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 2 1 + o1), a the largest eigevalue is at least + 2 1 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 1 + 4 + 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 1 + 2 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 + 2 1. By multiplyig the weight of every ege of a Ramauja graph by 2, we obtai a graph 1
2 that spectrally approximates the complete graph with approximatio factor κ = +2 1 2 1 = 1 + 4 + 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 1 + 2 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 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 ) 2 1 + 2 1 1 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 ) ) 2 1 + V k 1) k, 8) ) ) 2 1 + U k 1) k, 9)
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+1 2 1 + V 1) i+ 1 k 2 1) k 1 k 1 = a 2 V i 1) i 2 ) 1 + 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 2 1 + 2 1 1, 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 ) 2 1 15) 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 ) + 2 1 16) 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 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 ) 2 1 + 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 1 1 20) + 2 1 2 1 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) + 2 1 2 o1) 23) 1 κx T L K x + 2 1 2 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 +2 1 2. 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 + 2 1 2 1 = 1 + 4 ) 1 + O 25) 26)
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 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 1 + 2 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) ) ) 1 1 1 a κ 1 1 1. 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 1 + 1 1 1 2 1+κ 2 1+κ δ i +w i wi κ. 41)
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 1 + ) 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 2 1 + 1 2 )) 1+κ 1 1 2 1 O 46) 1+κ )) 1 + 1 2 1 O 47) 1 + κ )) 1 + 1 2 O 48) ) = 1 + 2 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 1 + 2 50) 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 ) 1 + 4 + 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 1 + 2 o the approximatio factor of ay large weighte graph of average combiatorial egree that approximates the complete graph.
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