Eigenvalues of graphs are useful for controlling many graph


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1 Spectra of radom graphs with give expected degrees Fa Chug, Liyua Lu, ad Va Vu Departmet of Mathematics, Uiversity of Califoria at Sa Diego, La Jolla, CA Edited by Richard V. Kadiso, Uiversity of Pesylvaia, Philadelphia, PA, ad approved February 20, 2003 (received for review December 9, 2002) I the study of the spectra of powerlaw graphs, there are basically two competig approaches. Oe is to prove aalogues of Wiger s semicircle law, whereas the other predicts that the eigevalues follow a powerlaw distributio. Although the semicircle law ad the power law have othig i commo, we will show that both approaches are essetially correct if oe cosiders the appropriate matrices. We will prove that (uder certai mild coditios) the eigevalues of the (ormalized) Laplacia of a radom powerlaw graph follow the semicircle law, whereas the spectrum of the adjacecy matrix of a powerlaw graph obeys the power law. Our results are based o the aalysis of radom graphs with give expected degrees ad their relatios to several ey ivariats. Of iterest are a umber of (ew) values for the expoet, where phase trasitios for eigevalue distributios occur. The spectrum distributios have direct implicatios to umerous graph algorithms such as, for example, radomized algorithms that ivolve rapidly mixig Marov chais. Eigevalues of graphs are useful for cotrollig may graph properties ad cosequetly have umerous algorithmic applicatios icludig low ra approximatios, iformatio retrieval (), ad computer visio. Of particular iterest is the study of eigevalues for graphs with powerlaw degree distributios (i.e., the umber of vertices of degree j is proportioal to j for some expoet ). It has bee observed by may research groups (2 8, ) that may realistic massive graphs icludig Iteret graphs, telephoecall graphs, ad various social ad biological etwors have powerlaw degree distributios. For the classical radom graphs based o the Erdös Réyi model, it has bee proved by Füredi ad Komlós that the spectrum of the adjacecy matrix follows the Wiger semicircle law (9). Wiger s theorem (0) ad its extesios have log bee used for the stochastic treatmet of complex quatum systems that lie beyod the reach of exact methods. The semicircle law has extesive applicatios i statistical ad solidstate physics (2, 22). I the 999 article by Faloutsos et al. (6) o Iteret topology, several powerlaw examples of Iteret topology are give, ad the eigevalues of the adjacecy matrices are plotted, which do ot follow the semicircle law. It is cojectured that the eigevalues of the adjacecy matrices have a powerlaw distributio with its ow expoet differet from the expoet of the graph. Faras et al. () looed beyod the semicircle law ad described a triagularlie shape distributio (see ref. 2). Recetly, M. Mihail ad C. H. Papadimitriou (upublished wor) showed that the eigevalues of the adjacecy matrix of powerlaw graphs with expoet are distributed accordig to a power law for 3. Here we ited to recocile these two schools of thought o eigevalue distributios. To begi with, there are i fact several ways to associate a matrix to a graph. The usual adjacecy matrix A associated with a (simple) graph has eigevalues quite sesitive to the maximum degree (which is a local property). The combiatorial Laplacia D A, with D deotig the diagoal degree matrix, is a major tool for eumeratig spaig trees ad has umerous applicatios (3, 4). Aother matrix associated with a graph is the (ormalized) Laplacia L I D /2 AD /2, which cotrols the expasioisoperimetrical properties (which are global) ad essetially determies the mixig rate of a radom wal o the graph. The traditioal radom matrices ad radom graphs are regular or almost regular, thus the spectra of all the above three matrices are basically the same (with possibly a scalig factor or a liear shift). However, for graphs with ueve degrees, the above three matrices ca have very differet distributios. I this article, we will cosider radom graphs with a geeral give expected degree distributio, ad we examie the spectra for both the adjacecy matrix ad the Laplacia. We first will establish bouds for eigevalues for graphs with a geeral degree distributio from which the results o radom powerlaw graphs the follow. The followig is a summary of our results.. The largest eigevalue of the adjacecy matrix of a radom graph with a give expected degree sequece is determied by m, the maximum degree, ad d, the weighted average of the squares of the expected degrees. We show that the largest eigevalue of the adjacecy matrix is almost surely ( o())max{d, m} provided some mior coditios are satisfied. I additio, suppose that the th largest expected degree m is sigificatly d 2. The the th largest eigevalue of the adjacecy matrix is almost surely ( o())m. 2. For a radom powerlaw graph with expoet 2.5, the largest eigevalue of a radom powerlaw graph is almost surely [ o()]m, where m is the maximum degree. Moreover, the largest eigevalues of a radom powerlaw graph with expoet have a powerlaw distributio with expoet 2 if the maximum degree is sufficietly large ad is bouded above by a fuctio depedig o, m, ad d, the average degree. Whe 2 2.5, the largest eigevalue is heavily cocetrated at cm 3 for some costat c depedig o ad the average degree. 3. We will show that the eigevalues of the Laplacia satisfy the semicircle law uder the coditio that the miimum expected degree is relatively large ( the square root of the expected average degree). This coditio cotais the basic case whe all degrees are equal (the Erdös Réyi model). If we weae the coditio o the miimum expected degree, we ca still have the followig strog boud for the eigevalues of the Laplacia, which implies strog expasio rates for rapidly mixig, 4 max i o i 0 w glog2, This paper was submitted directly (Trac II) to the PNAS office. To whom correspodece should be addressed. Achlioptas, D. & McSherry, F., ThirtyThird Aual ACM Symposium o Theory of Computig, July 6 8, 200, Crete, Greece. Fowles, C., Belogie, S., Chug, F. & Mali, J., Europea Coferece o Computer Visio, May 27 Jue 2, 2002, Copehage. Lu, L., Twelfth Aual ACMSIAM Symposium o Discrete Algorithms, Jauary 7 9, 200, Washigto, DC. MATHEMATICS PNAS May 27, 2003 vol. 00 o
2 where w is the expected average degree, is the miimum expected degree, ad g() is ay slowgrowig fuctio of. I applicatios, it usually suffices to have the i values (i 0) bouded away from 0. Our result shows that (uder some mild coditios) these eigevalues are actually very close to. The rest of the article has two parts. I the first part we preset our model ad the results cocerig the spectrum of the adjacecy matrix. The last part deals with the Laplacia. The Radom Graph Model The primary model for classical radom graphs is the Erdos Réyi model G p, i which each edge is idepedetly chose with the probability P for some give P 0 (see ref. 5). I such radom graphs the degrees (the umber of eighbors) of vertices all have the same expected value. Here we cosider the followig exteded radomgraph model for a geeral degree distributio (also see refs. 6 ad 7). For a sequece w (w, w 2,...,w ), we cosider radom graphs G(w) i which edges are idepedetly assiged to each pair of vertices (i, j) with probability w i w j, where w i. i Notice that we allow loops i our model (for computatioal coveiece), but their presece does ot play ay essetial role. It is easy to verify that the expected degree of i is w i. To this ed, we assume that max i w 2 i w such that p ij for all i ad j. This assumptio isures that the sequece w i is graphical [i the sese that it satisfies the ecessary ad sufficiet coditio for a sequece to be realized by a graph (8) except that we do ot require the w i values to be itegers]. We will use d i to deote the actual degree of v i i a radom graph G i G(w), where the weight w i deotes the expected degree. For a subset S of vertices, the volume Vol(S) is defied as the sum of weights i S ad vol(s) is the sum of the (actual) degrees of vertices i S. That is, Vol(S) is w i ad vol(s) is d i. I particular, we have Vol(G) i w i, ad we deote Vol(G). The iduced subgraph o S is a radom graph G(w), where the weight sequece is give by w i w i Vol(S) for all i S. The expected average degree is w i w i (). The secodorder average degree of G(w) is d ( is w 2 i i w i ) is w 2 i. The maximum expected degree is deoted by m. The classical radom graph G(, p) ca be viewed as a special case of G(w) by taig w to be (p, p,...,p). I this special case, we have d w m p. It is well ow that the largest eigevalue of the adjacecy matrix of G(, p) is almost surely [ o()]p provided that p log. The asymptotic otatio is used uder the assumptio that, the umber of vertices, teds to ifiity. All logarithms have the atural base. Spectra of the Adjacecy Matrix of Radom Graphs with Give Degree Distributio For radom graphs with give expected degrees (w, w 2,..., w ), there are two easy lower bouds for the largest eigevalue A of the adjacecy matrix A, amely, [ o()]d ad [ o()]m. It has bee prove that the maximum of the above two lower bouds is essetially a upper boud (also see ref. 9). Chug, F., Lu, L. & Vu, V., Worshop o Algorithms ad Models for the WebGraph, November 6, 2002, Vacouver, BC, Caada. Theorem. If d m log, the the largest eigevalue of a radom graph i G(w) is almost surely [ o()]d. Theorem 2. If m d log 2, the almost surely the largest eigevalue of a radom graph i G(w) is [ o()]m. If the th largest expected degree m satisfies m d log 2, the almost surely the largest eigevalues of a radom graph i G(w) is [ o()]m. Theorem 3. The largest eigevalue of a radom graph i G(w) is almost surely at most 7log maxm, d. We remar that the largest eigevalue A of the adjacecy matrix of a radom graph is almost surely [ o()]m if m is d by a factor of log 2, ad A is almost surely [ o()]d if m is d by a factor of log. I other words, A is (asymptotically) the maximum of m ad d if the two values of m ad d are far apart (by a power of log ). Oe might be tempted to cojecture that A omaxm, d. This, however, is ot true as show by a couterexample give previously (0). We also ote that with a more careful aalysis the factor of log i Theorem 3 ca be replaced by (log()) /2 ad the factor of log 2 ca be replaced by (log()) 3/2 for ay positive provided that is sufficietly large. We remar that the costat 7 i Theorem 3 ca be improved. We made o effort to get the best costat coefficiet here. Eigevalues of the Adjacecy Matrix of PowerLaw Graphs I this sectio we cosider radom graphs with powerlaw degree distributio with expoet. We wat to show that the largest eigevalue of the adjacecy matrix of a radom powerlaw graph is almost surely approximately the square root of the maximum degree m if 2.5 ad is almost surely approximately cm 3 if A phase trasitio occurs at 2.5. This result for powerlaw graphs is a immediate cosequece of a geeral result for eigevalues of radom graphs with arbitrary degree sequeces. We choose the degree sequece w (w, w 2,..., w ) satisfyig w i ci /() for i 0 i i 0. Here c is determied by the average degree, ad i 0 depeds o the maximum degree m, amely, c 2 d/(), d 2 i 0. m It is easy to verify that the umber of vertices of degree is proportioal to. The secodorder average degree d ca be computed as follows. 2 2 o if 3. 3 d d 2 d l2m o if 3. d m d o if 2 3. d Chug et al.
3 We remar that for 3, the secodorder average degree is idepedet of the maximum degree. Cosequetly, the powerlaw graphs with 3 are much easier to deal with. However, may massive graphs are powerlaw graphs with 2 3, i particular, Iteret graphs (9) have expoets betwee 2. ad 2.4, whereas the Hollywood graph (6) has expoet 2.3. I these cases, it is d that determies the first eigevalue. Theorem 4 is a cosequece of Theorems ad 2. Whe 2.5, the ith largest eigevalue i is i m i i i 0 /[(2)], for i sufficietly large. These large eigevalues follow the powerlaw distributio with expoet 2. (The expoet is differet from oe i Mihail ad Papadimitriou s upublished wor, because they use a differet defiitio for power law.) Theorem 4.. For 3 ad m d 2 log 3, almost surely the largest eigevalue of the radom powerlaw graph G is [ o()]m. 2. For ad m d (2)/(2.5) log 3/(2.5), almost surely the largest eigevalue of the radom powerlaw graph G is [ o()]m. 3. For ad m log 3/2.5, almost surely the largest eigevalue is [ o()]d. 4. For (dm log ) ad 2.5, almost surely the largest eigevalues of the radom powerlaw graph G with expoet have powerlaw distributios with expoet 2, provided that m is large eough (satisfyig the iequalities i ad 2). Spectrum of the Laplacia Suppose G is a graph that does ot cotai ay isolated vertices. The Laplacia L is defied to be the matrix L I D /2 AD /2, where I is the idetity matrix, A is the adjacecy matrix of G, ad D deotes the diagoal degree matrix. The eigevalues of L are all oegative betwee 0 ad 2 (see ref. 20). We deote the eigevalues of L by For each i, let i deote a orthoormal eigevector associated with i. We ca write L as L i i P i, where P i deotes the i projectio ito the eigespace associated with eigevalue i. We cosider M I L P 0 ( i )P i. i0 For ay positive iteger, we have TraceM 2 ( i ) 2. i0 Lemma. For ay positive iteger, we have max i M TraceM 2 /2. i 0 The matrix M ca be writte as M D /2 AD /2 P 0 D /2 AD /2 * 0 0 D /2 AD /2 volg D/2 KD /2, where 0 is regarded as a row vector (d vol(g),..., d vol(g)), * 0 is the traspose of 0, ad K is the all s matrix. Let W deote the diagoal matrix with the (i, i) etry havig value w i, the expected degree of the ith vertex. We will approximate M by C W /2 AW /2 VolG W/2 KW /2 W /2 AW /2 *, where is a row vector (w,...,w ). We ote that * * is strogly cocetrated at 0 for radom graphs with give expected degree w i. C ca be see as the expectatio of M, ad we shall cosider the spectrum of C carefully. A Sharp Boud for Radom Graphs with Relatively Large Miimum Expected Degree I this sectio we cosider the case whe the miimum of the expected degrees is ot too small compared to the mea. I this case, we are able to prove a sharp boud o the largest eigevalue of C. Theorem 5. For a radom graph with give expected degrees w,..., w where w log 3, we have almost surely C o 2 w. Proof: We rely o the Wiger highmomet method. For ay positive iteger ad ay symmetric matrix C which implies TraceC 2 C 2 C 2, E C 2 ETraceC 2, where is the eigevalue with maximum absolute value: C. If we ca boud E(Trace(C 2 )) from above, the we have a upper boud for E( (C) 2 ). The latter would imply a upper boud (almost surely) o (C) via Marov s iequality provided that is sufficietly large. Let us ow tae a closer loo at Trace(C 2 ). This is a sum where a typical term is c i i 2 c i2 i 3,...,c i2 i 2 c i2 i. I other words, each term correspods to a closed wal of legth 2 (cotaiig 2, ot ecessarily differet, edges) of the complete graph K o {,..., } (K has a loop at every vertex). O the other had, the etries c ij of C are idepedet radom variables with mea zero. Thus, the expectatio of a term is ozero if ad oly if each edge of K appears i the wal at least twice. To this ed, we call such a wal a good wal. Cosider a closed good wal that uses l differet edges e,...,e l with correspodig multiplicities m,..., m l (the m h values are positive itegers at least 2 summig up to 2). The (expected) cotributio of the term defied by this wal i E(Trace(C 2 )) is l m Ec h eh. [] h I order to compute E(c ij m ), let us first describe the distributio of c ij : c ij w i w j w i wj q ij w i w j with probability p ij w i w j ad c ij w i w j p ij w i w j with probability q ij p ij. This implies that for ay m 2, MATHEMATICS Chug et al. PNAS May 27, 2003 vol. 00 o. 635
4 Ec m ij q ij m p ij p ij m q ij w i w j m/2 p ij w i w j m/2 w i w j m/2 m2. [2] Here we used the fact that q m ij p ij (p ij ) m q ij p ij i the first iequality (the reader ca cosider this fact a easy exercise) ad the defiitio p ij w i w j i the secod equality. Let W l, deote the set of closed good wals o K of legth 2 usig exactly l differet vertices. Notice that each wal i W l, must have at least l differet edges. By Eqs. ad 2, the cotributio of a term correspodig to such a wal toward E(Trace(C 2 )) is at most l w 22l mi. It follows that ETraceC 2 W l, l0 22l. [3] I order to boud the last sum, we eed the followig result of Füredi ad Komlós (9). Lemma 2. For all l, W l,... l 2l 2 2l l l l 4l. I order to prove Theorem 5, it is more coveiet to use the followig cleaer boud, which is a direct corollary of Eq W l, l 4l 2ll 4l [5] Substitutig Eq. 5 ito 3 yields l 2 22l l 4l ETraceC 2 l0 2ll 4l l0 s l,. [4] [6] Now fix g()log, where g() teds to ifiity (with ) arbitrarily slowly. With this ad the assumptio about the degree sequece, the last sum i Eq. 6 is domiated by its highest term. To see this, let us cosider the ratio s, s l, for some l : s, 4 s l, 2 2ll 4l 2 l 4w 2 mi l 2 2l 4l 2 4w 2 mi 6 where i the first iequality we used the simple fact that 2l 2 22l 2 l! 22l. l, ETraceC 2 s l, os, l 0 o 4 o4. Because E( (C) 2 ) E(Trace(C 2 )) ad w, we have E C 2 o w 2 2. [7] By Eq. 7 ad Marov s equality P C w 2 2 P C 2 w 2 2 E C 2 2 w 2 2 o w w 2 2 o 2. Because (log ), we ca fid a () tedig to 0 with such that ( ) 2 o(), which implies that almost surely (C) [ o()](2w ) as desired. The lower boud o (C) follows from the semicircle law proved i the ext sectio. The Semicircle Law We show that if the miimum expected degree is relatively large, the the eigevalues of C satisfy the semicircle law with respect to the circle of radius r 2w cetered at 0. Let W be a absolute cotiuous distributio fuctio with (semicircle) desity w(x) (2) x 2 for x ad w(x) 0 for x. For the purpose of ormalizatio, cosider C or (2w ) C. Let N(x) be the umber of eigevalues of C or x ad W (x) N(x). Theorem 6. For radom graphs with a degree sequece satisfyig w, W (x) teds to W(x) i probability as teds to ifiity. Remar: The assumptio here is weaer tha that of Theorem 5 due to the fact that we oly eed to cosider momets of costat order. Proof: Because covergece i probability is etailed by the covergece of momets, to prove this Theorem 6 we eed to show that for ay fixed s, the sth momet of W (x) (with tedig to ifiity) is asymptotically the sth momet of s W(x). The sth momet of W (x) equals ()E(Trace(C or )). For s eve, s 2, the sth momet of W x is (2)!2 2!( )! (see ref. 0). For s odd, the sth momet of W x is 0 by symmetry. I order to verify Theorem 6, we eed to show that for ay fixed ETraceC or 2 o 2! 2 2!! [8] With a proper choice of g(), the assumptio (log 3 2 )w guaratees that (4 6 ) (), where () teds to ifiity with, which implies s, s l, [()] l. Cosequetly, ad ETraceC or 2 )) o. [9] 636 Chug et al.
5 Fig.. The large eigevalues of the adjacecy matrix follow the power law. Fig. 2. The Laplacia spectrum follows the semicircle law. We first cosider Eq. 8. Let us go bac to Eq. 3. Now we eed to use the more accurate estimate of W l, give by Eq. 4 istead of the weaer but cleaer oe i Eq. 5. Defie s l, l 22l... l 2 2l l 2l l l 4l. Oe ca chec, with a more tedious computatio, that the sum s l, l0 is still domiated by the last term, amely s l, os,. l0 It follows that E(Trace(C 2 )) [ o()]s,. O the other had, E(Trace(C 2 )) W,. Now comes the importat poit, for l, W l, is ot oly upperbouded by but i fact equals the righthad side of Eq. 4. Therefore, It follows that ETraceC 2 os,. ETraceC 2 or ow2 2 s, 2! o 2 2!!, which implies Eq. 8. Now we tur to Eq. 9. Cosider a term i Trace(C 2 ). If the closed wal correspodig to this term has at least differet edges, the there should be a edge with multiplicity oe, ad the expectatio of the term is 0. Therefore, we oly have to loo at terms with wals that have at most differet edges (ad at most differet vertices). It is easy to see that the umber of closed good wals of legth 2 with exactly l differet vertices is at most O( l ). The costat i O depeds o ad l (recall that ow is a costat), but for the curret tas we do ot eed to estimate this costat. The cotributio of a term correspodig to a wal with at most l differet edges is bouded by l w 22l mi. Thus E(Trace(C 2 )) is upperbouded by l0 c l 22l l [0] for some costat c. To compute the (2 )th momet of W (x), we eed to multiply E(Trace(C 2 )) by the ormalizig factor 2 2. It follows from Eq. 4 that the absolute value of the (2 )th momet of W (x) is upperbouded by l 22l l. 2 2 l0 l0 22l 2 [] Uder the assumptio of the theorem (2 ) o(). Thus, the last sum i Eq. is o(), completig the proof. Summary I this article we prove that the Laplacia spectrum of radom graphs with give expected degrees follows the semicircle law, provided some mild coditios are satisfied. We also show that the spectrum of the adjacecy matrix is essetially determied by its degree distributio. I particular, the largest eigevalues of the adjacecy matrix of a radom powerlaw graph follow a powerlaw distributio, provided that the largest degrees are large i terms of the secodorder average degree. Here we compute the spectra of a subgraph G of a simulated radom powerlaw graph with expoet 2.2. The graph G has 588 vertices with the average degree w ad the secod average degree d The largest eigevalue of its adjacecy matrix is 6.78, which is very close to the secodorder average degree d, as asserted by Theorem (see Fig. ). All the otrivial eigevalues of the Laplacia are withi 0.3 (2w ) from, as predicted by Theorems 5 ad 6 (see Fig. 2). This research was supported i part by Natioal Sciece Foudatio Grats DMS ad ITR (to F.C. ad L.L.) ad DMS (to V.V.) ad a A. Sloa fellowship (to V.V.). MATHEMATICS Chug et al. PNAS May 27, 2003 vol. 00 o. 637
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