On the subgraphs of percolated random geometric graphs and the associated random complexes
OOn the subgraphs of percolated random geometric graphs andthe associated random complexes
Anshui Li ∗ September 18, 2018
Abstract
In this paper, we investigate the induced subgraphs of percolated random geomet-ric graphs, and get some asymptotic results for the expected number of the subgraph.Moreover, we get the Poisson approximation for the counting by Stein’s method. Wealso present some similar results for the expectation of Betti number of the associatedpercolated random geometric complexes.
The idea of modeling networks using random graphs was first given by Gilbert (1961) in[7] where he considered a network formed by connecting points of a Poisson point processthat are sufficiently close to each other. The model Gilbert introduced was a different onefrom the Erd˝os-Renyi random graph models in [6] [5] [4]. In this model the vertices havesome (random) geometric layout and the edges are determined by the distances between thepositions of the vertices. We call graphs formed in this way random geometric graphs .Recently, quite a lot of work has been done on random geometric graphs , partly due to theimportance of these graph model as some theoretical models for ad hoc networks, e.g., see [9].Most of the theoretical results on random geometric graphs can be found in the monographwritten by Penrose [14].
Random geometric graphs model is as follows. Let f be some specific probability densityfunction on R d , and let X , X , ... be independent and identically distributed d -dimensionalrandom variables with common density f : R d → [0 , ∞ ). In the whole paper, we assume that f is measurable and bounded, which also satisfies (cid:82) R d f ( x ) dx = 1 . Let X n = { X , X , ..., X n } . We denote G ( X n ; r n ) the undirected graph with vertex set X n and with undirected edgesconnecting all those pairs { X i , X j } with (cid:107) X i − X j (cid:107) ≤ r ( n ) , in which (cid:107) (cid:5) (cid:107) denotes the Euclideandistance.The random connection model was introduced in the context of continuum percolationby Penrose [15]. Let g : R → [0 ,
1] be such that g ( x ) = g ( − x ) . The function g is calledthe connection function . For two vertices x , y , they are connected with probability g ( y − x ).Typically, it is also assumed that g only depends on the distance between x and y , i.e., g ( y − x ) = ˆ g ( (cid:107) y − x (cid:107) ) where ˆ g : R + → [0 ,
1] and (cid:107) (cid:5) (cid:107) denotes the Euclidean norm. The randomgeometric graph is a random connection model with ˆ g ( x ) = [0 ,r n ] ( x ). ∗ The author acknowledges support from CSC-UU scholarship grant. a r X i v : . [ m a t h . P R ] M a r ery recently, Penrose [16] investigated the connectivity of random connection model withvarious classes of connection functions, which are called soft random geometric graphs . Heshowed that as vertex number n → ∞ , the probability of full connectivity is governed by thatof having no isolated vertices, itself governed by a Poisson approximation for the number ofisolated vertices. He generalized this beautiful result to higher dimensions, and to a largeclass of connection probability function in d = 2.In this paper, we consider a specific connection function which is also mentioned in Penrose[16]: ˆ g ( x ) = p n [0 ,r n ] ( | x | ), for some p n ∈ [0 , soft random geometric graph gotten bythis connection function, is called percolated random geometric graph . To be more precise,a percolated random geometric graph is defined as a random graph with vertex set X n = { X , X , ..., X n } in which n vertices are chosen at random and independently from distributionin R with probability density f , and a pair of vertices with Euclidean distance r appears asan edge with probability p n , some function of n , independently for each such a pair, we denotethis graph G ( X n ; p n , r n ). In particular, for p = 1, we can get the classic random geometricgraph , which we denote G ( X n ; r ) . Hereafter, we always consider p = p ( n ) as a function of n .In this paper, we focus on the induced subgraph count problem on percolated randomgeometric graph G ( X n ; p, r ). Let Γ be a fixed connected graph on k vertices, k ≥
2. Considerthe number of induced subgraphs of G ( X n ; p, r ) isomorphic to Γ. In [14], the author alwaysassumes that the subgraph Γ is feasible , which means P r [( X k ; r ) ∼ = Γ] > r >
0. However, we will not make this assumption in this paper: the subgraphsare always feasible for percolated random geometric graphs . Surprisingly, we can attain theasymptotic results for the means of the Γ-subgraph counts on G ( X n ; p, r ) with the help of Γ-subgraph counts on G ( X n ; r ), given that Γ is a clique (i.e., a complete graph), see Corollary 2.5.But when it comes to general induced subgraph and component, we can only get a lower boundfor the asymptotic result for the means of the Γ-component counts on G ( X n ; p, r ) for a widerange of p n , see Theorem 2.3 and Corollary 2.4. The main reason behind this is that thereexist many subgraphs which are feasible for G ( X n ; p, r ) but not for G ( X n ; r ), which makesthe counting on G ( X n ; p, r ) more complicated.Recent years have seen an explosive growth in research of random geometric simplicialcomplexes . Random simplicial complexes may be viewed as higher dimensional generalizationsof random graphs. Simplicial complex analogues of the classic Erd¨os-Renyi model and theirtopological properties have been the subjects of many literatures in recent years. See forexample [11], [12], and [13]. It is also natural to generalize the random geometric graphs, anda lot of references can be found in the survey articles [10]. As Kahle mentioned in [10], twonatural ways of extending a geometric graph to a simplicial complexes are: the Cech complexand the Vietoris-Rips complex(see formal definitions in the following). Most of the results inthe researches of the topology of random geometric complexes are related to their homology.Briefly speaking, if X is a topology space, its degree i -homology, denote by H i ( X ) is a vectorspace. The dimension dim H ( X ) is the number of connected components of X , and for i > H i ( X ) contains information about i − dimensional ”holes”.Since we focus on ”counting” in this paper, we will also count the expected number of”holes” for the corresponding percolated random geometric complexes in this paper. We alsoget the expectation of Betti number of the percolated random geometric complex (see formaldefinitions below). 2ur argument is based on ”coupling” of two random graph models: G ( X n ; p, r ) and G ( X n ; r ); and then use the same technique in Chapter 3 in Penrose [14].The paper is organized as follows: In Section 2 we present our main results. In Section3 we prove the main results. Finally, we note some possible generalizations and remarks inSection 4. We first present one asymptotic result for the means of the Γ-subgraphs counts G n in [14]given by Penrose. Given a connected graph Γ on k vertices, and given A ⊆ R d , define theindicator functions h Γ ( Y ), h n,A, Γ ( Y ), ˆ h Γ ( Y ) and ˆ h n,A, Γ ( Y ) for finite Y ⊂ R d by h Γ ( Y ) := 1 { G ( Y ; r n ) ∼ =Γ } ,h n,A, Γ ( Y ) := 1 { G ( Y ; r n ) ∼ =Γ }∩{ LMP ( Y ) ∈ A } , and ˆ h Γ ( Y ) := 1 { G ( Y ; r n ,p ) ∼ =Γ } , ˆ h n,A, Γ ( Y ) := 1 { G ( Y ; r n ,p ) ∼ =Γ }∩{ LMP ( Y ) ∈ A } , in which LM P ( Y ) means the left-most point of set Y . It is easy to observe that h Γ ( Y ) = h n,A, Γ ( Y ) = ˆ h Γ ( Y ) = ˆ h n,A, Γ ( Y ) = 0 unless Y has k elements.Similarly, we define g Γ ( Y ) := 1 { G ( Y ; r n ) (cid:37) Γ } , and g n,A, Γ ( Y ) := 1 { G ( Y ; r n ) (cid:37) Γ }∩{ LMP ( Y ) ∈ A } , in which { G ( Y ; r n ) (cid:37) Γ } means Γ is a subgraph of G ( Y ; r n ), but not equals G ( Y ; r n ). Here-after we define g Γ ( Y ) = g n,A, Γ ( Y ) = 0 unless Y has k elements, which means we justneed to consider the graph G ( Y ; r n ) with order k .The reader should keep in mind that all the functions h Γ ( · ), h n,A, Γ ( · ), g Γ ( · ), g n,A, Γ ( · ) aredefined on random geometric graph G ( X n ; r ); and only functions ˆ h Γ ( · ), ˆ h n,A, Γ ( · ), are definedon percolated random geometric graph G ( X n ; r, p ).We set µ Γ ,A := k ! − (cid:90) A f ( x ) k dx (cid:90) ( R d ) k − h Γ ( { , x , ..., x k − } ) d ( x , ..., x k − ) , ˆ µ Γ ,A := k ! − (cid:90) A f ( x ) k dx (cid:90) ( R d ) k − ˆ h Γ ( { , x , ..., x k − } ) d ( x , ..., x k − ) ,µ (cid:48) Γ ,A := k ! − (cid:90) A f ( x ) k dx (cid:90) ( R d ) k − g Γ ( { , x , ..., x k − } ) d ( x , ..., x k − ) . We write µ Γ , ˆ µ Γ , µ (cid:48) Γ for µ Γ , R d , ˆ µ Γ , R d and µ (cid:48) Γ , R d respectively.Let G n,A (Γ) and G (cid:48) n,A (Γ) be the number of induced subgraphs of G ( X n ; r ) and G ( X n ; p, r )for which the left-most of the vertex set lies in A , respectively.3 heorem 2.1 (Penrose [14]) Suppose that Γ is a feasible connected graph of order k ≥ ,that A ⊆ R d is open with Leb ( ∂A ) = 0 , and that lim n →∞ ( r n ) = 0 . Then lim n →∞ r − d ( k − n n − k E ( G n,A (Γ)) = µ Γ ,A . Similar with the result above, we count the induced subgraph in the percolated randomgeometric graph G ( X n ; p, r ) and get one theorem below: Theorem 2.2
Suppose that Γ is a connected graph of order k ≥ , that A ⊆ R d is open with Leb ( ∂A ) = 0 , and that lim n →∞ ( r n ) = 0 . Then lim n →∞ r − d ( k − n n − k E ( G (cid:48) n,A (Γ)) = ˆ µ Γ ,A . However, if have more information about graph Γ, we can get more detailed results. Inthe following, we will present some results related to induced-graph Γ with order k ≥ m ≥ Theorem 2.3 (Counting of induced subgraph)
Suppose that Γ is a connected graph oforder k ≥ and size m , that A ⊆ R d is open with Leb ( ∂A ) = 0 , and that lim n →∞ ( r n ) = 0 . Thenif p n ≡ p , we have lim n →∞ p − m n − k r − d ( k − n E ( G (cid:48) n,A (Γ)) ≥ µ Γ ,A + (1 − p )( k ) µ (cid:48) Γ ,A . If lim n →∞ n p n → α ∈ (0 , ∞ ) , we have lim n →∞ p − m n − k r − d ( k − n E ( G (cid:48) n,A (Γ)) ≥ µ Γ ,A + e − α/ µ (cid:48) Γ ,A . If lim n →∞ n p n → , we have lim n →∞ p − m n − k r − d ( k − n E ( G (cid:48) n,A (Γ)) ≥ µ Γ ,A + µ (cid:48) Γ ,A . Corollary 2.4 (Counting of tree-subgraph )
Suppose that Γ is a connected graph of or-der k ≥ and size m = k − , that A ⊆ R d is open with Leb ( ∂A ) = 0 , and that lim n →∞ ( r n ) =0 . Thenif p n ≡ p , we have lim n →∞ E ( G (cid:48) n,A (Γ)) n (cid:18) θd ( n ) (cid:19) k − ≥ µ Γ ,A + (1 − p )( k ) − ( k − µ (cid:48) Γ ,A ; If n p n → α ∈ (0 , ∞ ) , we have lim n →∞ E ( G (cid:48) n,A (Γ)) n (cid:18) θd ( n ) (cid:19) k − ≥ µ Γ ,A + e − α/ µ (cid:48) Γ ,A ; If n p n → , we have lim n →∞ E ( G (cid:48) n,A (Γ)) n (cid:18) θd ( n ) (cid:19) k − ≥ µ Γ ,A + µ (cid:48) Γ ,A , in which d ( n ) = nθr dn p n . orollary 2.5 (Counting of clique-subgraph) Suppose that Γ is a clique of order k ≥ ,that A ⊆ R d is open with Leb ( ∂A ) = 0 , and that lim n →∞ ( r n ) = 0 . Then E ( G (cid:48) n,A (Γ)) = p ( k ) E ( G n,A (Γ)) . Moreover, we can get lim n →∞ p − ( k ) n − k r − d ( k − n E ( G (cid:48) n,A (Γ)) = µ Γ ,A . Next consider the component count in the thermodynamic limit where nr dn tends to aconstant. Given λ >
0, and given a feasible connected graph Γ of order k ≥
2, define p Γ ( λ ) := λ k − ( k − (cid:90) ( R d ) k − h Γ ( { , x , ..., x k − } ) × exp( − λV (0 , x , ..., x k − )) d ( x , ..., x k − )andˆ p Γ ( λ ) := λ k − ( k − (cid:90) ( R d ) k − ˆ h Γ ( { , x , ..., x k − } ) × exp( − λV (0 , x , ..., x k − )) d ( x , ..., x k − ) , and p (cid:48) Γ ( λ ) := λ k − ( k − (cid:90) ( R d ) k − g Γ ( { , x , ..., x k − } ) × exp( − λV (0 , x , ..., x k − )) d ( x , ..., x k − ) , where V ( y , ..., y m ) denotes the Lebesgue measure (volume) of the union of balls of unit radius(in the chosen norm) centered at y , ..., y m . If Γ consists of one single point (i.e. if k = 1), set p Γ ( λ ) = p (cid:48) Γ ( λ ) = ˆ p Γ ( λ ) := exp( − λθ ), in which θ is the volume of the unit ball in R d .Let J n,A (Γ) be the number of Γ-components of G ( X n ; r ) for which the left-most point ofthe vertex set lies in A ; and J (cid:48) n,A (Γ) be the number of Γ-components of G ( X n ; p, r ) for whichthe left-most point of the vertex set lies in A . Theorem 2.6 (Penrose [14])
Suppose that A ⊆ R d is open with Leb ( ∂A ) = 0 , that Γ is afeasible connected graph order k ∈ N , and that nr dn → ρ ∈ (0 , ∞ ) . Then lim n →∞ (cid:18) E ( J n,A (Γ)) n (cid:19) = k − (cid:90) A p Γ ( ρf ( x )) f ( x ) dx. For the percolated random geometric graphs, we have one similar result as follows.
Theorem 2.7
Suppose that A ⊆ R d is open with Leb ( ∂A ) = 0 , that Γ is a connected graphorder k ∈ N , and that nr dn → ρ ∈ (0 , ∞ ) . Then lim n →∞ (cid:18) E ( J n,A (Γ)) n (cid:19) = k − (cid:90) A ˆ p Γ ( ρf ( x )) f ( x ) dx. Same story as the counting of induced subgraph, we can get more detailed results if havemore information about the induced component.5 heorem 2.8 (Counting of Γ -component) Suppose that A ⊆ R d is open with Leb ( ∂A ) =0 , that Γ is a connected graph order k ∈ N and size m , and that nr dn → ρ ∈ (0 , ∞ ) . Thenif p n ≡ p , we have lim n →∞ (cid:32) E ( J (cid:48) n,A (Γ)) np mn (cid:33) ≥ k − (cid:90) A p Γ ( ρf ( x )) f ( x ) dx + k − (1 − p )( k ) − m (cid:90) A p (cid:48) Γ ( ρf ( x )) f ( x ) dx. If n p n → α ∈ (0 , ∞ ) , we have lim n →∞ (cid:32) E ( J (cid:48) n,A (Γ)) np mn (cid:33) ≥ k − (cid:90) A p Γ ( ρf ( x )) f ( x ) dx + k − e − α/ (cid:90) A p (cid:48) Γ ( ρf ( x )) f ( x ) dx. If n p n → , we have lim n →∞ (cid:32) E ( J (cid:48) n,A (Γ)) np mn (cid:33) ≥ k − (cid:90) A p Γ ( ρf ( x )) f ( x ) dx + k − (cid:90) A p (cid:48) Γ ( ρf ( x )) f ( x ) dx. Corollary 2.9 (Counting of clique-component)
Suppose that A ⊆ R d is open with Leb ( ∂A ) =0 , that Γ is a clique with order k ∈ N , and that nr dn → ρ ∈ (0 , ∞ ) . Then we have E ( J (cid:48) n,A (Γ)) ≥ p ( k ) n E ( J n,A (Γ)) . Moreover, we have lim n →∞ E ( J (cid:48) n,A (Γ)) np ( k ) n ≥ k − (cid:90) A p Γ ( ρf ( x )) f ( x ) dx. (1)In the following subsection, we will present the basic Poisson approximation theorem forthe induced Γ-subgraph count G (cid:48) n on percolated random geometric graph G ( X n ; p, r ) . Compareto the similar results for random geometric graphs in [14], the total variation distance betweenthe distribution of G (cid:48) n and corresponding Poisson distribution is tighter for percolated randomgeometric graphs. Theorem 2.10
Let Γ be a connected graph of order k ≥ and size m , and we define G (cid:48) n := G (cid:48) n, R d (Γ) . Suppose ( nr dn ) n ≥ is a bounded sequence. Let Z n be Poisson with parameter E ( G (cid:48) n ) . Then there exists a constant c such that for all n, d T V ( G (cid:48) n , Z n ) ≤ (cid:26) cnp m +2 − kn r dn if k ≥ cnp m − n r dn if ≤ k < If n k r d ( k − n → α ∈ (0 , ∞ ) , then G (cid:48) n D −→ P o ( λ ) with λ = α ˆ µ Γ .If n k r d ( k − n → ∞ and nr dn → , then (cid:16) n k r d ( k − n ˆ µ Γ (cid:17) − / ( G (cid:48) n − EG (cid:48) n ) D −→ N (0 , . .2 Counting on random geometric complexes In this section, we present some very preliminary results related to the percolated random geo-metric complexes , which are the corresponding results in percolated version for the expectationof Betti numbers of Vietoris-Rips complex as in Kahle [10].For completeness of this paper, we first review some definitions related to simplicialcomplexes. A set of k + 1 points, u , u , ..., u k , is affinely independent if the k vectors, u − U − , u − u , ..., u k − u , are linear independent. A k-simplex is the convex hull of k + 1affinely independent points. Writing σ for the k -simplex, we call k =dim σ its dimension, and u to u k its vertices . Simplifies of dimension 0 , , , vertices, edges,triangles, tetrahedra . A face of σ is a simplex spanned by a subset of the vertices of σ . Since aset of k + 1 elements has (cid:0) k +1 l +1 (cid:1) subsets of size l + 1, σ has this number of l -faces, for 0 ≤ l ≤ k .The total number of faces is k (cid:88) l =0 (cid:18) k + 1 l + 1 (cid:19) = 2 k +1 − , the number of subsets minus 1 means we do not count the empty set. We then define a simplicial complex as a finite collection of simplices, K , such that(i) for every simplex σ ∈ K , every face of σ is in K ;(ii) for every two simplifies σ, τ ∈ K , the intersection, σ ∩ τ , is either empty or a face ofboth simplices.If the intersection of two simplifies is a common face, then ( i ) implies that it is a simplex in K .The dimension of a simplicial complex K is the largest dimension of any simplex in K . A subcomplex of K is the simplices that is itself a simplicial complex. For more detains on simplicialcomplex and related properties, we recommend the brief monograph [3] by Edelsbrunner.The random geometric complexes studied are simplicial complexes built on independentand identically distributed random points in Euclidean space R d . In this section, we makemild assumptions about the common density f : f is bounded Lebesgue-measurable functionand (cid:90) R d f dx = 1 . The main object of study in this section is the percolated
Vietoris-Rips complexes on X , X , ..., X n , which is a sequence of independent and identically distributed d -dimensionalrandom variables with common density f , we denote the sequence by X n = { X , X , ..., X n } .The Vietoris-Rips complex was first introduced by Vietoris in order to extend simplicialhomology to a homology theory for metric spaces [17]. Eliyahu Rips applied the same complexto the study of hyperbolic groups, and Gromov popularized the name of Rips complex [8].Denote the closed ball of radius r centered at a point p by B ( p, r ) = { x | || x − p || ≤ r } , inwhich || · || is the Euclidean distance in R d .The formal definition of Vietoris-Rips complex goes as follows: Definition 2.11 (Random VR complex)
The random Vietoris-Rips complex R ( X n ; r ) isthe simplicial complex with vertex set X n and σ a face if B ( X i , r/ ∩ B ( X j , r/ (cid:54) = ∅ for every pair X i , X j ∈ σ . G ( X n ; r ).As we mentioned before, we want to study one percolated version of the random Vietoris-Rips complex. Roughly speaking, the underlying graph for the random Vietoris-Rips complexis the classic random graphs G ( X n ; r ), while the underlying graph for the percolated randomVietoris-Rips complex is the percolated random geometric graph G ( X n ; r, p ). Definition 2.12 (Percolated random VR complex)
Let G ( X n ; r, p ) be the percolated ran-dom geometric graph built on the random points X n = { X , X , ..., X n } , which are i.i.d withcommon density f . The percolated random Vietoris-Rips complex R ( X n ; r, p ) associated withgraph G ( X n ; r, p ) is the simplicial complex with vertex X n and σ a face if ( X i , X j ) ∈ E ( G ( X n ; r, p )) for every pair X i , X j ∈ σ . In other words, we build any k − simplex by its basic 2-faces, i.e., edges. A face σ exists ifall its 2-subfaces exist.In this paper, we only mention the similar result for the expectation of Betti number inthe subcritical regime. Theorem 2.13 (Betti number of random geometric VR complex,[10])
For d ≥ , k ≥ , and r n = o ( n − /d ) , the expectation of the k th Betti number E [ β k ] of the randomVietoris-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 the underlying density f . For the percolated random Vietoris-Rips complex, we have the similar results:
Theorem 2.14 (Betti number of percolated random VR complex)
For d ≥ , k ≥ , and r n = o ( n − /d ) , the expectation of the k th Betti number E [ β k ] of the percolated randomVietoris-Rips complex R ( X n ; r, p ) associated with graph G ( X n ; r, p ) satisfies E [ β k ] n k +2 r d (2 k +1) p k ( k − → C (cid:48) k as n → ∞ , where C (cid:48) k is a constant that depends only on k and the underlying density f . G ( X n ; p, r ) and G ( X n ; r ) Given a vertex set X n = { X , ..., X n } which are independently from a distribution on R withprobability density f , and two functions r n > , p n ∈ [0 , G ( X n ; p, r ) in the following two steps: • Put an edge between X i and X j if (cid:107) X i − X j (cid:107) ≤ r , for 1 ≤ i < j ≤ n , we get G ( X n ; r );8 For G ( X n ; r ) obtained above, we keep every edge with probability p (i.e., we delete itwith probability 1 − p ), independently with all other edges. Then we get G ( X n ; p, r ).From the procedure above, we can get that there are at least two ways to get the induced subgraphs in G ( X n ; p, r ): • If G ( X k ; r ) ∼ = Γ, we can keep it in the second step; • If G ( X k ; r ) (cid:37) Γ, we can delete the unwanted edges in the second step, and get G ( X k ; r, p ) ∼ =Γ.In short, all the induced subgraphs G ( X k ; r, p ) ∼ = Γ are born from some graphs G ( X k ; r )with more (or same) edges.It is easy to observe that: if ( X i , X j ) is one edge in G ( X n ; r ), then ( X i , X j ) is one edge in G ( X n ; p, r ) with probability p . As a consequence, we can get a lemma. Lemma 3.1 (Coupling Lemma)
Suppose that Γ be a fixed connected graph of order k ≥ and size m ≥ . Then P r [ G ( X k ; p, r ) ∼ = Γ] ≥ p m P r [ G ( X k ; r ) ∼ = Γ] . Proof:
If Γ is not feasible for G ( X n ; r ), i.e., P r [ G ( X k ; r ) ∼ = Γ] = 0, the statement of courseholds; If G ( X k ; r ) ∼ = Γ, then keep all the edges in G ( X k ; r ), we can get G ( X k ; p, r ) ∼ = Γ. Wecomplete the proof. (cid:4) Remark 3.2
From the Lemma 3.1, every Γ -subgraph with size m in G ( X n ; r ) , can contribute p mn to the expectation of number of Γ -subgraph in G ( X n ; p, r ) . Slightly modify the proof of Theorem 2.1 in [14], we can get the theorem.
It is easy to get
P r ( G ( X k ; r, p ) ∼ = Γ) = P r ( G ( X k ; r, p ) ∼ = Γ | G ( X k ; r ) ∼ = Γ) P r ( G ( X k ; r ) ∼ = Γ)+ P r ( G ( X k ; r, p ) ∼ = Γ | G ( X k ; r ) (cid:37) Γ) P r ( G ( X k ; r ) (cid:37) Γ)= p m P r ( G ( X k ; r ) ∼ = Γ)+ P r ( G ( X k ; r, p ) ∼ = Γ | G ( X k ; r ) (cid:37) Γ) P r ( G ( X k ; r ) (cid:37) Γ) ≥ p m P r ( G ( X k ; r ) ∼ = Γ) + p m (1 − p )( k ) − m P r ( G ( X k ; r ) (cid:37) Γ) ≥ p m P r ( G ( X k ; r ) ∼ = Γ) + p m (1 − p )( k ) P r ( G ( X k ; r ) (cid:37) Γ) . (2)The first equality means: the conditional probability P r ( G ( X k ; r, p ) ∼ = Γ | G ( X k ; r ) (cid:37) Γ)cannot be calculated easily, as it depends on both the structures Γ and G ( X k ; r ). However,we can get a lower bound for this probability by considering G ( X k ; r ) as a complete graph,and delete all the unwanted (cid:0) k (cid:1) − m edges.As E ( G (cid:48) n,A (Γ)) = (cid:0) nk (cid:1) P r ( G ( X k ; r, p ) ∼ = Γ), we have E [ G (cid:48) n,A (Γ)] ≥ (cid:0) nk (cid:1) p m ( P r ( G ( X k ; r ) ∼ = Γ) + (cid:0) nk (cid:1) p m (1 − p )( k ) P r ( G ( X k ; r ) (cid:37) Γ) . (3)9ollow the same idea of the proof of the Proposition 3.1 in [14], we can get that the first termand second term on the right side of ( 3 ) is asymptotic to n k p m r d ( k − n µ Γ ,A and n k p m (1 − p )( k ) r d ( k − n µ (cid:48) Γ ,A .(i) If p n ≡ p , we can rearrange the terms, and get the result;(ii) if n p n → α , which means (1 − p )( k ) ≥ (1 − p )( n ) ∼ e − p n n / ∼ e − α/ as n → ∞ ;(iii) if n p n → − p )( k ) ≥ (1 − p )( n ) ∼ e − p n n / ∼ n → ∞ .We complete our proof. let m = k −
1, use the same idea as proof of 2.3, we can get the Theorem 2.4.
If Γ is a clique with order k , we have P r ( G ( X k ; r ) (cid:37) Γ) = 0 , i.e., P r ( G ( X k ; r, p ) ∼ = Γ) = P r ( G ( X k ; r, p ) ∼ = Γ | G ( X k ; r ) ∼ = Γ) P r ( G ( X k ; r ) ∼ = Γ) . Use the same argument as proof of Theorem 2.3, we can finish the proof here.
Almost same argument as the proof of Theorem 2.6 in [14], we can get the theorem.
For the component counting, we have
P r ( G ( X k ; r, p ) ∼ = Γ) = P r ( G ( X k ; r, p ) ∼ = Γ | G ( X k ; r ) ∼ = Γ) P r ( G ( X k ; r ) ∼ = Γ)+ P r (cid:0) G ( X k ; r, p ) ∼ = Γ | G dis ( X k ; r ) (cid:37) Γ (cid:1) P r ( G dis ( X k ; r ) (cid:37) Γ)+
P r ( G ( X k ; r, p ) ∼ = Γ | G con ( X k ; r ) (cid:37) Γ) P r ( G con ( X k ; r ) (cid:37) Γ)= p m P r ( G ( X k ; r ) ∼ = Γ)+ P r (cid:0) G ( X k ; r, p ) ∼ = Γ | G dis ( X k ; r ) (cid:37) Γ (cid:1) P r ( G dis ( X k ; r ) (cid:37) Γ)+
P r ( G ( X k ; r, p ) ∼ = Γ | G con ( X k ; r ) (cid:37) Γ) P r ( G con ( X k ; r ) (cid:37) Γ) ≥ p m P r ( G ( X k ; r ) ∼ = Γ)+ p m (1 − p )( k ) − m P r ( G dis ( X k ; r ) (cid:37) Γ)+
P r ( G con ( X k ; r, p ) ∼ = Γ | G con ( X k ; r ) (cid:37) Γ) P r ( G con ( X k ; r ) (cid:37) Γ) , (4)in which, G dis ( X k ; r ) means that G ( X k ; r ) does not connect with any vertices in X n \ X k ; and G con ( X k ; r ) means that G ( X k ; r ) does connect with some vertex in X n \ X k .10o we get E ( J (cid:48) n,A (Γ)) = (cid:0) nk (cid:1) P r ( G ( X k ; r, p ) ∼ = Γ)= (cid:0) nk (cid:1) P r ( G ( X k ; r, p ) ∼ = Γ | G ( X k ; r ) ∼ = Γ) P r ( G ( X k ; r ) ∼ = Γ)+ (cid:0) nk (cid:1) P r (cid:0) G ( X k ; r, p ) ∼ = Γ | G dis ( X k ; r ) (cid:37) Γ (cid:1) P r ( G dis ( X k ; r ) (cid:37) Γ)+ (cid:0) nk (cid:1) P r ( G ( X k ; r, p ) ∼ = Γ | G con ( X k ; r ) (cid:37) Γ) P r ( G con ( X k ; r ) (cid:37) Γ) . (5)For the first term of the right side of ( 5), by Theorem 2.6, we can know the the asymptoticresult n − (cid:18) nk (cid:19) P r ( G ( X k ; r, p ) ∼ = Γ | G ( X k ; r ) ∼ = Γ) P r ( G ( X k ; r ) ∼ = Γ) → p m k − (cid:90) A p Γ ( ρf ( x )) f ( x ) dx. For the second term, we use the same argument as in the proof of Proposition 3.3 in [14],and get that n − (cid:18) nk (cid:19) P r (cid:16) G ( X k ; r, p ) ∼ = Γ | G dis ( X k ; r ) (cid:37) Γ (cid:17) P r ( G dis ( X k ; r ) (cid:37) Γ)is asymptotically bounded from below by p m (1 − p )( k ) − m k − (cid:90) A p (cid:48) Γ ( ρf ( x )) f ( x ) dx. Then use the same arguments as the 3.3, we finish the proof.
By Lemma 3.1, we can get E [ J (cid:48) n,A (Γ)] ≥ p ( k ) E [ J n,A (Γ)] . (6)By Theorem 2.6, which gives us ( 1). Before we prove this theorem, we present some notations related to dependency graphs andsome approximation results for sums of Bernoulli variables indexed by the vertices of a de-pendency graph.Suppose (
I, E ) is a graph with finite for countable vertex set I . For i, j ∈ I write i ∼ j if { i, j } ∈ E. For i ∈ I , let N i denote the adjacency neighborhood of i , that is, the set { i } ∪ { j ∈ I : j ∼ i } . We say that the graph ( I, ∼ ) is a dependency graph for a collectionof random variables ( ξ i , i ∈ I ) if for any two disjoint subsets I , I of I such that there areno edges connecting I to I , the collection of random variables ( ξ i , i ∈ I ) is independentof ( ξ j , j ∈ I ) . The notation of dependency graph is very helpful to cope with some problemrelated to near-independence random variables.
Theorem 3.3 (Arratia et al. 1989 [1])
Suppose ( ξ i , i ∈ I ) is a finite collection of Bernoullirandom variables with dependency graph ( I, ∼ ) . Set p i := E ( ξ i ) = P [ ξ i = 1] , and set p ij := E [ ξ i ξ j ] . Let λ := (cid:80) i ∈ I p i , and suppose λ is finite. let W := (cid:80) i ∈ I ξ i . Then d T V ( W, P o ( λ )) ≤ min(3 , λ − ) (cid:88) i ∈ I (cid:88) j ∈N i \{ i } p ij + (cid:88) i ∈ I (cid:88) j ∈N i p i p j . roof: [Proof of Theorem 2.10] Clearly we have G (cid:48) n = (cid:88) i ∈I n ξ i ,n , where i runs through the index set I n of all k -subsets i = { i , ..., i k } of { , , ..., n } , and ξ i ,n = 1 { G ( { X i , i ∈ i } ; p,r ) ∼ =Γ } . Then we use stein’s method to get the error bounds for the convergence.For each index i ∈ I n , let N i be the set of j ∈ I n such that i and j have at least one elementin common. Let ∼ be the associated adjacency relation on I n , that is i ∼ j if j ∈ N i and i (cid:54) = j . Then ξ i ,n is independent of ξ j ,n except when j ∈ N i . In this way, we get a dependencygraph ( I n , ∼ ) for ( ξ i ,n , i ∈ I n ) . By connectedness all vertices of any Γ-subgraph of G ( X n ; p, r ) lie within a distance ( k − r n of one another, and hence, with θ denoting the volume of the unit ball in R d , we have Eξ i ,n ≤ p m (cid:82) R d · · · (cid:82) R d h Γ ,n ( { x , ..., x k } ) f ( x ) k dx k ...dx + p m (cid:82) R d · · · (cid:82) R d h Γ ,n ( { x , ..., x k } ) × (cid:16)(cid:81) ki =1 f ( x i ) − f ( x ) k (cid:17) (cid:81) ki =1 dx i + (cid:80) ( k ) j = m +1 (cid:0) jm (cid:1) p m (1 − p ) j − m (cid:82) R d · · · (cid:82) R d g Γ ,n ( { x , ..., x k } ) f ( x ) k dx k ...dx + (cid:80) ( k ) j = m +1 (cid:0) jm (cid:1) p m (1 − p ) j − m (cid:82) R d · · · (cid:82) R d g Γ ,n ( { x , ..., x k } ) × (cid:16)(cid:81) ki =1 f ( x i ) − f ( x ) k (cid:17) (cid:81) ki =1 dx i ≤ p m (cid:82) B ( x ,kr n ) · · · (cid:82) B ( x ,kr n ) f ( x ) k − dx k ...dx (cid:82) R d h Γ ,n ( { x , ..., x k } ) f ( x ) dx + p m (cid:82) R d · · · (cid:82) R d h Γ ,n ( { x , ..., x k } ) × (cid:16)(cid:81) ki =1 f ( x i ) − f ( x ) k (cid:17) (cid:81) ki =1 dx i + p m − pp (cid:0) ( k ) m (cid:1) (cid:82) B ( x ,kr n ) · · · (cid:82) B ( x ,kr n ) f ( x ) k − dx k ...dx (cid:82) R d h Γ ,n ( { x , ..., x k } ) f ( x ) dx + p m − pp (cid:0) ( k ) m (cid:1) (cid:82) R d · · · (cid:82) R d g Γ ,n ( { x , ..., x k } ) × (cid:16)(cid:81) ki =1 f ( x i ) − f ( x ) k (cid:17) (cid:81) ki =1 dx i ≤ p m ( f max θ ( kr n ) d ) k − + p m − (cid:0) ( k ) m (cid:1) ( f max θ ( kr n ) d ) k − ≤ p m − ( f max θ ( kr n ) d ) k − (1 + C (cid:48) ) , (7)in which C (cid:48) = (cid:0) ( k ) m (cid:1) . Considering the third items on the right side of the first inequality, wesum up all the possibilities of the graphs which contains” strictly” subgraph Γ, and keep the m edges we want and delete all other edges unwanted; and then we bound (cid:80) ( k ) j = m +1 (cid:0) jm (cid:1) (1 − p ) j − m by (cid:0) ( k ) m (cid:1) (cid:80) ∞ j =1 (1 − p ) j .We can also get card ( N i ) = (cid:18) nk (cid:19) − (cid:18) n − kk (cid:19) = k ! − k n k − + O ( n k − ) , which leads to (cid:88) i ∈I n (cid:88) j ∈N i Eξ i ,n Eξ j ,n ≤ c (cid:48) p m − n k − r d ( k − n = c (cid:48) p m − n k +1 r dkn ( nr dn ) k − . The next step is to bound Eξ i ,n ξ j ,n when i ∼ j and i (cid:54) = j . Here we have h = | i ∩ j |∈ { , ..., k − } .
12y the same arguments as we bound Eξ i ,n , we can get E [ ξ i ,n ξ ξ j ,n ] ≤ C (cid:48)(cid:48) p m − h +1 ( f max θ (2 kr n ) d ) k − h − . (8)Given h ∈ { , , ..., k − } , the number of pairs ( i , j ) ∈ I n × I n with h elements in commonis (cid:18) nk (cid:19)(cid:18) kh (cid:19)(cid:18) n − kk − h (cid:19) = Θ( n k − h ) . Finally, we get (cid:88) i ∈I n (cid:88) j ∈N i \{ i } Eξ i ,n ξ j ,n ≤ C (cid:80) k − h =1 p m − h +1 n k − h r d (2 k − h − n = C (cid:80) k − h =1 p m − h +1 n k +1 r dkn ( nr dn ) k − h − ≤ Cp m − k +2 n k +1 r dkn (cid:80) k − h =1 ( nr dn ) k − h − = c (cid:48)(cid:48) p m − k +2 n k +1 r dkn , (9)the last equality holds as the condition: ( nr dn ) n ≥ is a bounded sequence.From the bound ( 8) and ( 9) and Theorem 3.3, we have d T V ( G (cid:48) n , Z n ) = (cid:26) cnp m +2 − kn r dn if k ≥ cnp m − n r dn if 2 ≤ k < λ ) to the normal when λ → ∞ gives us the remaining assertionof the theorem. (cid:4) Proof:
From the definition of percolated random Vietoris-Rips complex, if a simplex σ exists,which means each of its 2-faces exists, i.e., this underlying subgraph with vertex set same asof σ is a complete graph. And the 1-skeleton of the cross polytope O k has 2 (cid:0) k (cid:1) = 2 k ( k − E [˜ o (cid:48) k ] = Θ( n k +2 r (2 k +1) d p k ( k − ) . Then slightly modify the proof of Theorem 2.13 in [10], we finish the proof. (cid:4)
One of the difficulties on the counting of induced subgraphs of random geometric graphsarises from the complicated geometric structures. In the percolated random geometric graphs G ( X n ; p, r ), there are more Γ-subgraphs than we can get directly from G ( X n ; r ) by keepingedges. In other words, there exist many subgraphs which are feasible in G ( X n ; p, r ) but notin G ( X n ; r ) , however, if some subgraph Γ is not feasible for G ( X n ; p, r ), of course, it is notfeasible for G ( X n ; r ) , either. We disturb the geometric structures of G ( X n ; r ) by deleting theunwanted edges with certain probability 1 − p . Roughly speaking, there are ”fewer” edges in G ( X n ; p, r ), but ”more” induced subgraphs with some positive probability.13n this paper, we only explore the counting of induced subgraphs on percolated randomgeometric graph , which is the simplest soft random geometric graph . Can we extend thecounting to the general soft random geometric graph with other more general connectionfunctions(e.g., see [16])? We would be very interested to see more results related to thistopic.Moreover, as we mentioned already, random geometric simplicial complexes is extensivelystudied in these years. There are of course a lot of interesting and challenging open problemsin this areas, see the last part of [2] for some of them. We would like to explore more infurther directions. References [1] Richard Arratia, Larry Goldstein, and Louis Gordon. Two moments suffice for poissonapproximations: the chen-stein method.
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Anshui LiDepartment of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang, P.R.China, 310000.E-mails: [email protected] ..