ON THE APPROXIMATION OF LAPLACIAN EIGENVALUES IN GRAPH DISAGGREGATION

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1 O THE APPROXIMATIO OF LAPLACIA EIGEVALUES I GRAPH DISAGGREGATIO XIAOZHE HU, JOH C. URSCHEL, AD LUDMIL ZIKATAOV Abstract. Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broen into multiple small-degree vertices, in order to allow for more efficient computation in parallel. In particular, we consider computations involving the graph Laplacian, which has significant applications, including diffusion mapping and graph partitioning, among others. We prove results regarding the spectral approximation of the Laplacian of the original graph by the Laplacian of the disaggregated graph. In addition, we construct an alternate disaggregation operator whose eigenvalues interlace those of the original Laplacian. Using this alternate operator, we construct a uniform preconditioner for the original graph Laplacian. 1. Introduction A variety of real-world graphs, including web networs [1], social networs [14], and bioinformatics networs [9], exhibit a degree power law. amely, the fraction of nodes of degree, denoted by P, follows a power distribution of the form P γ, where γ is typically in the range < γ < 3. etwors of this variety are often referred to as scale-free networs. The pairing of a few high-degree vertices with many low-degree vertices on large scale-free networs maes computations such as Laplacian matrix-vector products and solving linear and eigenvalue equations challenging. The computation of the minimal nontrivial eigenpair can become prohibitively expensive. This eigenpair has many important applications, such as diffusion mapping and graph partitioning [, 1, 15, 16]. Breaing the few high degree nodes into multiple smaller degree nodes is a way to address this issue, especially when large-scale parallel computers are available. This technique, called graph disaggregation, was introduced by Kuhlemann and Vassilevsi [10, 6]. In this process, each of the high-degree vertices of the networ is replaced by a graph, such as a cycle or a clique, where each incident edge of the original node now connects to a node of the cycle or clique see Figure 1.1. Independently, Lee, Peng, and Spielman investigated the concept of graph disaggregation, referred to as vertex splitting, in the setting of combinatorial spectral sparsifiers [11]. They proved results for graphs disaggregated from complete graphs and expanders, and used the Schur complement of the disaggregated Laplacian with respect to the disaggregated vertices to approximate the original Laplacian. The basic motivating assumption in such constructions is that the spectral structure of the graph Laplacian induced by the disaggregated graph approximates the spectral structure of the original graph well. In [10, 6] Kuhlemann and Vassilevsi too a numerical approach. We extend, expand upon, and prove precise and rigorous theoretical results regarding this technique. First, we 010 Mathematics Subject Classification. 05C85; 65F15; 65F08; 68R10. Key words and phrases. Spectral Graph Theory; Graph Laplacian; Disaggregation; Spectral Approximation; Preconditioning. 1

2 XIAOZHE HU, JOH C. URSCHEL, AD LUDMIL ZIKATAOV Figure 1.1. Example of disaggregation: original graph left; disaggregate using cycle middle; disaggregate using clique right. loo at the case of a single disaggregated vertex and establish bounds on the error in spectral approximation with respect to the Laplacians of the original and disaggregated graph, as well as results related to the Cheeger constant. We investigate a conjecture made in [10] and give strong theoretical evidence that it does not hold in general. Then, we treat the more general case of disaggregation of multiple vertices and prove analogous results. Finally, we construct an alternative disaggregation operator whose eigenvalues interlace with those of the original graph Laplacian, and, hence, provide excellent approximation to the spectrum of the latter. We then use this new disaggregation operator to construct a uniform preconditioner for the graph Laplacian of the original graph. We prove that the preconditioned graph Laplacian can be made arbitrarily close to the identity operator if we require that the weights of the internal disaggregated edges are sufficiently large.. Single Vertex Disaggregation Consider a weighted, connected, undirected graph G = V, E, ω, V = n. Let e = i, j denote an edge that connects vertices i and j, and, and denote the standard l -inner product and the corresponding induced norm. The associated weighted graph Laplacian A R n n is given by Au, v = ω e u i u j v i v j, ω e = a ij, e=i,j E where we denote the i, j-th element of A by a ij. Without loss of generality, let us disaggregate the last vertex v n of the graph G. Then, the Laplacian can be written in the following bloc form A0 a A = n a T, n a nn where a nn is the degree of v n. Here, we assume that the graph is simply connected and the associated Laplacian A has eigenvalues and corresponding eigenvectors 0 = λ 1 A < λ A λ n A 1 n = ϕ 1 A, ϕ A,, ϕ n A, where 1 n = 1,, 1 }{{} T. n The eigenpair λ A, ϕ A has special significance, and therefore λ A is referred to as the algebraic connectivity, denoted ag, and ϕ A is referred to as the Fiedler vector.

3 O THE APPROXIMATIO OF LAPLACIA EIGEVALUES I GRAPH DISAGGREGATIO 3 We can also write a given nontrivial eigenpair λa, ϕa, λa 0, ϕa = 1, in bloc notation, namely ϕ0 ϕa =. ϕ n We have the relations ϕ 0, 1 n0 + ϕ n = 0, A 0 ϕ 0 ϕ n a n = λaϕ 0, a nn ϕ n a T nϕ 0 = λaϕ n, where n 0 = n 1. Suppose that the vertex v n is disaggregated into d vertices, with an unspecified connected structure between the disaggregated elements. We will denote this graph by G D. This induces a disaggregated graph Laplacian A D R, = n 0 + d, with eigenvalues 0 = λ 1 A D < λ A D λ A D and corresponding eigenvectors 1 = ϕ 1 A D, ϕ A D,, ϕ A D. We can write A D in bloc form A0 A A D = 0n A T. 0n A n We have the relations a nn = a T n1 n0, A T 0n1 n0 = A n 1 d, a n = A 0 1 n0 = A 0n 1 d. Let us introduce the prolongation operator P : R n R, In0 n.1 P = d The following result is immediate. Lemma 1. Let A and A D be the graph Laplacian of the original graph G and the disaggregated and simply connected graph G D, respectively. If P is defined as.1, then we have A = P T A D P. We aim to show that the algebraic connectivity of A D is bounded away from the algebraic connectivity of the original graph A. To do so, suppose we have an eigenpair λ, ϕ of the Laplacian of the original graph G. We prolongate the eigenvector ϕ to the disaggregated graph G D and obtain an approximate eigenvector by the procedure. ϕ = P ϕ s1 = ϕ0 ϕ n 1 d s1, where s = d 1 ϕ n. This gives ϕ, 1 = 0. We consider ϕ to be an approximation of ϕ on the non-trivial eigenspace of the disaggregated operator A D. We have the following relation between the eigenvalue λ of A and the Rayleigh quotient of ϕ with respect to A D.

4 4 XIAOZHE HU, JOH C. URSCHEL, AD LUDMIL ZIKATAOV Lemma. Let λ, ϕ, ϕ = 1, be an eigenpair of the graph Laplacian A associated with a simply connected graph G, and ϕ be defined by.. We have Proof. We have and RQ ϕ := A D ϕ, ϕ ϕ, ϕ = λ. 1 + d 1n ϕ n ϕ, ϕ = P ϕ s1, P ϕ s1 = P ϕ, P ϕ s P ϕ, 1 + s 1, 1 This completes the proof. = ϕ 0, ϕ 0 + ϕ n 1 d, 1 d s ϕ 0, 1 n0 + ϕ n 1 d, 1 d + s = ϕ 0, ϕ 0 + ϕ n + d 1ϕ n sd 1ϕ n + s ] d 1 d 1 = 1 + [d 1 + ϕ n d 1n = 1 + ϕ n A D ϕ, ϕ = A D P ϕ s1, P ϕ s1 = A D P ϕ, P ϕ s A D 1, P ϕ + s A D 1, 1 = P T A D P ϕ, ϕ = Aϕ, ϕ = λ. The following result quicly follows by applying Lemma to the Fielder vector. Theorem 1. Let ϕ = ϕ 0, ϕ n T, ϕ = 1, be the Fiedler vector of the graph Laplacian A associated with a simply connected graph G. Let A D be the graph Laplacian corresponding to the disaggregated and simply connected graph G D resulting from disaggregating one vertex into d > 1 vertices. We have ag ag D 1 + d 1n ϕ n. Proof. oting that ϕ is orthogonal to 1, we have which completes the proof. ag D A D ϕ, ϕ ϕ, ϕ = λ 1 + d 1n ϕ n = ag, 1 + d 1n ϕ n If the characteristic value of the disaggregated vertex is non-zero, then the algebraic connectivity of the disaggregated graph stays bounded away from that of the original graph, independent of the structure of A n. Therefore, as the weight on the internal edges approaches infinity, the approximation stays bounded away. In [10], the authors made the following conjecture. Conjecture 1. Under certain conditions the Laplacian eigenvalues of the graph Laplacian of the disaggregated graph approximate the eigenvalues of the graph Laplacian of the original graph, provided that the weight on the internal edges of the disaggregation is chosen to be large enough.

5 O THE APPROXIMATIO OF LAPLACIA EIGEVALUES I GRAPH DISAGGREGATIO 5 Theorem 1 directly implies that Conjecture 1 is false when the characteristic value of the disaggregated vertex is non-zero, which, for a random power law graph, occurs with probability one. We also have the following result, providing an estimate of how close the approximation ϕ is to the invariant subspace with respect to A D. Lemma 3. Let A and A D be the original graph and disaggregated graph, λ, ϕ be an eigenpair of the graph Laplacian A associated with a simply connected graph G, and ϕ be defined by.. We have dnd + n A D ϕ RQ ϕ ϕ A T 0n1 n0 ϕ 0 /ϕ n + λ + dn λ ϕ n ϕ n. Proof. We recall that and ϕ = 1 + 1/ d 1n ϕ n d 1n ϕ n λ RQ ϕ = λ. 1 + d 1n ϕ n We also have A0 ϕ A D ϕ = A D P ϕ s1 = A D P ϕ = 0 ϕ n A 0n 1 d A0 ϕ ϕ n A n 1 d A T = 0 ϕ n a n 0nϕ 0 A T 0nϕ n 1 n0 ϕ = λp ϕ + A T = λ ϕ + sλ1 0nϕ n 1 n0 ϕ 0 λϕ n 1 + d A T 0nϕ n 1 n0 ϕ 0 λϕ n 1 d = λ ϕ + λϕ [ n d 1 n + 0 1d] A T, 0nϕ n 1 n0 ϕ 0 giving A D ϕ RQ ϕ ϕ A D ϕ λ ϕ + λ RQ ϕ ϕ d A T 1 n 0 + dn 0nϕ n 1 n0 ϕ 0 + ϕ n λ + A T 0n1 n0 ϕ 0 /ϕ n + dnd + n λ + dn λ ϕ n d 1n 1 + d 1n ϕ n In many applications, we are only concerned with minimal Laplacian eigenpairs. For minimal eigenvalues of scale-free graphs, we have λ = O1 and ϕ n = O 1/. In this way, often the largest source of error comes from the term A T 0nϕ n 1 n0 ϕ 0. Heuristically, the error of this term is typically best controlled when d is relatively small and each new disaggregate is connected to roughly the same number of exterior vertices. ext, we consider the eigenvalues of the normalized Laplacian. This gives us insight into how the Cheeger constant changes after disaggregating a vertex. Again, suppose we have an eigenpair ν, φ of the normalized graph Laplacian D 1 A, where D is the degree matrix of G, namely D = diaga 11, a,, a nn. Again, we prolongate the eigenvector φ and obtain ϕ n. ϕ nλ

6 6 XIAOZHE HU, JOH C. URSCHEL, AD LUDMIL ZIKATAOV an approximate eigenvector of the disaggregated normalized graph Laplacian D 1 D A D, where D D is the degree matrix of G D, in a similar fashion.3 φ = P φ s1, where s = P φ, 1 DD 1, 1 DD. We may write D D in the following way D D = diaga D 1,1,, a D n 0,n 0, a D n,n, a D n+1,n+1,, a D, = diaga D 1,1,, a D n 0,n 0, ω n, ω n+1,, ω + diag0,, 0, d ex n, d ex n+1,, d ex =: D 1 D + D ex D, where ω n, ω n+1, ω are the weights of the edges incident with vertex v n on the original graph note i=n ω i = a n,n and d ex i = a D i,i ω i, i = n, n + 1,,. We may also rewrite the shift s as s = P φ, 1 DD 1, 1 DD = i=n dex i φ n ad i,i Let ω total H denote the total weights of a graph H, and let G a be the disaggregated local subgraph. Similarly, we consider φ as an approximation of the eigenvectors of the disaggregated normalized graph Laplacian. We have the following lemma. Theorem. Let ν, φ be an eigenpair of the normalized graph Laplacian associated with a simply connected graph G and φ be defined by.3. We have.4 A D φ, φ D D φ, φ = ν 1 + ω totalg ω total G a, ω total G D φ n and.5 ν D = αν, α = 1 + ω 1 totalg ω total G a φ ω total G D n 1. Proof. We have and A D φ, φ = AD P φ s1, P φ s1 = A D P φ, φ = Aφ, φ = ν D D φ, φ = D D P φ, P φ s D D 1, P φ = DDP 1 φ, P φ + DD ex P φ, P φ s D D 1, P φ = Dφ, φ + = 1 + = 1 + d ex i φ n i=n i a D i,i i=n dex i i a D i,i n a i,i i=n dex i φ n. ad i,i. i=n dex i φ n i a D i,i i=n dex i φ n

7 O THE APPROXIMATIO OF LAPLACIA EIGEVALUES I GRAPH DISAGGREGATIO 7 oting that a i,i = ω total G, ad i,i = ω total G D, and i=n dex i obtain.4. Moreover,.5 follows directly from.4. The Cheeger constant of a weighted graph is defined as follows [8, 4] where Ū = V \U, hg = volu = i U min =U V EU, Ū minvolu, volū, δ i, δ i = e=i,j E ω e, EU, Ū = = ω total G a, we e=i,j E i U,j Ū Due to the Cheeger inequality [8, 4], the Cheeger constant hg and ν are related as follows hg ν hg. Theorem 3. For the Cheeger constant of the original graph G and the disaggregated and simply connected graph G D, we have where α is defined by.5. If hg hg D 1 1 αhg, 4α, then hg 4α +1 D hg. Proof. Based on.6 and.5, we have hg D 1 1 ν D = 1 1 αν 1 1 αhg. ω e. Basic algebra shows that hg D hg if hg 4α. 4α Graph Disaggregation We now move on to the more general case of multiple disaggregated vertices. Without loss of generality, for a graph Laplacian A R n n, suppose we are disaggregating the first m vertices. This gives us the disaggregated Laplacian A D R. Here, is the number of vertices in the disaggregated graph, given by = n m + d = n m + n d, n d = d. =1 ote that we have m groups of vertices associated with the disaggregation, which we can also number consecutively 3.1 {1,..., } = {1,..., d }{{} 1,... d 1 + 1,..., d 1 + d,..., n }{{} d + 1,..., }. d 1 Similar to the case of a single disaggregated vertex, we can establish a relationship between A and A D through a prolongation matrix P : R n R, given by Pm 0 3. P =, where n 0 I 0 = n m. n0 n 0 d =1

8 8 XIAOZHE HU, JOH C. URSCHEL, AD LUDMIL ZIKATAOV Here, P m R n d m, and 1 d P m = d dm ote that P T mp m = diagd 1,... d m. We have the following lemma, which can be easily verified by simple algebraic calculation. Lemma 4. Let A and A D be the graph Laplacian of the original graph G and the disaggregated and simply connected graph G D, respectively. If P is defined as 3., then we have 3.3 A = P T A D P. If we loo at the disaggregated graph Laplacian A D directly, we can obtain a similar bound on the algebraic connectivity as shown in Theorem 1. Let λ, ϕ be an eigenpair of A. We can define an approximated eigenvector of A D by prolongating ϕ as follows 3.4 ϕ = P ϕ s1, where s = 1 d i 1ϕ i. It is easy to chec that ϕ, 1 = 0. ow we have the following lemma about the Rayleigh quotient of ϕ with respect to A D. Lemma 5. Let λ, ϕ be an eigenpair of the graph Laplacian A associated with a simply connected graph G and ϕ be defined by 3.4. We have RQ ϕ := A D ϕ, ϕ ϕ, ϕ λ m d. i 1n + n d md i ϕ i Moreover, if for d max i Proof. We note that = max i d i, we have n + n d md max i > 0, then RQ ϕ < λ. A D ϕ, ϕ = A D P ϕ s1, P ϕ s1 = A D P ϕ, P ϕ s A D 1, P ϕ + s A D 1, 1 = P T A D P ϕ, ϕ = Aϕ, ϕ = λ.

9 O THE APPROXIMATIO OF LAPLACIA EIGEVALUES I GRAPH DISAGGREGATIO 9 Denoting ϕ by ϕ = ϕ 1, ϕ,, ϕ m, ϕ 0 T, we have ϕ, ϕ = P ϕ s1, P ϕ s1 = P ϕ, P ϕ s P ϕ, 1 + s 1, 1 = ϕ 0, ϕ 0 + ϕ i 1 di, 1 di s ϕ 0, 1 n0 + ϕ i 1 di, 1 di + s = ϕ 0, ϕ 0 + ϕ i + d i 1ϕ i s d i 1ϕ i + s = 1 + d i 1ϕ i 1 m d i 1ϕ i 1 + d i 1ϕ i m d i 1 ϕ i = 1 + d i 1 md i 1 ϕ i = d i 1n + n d md i ϕ i. This completes the proof. From the Rayleigh quotient and applying the above lemma to the Fielder vector, we have the following theorem concerning the algebraic connectivity. Theorem 4. Let ϕ be the Fiedler vector of the graph Laplacian A associated with a simply connected graph G and A D be the graph Laplacian corresponding to the disaggregated and simply connected graph G D. Suppose we have disaggregated m vertices and each of those vertices are disaggregated into d i > 1 vertices, i = 1,,, m. We have ag ag D d i 1n + n d md i ϕ i. Proof. The proof is similar to the proof of Theorem 1 and uses Lemma 5. It is possible to perform a more careful estimate, using the fact that disaggregating m vertices at once is equivalent to disaggregating m vertices one by one. Denote n 0 = n and n i = n i d i, i = 1,,, m. ote that n i = n i + i =1 d, i = 1,,, m and = n m. Recursively applying Theorem 1, we have the following result. Theorem 5. Let ϕ be the Fiedler vector of the graph Laplacian A associated with a simply connected graph G and A D be the graph Laplacian corresponding to the disaggregated and simply connected graph G D, respectively. Suppose we disaggregated m vertices and each of those are disaggregated into d i > 1 vertices, i = 1,,, m. We have ag ag D 1 + d i 1n i 1 ϕ i n i.

10 10 XIAOZHE HU, JOH C. URSCHEL, AD LUDMIL ZIKATAOV Proof. Let G i D be the resulting graph after disaggregating the i-th vertex in the graph Gi 1 D and note that G D = G m D. We have ag ag D = ag ag 1 D ag1 D ag D agm 1 D ag m D. The result follows immediately from applying Theorem 1 on each pair of graphs G i D and G i+1 D. Remar 1. A direct consequence of Theorem 5 is ag D ag. Similarly, we also have the following result concerning the minimal eigenvalue of the normalized graph Laplacian after disaggregating several vertices. Denote G 0 D = G, and denote the graph after disaggregating vertex i by G i D. ote that Gm D = G D. The local subgraph corresponding to disaggregating vertex i is denoted by G i a. Theorem 6. Let ν be the second smallest eigenvalue of the normalized graph Laplacian associated with a simply connected graph G and ν D be the second smallest eigenvalue of the normalized graph Laplacian corresponding to the disaggregated and simply connected graph G D. Suppose we disaggregated m vertices and each of them are disaggregated into d i > 1 vertices, i = 1,,, m. We have ν m 1 + ω totalg i 1 D ω totalg i a ω total G i D φ i. ν D Consequently, we have ν D = αν, where [ m 3.5 α := 1 + ω totalg i 1 D ω ] totalg i 1 a ω total G i D φ i 1. Proof. The result follows by applying Theorem recursively. Based on the estimates on the eigenvalues of normalized graph Laplacian, we can estimate the Cheeger constants as follows. Theorem 7. For the Cheeger constant of the original graph G and the disaggregated and simply connected graph G D, we have where α is defined by 3.5. If hg hg D 1 1 αhg, 4α, then hg 4α +1 D hg. Proof. The proof is the same as the proof of Theorem Preconditioning Using Disaggregated Graph We aim to show eigenvalue interlacing between A and a new operator, which is obtained by scaling A D appropriately. We can rescale P by introducing P = D s P, where 4.1 D s = diagd 1/ 1 I d1 d 1,..., d 1/ m I dm d m, I n0 n0 is a diagonal scaling matrix, giving us P T P = I. Based on the scaled prolongation, we are able to show the eigenvalues of diagonal scaled matrix 4. Ã D := Ds 1 A D Ds 1

11 O THE APPROXIMATIO OF LAPLACIA EIGEVALUES I GRAPH DISAGGREGATIO 11 interlaces with A. First, let us recall the interlacing theorem. Theorem 8 Interlacing Theorem [5], Vol. 1, Chap. I. Let S R n be such that S T S = I n n, n < and let B R be symmetric, with eigenvalues λ 1 λ... λ n. Define A = S T BS and let A have eigenvalues µ 1 µ... µ m. Then λ i µ i λ n m+i. From here, we have the following. Theorem 9. Let A have eigenvalues λ 1 A λ A... λ n A and ÃD = Ds 1 A D Ds 1 have eigenvalues λ 1 ÃD λ ÃD... λ ÃD. Then Proof. From the above Lemma, we have A = P T A D P = P T D s D 1 s λ i ÃD λ i A λ n+i ÃD. A D D 1 s D s P = P T D 1 s A D D 1 s P = P T à D P. As P T P = In n, by the Interlacing Theorem 8, the eigenvalues of A and ÃD interlace. We now discuss how to use the disaggregated graph G D to solve the graph Laplacian on the original graph G. Here, we will use ÃD as the auxiliary problem and design a preconditioner based on the Fictitious Space Lemma [13] and auxiliary space framewor [18]. Because A and ÃD are both symmetric positive semi-definite, we first state the refined version of the Fictitious Space Lemma proposed in [7]. Theorem 10 Theorem 6.3 and 6.4 in [7]. Let Ṽ and V be two Hilbert spaces and Π : Ṽ V be a surjective map. Suppose that à : Ṽ Ṽ and A : V V are symmetric semi-definite operators. Moreover, suppose 4.3 Πà = A, 4.4 Π ṽ A c 1 ṽ Ã, ṽ Ṽ, 4.5 for any v V there exists ṽ Ṽ such that Π ṽ = v and ṽ à c 0 v A, then for any symmetric positive definite operator B : Ṽ Ṽ, we have that for B = Π B Π T, c1 κba κ BÃ. c 0 Applying the above theory to our disaggregation framewor, we tae A = A, à = ÃD, and Π = P T. oting that the null space of ÃD is spanned by D s 1, we have P T D s 1 = P T D s1 = 1 n which verifies 4.3. aturally, P T is surjective. Using a preconditioner B D of ÃD, we can define a preconditioner B = P T BD P for A. We give the following results concerning the quality of the preconditioner B. Corollary 1. Let A be the graph Laplacian corresponding to the graph G and A D be the graph Laplacian corresponding to the disaggregated and simply connected graph G D. Let D s be defined by 4.1 and ÃD be defined by 4.. If 4.6 P T ṽ A c 1 ṽ ÃD, ṽ Ṽ

12 1 XIAOZHE HU, JOH C. URSCHEL, AD LUDMIL ZIKATAOV and for any v V, there exist a ṽ Ṽ such that P T ṽ = v and 4.7 ṽ ÃD c 0 v A. Then for the preconditioner B = P T BD P, we have c1 κba κ B D Ã D. c 0 We need to verify that conditions 4.6 and 4.7 hold for P = D s P. For condition 4.7, we choose ṽ = P T v for any v V, giving P T ṽ = P T P v = v since P T P = I. ote that 4.8 ṽ Ã D = ÃD P v, P v = v A, which implies condition 4.7 holds with c 0 = 1. To show that condition 4.6 holds, we use the following result. Lemma 6. Let A R n n be a graph Laplacian corresponding to a connected graph with n vertices. For all i {1,..., n} and u R n we have where A i = A + e i e T i. 1 n u, 1 n u, e i = 1 n A 1 i 1 n, Au, Proof. First, we note that for all i {1,..., n} the matrices A i are invertible, because they all are irreducibly diagonally dominant M-matrices. We refer to Varga [17] for this classical result. ext, observe that A i 1 n = e i, and hence, A 1 i e i = 1 n. Therefore, we have that A 1 i 1 n, Au = A 1 i 1 n, A i e i e T i u = 1 n, u u, e i A 1 i 1 n, e i = 1 n, u u, e i 1 n, A 1 i e i = 1 n, u u, e i 1 n, 1 n. As 1 n, 1 n = n, this completes the proof. The result shown in Lemma 6 is also found in [3, Lemma 3.], but is included for completeness. We now apply Lemma 6 to each disaggregated local subgraph G a = Va, Ea, ωa, = 1,,, m, with u = ṽ, the restriction of ṽ on G a. For j Va, we have, 4.9 ṽ j = 1 ṽ p 1 L 1,j d d 1 d, L ṽ, p V a where L is the unweighted graph Laplacian of the local graph G a and L,j is defined in accordance with Lemma 6: L,j = L + e j e T j, for j V a. Setting W j := 1 L 1 d,j 1 d L, j Va, and denoting E 0 D := {e = i, j E D, i, j V 0 }, E 1 D := {e = i, j E D, i V 0, j V a, = 1,,, m}, E D := {e = i, j E D, i V a, j V l a,, l = 1,,, m, l}, we are ready to present the following lemma related to the condition 4.6.

13 O THE APPROXIMATIO OF LAPLACIA EIGEVALUES I GRAPH DISAGGREGATIO 13 Lemma 7. For each disaggregated local subgraph G a, if, for an edge e = p, q Ea, we assign a weight ω e such that 4.10 ω e W e := 1 + ɛ 1 then we have e=i,j E 1 D i V 0, j V a m ω e W j + l=1 e=i,j E D i V a, j V l a 4.11 P T ṽ A 1 + ɛ ṽ Ã D, ṽ Ṽ, where ɛ > 0. Proof. We denote ũ = P T P ṽ, and we have, P T ṽ A = ÃDũ, ũ = = e=i,j E 0 D + e=i,j E 1 D =: I 0 + I 1 + I. e=i,j E D ω e ũ i ũ j ω e ũ i ũ j + ω e ũ i ũ j + ω e W i + =1 e=i,j Ea e=i,j E D Here, we have set I 0 = e=i,j E ω D 0 e ṽ i ṽ j, I 1 = ṽ i 1 d and I = =1 e=i,j E 1 D i V 0, j V a =1 l=1 e=i,j ED i Va, j V a l ω e ω e p V a ṽ p 1 ṽ p 1 d d l p V a e=i,j E D i V l a, j V a ω e ũ i ũ j ω e ũ i ũ j, q V l a ṽ q. ω e W j, ext, we estimate I 1 and I on the right-hand side. For e = i, j ED 1, i V 0 and j Va, using 4.9, we have ṽ i 1 ṽ p = ṽ i ṽ j 1 L 1,j d d 1 d, L ṽ p V a 1 + ɛ ṽ i ṽ j ɛ 1 1 L 1 d,j 1 d L ṽ L = 1 + ɛ ṽ i ṽ j + [ ] 1 + ɛ 1 W j ṽp ṽ q. e =p,q E a

14 14 XIAOZHE HU, JOH C. URSCHEL, AD LUDMIL ZIKATAOV Then I 1 =1 = 1 + ɛ + e=i,j E 1 D i V 0, j V a ω e =1 e=i,j ED 1 i V 0, j Va =1 e =p,q Ea 1 + ɛ ṽ i ṽ j + ω e ṽ i ṽ j 1 + ɛ 1 e=i,j E 1 D i V 0, j V a e =p,q E a ω e W j ṽ p ṽ q. [ ] 1 + ɛ 1 W j ṽp ṽ q ext, using 4.9, for e = i, j E D, i V a and j V l a we have 1 d p V a ṽ p 1 d l q V l a ṽ q = ṽ i ṽ j + 1 L 1,i d 1 d, L ṽ 1 L 1 l,j d 1 d l, L l ṽ l l 1 + ɛ ṽ i ṽ j ɛ 1 1 L 1 d,i 1 d L ṽ L ɛ 1 1 L 1 d l,j 1 d l Ll ṽ l L l l = 1 + ɛ ṽ i ṽ j + [ 1 + ɛ 1 W] i ṽp ṽ q + e =p,q E a [ 1 + ɛ 1 W j l ] ṽp ṽ q. e =p,q E l a Then I =1 l=1 e=i,j ED i Va, j Va l + e =p,q E l a ω e 1 + ɛ ṽ i ṽ j + [ ] 1 + ɛ 1 W j l ṽp ṽ q. e =p,q E a [ 1 + ɛ 1 W i ] ṽp ṽ q

15 O THE APPROXIMATIO OF LAPLACIA EIGEVALUES I GRAPH DISAGGREGATIO 15 Therefore, we have Hence, I 1 + ɛ + I 1 + ɛ + + =1 l=1 e=i,j ED i Va, j V a l =1 e =p,q Ea l=1 e =p,q Ea l =1 l=1 e=i,j ED i Va, j Va l =1 e =p,q Ea l=1 =1 ω e ṽ i ṽ j 1 + ɛ 1 l=1 ω e ṽ i ṽ j e=i,j E D i V a, j V l a e=i,j E D i V a, j V l a e=i,j E D i V a, j V l a 1 + ɛ 1 ω e W i ṽ p ṽ q 1 + ɛ 1 ω e W j l ṽ p ṽ q. ω e W i + l=1 e=i,j E D i V l a, j V a ω e W j ṽ p ṽ q. ow, we use the definition of W e 4.10 and the estimates on I 1 and I to obtain that P T ṽ A ω e ṽ i ṽ j ɛ ω e ṽ i ṽ j +1 + ɛ e E 0 D =1 l=1 e=i,j ED i Va, j Va l =1 ω e ṽ i ṽ j + e=i,j E 1 D i V 0, j V a =1 e =p,q Ea W e ṽ p ṽ q. Due to 4.10, we have that ω e W e and 4.11 follows. This completes the proof. Lemma 7 shows that the constant c 1 can be made arbitrarily close to 1 if the weights on the internal edges of the disaggregation are chosen to be large enough. As an immediate consequence, we have the following theorem for the preconditioner B. Theorem 11. Under the assumptions of Corollary 1 and Lemma 7, for the preconditioner B = P T BD P, we have 4.1 κba 1 + ɛ κ B D Ã D. Proof. The relation 4.1 follows from Corollary 1 since c 0 = 1 in 4.8 and c 1 = 1 + ɛ 1/ in Lemma 7.

16 16 XIAOZHE HU, JOH C. URSCHEL, AD LUDMIL ZIKATAOV Finally, since ÃD := Ds 1 A D Ds 1, if we have a preconditioner B D for A D and define B D = D s B D D s, then it is easy to verify that κ B D Ã D = κb D A D. We have the following theorem showing that the preconditioned operator BA has a condition number comparable to the condition number of B D A D. Theorem 1. Under the assumptions of Corollary 1 and Lemma 7 and let B D = D s B D D s, for the preconditioner B = P T BD P, we have 4.13 κba 1 + ɛ κb D A D. Proof follows from Theorem 11 and the fact that κ B D Ã D = κb D A D. Clearly, Theorems 4.1 and 4.13 imply that, when the weights on the internal edges of the disaggregation are chosen to be large enough, preconditioners for disaggregated graph provide effective preconditioners for the original graph, which indirectly supports the technique suggested in [10]. Acnowledgements The authors would lie to than Louisa Thomas for improving the style of the presentation. The wor of Ludmil Ziatanov was supported in part by the ational Science Foundation under grants DMS and DMS and by the Department of Mathematics at Tufts University. References [1] Albert-László Barabási and Réa Albert. Emergence of scaling in random networs. Science, :509 51, [] Stephen Barnard and Horst Simon. Fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems. Concurrency: Practice and Experience, 6: , [3] James Brannic, Yao Chen, Johannes Kraus, and Ludmil Ziatanov. Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs. SIAM Journal on umerical Analysis, 513: , 013. [4] Fan Chung. Spectral graph theory, volume 9. American Mathematical Soc., [5] Richard Courant and David Hilbert. Methoden der mathematischen Physi. Berlin, 194. [6] Pasqua D Ambra and Panayot Vassilevsi. Compatible matching adaptive AMG preconditioners for Laplacian matrices on general graphs. Tech. Rep. LLL-TR , Lawrence Livermore ational Laboratory, 015. [7] Blanca Ayuso de Dios, Franco Brezzi, L. Donatella Marini, Jinchao Xu, and Ludmil T. Ziatanov. A simple preconditioner for a discontinuous Galerin method for the Stoes problem. Journal of Scientific Computing, 583: , 014. [8] Shmuel Friedland and Reinhard abben. On Cheeger-type inequalities for weighted graphs. Journal of Graph Theory, 411:1 17, 00. [9] Yongmei Ji, Xing Xu, and Gary Stormo. A graph theoretical approach for predicting common RA secondary structure motifs including pseudonots in unaligned sequences. Bioinformatics, 010: , 004. [10] Verena Kuhlemann and Panayot Vassilevsi. Improving the communication pattern in matrix-vector operations for large scale-free graphs by disaggregation. SIAM Journal on Scientific Computing, 355:S465 S486, 013. [11] Yin Tat Lee, Richard Peng, and Daniel A Spielman. Sparsified cholesy solvers for sdd linear systems. arxiv preprint arxiv: , 015. [1] Boaz adler, Stephane Lafon, Ronald Coifman, and Ioannis Kevreidis. Diffusion maps, spectral clustering and eigenfunctions of foer-planc operators. In in Advances in eural Information Processing Systems 18, pages MIT Press, 005.

17 O THE APPROXIMATIO OF LAPLACIA EIGEVALUES I GRAPH DISAGGREGATIO 17 [13] Sergey epomnyaschih. Decomposition and fictitious domains methods for elliptic boundary value problems. In Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations orfol, VA, 1991, pages 6 7. SIAM, Philadelphia, PA, 199. [14] Mar E.J. ewman. Finding community structure in networs using the eigenvectors of matrices. Phys. Rev. E 3, 743:036104, 19, 006. [15] John C Urschel, Jinchao Xu, Xiaozhe Hu, and Ludmil Ziatanov. A cascadic multigrid algorithm for computing the fiedler vector of graph Laplacians. Journal of Computational Mathematics, 33, 015. [16] John C Urschel and Ludmil Ziatanov. Spectral bisection of graphs and connectedness. Linear Algebra and its Applications, 449:1 16, 014. [17] Richard S. Varga. Matrix iterative analysis, volume 7 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, expanded edition, 000. [18] Jinchao Xu. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing, 563:15 35, Department of Mathematics, Tufts University, Medford, MA address: Xiaozhe.Hu@tufts.edu Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA address: urschel@mit.edu Department of Mathematics, The Pennsylvania State University, University Par, PA address: ludmil@psu.edu