first introduced by Vietoris in order to extend simplicialhomology to a homology theory for more general metric spaces [28].Eliyahu Rips applied the same complex to the study of hyperbolic MATTHEW KAHLE groups, and Gromov popularized the name Rips complex [14]. Thename “Vietoris-Rips complex” is apparently due to Hausmann [17].Denote the closed ball of radius r centered at a point p by B ( p, r ) = { x | d ( x, p ) ≤ r } . Definition 1.4.
The random ˇCech complex C ( X n ; r ) is the simplicialcomplex with vertex set X n , and σ a face of C ( X n ; r ) if \ x i ∈ σ B ( x i , r/ = ∅ . Definition 1.5.
The random Vietoris-Rips complex R ( X n ; r ) is thesimplicial complex with vertex set X n , and σ a face if B ( x i , r/ ∩ B ( x j , r/ = ∅ for every pair x i , x j ∈ σ . Equivalently, the random Vietoris-Rips complex is the clique com-plex of G ( X n ; r ).We are interested in the topological properties, in particular the van-ishing and non-vanishing, and expected rank of homology groups, of therandom ˇCech and Vietoris-Rips complexes, as r varies. Qualitativelyspeaking, the two kinds of complexes behave very similarly. Howeverthere are important quantitative differences and one of the goals of thisarticle is to point these out.Throughout this article, we use Bachmann-Landau big- O , little- O ,and related notations. In particular, for non-negative functions g and h , we write the following. • g ( n ) = O ( h ( n )) means that there exists n and k such that for n > n , we have that g ( n ) ≤ k · h ( n ). (i.e. g is asymptoticallybounded above by h , up to a constant factor.) • g ( n ) = Ω( h ( n )) means that there exists n and k such that for n > n , we have that g ( n ) ≥ k · h ( n ). (i.e. g is asymptoticallybounded below by h , up to a constant factor.) • g ( n ) = Θ( h ( n )) means that g ( n ) = O ( h ( n )) and g ( n ) = Ω( h ( n )).(i.e. g is asymptotically bounded above and below by h , up toconstant factors.) • g ( n ) = o ( h ( n )) means that for every ǫ >
0, there exists n such that for n > n , we have that g ( n ) ≤ ǫ · h ( n ). (i.e. g isdominated by h asymptotically.) • g ( n ) = ω ( h ( n )) means that for every k >
0, there exists n suchthat for n > n , we have that g ( n ) ≥ k · h ( n ). (i.e. g dominates h asymptotically.) ANDOM GEOMETRIC COMPLEXES 5
When we discuss homology H k we mean either simplicial homologyor singular homology, which are isomorphic. Our results hold withcoefficients taken over any field.Finally, we use µ ( S ) to denote Lebesgue measure for any measurableset S ⊂ R d , and k x k to denote the Euclidean norm of x ∈ R d .2. Summary of results
It is known from the theory of random geometric graphs [26] thatthere are four main regimes of parameter (sometimes called regimes ),with qualitatively different behavior in each. The same is true for thehigher dimensional random complexes we build on these graphs. Thefollowing is a brief summary of our results.In the subcritical and critical regimes, our results hold fairly gen-erally, for any distribution on R d with a bounded measurable densityfunction.In the subcritical regime, r = o ( n − /d ), the random geometric graph G ( X n ; r ) (and hence the simplicial complexes we are interested in) con-sists of many disconnected pieces. Here we exhibit a threshold for H k ,from vanishing to non-vanishing, and provide an asymptotic formulafor the k th Betti number E [ β k ], for k ≥ r = Θ( n − /d ), the components of the randomgeometric graph start to connect up and the giant component emerges.In other words, this is the regime wherein percolation occurs, and it issometimes called the thermodynamic limit. Here we show that E [ β k ] =Θ( n ) and Var[ β k ] = Θ( n ) for every k .The results in the subcritical and critical regimes hold fairly gen-erally, for any distribution on R d with a bounded measurable densityfunction. In the supercritical and connected regimes, our results are foruniform distributions on smoothly bounded convex bodies in dimension d . In the supercritical regime, r = ω ( n − /d ). We put an upper boundon E [ β k ] to show that it grows sub-linearly, so the linear growth of theBetti numbers in the critical regime is maximal. Here our results arefor the Vietoris-Rips complex, and the method is a Morse-theoretic ar-gument. The combination of geometric probability and discrete Morsetheory used for these bounds is the main technical contribution of thearticle.The connected regime, r = Ω((log n/n ) /d ), is where G ( X n ; r ) isknown to become connected [26]. In this case we show that the ˇCechcomplex is contractible and the Vietoris-Rips complex is approximatelycontractible, in the sense that it is k -connected for any fixed k . (This MATTHEW KAHLE means that the homotopy groups π i vanish for i ≤ k , which implies inturn that the homology groups H i vanish for i ≤ k as well.)Despite non-monotonicity, we are able to exhibit thresholds for van-ishing of H k . For every k ≥
1, there is an interval in which H k = 0and outside of which H k = 0, so every higher homology group passesthrough two thresholds.The rest of the article is organized as follows. In Section 3 we con-sider the subcritical regime of radius, in Section 4 the critical regime,in Section 5 the supercritical regime, and in Section 6 the connectedregime. In Sections 5 and 6 we assume that the underlying distributionis uniform on a smoothly bounded convex body mostly as a matter ofconvenience, but similar methods should apply in a more general set-ting. In Section 7 we discuss open problems and future directions.3. Subcritical
In this regime, we exhibit a vanishing to non-vanishing threshold forhomology H k , and in the non-vanishing regime compute the asymp-totic expectation of the Betti numbers β k , for k ≥
1. (The case k = 0,the number of path components, is examined in careful detail by Pen-rose [26], Ch. 13.) As a corollary, we also obtain information aboutthe threshold where homology passes from vanishing to non-vanishinghomology. We emphasize that the results in this section do not dependin any essential way on the distribution on R d , although we make themild assumption that the underlying density function f is bounded andmeasurable.3.1. Expectation.Theorem 3.1. [Expectation of Betti numbers, Vietoris-Rips complex]For d ≥ , k ≥ , ǫ > , and r n = O ( n − /d − ǫ ) , the expectation of the k th Betti number E [ β k ] of the random Vietoris-Rips complex R ( X n ; r ) satisfies E [ β k ] n k +2 r d (2 k +1) → C k , as n → ∞ where C k is a constant that depends only on k and theunderlying density function f . (We note that this result holds for all k , even when k ≥ d .)Using similar methods, we also prove the following about the randomˇCech complex. ANDOM GEOMETRIC COMPLEXES 7
Theorem 3.2. [Expectation of Betti numbers, ˇCech complex] For d ≥ , ≤ k ≤ d − , ǫ > , and r = O ( n − /d − ǫ ) , the expectation of the k thBetti number E [ β k ] of the random ˇCech complex C ( X n ; r ) satisfies E [ β k ] n k +2 r d ( k +1) → D k , as n → ∞ where D k is a constant that depends only on k and theunderlying density function f . One feature that distinguishes the ˇCech complex from the Vietoris-Rips complex is that a Cech complex is always homotopy equivalent towhatever it covers (this follows form the nerve theorem, i.e. Theorem10.7 in [4]). So in particular H k = 0 when k ≥ d .In both cases we will see that almost all of the homology is con-tributed from a single source: whatever is the smallest possible vertexsupport for nontrivial homology. For the Vietoris-Rips complex thiswill be the boundary of the cross-polytope, and for the ˇCech complexthe empty simplex. Definition 3.3.
The ( k + 1) -dimensional cross-polytope is defined tobe the convex hull of the k + 2 points {± e i } , where e , e , . . . , e k +1 arethe standard basis vectors of R k +1 . The boundary of this polytope is a k -dimensional simplicial complex, denoted O k . Simplicial complexes which arise as clique complexes of graphs aresometimes called flag complexes . A useful fact in combinatorial topol-ogy is the following; for a proof see [19].
Lemma 3.4. If ∆ is a flag complex, then any nontrivial element of k -dimensional homology H k (∆) is supported on a subcomplex S with atleast k + 2 vertices. Moreover, if S has exactly k + 2 vertices, then S is isomorphic to O k . We also use results for expected subgraph counts in geometric ran-dom graphs.Recall that a subgraph H ≤ G is said to be an induced subgraph iffor every pair of vertices x, y ∈ V ( H ), we have { x, y } is an edge of H if and only if { x, y } is an edge of G . Definition 3.5.
A connected graph is feasible if it is geometricallyrealizable as an induced subgraph.
For example the complete bipartite graph K , is not feasible, sinceit is not geometrically realizable as an induced subgraph of a geometricgraph in R , since there must be at least one edge between the sevendegree-one vertices. MATTHEW KAHLE
Denote the number of induced subgraphs of G ( X n ; r ) isomorphicto H by G n ( H ), and the number of components isomorphic to H by J n ( H ). Recall that f is the underlying density function. For a feasiblesubgraph H of order k , and Y ∈ ( R d ) k define the indicator function h H ( Y ) on sets Y of k elements in R d by h H (( Y )) = 1 if the geometricgraph G ( Y,
1) is isomorphic to H , and 0 otherwise. Let µ H = k ! − Z R d f ( x ) k dx Z ( R d ) k − h H ( { , x , . . . , x k − } ) d ( x , . . . x k − ) . Penrose proved the following [26].
Theorem 3.6 (Expectation of subgraph counts, Penrose) . Supposethat lim n →∞ ( r ) = 0 , and H is a connected feasible graph of order k ≥ .Then lim n →∞ r − d ( k − n − k E ( G n ( H )) = lim n →∞ r − d ( k − n − k E ( J n ( H )) = µ H . Together with our topological and combinatorial tools, Theorem 3.6will be sufficient to prove Theorem 3.1. To prove Theorem 3.2 wealso require a hypergraph analogue of Theorem 3.6, established by theauthor and Meckes in Section 3 of [20], which we state when it is needed.
Proof of Theorem 3.1.
The intuition is that in the sparse regime, al-most all of the homology is contributed by vertex-minimal spheres.
Definition 3.7.
For a simplicial complex ∆ , let o k (∆) (or o k if contextis clear) denote the number of induced subgraphs of ∆ combinatoriallyisomorphic to the -skeleton of the cross-polytope O k , and let e o k (∆) denote the number of components of ∆ combinatorially isomorphic tothe -skeleton of the cross-polytope O k . Definition 3.8.
Let f = ik (∆) denote the number of k -dimensional faceson connected components with exactly i vertices. Similarly, let f ≥ ik (∆) denote the number of k -dimensional faces on connected componentscontaining at least i vertices. A dimension bound paired with Lemma 3.4 yields(3.1) e o k ≤ β k ≤ e o k + f ≥ k +3 k . One could work with f ≥ k +3 k directly, but it turns out to be sufficientto overestimate f ≥ k +3 k as follows. For each k -dimensional face in acomponent with at least 2 k +3 vertices, extend to a connected subgraphwith exactly 2 k + 3 vertices and (cid:0) k +12 (cid:1) + k + 2 edges. ANDOM GEOMETRIC COMPLEXES 9
Figure 1.
The case k = 2: the seventeen isomorphismtypes of subgraphs which arise when extending a 3-cliqueto a connected graph on 7 vertices with 7 edges. Eachsubgraph isomorphic to one of these can contribute atmost 1 to the sum bounding the error term f ≥ .For example, let k = 2; then(3.2) e o ≤ β ≤ e o + f ≥ . Up to isomorphism, the seventeen graphs that arise when extending a2-dimensional face (i.e. a 3-clique) to a minimal connected graph on 7vertices are exhibited in Figure 1.In particular, f ≥ ≤ P i =1 s i , where s i counts the number of sub-graphs isomorphic to graph i for some indexing of the seventeen graphsin Figure 1.Moreover, as noted in [26], the number of occurences of a given sub-graph Γ on v vertices is a positive linear combination of the inducedsubgraph counts for those graphs on v vertices which have Γ as a sub-graph.For an example of this, let G H denote the number of induced sub-graphs of G isomorphic to H , and let e G H denote the number of sub-graphs (not necessarily induced) of G isomorphic to H . If P is thepath on 3 vertices and K is the complete graph on 3 vertices, then e G P = 3 G K + G P . So for each i we can write s i as a positive linear combination ofinduced subgraph counts, and every type of induced subgraphs hasexactly 7 vertices. We take expectation of both sides of Equation 3.2, applying linearityof expectation, to obtain E [ e o ] ≤ E [ β ] ≤ E [ e o ] + E [ f ≥ ] ≤ E [ e o ] + E [ X i =1 s i ] ≤ E [ e o ] + X i =1 E [ s i ] . For each i , E [ s i ] = O ( n r d ), by Theorem 3.6. On the other hand, E [ e o ] = Θ( n r d ), also by Theorem 3.6. Since we are assuming that nr d → n → ∞ , we have shown that E [ f ≥ ] = o ( E [ e o ]). We con-clude that E [ β ] /E [ e o ] → n → ∞ . This gives E [ β ] = Θ( n r d ),as desired.The proof for k ≥ k + 3 vertices that can arise from the algorithm above is a constantthat only depends on k , so denote this constant by c k .So in general we will have E [ e o k ] ≤ E [ β k ] ≤ E [ e o k ] + E [ f ≥ k +3 k ] ≤ E [ e o k ] + E [ c k X i =1 s i ] ≤ E [ e o ] + c k X i =1 E [ s i ] . For each i = 1 , , . . . , c k we have E [ s i ] = O ( n k +3 r (2 k +2) d ) , and on the other hand E [ e o k ] = Θ( n k +2 r (2 k +1) d ) . Since nr d →
0, we conclude that E [ β k ] /E [ e o k ] →
1, and E [ β k ] = Θ( n k +2 r (2 k +1) d ) . The case k = 1 is slightly different. There are several ways of ex-tending a 2-clique (i.e. an edge) to a connected graph on 5 vertices and4 edges. In this case the graph must be a tree, and there are threeisomorphism types of trees on five vertices, shown in Figure 2. But inthis case counting these subgraphs will result in an underestimate for ANDOM GEOMETRIC COMPLEXES 11
Figure 2.
The case k = 1: the three isomorphism typesof trees on five vertices. Each subgraph isomorphic to oneof these can contribute at most 4 to the sum boundingthe error term f ≥ . Figure 3.
The regular 2 k -gons prove that the 1-skeleton of the cross-poytope O k is geometrically feasiblein the plane for every k . If r is slightly shorter than thelength of the main diagonal, components combinatoriallyisomorphic to this contribute to β k in the Vietoris-Ripscomplex. f ≥ . However, each tree has only four edges, and so one can obtain thebound f ≥ ≤ t + t + t ) , where t , t , t count the number of subgraphs isomorphic to the threetrees in Figure 2. The argument is then the same as in the case k ≥ d = 2. But theregular 2 k -gons provide examples of geometic realizations of the 1-skeleton of O k for every k , as in Figure 3. (This fact was previouslynoted by Chambers, de Silva, Erickson, and Ghrist in [9].) (cid:3) Proof of Theorem 3.2.
The argument for the ˇCech complex proceedsalong the same lines, mutatis mutandis, but with one important differ-ence. Again the dominating contribution to β k will come from vertex-minimal k -dimensional spheres, but for a ˇCech complex the smallestpossible vertex support that a simplicial complex with nontrivial H k can have is k + 2 vertices, coming from the boundary of a ( k + 1)-dimensional simplex.Let f S k denote the number of connected components isomorphic tothe boundary of a ( k + 1)-dimensional simplex. By the same argumentas before we have E [ f S k ] ≤ E [ β k ] ≤ E [ f S k ] + E [ f ≥ k +3 k ] . Deciding whether some set of k + 2 vertices span the boundary ofa ( k + 1)-dimensional simplex depends on higher intersections, so inparticular when k > r = o ( n − /d ) then E [ f S k ] = Θ( n k +2 r ( k +1) d ). On the other handwe have E [ f ≥ k +3 k ] = O ( n k +3 r ( k +2) d ). As before, since r = o ( n − /d ) thisis enough to give that lim n →∞ E [ β k ] /E [ f S k ] = 1 , and then E [ β k ] = Θ( n k +2 r ( k +1) d ) as desired. (cid:3) Vanishing / non-vanishing threshold.
To state the followingtheorems we assume that d ≥ k ≥ r is stillin the sparse regime, i.e. that r = o ( n − /d ). Theorem 3.9 (Threshold for non-vanishing of H k in the random Vi-etoris-Rips complex) . (1) If r = o (cid:16) n − k +2 d (2 k +1) (cid:17) , then a.a.s. H k ( V R ( n ; r )) = 0 , and (2) if r = ω (cid:16) n − k +2 d (2 k +1) (cid:17) , then a.a.s. H k ( V R ( n ; r )) = 0 .Proof. The first statement follows directly from Lemma 3.4 and Theo-rem 3.6; i.e. if r is too small then the connected components are simplytoo small to support nontrivial homology.For the second statement, we have from Theorem 3.1 that given thishypothesis on r we have that E [ β k ] → ∞ . This by itself is not enoughto establish that β k = 0 a.a.s. However it is established in Section 4of [20] that Var[ β k ] is of the same order of magnitude as E [ β k ], so thisfollows from Chebyshev’s inequality, as in [1], Chapter 4. (cid:3) ANDOM GEOMETRIC COMPLEXES 13
The corresponding result for ˇCech complexes is the following.
Theorem 3.10 (Threshold for non-vanishing of H k in the randomˇCech complex) . (1) If r = o (cid:16) n − k +2 d ( k +1) (cid:17) , then a.a.s. H k ( V R ( n ; r )) = 0 , and (2) if r = ω (cid:16) n − k +2 d ( k +1) (cid:17) , then a.a.s. H k ( V R ( n ; r )) = 0 .Proof. The proof is identical. The needed result for bounding the vari-ance of Var[ β k ] is established in Section 3 of [20]. (cid:3) Critical
The situation in the critical regime (or thermodynamic limit) is moredelicate to analyze. We are still able to compute the right order ofmagnitude for E [ β k ]: it grows linearly for every k . Theorem 4.1.
For either the random Vietoris-Rips and ˇCech com-plexes on a probability distribution on R d with bounded measurable den-sity function, if r = Θ( n − /d ) and k ≥ is fixed, then E [ β k ] = Θ( n ) .Proof. The proof is the same as in the previous section. For example,for the Vietoris-Rips complex we still have E [ e o k ] ≤ E [ β k ] ≤ E [ e o k ] + E [ f ≥ k +3 k ] . Penrose’s results for component counts extend in to the thermodynamiclimit, so in particular E [ e o k ] = Θ( n ) and E [ f ≥ k +3 k ] = O ( n ). The desiredresult follows. (cid:3) The thermodynamic limit is of particular interest since this is theregime where percolation occurs for the random geometric graph [26].Bollob´as recently exhibited an analogue of percolation on the k -cliquesof the Erd˝os-R´enyi random graph [5]. It would be interesting to knowif analogues of his result occurs in the random geometric setting.For example, define a graph with vertices for k -dimensional faces,with edges between a pair whenever they are both contained in thesame ( k + 1)-dimensional face. Does there exist a constant C k > n →∞ nr d > C k there is a.a.s. a unique k -dimensional “giant component” (suitably de-fined), and whenever lim n →∞ nr d < C k , all the components are a.a.s. “small”?5. Supercritical
For this section and the next we assume that the underlying dis-tribution is uniform on a smoothly bounded convex body. (Recallthat a smoothly bounded convex body is a compact, convex set, withnonempty interior.) This assumption is not only a matter of conve-nience – it would seem that some assumption on density must be madeto make topological statements in the denser regimes.For example, the geometric random graph becomes connected once r = Ω((log n/n ) /d ) for a uniform distribution on a convex body, but fora standard multivariate normal distribution r must be much larger, r =Ω((log log n/ log n ) / ), before the geometric random graph becomesconnected [26].The supercritical regime is where r = ω ( n − /d ). In this section wegive an upper bound on the expectation of the Betti numbers for therandom Vietoris-Rips complex in this regime. This upper bound issub-linear so this shows that the Betti numbers are growing the fastestin the thermodynamic limit.The main tool is discrete Morse theory – see the Appendix for thebasic terminology and the main theorem. A much more complete (andvery readable) introduction to discrete Morse theory can be found in[13]. Theorem 5.1.
Let R ( X n ; r ) be a random Vietoris-Rips complex on n points taken i.i.d. uniformly from a smoothly bounded convex body K in R d . Suppose r = ω ( n − /d ) , and write W = nr d . Then E [ β k ] = O ( W k e − cW n ) for some constant c > , and in particular E [ β k ] = o ( n ) . Here c depends on the convex body K but not on k . In fact it isapparent from the proof that c depends only on the volume of K andnot on its shape.Recall that µ ( S ) denotes the Lebesgue measure of S ⊂ R d , and k x k denotes the Euclidean norm of x ∈ R d . We require a geometric lemmain order to prove the main theorem. ANDOM GEOMETRIC COMPLEXES 15
Lemma 5.2 (Main geometric lemma) . There exists a constant ǫ d > such that the following holds. Let l ≥ and { y , . . . , y l } ⊂ R d be an ( l + 1) -tuple of points such that k y k ≤ k y k ≤ . . . ≤ k y l k , and k y k ≥ / . If k y − y k > and k y i − y j k ≤ for every other ≤ i < j ≤ l , then the intersection I = l \ i =1 B ( y i , ∩ B (0 , k y || ) satisfies µ ( I ) ≥ ǫ d . As the notation suggests, ǫ d depends on d but holds simultaneouslyfor all l . Proof of Lemma 5.2.
Let y m = ( y + y ) / y y . By assumption that k y − y k >
1, we have k y m − y k = k y m − y k > /
2. We now wish to check that y m is still not too faraway from any y i with 2 ≤ i ≤ l .Let θ be the positive angle between y − y and y − y . Since k y − y k ≤ k y − y k ≤
1, and k y − y k >
1, the law of cosinesgives that( y − y ) · ( y − y ) = k y − y kk y − y k cos θ = 12 ( k y − y k + k y − y k − k y − y k ) < k y m − y k = ( y m − y ) · ( y m − y )= (( y + y ) / − y ) · (( y + y ) / − y )= (( y − y ) / y − y ) / · (( y − y ) / y − y ) / / k y − y k + k y − y k + 2( y − y ) · ( y − y )) < (1 / / / , so k y m − y k < √ / . The same argument works as written with y replaced by y i with 3 ≤ i ≤ l . Now set ρ = 1 − √ /
2. By the triangle inequality B ( y m , ρ ) ⊂ B ( y i ,
1) for 1 ≤ i ≤ l . So we have that B ( y m , ρ ) ∩ B (0 , k y k ) ⊂ l \ i =1 B ( y i , ∩ B (0 , k y k ) . By the triangle inequality we have that k y m k ≤ k y k , and it followsthat µ ( B ( y m , ρ ) ∩ B (0 , k y k )) ≥ µ ( B ( y , ρ ) ∩ B (0 , k y k )) . Since k y k ≥ /
2, the quantity µ ( B ( y , ρ ) ∩ B (0 , k y k )) is boundedaway from zero, and in fact it attains its minimum when k y k = 1 / ǫ d equal to this minimum value of µ ( B ( y , ρ ) ∩ B (0 , k y k )), and thestatement of the lemma follows. (cid:3) Scaling everything in R d by a linear factor of r we rewrite the lemmain the form in which we will use it. Lemma 5.3. [Scaled geometric lemma] There exists a constant ǫ d > such that the following holds for every r > . Let l ≥ and { y , . . . , y k } ⊂ R d be an ( l + 1) -tuple of points, , such that k y k ≤ k y k ≤ . . . ≤ k y l k and (1 / r ≤ k y k . If k y − y k > r and k y i − y j k ≤ r for every other ≤ i < j ≤ l , then the intersection I = l \ i =1 B ( y i , r ) ∩ B (0 , k y k ) satisfies µ ( I ) ≥ ǫ d r d . We are ready to prove the main result of the section.
Proof of Theorem 5.1.
By translation and rescaling if necessary, as-sume without loss of generality that B (0 , ⊂ K . Since with prob-ability 1 no two points are the same distance to the origin, index thepoints X n = { x , . . . , x n } by distance to 0, i.e. k x k < k x k < · · · < k x n k . Now we define a discrete vector field V on R ( X n ; r ) in the sense ofdiscrete Morse theory, as discussed in the Appendix.Whenever possible pair face S = { x i , x i , . . . , x i j } with face { x i }∪ S with i < i and i as small as possible. This can be done in anyparticular order or simultaneously, and still each face gets paired at ANDOM GEOMETRIC COMPLEXES 17 most once, as follows. A face S can not get paired with two differenthigher dimensional faces { x a } ∪ S and { x b } ∪ S , since S will prefer thevertex with smaller index min { a, b } . On the other hand, it is also notpossible for S to get paired with both a lower dimensional face and ahigher dimensional face: Suppose S gets paired with { x a } ∪ S . Then k x a k < k s k for every s ∈ S , and no codimension 1 face F ≺ S couldalso get paired with S , since F would prefer to get paired with { x a }∪ F .Hence each face is in at most one pair and V is a well defined discretevector field. Moreover, the indices are decreasing along any V -path, sothere are no closed V -paths. Therefore V is a discrete gradient vectorfield.Let us bound the probability p k that a set of k + 1 vertices span a k -dimensional face in the Vietoris-Rips complex. Given the first vertex v , the other vertices would all have to fall in B ( v, r ), so p k = O ( r dk ).Recall that we defined W = nr d and we rewrite this bound as p k = O (cid:0) ( W/n ) k (cid:1) . Given that a set of k + 1 vertices { x i , x i , . . . , x i k +1 } span a k -dimensional face F , how could F be critical (or unpaired) with respectto V ? It must be that there is no common neighbor x a of these verticeswith a < i or else F would be paired up by adding the smallest indexsuch point. On the other hand F would be paired with { x i , . . . , x i k +1 } ,unless x i , . . . , x i k +1 had a common neighbor with smaller index than x i . So assuming that F is unpaired call this common neighbor x i .We have satisfied the hypothesis of Lemma 5.3 with l = k + 1 and y m = x i m . (If k y k < (1 / r then either k y − y k < r or k y o k > k y k ,a contradiction to our assumptions.) So let I = k +1 \ j =1 B ( x i k , r ) ∩ B (0 , k x i k ) , and we know from the lemma that µ ( I ) ≥ ǫ d r d with ǫ d > I then F would be paired; indeed if x a ∈ I then x a would be a common neighbor of all the vertices in F ,with a < i .The probability that a uniform random point in K falls in region I is µ ( I ) /µ ( K ) ≥ ǫ d r d /µ ( K ), where µ ( K ) is the volume of the ambientconvex body. By independence of the random points, we have that theprobability p c that F is critical (given that it is a face) is at most p c ≤ (cid:18) − ǫ d µ ( K ) r d (cid:19) n − k − . Now (cid:18) − ǫ d µ ( K ) r d (cid:19) n − k − ≤ exp( − ǫ d µ ( K ) r d ( n − k − O (exp( − cW )) , where c is any constant such that0 < c < ǫ d µ ( K ) . Let C k denote the number of critical k -dimensional faces, and wehave that E [ C k ] ≤ (cid:18) nk + 1 (cid:19) p f p c ≤ (cid:18) nk + 1 (cid:19) (cid:18) Wn (cid:19) k e − cW = O ( W k e − cW n ) . Since β k ≤ C k in every case we have E [ β k ] ≤ E [ C k ], and then E [ β k ] = O ( W k e − cW n ) , as desired. (cid:3) Connected
As in the previous section, we assume that the underlying distribu-tion is uniform on a smoothly bounded convex body K , but we nowrequire r to be slightly larger, r = Ω((log n/n ) /d ). In this regime, thegeometric random graph is known to be connected [26], and we showhere that the ˇCech complex is contractible, and the Vietoris-Rips com-plex “approximately contractible” (in the sense of k -connected for anyfixed k ). Theorem 6.1 (Threshold for contractibility, random ˇCech complex) . For a uniform distribution on a smoothly bounded convex body K in R d ,there exists a constant c , depending on K , such that if r ≥ c (log n/n ) /d then the random ˇCech complex C ( X n ; r ) is a.a.s. contractible. This is best possible up to the constant in front, since there alsoexists a constant c ′ such that if r ≤ c ′ (log n/n ) /d , then the randomˇCech complex is a.a.s. disconnected [26]. ANDOM GEOMETRIC COMPLEXES 19
Definition 6.2.
Let A = { A , A , . . . , A k } be a cover of a topologicalspace T . Then the nerve of the cover A , is the (abstract) simplicialcomplex N ( A ) on vertex set [ k ] = { , , . . . , k } with σ ⊂ [ k ] a facewhenever T i ∈ σ A i = ∅ . The proof depends on the following result (Theorem 10.7 in [4]).
Theorem 6.3 (Nerve Theorem) . If T is a triangulable topologicalspace, and A = ( A i ) i ∈ [ k ] is a finite cover of T by closed sets, suchthat every nonempty section A i ∩ A i ∩ · · · ∩ A i t is contractible, then T and the nerve N ( T ) are homotopy equivalent.Proof of Theorem 6.1. Once r is sufficiently large the balls { B ( x i , r/ } cover the smoothly bounded convex body K , and then Theorem 6.3gives that it is contractible. So to prove the claim it suffices to showthat there exists a constant c > r ≥ c (log n/n ) /d ,the balls of radius r/ K . There is no harm in assumingthat r → n → ∞ since the statement is trivial otherwise.Let Z d denote the d -dimensional cubical lattice, and λ Z d the samelattice linearly scaled in every direction by a factor λ >
0. With theend in mind we set λ = r/ (4 √ d ). (Note that since r = r ( n ), λ is alsoa function of n .) Since K is bounded, only a finite number N of theboxes of side length λ intersect it. More precisely, it is easy to see that N = µ ( K ) /λ d + O (1 /λ d − ) . As n → ∞ and λ → N boxes are containedin K , but some are on the boundary. Denote by S K the set of boxescompletely contained in K . Suppose every box in S K contains at leastone point in X n . Then the balls of radius r/ K , as follows.First of all, each box has diameter λ √ d = r/
4. So a ball of radius r/ S K , this is sufficient.For a box B ∈ S K , let p o denote the probability that box B ∩ X n = ∅ .By uniformity of distribution this is the same for every B , and byindependence of the points we have that p o = (1 − λ d /µ ( K )) n ≤ exp( − λ d n/µ ( K ))= exp( − ( r/ √ d ) d n/µ ( K ))= exp( − Cr d n ) , where C = 14 d d d/ µ ( K )is constant.Setting r = c k (log n/n ) /d we have that p o ≤ exp( − Cc dk log n )= n − Cc dk . There are at most N boxes in S K and N = µ ( K ) /λ d + O (1 /λ d − )= (1 + o (1)) /Cr d , so applying a union bound, the probability p f that at least one box in S K fails to contain any points from X n is bounded by p f ≤ N p o ≤ o (1) Cr d n − Cc dk = 1 + o (1) Cc dk log n n − Cc dk . So choosing c k > (1 /C ) /d is sufficient to ensure that K is a.a.s. coveredby the n random balls of radius r/
2, and the desired result follows. (cid:3)
The situation for the Vietoris-Rips complex is a bit more subtlesince the nerve theorem is not available to us. Nevertheless, we useMorse theory to show in the connected regime that the Vietoris-Ripscomplex becomes “approximately contractible,” in the sense of highlyconnected.
Definition 6.4.
A topological space T is k -connected if every map froman i -dimensional sphere S i → T is homotopically trivial for ≤ i ≤ k . For example, 0-connected means path-connected, and 1-connectedmeans path-connected and simply-connected. The Hurewicz Theoremand universal coefficients for homology gives that if T is k -connected,then e H i ( T ) = 0 for i ≤ k , with coefficients in Z or any field [16]. Theorem 6.5 ( k -connectivity of the random Vietoris-Rips complex) . For a smoothly bounded convex body K in R d , endowed with a uniformdistribution, and fixed k ≥ , if r ≥ c k (log n/n ) /d then the randomVietoris-Rips complex R ( X n ; r ) is a.a.s. k -connected. (Here c k > isa constant depending only on the volume µ ( K ) and k .) ANDOM GEOMETRIC COMPLEXES 21
Proof of Theorem 6.5.
The proof is identical to the proof of Theorem5.1, but now r is bigger and we obtain a stronger result. We place adiscrete gradient vector field on R ( X n ; r ) in the same way describedbefore, and repeat the same argument. If C k denotes the number ofcritical k -dimensional faces, c is the constant in the statement of The-orem 5.1, and W = nr d , then we have E [ C k ] = O (cid:0) W k e − cW n (cid:1) = O (cid:16) ( nr d ) k e − cnr d n (cid:17) = O (cid:16) ( c dk log n ) k n − cc dk (cid:17) , since nr d = c dk log n . So we have that E [ C k ] → c k > (1 /c ) /d .The same argument holds simultaneously for all smaller values of k ≥ ≤ k is thevertex closest to the origin. By Theorem 7.2 in the Appendix, R ( X n ; r )is a.a.s. homotopy equivalent to a CW-complex with one 0-cell and noother cells of dimension ≤ k . This implies that R ( X n ; r ) is k -connectedby cellular approximation [16]. (cid:3) At the moment we do not know if there is a sufficiently large constant t > r ≥ t (log n/n ) /d , the random Vietoris-Ripscomplex R ( X n ; r ) is a.a.s. contractible. In fact it is not even clearthat making r = ω (cid:0) (log n/n ) /d (cid:1) is sufficient for this; this ensures that R ( X n ; r ) is a.a.s. k -connected for every fixed k , but our results heredo not rule out the possibility that there is nontrivial homology indimension k where k → ∞ as n → ∞ .6.1. Non-vanishing to vanishing threshold.
Given a lemma aboutgeometric random graphs which we state without proof, we have asecond threshold where k th homology passes back from non-vanishing.First the statement of the lemma. (We are still assuming that the un-derlying distribution is uniform on a smoothly bounded convex body.) Lemma 6.6.
Suppose H is a feasible subgraph, that r = Ω( n − /d ) , andthat r = o (cid:0) (log n/n ) /d (cid:1) . Then the geometric random graph X ( n ; r ) a.a.s. has at least one connected component isomorphic to H . This lemma should follow from the techniques in Chapter 3 of [26].Given the lemma, we have the following intervals of vanishing and non-vanishing homology for
V R ( n ; r ). Theorem 6.7 (Intervals of vanishing and non-vanishing, random Vi-etoris-Rips complex) . Fix k ≥ . For a random Vietoris- Rips complexon a uniform distribution on a smoothly bounded convex body in R d , (1) if r = o (cid:16) n − k +2 d (2 k +1) (cid:17) or r = ω (cid:0) (log n/n ) /d (cid:1) then a.a.s. H k = 0 , and (2) if r = ω (cid:16) n − k +2 d (2 k +1) (cid:17) and r = o (cid:0) (log n/n ) /d (cid:1) then a.a.s. H k = 0 . Similarly for C ( n, r ) , we have the following. Theorem 6.8 (Intervals of vanishing and non-vanishing, random ˇCechcomplex) . Fix k ≥ . For a random ˇCech complex on a uniform dis-tribution on a smoothly bounded convex body, (1) if r = o (cid:16) n − k +2 d ( k +1) (cid:17) or r = ω (cid:0) (log n/n ) /d (cid:1) then a.a.s. H k = 0 , and (2) if r = ω (cid:16) n − k +2 d ( k +1) (cid:17) and r = o (cid:0) (log n/n ) /d (cid:1) then a.a.s. H k = 0 .Proof. In both cases, (1) follows from the results we have established inthe sparse regime. The point of the Lemma is that as long as r falls inthis intermediate regime, there is a.a.s. at least one connected compo-nent homeomorphic to the sphere S k , hence contributing to homology H k . (cid:3) Further directions
From the point of view of applications to topological data analysis,the thing that is most needed is results for statistical persistent ho-mology [8]. Bubenik and Kim computed persistent homology for i.i.d.uniform random points in the interval [7] applying the theory of orderstatistics, but so far these are some of the only detailed results for per-sistent homology of randomly sampled points. (More recently Bubenik,Carlsson, Kim, and Luo discussed recovering persistent homology of amanifold with respect to some fixed function by data smoothing withkernels, and then applying stability for persistent homology [6].)
ANDOM GEOMETRIC COMPLEXES 23
The theorems in this article have implications for statistical persis-tent homology. In particular, we have bounded the number of nontriv-ial homology classes, and since almost all of the homology comes fromvertex minimal spheres, almost all classes should not persist for long.What one might like is to rule out homology classes that persist fora long time altogether. Such a theorem would be an important steptoward quantifying the statistical significance of persistent homology.All the results here are stated for Euclidean space, but we believethis is mostly a matter of convenience. Analogous results for homologyshould hold for d -dimensional compact Riemannian manifolds. Themanifold will contribute its own homology in the supercritical regime,but for most functions r = r ( n ) this will be overwhelmed by noise,since E [ β k ] → ∞ and the homology of the manifold itself will be finitedimensional. In contrast, one would expect persistent homology todetect the homology of the manifold itself.Although we have bounded Betti numbers here, coefficients have notcome into play. It seems more refined tools are needed to detect thetorsion in Z -homology of random complexes. (This comes up for otherkinds of random simplicial complexes as well; see for example [2].)Finally, we comment that the topological properties studied hereare not monotone, the results suggest strongly that they are roughlyunimodal. But can this be made more precise? For example, can oneshow that for sufficiently large n , E [ β k ] is actually a monotone functionof r ? Similar statistically unimodal behavior in random homology hasbeen previously observed in [18] and [19]. Acknowledgements
I gratefully acknowledge Gunnar Carlsson and Persi Diaconis fortheir mentorship and support, and for suggesting this line of inquiry.I would especially like to thank an anonymous referee for a carefulreading of an earlier version of this article and for several suggestionswhich significantly improved it. I also thank Yuliy Barishnikov, PeterBubenik, Mathew Penrose, and Shmuel Weinberger for helpful conver-sations, and Afra Zomorodian for computing and plotting homology ofa random geometric complex.This work was supported in part by Stanford’s NSF-RTG grant.Some of this work was completed at the workshop in ComputationalTopology at Oberwolfach in July 2008.
Appendix: discrete Morse theory
In this section we briefly introduce terminology of discrete Morsetheory and state the main theorem. For a more complete introductionto the subject we refer the reader to [13].For two faces σ, τ of a simplicial complex, we write σ ≺ τ if σ is aface of τ of codimension 1. Definition 7.1.
A discrete vector field V of a simplicial complex ∆ isa collection of pairs of faces of ∆ { α ≺ β } such that each face is in atmost one pair. Given a discrete vector field V , a closed V -path is a sequence of faces α ≺ β ≻ α ≺ β ≻ . . . ≺ β n ≻ α n +1 , with α i +1 = α i such that { α i ≺ β i } ∈ V for i = 0 , . . . , n and α n +1 = α o .(Note that { β i ≻ α i +1 } / ∈ V since each face is in at most one pair.)We say that V is a discrete gradient vector field if there are no closed V -paths.Call any simplex not in any pair in V critical . Then the main theoremis the following [13]. Theorem 7.2 (Fundamental theorem of discrete Morse theory) . Sup-pose ∆ is a simplicial complex with a discrete gradient vector field V .Then ∆ is homotopy equivalent to a CW complex with one cell of di-mension k for each critical k -dimensional simplex. Simply counting cells is an extremely coarse measure of the topologya complex, but it can be enough to completely determine homotopytype; for example a CW complex with one 0-cell and all the rest of itscells d -dimensional is a wedge of d -spheres.In all cases, if f k is the number of cells of dimension k , then thedefinition of cellular homology gives that β k ≤ f k , and this is themain fact that we exploit in Sections 5 and 6 to bound the expecteddimension of homology. References [1] Noga Alon and Joel H. Spencer.
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