Sparsification using Regular and Weighted. Graphs

Transcription

1 Sparsificatio usig Regular a Weighte 1 Graphs Aly El Gamal ECE Departmet a Cooriate Sciece Laboratory Uiversity of Illiois at Urbaa-Champaig Abstract We review the state of the art results o spectral approximatio of complete graphs by weighte graphs. First, we show that the seco eigevalue of the laplacia matrix of ay -regular graph with vertices has to be at most o1), a the largest eigevalue is at least o1). We the use these bous to erive a fuametal limit o how well a -regular graph ca spectrally approximate the complete graph. If a -regular graph approximates the complete graph with approximatio factor κ, the κ O 1 ). It is cojecture i [5] that the same bou hols for ay weighte graph with average combiatorial egree that approximates the complete graph. We review a first step towars provig the cojecture. If a weighte graph with vertices a average combiatorial egree approximates the complete graph with approximatio factor κ, the κ o1). I. INTRODUCTION Sparsificatio of a graph is the process of costructig aother graph that has fewer eges, with a slight impact o a specific property. There are ifferet otios i the litterature for graph sparsificatio accorig to the efiitio of the preserve property. I [1], this property was efie as the istace betwee every pair of vertices, a i [2], it was efie as the weight of the bouary of every subset of the graph vertices. I this review, we cosier graph sparsifiers that resemble the origial graph accorig to the spectral otio of similarity itrouce i [3] a [4]. I [5], sparsificatio of geeral graphs by spectral approximatio was cosiere. Here, we restrict ourselves to the spectral approximatio of complete graphs. It is kow that all the o-zero eigevalues of the laplacia matrix of the complete graph with vertices are equal to, a the smallest eigevalue is zero with the all oes vector 1 as its eigevector, a hece, it follows that for ay graph G that approximates the complete graph with approximatio factor κ, x T L G x κ, x : x = 1, x 1 1) I particular, all the laplacia eigevalues of ay spectral sparsifier of the complete graph lie betwee a κ. Moreover, the coverse also hols sice 1 T L G 1 = 0 for ay graph G with vertices. I [6] a [7], costrcutios are provie for Ramauja graphs that are -regular graphs whose laplacia eigevalues lie betwee 2 1 a By multiplyig the weight of every ege of a Ramauja graph by 2, we obtai a graph 1

2 2 that spectrally approximates the complete graph with approximatio factor κ = = O 1 ). We show i this review the proof that this is the lowest possible approximatio factor for ay -regular graph that approximates the complete graph. We the show that the weaker bou of hols asymptotically for the approximatio factor of ay weighte graph with average combiatorial egree that approximates the complete graph. A. Documet Orgaizatio We efie the system moel a otatio i Sectio II. I Sectio III, we erive bous o the seco smallest a largest eigevalues of the laplacia matrix of ay -regular graph. We the use these bous i Sectio IV to erive a asymptotic lower bou o the approximatio factor of ay -regular graph that approximates the complete graph. We the show i the same sectio the erivatio of a weaker asymptotic lower bou o the approximatio factor of ay weighte graph with average combiatorial egree that approximates the complete graph. Fially, we raw cocluig remarks i Sectio V. II. SYSTEM MODEL AND NOTATION We oly cosier uirecte graphs i this review. For ay graph G, we let V G eote the set of its vertices, a E G eote the set of its eges, where each ege is represete by the pair of its e poits. We let eote the umber of vertices i the graph. For a weighte graph G, a for ay u, v) E G, we let wu, v) eote the weight of the ege betwee vertices u a v. We let A G eote the ajacecy matrix of the graph G, D G eote the iagoal matrix whose iagoal etries have the egrees of the vertices of G, a L G = D G A G eote the laplacia matrix of G. For a matrix M, λ 1 M), λ 2 M),..., λ M) eote the eigevalues of the matrix, where λ i M) λ i+1 M), i {1, 2,..., 1}. We recall that the combiatorial egree of a vertex i a graph G is the umber of its eighbors i G, a that all the vertices of -regular graphs have exactly eighbors. Also, ay weighte graph whose eges have the same weight is sai to be uiformly weighte. A graph H is sai to approximate aother graph G with approximatio factor κ, if a oly if the followig hols. x T L G x x T L H x κx T L G x, x R 1 2) Throughout the sequel, we ame a graph G, Z + oly if it has vertices. Fially, We let 1 eote the all oes vector of legth, a φ eote the empty set. III. MAXIMUM EXPANSION OF REGULAR GRAPHS I this sectio, we erive asymptotic upper a lower bous o the smallest a largest o-zero eigevalues of the laplacia matrix of ay -regular graph, resepctively. The erive bous will lea to a fuametal limit o how well a -regular graph ca spectrally approximate the complete graph. We erive this fuametal limit i

3 3 the ext sectio. Here, we ote that a upper bou o the seco smallest eigevalue of the laplacia matrix of ay graph implies a fuametal limit o its ege expasio property, as follows by Cheegar s iequality [8]. Theorem 1: Let G be a graph with maximum egree. If there exists two eges i G with istace at least 2k + 2, the λ 1 L G ) k + 1 Proof: Fix a orerig for the two eges with istace at least 2k + 2 i G, Let V 0 be the set of the two epoit vertices of the first ege, a U 0 be the set of the two epoit vertices of the seco ege. Also, for i {1, 2,..., k}, efie V i as the set of vertices at istace i from the first ege, a U i as the set of vertices at istace i from the seco ege. As the istace betwee the two eges is at least 2k + 2, it follows that V i k j=0 U j = φ, a U i k j=0 V j = φ for all i {0, 1,..., k}. Sice the laplacia matrix is symmetric, we kow from the variatioal efiitio of eigevalues i terms of Rayleigh quotiets that the followig hols. Hece, λ 1 L G ) xt L G x x T x λ 1 L G ) = x T L G x mi x R 1 :x 1 x T x for every -vector x such that x 1. We ow costruct a -vector that gives the esire upper bou. We view the vector x as a fuctio f x : {1, 2,..., } R, where the value of f x i) is the same as that of the i th a b elemet of the vector x. For 0 i k, f x v) =, v V 1) i/2 i, a f x v) =, v U 1) i/2 i, where a, b R. Also, f x v) = 0, v / k V i k U i. We set a a b such that a > 0, b < 0, a v=1 f xv) = 0. It follows that x 1 as esire. We ca ow see the followig. 3) 4) x T x = a 2 k V i 1) i + b2 k U i 1) i 5) We ow erive a upper bou o x T L G x. Recall that there are o eges coectig a vertex i V i to a vertex i U j for all i, j {0, 1,..., k}, a that the maximum egree of vertices i G is, implyig that there are at most 1 eges coectig a vertex i V i to a vertex i V i+1 for all i {0, 1,..., k 1}. Also, by efitio of the sets V i a U i for all i {0, 1,..., k}, we kow that o eges exist betwee a vertex i V i a a vertex i V j for all j / {i 1, i + 1}. x T L G x = u,v) E G f x u) f x v)) 2 6) A + B 7) where a k 1 A = a 2 k 1 B = b 2 V i 1) U i 1) 1 1) 1 i/2 1) i+1)/2 1 1) 1 i/2 1) i+1)/2 ) ) V k 1) k, 8) ) ) U k 1) k, 9)

4 4 Let à = a 2 k V i 1), the we obtai the followig bou for A. i k 1 ) ) A = a 2 1 V i 1) 1) i + 1 1) i V 1) i+ 1 k 2 1) k 1 k 1 = a 2 V i 1) i 2 ) ) V k 1) 1) k + 2 V k 1 1) 1) k where a) follows as = 2 1)à + a2 2 V k 1 1) 1) k 12) a) 2 1)à + 2 à 1 1) k + 1 V i 1) i is o-icreasig i i. This is true sice V i 1) V i+1, i {0, 1,..., k 1} as each vertex i V i is coecte oly to at most 1 vertices i V i+1, a each vertex i V i+1 has to be coecte to at least oe vertex i V i. I a similar fashio, we ca show that B B , 14) k + 1 where B = a 2 k U i 1). Sice the same bou hols for à a B B, the it hols for A+B A+B = i A Ã+ B x T x. The statemet follows from equatio 4) sice A + B is a upper bou o x T L G x. Sice the iameter of ay graph o vertices with maximum egree is at least result implies that for ay sequece G +1, G +2,... of -regular graphs, the followig hols. 10) 11) 13) log log 1) O1) [9], the above lim sup λ 1 L Gi ) ) i The fuctio f x costructe i the proof of Theorem 1 is calle a test fuctio. Oe ca also costruct a appropriate test fuctio to erive a lower bou o the largest eigevalue of the laplacia matrix of a -regular graph [5]. More precisely, test fuctios ca be fou to prove that the followig theorem hols. Theorem 2: For ay sequece of -regular graphs G +1, G +2,..., the followig hols. lim if i λ 1L Gi ) ) IV. APPROXIMATION OF THE COMPLETE GRAPH From Theorem 1 a Theorem 2, we ca coclue that λ 1L G ) λ 1L G ) +2 1+o1) 2 1+o1) for ay -regular graph G. I this sectio, we first use this fact to erive a bou o how well a -regular graph with uiform weights ca approximate the complete graph. We the iscuss the extesio of the erive bou to all weighte graphs with average combiatorial egree. We recall that a graph G with vertices is sai to approximate the complete graph K with a factor κ > 1, if a oly if the followig hols. x T L K x x T L G x κx T L K x, x R 1 17)

5 5 I particular, this hols for ay -vector x R 1 : x = 1, x 1. But we kow that all o-zero eigevalues of the laplacia matrix of the complete graph equal to. More precisely, λ i L K ) =, i {1, 2,..., 1}. It follows by the variatioal efiitio of eigevalues that, It follows from 17) a 18) that, max x T L K x = mi x T L K x = 18) x R 1 : x =1,x 1 x R 1 : x =1,x 1 λ 1 L G ) = mi x T L K x x R 1 : x =1,x 1 19) We are itereste i the approximatio of complete graphs by -regular graphs, a we kow from 15) that λ 1 L G ) ɛ, for ay -regular G, where ɛ 0 as the umber of vertices i the graph. I orer to obtai a sequece of -regular graphs with equal ege weights, it follows from 19) that this ege weight must be at least 2 1. We set the ege weight at this miimum value to tighte the erive bou. Now, let G +1, G +2,... be a sequece of -regural graphs with a uiform ege weight of from 15) that, Also, it follows from Theorem 2 that, lim if i lim sup i λ 1 L Gi ) λ 1 L Gi ) 2, the it follows ) Assume that the sequece of graphs asymptotically approximate the complete graph, the we kow that for a large eough graph G i the sequece, λ 1 L G ) = max x R 1 : x =1,x 1 x T L G x Let x be the maximizig argumet i 22), the it follows from 17) that, 21) 22) o1) 23) 1 κx T L K x o1), 24) 1 We also kow from 18) that x T L K x = 1. It follows that the sequece asymptotically approximate the complete graph oly if κ More precisely, we have prove the followig theorem. 1 Theorem 3: A sequece of uiformly weighte -regular graphs ca approximate complete graphs with a approximatio factor of κ, oly if, κ = ) 1 + O 25) 26)

6 6 I [5], it is cojecture that the bou i Theorem 3 hols for ay weighte graph with average egree. Towars the goal of provig this cojecture, the statemet i Theorem 4 below was prove. Before showig the proof of Theorem 4, we prove the followig auxiliary lemmas. Lemma 1: Let G be a weighte graph with vertices that approximates the complete graph K with approximatio factor κ, a let i, 1 i be the weighte egrees of its vertices, the the followig hols. 1 1 ) i κ 1 1 ), i {1, 2,..., } 27) Proof: For ay vector x R 1, x T L G x = u,v) E G wu, v) f x u) f x v)) 2, where f x : {1, 2,..., } R is the fuctio efiig the vector x. By the assumptio, we kow that the followig hols. I particular, it hols for ay vector x i efie by the followig fuctio, 1 f xi v) =, if v is the ith vertex of G 1, otherwise, x T L K x x T L G x κx T L K x 28) It is easy to see that x T i L Gx i = i, x i 2 = 1, a x i 1, i {1, 2,..., }. It follows from 28) that, x T i L K x i i κx T i L K x i 29) Fially, we kow from the variatioal efiitio of o-zero eigevalues of the complete graph K that x T L K x = x 2 for ay -vector x such that x 1, a hece, the statemet follows. Lemma 2: If a graph G with vertices approximates the complete graph K with approximatio factor κ, the the followig hols for ay two -vectors x a y such that x 1 a y 1, κ yt L G y x 2 x T L G x y 2 30) Proof: Sice G approximates K with approximatio factor κ, it follows that for ay two -vectors x a y y T L G y κy T L K y, 31) a x T L G x x T L K x. 32) It follows from 31) a 32) that y T L G y x 2 x T L G x y 2 κ yt L K y x 2 x T L K x y 2. 33) If we assume that x 1 a y 1, the it follows from the variatioal efiitio of the eigevalues of K that y T L K y = y 2, 34) a x T L K x = x 2, 35) a hece, the statemet follows from 33).

7 7 Theorem 4: Let G be a weighte graph with vertices a at least oe vertex with combiatorial egree. If G approximates the complete graph K with approximatio factor κ, the, ) κ O Proof: We use Lemma 2 by costructig -vectors x a y such that x 1 a y 1, a showig that the esire lower bou applies to yt L G y x 2 x T L G x y 2. Let v 0 be a vertex with combiatorial egree, a let its eighbors be v 1, v 2,..., v, a the weight of the ege betwee v 0 a v i is w i, for all 1 i. Also, assume that the sum of weights of the eges betwee the vertex v i a all vertices outsie {v 0, v 1,..., v } is δ i. We first costruct -vectors x a y, a the take their projectios o the subspace orthogoal to 1. We view the vectors x a y as fuctios f x : {1, 2,..., } R a f y : {1, 2,..., } R a costruct them as follows, 1, if v = v 0 f x v) = 1, if v = v i, i {1, 2,..., } 0, otherwise, 1, if v = v 0 f y v) = 1, if v = v i, i {1, 2,..., } 0, otherwise. We kow that x T L G x = u,v) E G wu, v) f x u) f x v)) 2 for ay vector x R 1, where f x is the fuctio efiig the vector x. We ow compute the followig, Similarly, we ca show that It follows that x T L G x = = y T L G y = w i 1 1 ) 2 + w i + w i + y T 1 L G y 1 + x T L G x = 1 1 δ i + w i δ i + w i 2 36) δ i 1 0) 2 37) + 2 w i 38) 2 wi wi+ δ i +w i 2 wi wi+ δ i +w i From Lemma 1, we kow that all weighte egrees of vertices i G lie betwee This implies that δ i+w i w i 39) 40) ) ) a κ κ 1 1 ), sice δ i + w i is the sum of weighte egrees of the vertices v 1, v 2,..., v. Also, w i 1 1 ), sice w i is the weight of vertex v 0, a hece, It follows from 40) that y T L G y x T L G x κ 2 1+κ δ i +w i wi κ. 41)

8 8 that We ow let x a y be the projectios of x a y o the subspace orthogoal to 1, respectively. It follows a hece, as x 2 = x 2 1 < x, > 2 = ) 2, 42) y 2 = y 2 1 < y, > 2 = 2 1 ) 2, 43) ) x 2 y 2 = 1 O Sice 1 T L G 1 = 0, it follows that x T L G x = x T L G x a y T L G y = y T L G y. We fially get the esire lower bou by usig Lemma 2 as follows. 44) κ y T L G y x 2 45) x T L G x y )) 1+κ O 46) 1+κ )) O 47) 1 + κ )) O 48) ) = O 49) We fially obtai the followig corollary. Corollary 1: Let G +1, G +2,... be a sequece of weighte graphs, where for every Z +, the graph G has vertices, average combiatorial egree, a approximates the complete graph K with approximatio factor κ, the, lim sup κ ) Proof: The statemet follows from Theorem 4 by observig that each graph i the sequece has a vertex whose ) combiatorial egree is at most, a that ay O term vaishes as. V. CONCLUSION I this review, we cosiere the spectral approximatio of the complete graph by regular a weighte graphs. We first showe that for ay sequece of -regular graphs G +1, G +2,..., the ratio lim if i λ 1L Gi ) λ 1L Gi ) O 1 ). We the showe that this implies that the same bou hols for the approximatio factor of ay large -regular graph that approximates the complete graph. Fially, we showe the weaker asymptotic lower bou of o the approximatio factor of ay large weighte graph of average combiatorial egree that approximates the complete graph.

9 9 REFERENCES [1] P. Chew, There is a plaar graph almost as goo as the complete graph, I Proc. SoCG, pp , [2] A. Beczur, D. Karger, Approximatig s-t miimum cuts i O 2 ) time, I Proc. STOC, pp.47-55, [3] D. Spielma, S. Teg, Nearly-liear time algorithms for graph partitioig, graph sparsificatio, a solvig liear systems, I Proc. STOC, pp.81-90, [4] D. Spielma, S. Teg, Spectral sparsificatio of graphs, available at [5] J. Batso, D. Spielma, N. Srivastava Twice-Ramauja Sparsifiers, available at Aug [6] A. Lubotzky, R. Phillips, P. Sarak, Ramauja graphs, Combiatorica, vol. 8, o. 3, pp , [7] G. Margulis, Explicit group theoretical costructios of combiatorial schemes a their applicatio to the esig of expaers a cocetrators, Problems of Iformatio Trasmissio, vol. 24, o. 1, pp.39-46, [8] J. Cheegar, A lower bou o the smallest eigevalue of the Laplacia, Problems i aalysis, Priceto Uiv. Press, pp , Priceto, NJ, [9] A. Nilli, O the seco eigevalue of a graph, i Discrete Mathematics, vol. 91, o. 2, pp , 1991.