Spectral Methods for Testing Clusters in Graphs

Transcription

1 Spectral Methods for Testing Clusters in Graphs Sandeep Silwal, Jonathan Tidor Mentor: Jonathan Tidor August, 8 Abstract We study the problem of testing if a graph has a cluster structure in the framework of graph property testing Given some ɛ, we say that a graph with a degree bound d is k, φ) clusterable if it can be partitioned into at most k parts such that the inner conductance of each part is at least φ and the outer conductance of each part is at most c d,k φ We present an algorithm that accepts all graphs that are k, φ) clusterable with probability at least and rejects all graphs that are ɛ-far from k, φ )-clusterable for φ c d,k µφ ɛ where < µ < C for some constant C This improves upon the work of Czumaj, Peng, and Sohler in [] by removing a log n factor from the denominator of φ Our technique combines the ideas present in [, 7] in addition to a new spectral perspective Currently this edition of the paper only has a proof for the case of k = but the proof for all k will be given in a future edition of the paper Introduction Background on Property Testing In this paper, we study property testing of graphs in the bounded degree model We are given a graph G = V, E) on n vertices where all the vertices have degree at most d Given a graph property P, we say that G is ɛ-far from satisfying P if ɛdn edges need to be added or removed from G for G to satisfy P A property testing algorithm for P is an algorithm that accepts every graph G with P with probability at least and rejects every graph that is ɛ-far from having P with probability at least We assume that the representation of G is given as an oracle which allows us to find the ith neighbor of any vertex v if i d If i is larger than the degree of v, then a special symbol is returned The complexity of a graph property testing algorithm is measured in terms of its query complexity, or the number of queries the algorithm accesses from the oracle Therefore, the goal of property testing algorithms is to find an algorithm with an efficient query complexity This framework of property testing of graphs was developed by Goldreich and Ron [4] has been applied to study various properties such as bipartiteness [5] and -colorability [4] See [4] and [9] for some more examples Our paper deals with a slightly different definition of property testing We test for a property P that is parameterized by a parameter α such that Pα) Pα ) if α α We

2 then accept graphs with property Pα) with probability at least and reject graphs that are ɛ-far from having property Pα ) with probability at least where α < α Testing k-clusterability We are interested in the property of k clusterability as introduced by Czumaj, Peng, and Sohler in [] Roughly, a graph is k-clusterable if it can be partitioned into at most k clusters where vertices in the same cluster are well connected while vertices in different clusters are well-separated The quality of the clusters will be measured in terms of their inner and outer conductance, which are defined below The idea of using conductance for graph clustering has been studied in numerous works, such as [] More formally, if S V such that S V /, the conductance of S is defined as es,v \S) d S φ G S) = where es, V \ S) is the number of edges between S and V \ S) The conductance of G is defined to be the minimum conductance over all subsets S V/ and is denoted as φg) Now for any S V, let G[S] denote the induced subgraph of G on the vertex set defined by S We will let φg[s]) denote the conductance of this subgraph To avoid confusion, if S V, we will call φ G S) the outer conductance of S and φg[s]) the inner conductance We say G is k, φ)-clusterable if there exits a partition of V into at most k subsets C i such that φg[c i ]) φ but φc i ) c d,k φ for all i where c d,k is a constant that depends only on d and k Czumaj et al have created an algorithm that accepts all k, φ)-clusterable graphs with probability at least and rejects all graphs that are ɛ-far from k, φ )-clusterable graphs where φ = c φ ɛ 4 d,k, where c log n d,k depends only on d, k Our work improves upon this result by removing the log n dependency We present an algorithm that accepts all k, φ)-clusterable graphs with probability at least and rejects all graphs that are ɛ-far from k, φ )-clusterable with probability at least where φ = c d,k φ ɛ In this writeup, we only present the case of k = and our paper on a general k will be available on the arxiv shortly We rely on spectral tools to remove the log n dependency Specifically, our algorithm works by sampling a minor of a power of M, the lazy random walk matrix We then reject the graph if all eigenvalues of this matrix are large and accept otherwise Our main result is the following Theorem Let c d,k be a constant depending on d and k Then, Algorithm accepts every k, φ)-clusterable graph of maximum degree at most d with probability at least and and rejects every graph of maximum degree at most d that is ɛ-far from being k, φ )-clusterable with probability at least if φ µc d,k φ ɛ and < µ < C for some constant C As stated above, the current writeup will only present the case of k = with the analysis for a general k in a forthcoming paper Related Work Testing k-clusterability was inspired by property testing of expansions which has been well studied A graph is called an α expander if if every S V of size at most V/ has a neighborhood of size at least α S Czumaj and Sohler [] showed that an algorithm of Goldreich and Ron [6] can distinguish between α expanders of degree bound d and graphs

3 which are ɛ far from having expansion at least Ωα / logn)) This work was subsequently improved by Kale and Seshadhri [7] and then by Nachmias and Shapira [8] who showed that the algorithm of Goldreich and Ron can actually distinguish graphs which are α expanders between graphs which are ɛ-far from Ωα ) expanders The work of Czumaj et al for testing k-clusterability also uses property testing of distributions, such as testing the l norm of a discrete distribution and testing the closeness of two discrete distributions are both used For some work on testing the norm of a discrete distribution and testing closeness of discrete distributions, see [] and [] respectively 4 Definitions In this section we introduce some definitions and tools that we will be used in our analysis Given a graph G with degree bound d, we let M denotes its lazy random walk matrix and I M denotes its Laplacian matrix L Let = λ λ λ n denote the eigenvalues of L and let v,, v n denote the corresponding orthonormal eigenvectors Let ν ν ν n denote the eigenvalues of M where ν i = λ i for i n For u G, we will denote p t u as the probability distribution of the endpoint of a length t random walk that starts at vertex u That is, p t u = u M t = n i= v i u) λ ) t i v i It will also be convinient to expander our definition of k, φ)-clusterable For an undirected graph G, and parameters k, φ in, φ out, we define G to be k, φ in, φ out )-clusterable if there exists a partition of V into h subsets C, C h such that h k and for each i, i h, φg[c ] ) φ in and φ G C i ) φ out Hence in our case, we want to accept graphs that are k, φ, c d,k φ )-clusterable and reject graphs that are ɛ far from k, µφ ɛ, c d,k φ4 ɛ 4 ) clusterable In this paper, we will denote to denote the l norm We will also need the following result from [] which roughly states that eigenvalues are stable under a good approximation Theorem Let H = A + P and suppose H has eigenvalues µ µ n and A has eigenvalues ν ν n Furthermore, suppose P ɛ Then, µ i ν i ɛ for all i n Our algorithm relies on estimating dot products and norms of various distributions Therefore, we need the following results about distribution property testing Theorem [, Theorem ] Let δ, ξ > and let p, q be two distributions over a set n b ) with b max p, q ) Let r > Ω log Then there exists an algorithm denoted by ξ η l -Inner-Product-Estimator that takes as input r samples from each distribution p, q, and returns an estimate of p, q that is accurate to within Oξ) with probability at least η Theorem 4 [, Lemma ] Let G = V, E) with V = n Let v V, σ >, and r 6 n There exists an algorithm denoted as l -Norm-Tester that takes as input r samples from p t v and accepts the distribution of p t v σ and rejects the distribution if 4 p t v > σ, with probability at least 6 n r

4 The Algorithm In this section, we describe our algorithm referenced in Theorem As stated section, our algorithm performs lazy random walks and uses the distribution testing results and 4 to approximate a minor of a power of M For all v in V, we define q t v = p t v We n also define the minor A u,v of M t J as n Our algorithm is the following: A u,v = [ qu ] q u, q v q v, q u q v Algorithm : Cluster-TestG, S, N, t, σ, µ, ξ) for S rounds do Pick a pair of vertices u and v uniformly at random from G Run N random walks of length t 4 Using the results of as an input, test if either p t u σ or p t v σ through the l Norm-Tester as defined in 4 If so, abort and reject 5 Using the results of as an input and the l -Inner-Product-Estimator, approximate each entry of A u,v within ξ Call the approximation Ãu,v 6 Abort and reject if both the eigenvalues of Ãu,v are larger than n +µ 7 Accept Proof of Theorem We will now prove our main result, theorem In the Cluster-Test, we set θ <, S = , < µ < C where C = min, ɛ 4 θ 5c 45 c )), γ = η =, r = 6 n, N = On /+µ ), t = 5 8S η 64c 5 logn), σ = 4 φ γn Completeness: Accepting k, φ) clusterable Graphs We will show that the Cluster-Test algorithm with the parameters defined above accepts G with probability greater than if G is, φ, c dφ ) clusterable We will do this by showing that if G is -clusterable, then the vectors q t v are essentially one dimensional Hence, the matrices A u,v should be close to rank which by Theorem, at least one eigenvalue of A u,v should be sufficiently small First we need a result from [] that states that there is a gap between the kth and the k + th eigenvalue of M if G is k, φ in, φ out )-clusterable Lemma [, Lemma 5] If G is a weighted d-regular graph and k, φ in, φ out )-clusterable then there exists h, h k, and a constant c 5 such that λ i φ out for any i h and λ i φ in for any i h + c 5 h4 4

5 We will also use the following result from [] which states that k, φ in, φ out )-clusterable are accepted with sufficiently high probability in step 4 of Algorithm Lemma [, Lemma 4] Let < γ < If G is k, φ in, φ out )-clusterable, then there V V with V γ) V such that for any u V and any t > c 4k 4 logn), for some φ in universal constant c 4, the following holds: p t u k γn The next two lemmas tell us that for k, φ in, φ out )-clusterable graphs, q u = p u n is well approximated by its projection onto v for all vertices u in G Lemma Let G be, φ in, φ out )-clusterable For a vertex u, let p t u be the projection of p ) v t onto the space spanned by v,, v n Then p t u p t u φ in c 5 Proof Write p t u in the eigenbasis of L Then, p t u p t u = n i= λ i λ φ in c 5 ) t v i u) ) t n v i u) i= ) t We can rewrite the conclusion of lemma in a slightly more useful statement Lemma 4 Let u and v be two vertices Then we can find e u and e v such that both q t u + e ) u t and q t v + e v lie on a line through the origin and max e u, e v ) < φ in c 5 Lemma 5 Let G be, φ in, φ out )-clusterable and let u, v be any two vertices in G Then A u,v has an eigenvalue less than ) t φ in c 5 Proof We know that A u,v is a gram matrix and hence positive semi-definite Therefore, we can write A u,v = κ w w T + κ w w T where w, w = Now let M be a n matrix with columns q u and q v and define M = κ w w T + κ w w T Denote the columns of M as q u, q v By the orthogonality of w and w, we have that M T M = M T M Hence, U = M M is an orthogonal matrix and the image of q u is q u and similarly, the image of q v is q v ) t Now by Lemma 4, we know we can find e u and e v such that e u, e v < φ in c 5 such that M plus the matrix with columns e ) u, e v is rank Since U is orthogonal, we can t find e u, e v such that e u, e v < φ in c such that M plus the matrix with columns 5 e u, e v is rank By Weyl s inequality Theorem ), this means that one of κ, κ must be smaller than ) t φ in c 5 5

6 Combining the above lemmas, we get the following lemma Lemma 6 For the parameters specified in section, Algorithm accepts, φ, c d φ )- clusterable graphs with probability greater than Proof From Lemma, we know that in step 4 of the algorithm, we accept with probability γ) After this step, we know that by Lemma 5 that A u,v has an eigenvalue that is at most ) t φ in c n When we estimate A u,v with Ãu,v in step 5 of Algorithm 5, the eigenvalues of A shift by at most by Weyl s inequality [] Then by the union n +µ bound, the probability that we reject is at most S γ) + γ η) < < Soundness: Rejecting Graphs ɛ-far from k, c d,k µφ )-clusterable We will show that Algorithm rejects G with probability greater than if G is ɛ- far from, µφ ɛ, c d µ φ 4 ɛ 4 )-clusterable We will do this by showing that if u, v) is a representative pair of vertices of G, then we need to perturb q u and q v by vectors of sufficiently large norm for q u and q v to lie on a line through the origin This will imply that both of the eigenvalues of A u,v are large We will also show that there a positive fraction of all pairs of vertices are representative and hence Algorithm will find a representative pair with high probability We need the following inverse theorem from [] which states that if G is ɛ-far from k, φ in, φ out )-clusterable then we can partition G into subsets of vertices that have sparse cuts between them Lemma 7 [, Lemma 45] If G is ɛ-far from k, φ in, φ out )-clusterable with φ in α 45 ɛ, then there exists a partition of V into k + subsets V,, V k+ such that for each i, i k +, V i = Ωɛ V /k) and φ G V i ) c 45 φ in ɛ Given Lemma 7 as a starting point, we now show that there are sufficiently many pairs of vertices u, v) such that that q u and q v are far from being onedimensional More precisely, we show that there is a positive fraction of pairs u, v) such that if the origin, q u + e u and q v + e v are collinear, then we must have max e u, e v ) be sufficiently large In the first step of our analysis which is Lemma 8, we borrow some of the tools from [7] Lemma 8 Let S and S be two disjoint subsets of vertices such that the cut S i, V \ S i ) has conductance less than δ Suppose that S + S n and let pt S = v S p t v, p t S = v S p t v, q t S = p t S, and n qt S = p t S Let Π denote the projection onto the n eigenvectors of M with eigenvalues greater than δthen if α + β, α, β, and α + β, απq S + βπq S S + S ) 6

7 Proof Let s = S, s = S Let α R and define the vector f as α s if v S β fv) = s if v S otherwise and let u = f α+β = n αq S + βq S Write u in the eigenbasis of M as u = i c iv i We have u = i βi = α + β α + β) s s n α s + β s n Therefore, u T Lu = u i c i λ i = i<j M ij u i u j ) From above, it follows that i i α δds d s + d ) α = δ + β s s β s δds c i λ i > α + β s s n δ ) α + β s s Call λ i > 4δ heavy Let H be the set of indices of the heavy eigenvalues Letting x = i H β i, we have ) x + c i x 4δ) > α + β s s n δ ) α + β s s which implies that By Cauchy-Schwartz, Now x > 4 α s + β s α s + β s α + β s + s ) n s + s s + s ) n = x s + s ) 7

8 Hence, απq S + βπq S S + S ) We will now show that the conclusions of Lemma 8 also hold if we consider a large subset of S and S Lemma 9 Let S and S be two disjoint subsets of vertices that satisfy the conditions of Lemma 8 Let T i S i and T i = θ) S i where θ is a sufficiently small constant for each i Let q t T = v T q t v and q t T = v T q t v Furthermore, define α, β, and Π as in Lemma 8 Then απq T + βπq T ) θ S + S Proof Let q t S = v S q t v and q t S = v S q t v Using the fact that T i = θ) S i for each i, we can compute that αq S + βq S ) ) αq T + βq T θ α = + β θ s s ) α θ + β s s Write αq t S + βq t S = i c iv i and αq t T + βq t T = i w iv i and let H denote the set of eigenvalues larger than δ just like in Lemma 8 We have αq S + βq S αq T βq T β i w i ) Let S = s +s From lemma 8 and the triangle inequality, as desired wi > i H βi i H i H S > ) θs = S θ), i H β i w i ) Lemma 9 states that the line segment from q T to q T and the line segment from q T to q T does not pass close to the origin We now need a similar lemma that states that this happens for not only the averages q T and q T, but also for a large number of pairs of vertices 8

9 Lemma Let the sets S and S constant θ, and matrix Π satisfy the conditions of lemma 9 Then there are at least θ S S pairs u, v) where u S, v S, such that the following holds: ) min απq u + α)πq v θ α 4 θ S + S ) Proof Call a pair u, v) bad if it does not satisfy ) and good otherwise Suppose for the sake of contradiction that there are more than θ ) S S bad pairs We will now show that there is a set T S where T θ) S such that for all u T, there are more than θ) S vertices v in S such that the pair u, v) is bad This must be true because otherwise, the number of bad pairs is at most θ S S + θ) S S < θ ) S S Hence, such a set T must exit Now for every u T, let T u denote the set of vertices in S such that u, v) is a bad for pair for all v T u and let q t T u = v T u q t v We now claim that min α απq u + α)πq T u 4 θ ) θ S + S ) Suppose for the sake of contradiction that inequality ) is not true Consider the plane that passing through the origin O, Πq t u and Πq t T u see figure for reference) If ) does not hold, then the line segment L connecting Πqu t and ΠqT t u lies entirely outside the circle C centered at the origin with radius equal to the right hand side of ) By the hyperplane separation theorem, we know that there is a line l that separates L and C Furthermore, we can guarantee that l never intersects C Consider the two half spaces formed by l From the definition of Πq Tu, we know that there is a point v T u such that Πq t v lies on the half space not containing C Hence, the line segment from Πq t u and Πq t v does not pass C However, contradictions our assumption on the set T u Hence, the first inequality in ) must hold Now let q t S = v S qv t Letting β = α, we have απq u + βπq S = απq u + βπq T u + βπq S q T u ) απq u + βπq T u + β q Tu q S We first bound the second term Then using ), we have min α απq u + βπq S β p S p T u = θ θ) S θ S 4θ S + S = ) θ 4 ) θ S + S 9 S + S 4θ S + S ) + 8θ S + S

10 Note that u was an arbitrary vertex in T By letting qt t geometric argument as above, we have that = u T qt u and using the same min α απq T + α)πq S ) θ S + S However, this is a contradiction to lemma 9 so we are done Hence, there must be at least θ S S pairs u, v) where u S, v S, such that ) holds The conclusion of Lemma also holds true if we let β = α In this case, the argument is similar except we consider the line segment passing through Πq t u and Πq t T u Now we need to show that there are sufficiently many pairs u, v) where u S and v S such that both the line segment from q u to q v and the line segment from q u to q v does not pass close to the origin Lemma Let the sets S and S constant θ, and matrix Π satisfy the conditions of Lemma 9 Then there are at least θ S S pairs u, v) where u S, v S, such that both the following hold: min απq u + α)πq v 4 θ min απq u + α )Πq v 4 θ α α ) θ ) θ S + S, S + S Proof The proof of this lemma will be given in a future edition of the paper ) Πq t v Πq t u Πq t T u l O C Figure : If the line segment connecting Πq t u and Πq t T u does not intersect circle C then there must be a separating line H Furthermore, there is a point Πq t v, where v T u, that lies on the opposite side of C with respect to H Hence the segment connecting Πq t u and Πq t v does not intersect C We now combine Lemmas 7 and 9 Furthermore, from our choice of constants in Section, we arrive at the following lemma

11 Lemma Let c = 5c 45 c 5 Call a pair u, v) representative if both ) min αq t u + α)q t v = Ω α n /+cµ and min α αq t u + α) q t v) ) = Ω n /+cµ A randomly chosen pair is representative with probability at least θ Lastly, we will show that if a pair u, v) is representative, then we need to preturb q u and q v by vectors with large norm for the origin, q, and q to be colliniear Lemma Suppose and min αq + α)q R α min αq + α) q ) R α Let e and e be such that the origin, q +e, and q +e are collinear Then, max e u, e v ) R Proof See figure for a reference Consider the plane that passes through the origin O, q, and q Let C denote the circle centered at the origin with radius R and let C denote the circle centered at the origin that passes through q Let q denote the projection of q onto C The first inequality in the lemma implies that the line segment connecting q and q does not intersect C Then by considering the tangents from q to C, we see that there is a sector S on C, formed by the intersection of the tangents with C, such that q cannot lie on this sector The angle of this sector is precisely 4 arcsinr/ q ) := 4φ The second statement tells us that the line segment connecting q and q does not intersect C Hence, by reflecting q across the origin, we see that there is a sector S identical to S such that q is at the midpoint of this sector Furthermore, q also cannot lie on this sector So far, we have shown that q cannot be close to diametrically opposite of q and that q and q cannot be close on C Now let L be an arbitrary line through the origin and let q denote the intersection of L with C We claim that max q Oq, q Oq ) > φ Indeed, without loss of generality, we can assume that q lies on the upper half circle with respect to q, ie, q O q π We now consider two cases If q lies on the smaller arc connecting q and q, then it is clear that the smallest max q Oq, q Oq ) can be is when q lies on the boundary of S which implies max q Oq, q Oq ) > φ In the case that q lies on the larger arc connecting q and q, it is also clear the smallest max q Oq, q Oq ) can be is when q lies on the boundary of S In this case, it is also true that max q Oq, q Oq ) > φ Thus, to project q and q to any line that passes through the origin, we must have that one of e, e > sinφ) q = R, as desired Lemma 4 If a pair u, v) is representative then both of the eigenvalues of à u,v are larger than Ω n /+cµ )

12 S S q O C Figure : q cannot like on the sectors S and S Proof By Theorem, the eigenvalues of à u,v and A u,v differ by less than Hence it n +µ suffices to show that the eigenvalues of A u,v are larger than Ω ) n /+cµ We know that A u,v is a gram matrix and hence positive semi-definite Therefore, we can write A u,v = κ w w T + κ w w T where w, w = Now let M be a n matrix with columns q u and q v and define M = κ w w T + κ w w T Denote the columns of M as q u, q v By the orthogonality of w and w, we have that M T M = M T M Hence, U = M M is an orthogonal matrix and the image of q u is q u and similarly, the image of q v is q v If κ < Ω/n /+cµ ), then we can find e u, e v such that e u, e v < κ such that M plus the matrix with e u, e v in its columns is rank Since U is an orthogonal matrix, we can thus find e u, e v such that e u, e v < κ and M plus the matrix with columns e u and e v is rank However, this contradicts Lemmas and so we are done We finally show that Algorithm passes the soundess case Lemma 5 If G is ɛ-far from ɛ- far from, µφ ɛ, c d µ φ 4 ɛ 4 )-clusterable then Algorithm rejects G with probability greater than Proof A trial is rejected if we find a representative pair Hence, the probability that a trial is rejected is at least θ S S η) 4 θ ɛ 4 Hence, the probability that all trials except n 64 5 is at most ) θ ɛ 4 S < exp ) < Future Work The authors plan to release a future addition of this paper that gives a full proof of Lemma 9 and presents the analysis of Algorithm for all values of k

13 5 Acknowledgements The authors would like to thank the MIT UROP+ program for the opportunity to work on this project We will also like to thank the Raymond Stevens Fund for financially supporting this project References [] Siu-On Chan, Ilias Diakonikolas, Gregory Valiant, and Paul Valiant Optimal algorithms for testing closeness of discrete distributions In Proceedings of the Twenty-fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 4, pages 9, Philadelphia, PA, USA, 4 Society for Industrial and Applied Mathematics [] Artur Czumaj, Pan Peng, and Christian Sohler Testing cluster structure of graphs CoRR, abs/5494, 5 [] Artur Czumaj and Christian Sohler Testing expansion in bounded-degree graphs Combinatorics, Probability and Computing, 95-6):69?79, [4] Goldreich and Ron Property testing in bounded degree graphs Algorithmica, ): 4, Feb [5] Oded Goldreich and Dana Ron A sublinear bipartiteness tester for bounded degree graphs In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC 98, pages 89 98, New York, NY, USA, 998 ACM [6] Oded Goldreich and Dana Ron On Testing Expansion in Bounded-Degree Graphs, pages Springer Berlin Heidelberg, Berlin, Heidelberg, [7] Satyen Kale and C Seshadhri An expansion tester for bounded degree graphs In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, pages 57 58, Berlin, Heidelberg, 8 Springer Berlin Heidelberg [8] Asaf Nachmias and Asaf Shapira Testing the expansion of a graph Information and Computation, 84):9 4, [9] Dana Ron Algorithmic and analysis techniques in property testing Foundations and Trends in Theoretical Computer Science, 5):7 5, [] Satu Elisa Schaeffer Survey: Graph clustering Comput Sci Rev, ):7 64, August 7 [] Hermann Weyl Das asymptotische verteilungsgesetz der eigenwerte linearer partieller differentialgleichungen mit einer anwendung auf die theorie der hohlraumstrahlung) Mathematische Annalen, 74):44 479, Dec 9