Modularity spectra, eigen-subspaces, and structure of weighted graphs

Modula i y spec a, eigen-subspaces, and s uc u e o weigh ed g aphs Ma ianna Bolla ∗ Ins i u e o Ma hema ics, Budapes Uni e si y o Technology and Economics and In a-Uni e si y Cen e o Telecommunica ions and In o ma ics, Deb ecen Abs ac The ole o he no malized modula i y ma ix in inding homogeneous cu s will be p esen ed. We also discuss he es abili y o he s uc u al eigen alues and ha o he subspace spanned by he co esponding eigen ec o s o his ma ix. In he p esence o a spec al gap be ween he k−1 la ges absolu e alue eigen alues and he emainde o he spec um, his in u n implies he es abili y o he sum o he inne a iances o he kclus e s ha a e ob ained by applying he k-means algo i hm o he app op ia ely chosen e ex ep esen a i es. Key wo ds: No malized modula i y, Volume egula i y, Spec al clus e ing, Tes able weigh ed g aph pa ame e s, S uc u al eigen alues, Spec al subspaces 1 In oduc ion The pu pose o his pape is o summa ize he spec al p ope ies and es a- bili y o he spec um and spec al subspaces o he no malized modula i y ma ix in oduced in [9] o ind egula e ex pa i ions. We will gene alize he Laplacian based spec al clus e ing me hods o eco e so-called olume egula clus e pai s such ha he in o ma ion low be ween he pai s and wi hin he clus e s is as homogeneous as possible. Fo his pu pose, we ake in o conside a ion bo h ends o he no malized Laplacian spec um, i.e., la ge absolu e alue, so-called s uc u al eigen alues o ou no malized modula i y ma ix in oduced jus o his con enience. ∗Resea ch suppo ed by he Hunga ian Na ional Resea ch G an OTKA-KTIA 77778 and by he T´ AMOP-4.2.2.C-11/1/KONV-2012-0001 p ojec , suppo ed by he Eu opean Union, co- inanced by he Eu opean Social Fund. Email add ess: [email p o ec ed] (Ma ianna Bolla). P ep in submi ed o Else ie 30 Ap il 2013 In Theo em 3, we es ima e he cons an o olume egula i y in e ms o he gap be ween he s uc u al and o he eigen alues, and he k- a iance o he op- imal e ex ep esen a i es cons uc ed by he eigen ec o s co esponding o he s uc u al eigen alues. He e we gi e a mo e de ailed p oo o his s a emen han in [10]. This heo em implies ha o a gene al edge-weigh ed g aph, he exis ence o k−1 s uc u al eigen alues o he no malized modula i y ma ix, sepa a ed om 0, is indica ion o a k-clus e s uc u e such ha he clus e - pai s a e olume egula wi h cons an depending on he spec al gap and he abo e k- a iance. The clus e s hemsel es can be eco e ed by applying he k-means algo i hm o he e ex ep esen a i es. Hence, Theo em 3 implies ha spec al clus e ing o he e ices in o kpa s gi es sa is ac o y pa i ion in he sense o olume egula i y. Fu he mo e, in Theo ems 8 and 10, we p o e he es abili y o he s uc u al eigen alues and he co esponding eigen-subspace o he no malized modula - i y ma ix in he sense o [12]. In iew o his, spec al clus e ing me hods can be pe o med on a smalle pa o he unde lying g aph and gi e good app oxima ion o he clus e s uc u e. 2 P elimina ies Th oughou he pape , we use he gene al amewo k o an edge-weigh ed g aph. Le G=Gn= (V, W) be an edge-weigh ed g aph on e ex-se V (|V|=n) and n×nsymme ic weigh -ma ix Wo non-nega i e eal en ies and ze o diagonal. We will call he numbe s di=Pn j=1 wij (i= 1, . . . , n) gene alized deg ees, and he diagonal ma ix D=diag (d1, . . . , dn)deg ee ma ix. In his and he nex sec ion, wi hou loss o gene ali y, Vol(V) = 1 will be assumed, whe e he olume o he e ex-subse U⊆Vis Vol(U) = Pi∈Udi. In he sequel, we only conside connec ed g aphs, which means ha Wis i educible. In [9], we de ined he no malized e sion o he modula i y ma ix (in oduced in [21]) as MD=D−1/2WD−1/2−√d√dT, whe e √d= (√d1,...,√dn)T, and we called i no malized modula i y ma ix. The spec um o his ma ix is in he [-1,1] in e al, and 0 is always an eigen alue wi h uni -no m eigen ec o √d. Indeed, in [5] we p o ed ha 1 is a single eigen alue o D−1/2WD−1/2wi h co esponding uni -no m eigen ec o √d, p o ided ou g aph is connec ed. This becomes a ze o eigen alue o MDwi h he same eigen ec o , whence 1 canno be an eigen alue o MDi Gis connec ed. In ac , he in oduc ion o his ma ix is a he echnical, he spec al gap, u he , Lemma 1 and Theo em 3 can be e be o mula ed wi h i . I can also be ob ained om he no malized Laplacian by sub ac ing i om he iden i y and dep i ing o i s i ial ac o . No malized Laplacian was used o spec al clus e ing in se e al 2 pape s (e.g., [3,5,6,14,20]), he idea o which can be summa ized by means o he spec al decomposi ion o he no malized modula i y ma ix. We in oduce he ollowing no a ion: he weigh ed cu be ween he e ex-subse s X, Y ⊆V is w(X, Y ) = Pi∈XPj∈Ywij. We will equen ly e e o he ollowing ac s. (a) The spec al decomposi ion o MDsol es he ollowing quad a ic placemen p oblem. Fo a gi en posi i e in ege k(1 < k < n), we wan o minimize Qk=Pi<j wijk i− jk2on he condi ions n X i=1 di i T i=Ik−1and n X i=1 di i=0(1) whe e he ec o s 1,..., na e (k−1)-dimensional ep esen a i es o he e ices, which o m he ow ec o s o he n×(k−1) ma ix X. Deno e he eigen alues o MD, in dec easing o de , by 1 > λ1≥ ··· ≥ λn≥ −1 wi h co - esponding uni -no m, pai wise o hogonal eigen ec o s u1,...,un. In [5], we p o ed ha he minimum o Qksubjec o (1) is k−1−Pk−1 i=1 λiand is a - ained by he ep esen a ion such ha he op imum e ex ep esen a i es ∗ 1,..., ∗ na e ow ec o s o he ma ix X∗= (D−1/2u1,...,D−1/2uk−1). Ins ead o X, he augmen ed n×kma ix ˜ Xcan as well be used, which is ob ained om Xby inse ing he column x0=1o all 1’s. In ac , x0=D−1/2u0, whe e u0=√dis he eigen ec o co esponding o he eigen alue 1 o D−1/2WD−1/2. Then Qk= (D1/2˜ X)T(In−D−1/2WD−1/2)(D1/2˜ X), and minimizing Qkon he cons ain (1) is equi alen o minimizing he abo e exp ession subjec o ˜ XTD˜ X=Ik. This p oblem is he con inuous elaxa ion o minimizing Qk(Pk) = (D1/2˜ X(Pk))T(In−D−1/2WD−1/2)(D1/2˜ X(Pk)) o e he se o k-pa i ions Pk= (V1, . . . , Vk) o he e ices such ha Pkis plan ed in o ˜ Xin he way ha he columns o ˜ X(Pk) a e so-called no mal- ized pa i ion- ec o s belonging o Pk. Namely, he coo dina es o he i h column a e ze os, excep hose indexing e ices o Vi, which a e equal o 1 √Vol(Vi)(i= 1, . . . , k). In ac , his is he no malized cu p oblem, which is discussed in [20] o k= 2, u he , in [3] and [6] o a gene al k, and he solu ion is based on he abo e con inuous elaxa ion. (b) Now, le us maximize he no malized Newman–Gi an modula i y o Gin- duced by Pk, de ined in [9] as Mk(Pk) = k X a=1 1 Vol(Va)X i,j∈Va (wij −didj) = k X a=1 w(Va, Va) Vol(Va)−1 o e he se Pko he k-pa i ions o V. I is easy o see ha Mk(Pk) = 3 k−1−Qk(Pk), and hence, he abo e ask has he same spec al elaxa ion as he no malized cu p oblem. Le Mk= maxPk∈PkMk(Pk) deno e he maximum k-way no malized Newman-Gi an modula i y o he weigh ed g aph G. (c) Finally, om he abo e conside a ions i is s aigh o wa d ha Mk≤ Pk−1 i=1 λi, o equi alen ly, he minimum no malized k-way cu is a leas he he sum o he k−1 smalles posi i e no malized Laplacian eigen alues. As o he minimum no malized k-way cu , in [6] we also ga e an uppe es ima e by cons an imes he sum o he k−1 smalles posi i e no malized Laplacian eigen alues, which cons an depends on he so-called k- a iance o he e ex ep esen a i es de ined in he ollowing way. S2 k(X) = min Pk∈Pk S2 k(X, Pk) = min Pk=(V1,...,Vk) k X a=1 X j∈Va djk j−cak2(2) whe e ca=1 Vol(Va)Pj∈Vadj jis he weigh ed cen e o clus e Vaand 1, . . . , n∈Rk−1a e ows o X. (The augmen ed ˜ Xwould gi e he same k- a iance.) The cons an o ou es ima ion depended on S2 k(X∗), and i was close o 1 i his k- a iance o he op imum (k−1)-dimensional e ex ep e- sen a i es was small enough. No e ha S2 k(X, Pk) is he objec i e unc ion o he weigh ed k-means algo i hm. In his way, we showed ha la ge posi i e eigen alues o he no malized modu- la i y ma ix a e esponsible o clus e s wi h high in a- and low in e -clus e densi ies. Likewise, maximizing Qk(Pk) ins ead o minimizing o e Pk, small nega i e eigen alues o he no malized modula i y ma ix a e esponsible o clus e s wi h low in a- and high in e -clus e densi ies (see [9]). Ou idea is ha aking in o accoun eigen alues om bo h ends o he no malized modu- la i y spec um, we can eco e so-called egula clus e pai s. Fo his pu pose, we use he no ion o olume egula i y o be in oduced in he nex sec ion. 3 No malized modula i y and olume egula i y Wi h he no malized modula i y ma ix, he well-known Expande Mixing Lemma ( o simple g aphs see, e.g., [17]) is o mula ed o edge-weigh ed g aphs in he ollowing way (see [8]). Lemma 1 P o ided Vol(V) = 1, o all X, Y ⊆V, |w(X, Y )−Vol(X)Vol(Y)| ≤ kMDk·qVol(X)Vol(Y), whe e kMDkdeno es he spec al no m o he no malized modula i y ma ix o G= (V, W). 4 Since he spec al gap o Gis 1 −kMDk, a la ge spec al gap indica es small disc epancy as a quasi- andom p ope y discussed in [15]. I he e is a gap no a he ends o he spec um, we wan o pa i ion he e ices in o clus e s so ha a ela ion simila o he abo e p ope y o he edge-densi ies be ween he clus e pai s would hold. Fo his pu pose, we use a sligh ly modi ied e sion o he olume egula i y’s no ion in oduced in [2]. De ini ion 2 Le G= (V, W)be an edge-weigh ed g aph wi h Vol(V)=1. The disjoin pai A, B ⊆Vis α- olume egula i o all X⊆A,Y⊆Bwe ha e |w(X, Y )−ρ(A, B)Vol(X)Vol(Y)| ≤ αqVol(A)Vol(B), whe e ρ(A, B) = w(A,B) Vol(A)Vol(B)is he ela i e in e -clus e densi y o (A, B). In he ideal k-clus e case, le us conside he ollowing gene alized andom simple g aph model: gi en he pa i ion (V1, . . . , Vk) o V(|V|=n), e ices i∈Vaand j∈Vba e connec ed wi h p obabili y pab, independen ly o each o he , 1 ≤a, b ≤k. We can hink o he p obabili y pab as he in e -clus e densi y o he pai (Va, Vb). Since gene alized andom g aphs can be iewed as edge-weigh ed g aphs wi h a special block-s uc u e bu dened wi h andom noise, based on [7], we a e able o gi e he ollowing spec al cha ac e iza ion o hem. Fixing k, and ending wi h n o in ini y in such a way ha he clus e sizes g ow a he same a e, he e exis s a posi i e numbe θ < 1, independen o n, such ha o e e y 0 < τ < 1/2 he e a e exac ly k−1 eigen alues o MDg ea e han θ−n−τ, while all he o he s a e a mos n−τin abso- lu e alue. Fu he , he k- a iance o he e ex ep esen a i es cons uc ed by he k−1 ans o med s uc u al eigen ec o s is O(n−2τ), and he clus e pai s a e α- olume egula wi h any small α, almos su ely. No e ha gen- e alized quasi andom g aphs de ined in [18] a e de e minis ic coun e pa s o gene alized andom g aphs wi h he same spec al p ope ies. Theo em 3 Le G= (V, W)be a connec ed edge-weigh ed g aph on n e - ices, wi h gene alized deg ees d1, . . . , dnand deg ee ma ix D. Assume ha Vol(V) = 1, and he e a e no dominan e ices, i.e., di= Θ(1/n),i= 1, . . . , n, as n→ ∞. Le he eigen alues o MD, enume a ed in dec easing absolu e alues, be 1≥ |µ1| ≥ ··· ≥ |µk−1|> ε ≥ |µk| ≥ ··· ≥ |µn|= 0. The pa i ion (V1, . . . , Vk)o Vis de ined so ha i minimizes he weigh ed k- a iance S2 k(X∗)o he op imum e ex ep esen a i es – de ined in (2) – ob ained as ow ec o s o he n×(k−1) ma ix X∗o column ec o s D−1/2ui, whe e uiis he uni -no m eigen ec o co esponding o µi(i= 1, . . . , k −1). Assume ha he e is a cons an 0< K ≤1 ksuch ha |Vi| ≥ Kn,i= 1, . . . , k. Wi h he no a ion s=qS2 k(X∗), he (Vi, Vj)pai s a e O(√2ks +ε)- olume egula (i6=j)and o he clus e s Vi(i= 1, . . . , k) he ollowing holds: o 5 all X, Y ⊂Vi, |w(X, Y )−ρ(Vi)Vol(X)Vol(Y)|=O(√2ks +ε)Vol(Vi), whe e ρ(Vi) = w(Vi,Vi) Vol2(Vi)is he ela i e in a-clus e densi y o Vi. No e ha , in Sec ion 2, we indexed he eigen alues o MDin non-inc easing o de and deno ed hem by λ’s. The se o all λi’s is he same as ha o all µi’s. None heless, we need a di e en no a ion o he eigen alues indexed in dec easing o de o hei absolu e alues. Recall ha 1 canno be an eigen alue o MDi Gis connec ed. Consequen ly, |µ1|= 1 can be i and only i µ1=−1, i.e., i Gis bipa i e. Fo example, i he condi ions o he abo e heo em hold wi h k= 2 and µ1=−1 (|µi| ≤ ε,i≥2), hen ou g aph is a bipa i e expande discussed in [1] in de ails. Fo he p oo we need he de ini ion o he cu no m o a ma ix (see e.g., [16]) and he ela ion be ween i and he spec al no m. De ini ion 4 The cu no m o he eal ma ix Awi h ow-se Row and column- se Col is kAk= max R⊂Row, C⊂Col X i∈RX j∈C aij . Lemma 5 Fo e e y m×n eal ma ix A, kAk≤√mnkAk, whe e he igh hand side con ains he spec al no m, i.e. he la ges singula alue o A. PROOF. kAk= max x∈{0,1}m,y∈{0,1}n|xTAy|= max x∈{0,1}m,y∈{0,1}n (x kxk)TA(y kyk)·kxk·kyk| ≤√mn max kxk=1,kyk=1 |xTAy|=√mnkAk, since o x∈ {0,1}m,kxk ≤ √m, and o y∈ {0,1}n,kyk ≤ √n.2 The de ini ion o he cu no m and he esul o he abo e lemma na u ally ex ends o symme ic ma ices wi h m=n, he spec al no m o which is he maximum o absolu e alues o hei eigen alues. PROOF. (Theo em 3). Recall ha he spec um o D−1/2WD−1/2di e s om ha o MDonly in he ollowing: i con ains he eigen alue µ0= 1 wi h 6 co esponding uni -no m eigen ec o u0=√dins ead o he eigen alue 0 o MDwi h he same eigen ec o . I Gis connec ed, 1 is a simple eigen alue. The op imum (k−1)-dimensional ep esen a i es o he e ices a e ow ec o s o he ma ix X∗= (x∗ 1,...,x∗ k−1), whe e x∗ i=D−1/2ui(i= 1, . . . , k −1). The ep esen a i es can as well be ega ded as k-dimensional ones, as inse ing he ec o x∗ 0=D−1/2u0=1will no change he k- a iance s2=S2 k(X∗). Assume ha he minimum k- a iance is a ained on he k-pa i ion (V1, . . . , Vk) o he e ices. By an easy analysis o a iance a gumen (see [5]) i ollows ha s2= k−1 X i=0 dis 2(ui, F),(3) whe e F=Span {D1/2z1,...,D1/2zk}wi h he so-called no malized pa i ion ec o s z1,...,zko coo dina es zji =1 √Vol(Vi)i j∈Viand 0, o he wise (i= 1, . . . , k). No e ha he ec o s D1/2z1,...,D1/2zk o m an o hono mal sys em. By conside a ions p o ed in [5], we can ind ano he o hono mal sys em 0,..., k−1∈Fsuch ha s2≤ k−1 X i=0 kui− ik2≤2s2(4) ( 0=u0, since u0∈F). We app oxima e he ma ix D−1/2WD−1/2= Pn−1 i=0 µiuiuT iby he ank kma ix Pk−1 i=0 µi i T iwi h he ollowing accu acy (in spec al no m):      n−1 X i=0 µiuiuT i− k−1 X i=0 µi i T i    ≤ k−1 X i=0 |µi|·  uiuT i− i T i  +     n−1 X i=k µiuiuT i     (5) which can be es ima ed om abo e wi h Pk−1 i=0 sin αi+ε≤Pk−1 i=0 kui− ik+ε≤ √2ks+ε, whe e αiis he angle be ween uiand i, and o i , sin αi 2=1 2kui− ik holds, i= 0, . . . , k −1. Based on hese conside a ions and ela ion be ween he cu no m and he spec al no m (see Lemma 5), he densi ies o be es ima ed in he de ining o mula o olume egula i y can be w i en in e ms o s epwise cons an ec o s in he ollowing way. The ec o s yi:= D−1/2 ia e s epwise cons an s on he pa i ion (V1, . . . , Vk), i= 0, . . . , k −1. The ma ix Pk−1 i=0 λiyiyT iis he e o e a symme ic block-ma ix on k×kblocks belonging o he abo e pa i ion o he e ices. Le ˆwab deno e i s en ies in he (a, b) block (a, b = 1, . . . , k). Using (5), he ank kapp oxima ion o he ma ix Wis pe o med wi h he ollowing accu acy o he pe u ba ion E: kEk=     W−D( k−1 X i=0 µiyiyT i)D     =     D1/2(D−1/2WD−1/2− k−1 X i=0 µi i T i)D1/2     . 7 The e o e, he en ies o W– o i∈Va,j∈Vb– can be decomposed as wij =didjˆwab +ηij, whe e he cu no m o he n×nsymme ic e o ma ix E= (ηij) es ic ed o Va×Vb(o he wise i con ains en ies all ze os) and deno ed by Eab, is es ima ed as ollows: kEabk≤nkEabk ≤ n·kD1/2 ak·(√2ks +ε)·kD1/2 bk ≤n· u u c1 Vol(Va) |Va|· u u c1 Vol(Vb) |Vb|·(√2ks +ε) =c1·sn |Va|·sn |Vb|·qVol(Va)qVol(Vb)(√2ks +ε) ≤c1·1 KqVol(Va)qVol(Vb)(√2ks +ε) =cqVol(Va)qVol(Vb)(√2ks +ε). He e he diagonal ma ix Dacon ains he diagonal pa o D es ic ed o Va, o he wise ze os, and he cons an cdoes no depend on n. Consequen ly, o a, b = 1, . . . , k and X⊆Va,Y⊆Vb: |w(X, Y )−ρ(Va, Vb)Vol(X)Vol(Y)|= X i∈XX j∈Y (didjˆwab +ηij)−Vol(X)Vol(Y) Vol(Va)Vol(Vb)X i∈VaX j∈Vb (didjˆwab +ηij) = X i∈XX j∈Y ηij −Vol(X)Vol(Y) Vol(Va)Vol(Vb)X i∈VaX j∈Vb ηij≤2c(√2ks +ε)qVol(Va)Vol(Vb), ha gi es he equi ed s a emen bo h in he a6=band a=bcase. 2 No e ha in he k= 2 special case, due o a heo em p o ed in [5], he 2- a iance o he op imum 1-dimensional ep esen a i es can be di ec ly es i- ma ed om abo e by he gap be ween he wo la ges absolu e alue eigen- alues o MD, and hence, he s a emen o Theo em 3 simpli ies, see [8]. Fo a gene al k, we can make he ollowing conside a ions. Assume ha he no malized modula i y spec um (wi h dec easing absolu e alues) o G= (V, W) sa is ies 1≥ |µ1| ≥ ··· ≥ |µk−1| ≥ θ > ε ≥ |µk| ≥ ··· ≥ |µn|= 0. Ou pu pose is o es ima e swi h he gap δ:= θ−ε. We will use he no a ion o he p oo o Theo em 3 and apply he esul s o [4] o he pe u ba ion o 8 spec al subspaces o he symme ic ma ices A= n−1 X i=0 µiuiuT iand B= k−1 X i=0 µi i T i in he ollowing si ua ion. The subse s S1={µk, . . . , µn−1}and S2={µ0, . . . , µk−1} o he eigen alues o D−1/2WD−1/2a e sepa a ed by an annulus, whe e dis (S1, S2) = δ > 0. Deno e by PAand PB he p ojec ions on o he spec al subspaces o Aand Bspanned by he eigen ec o s co esponding o he eigen alues in S1 and S2, espec i ely: PA(S1) = n−1 X j=k ujuT j,PB(S2) = k−1 X i=0 i T i. Then Theo em VII.3.4 o [4] implies ha kPAPBkF≤1 δkPA(A−B)PBkF,(6) whe e k.kFdeno es he F obenius no m. On he le hand side, kPAPBkF= qPk−1 i=0 sin2αi, and in iew o kui− ik= 2 sin αi 2and (4), his is be ween √3 2s and s. On he igh hand side, PAAPB−PABPB= (PAA)PB−PA(PBB) = k−1 X i=0 n−1 X j=k (µj−µi)uT j(ui− i)uj T i, whe e he F obenius no m o he ank 1 ma ices uj T iis 1, and he inne p oduc uT j(ui− i) is he smalle i he ui’s and he i’s a e he close (i= 1, . . . , k −1). The e o e, by he inequali y (6), sis he smalle i δis he la ge and he |µj−µi|di e ences o i= 0, . . . , k −1; j=k, . . . , n −1 a e close o δ. I |µk|=εis small, hen |µ1|,...,|µk−1|should be close o each o he (µ0= 1 does no play an impo an ole because o u0= 0). 4 Tes abili y o he no malized modula i y spec um and eigen- subspaces Au ho s o [12] de ined he es abili y o simple g aph pa ame e s and p o ed equi alen no ions o his es abili y. They also an icipa ed ha hei esul s emain alid i hey conside weigh ed g aph sequences (Gn) wi h edge-weigh s in he [0,1] in e al and no dominan e ex-weigh s αi(Gn)>0 (i= 1, . . . , n), i.e., maxiαi(Gn) αGn→0 as n→ ∞, whe e αGn=Pn i=1 αi(Gn). To his end, in [11], we sligh ly modi ied he de ini ion o a es able g aph pa ame e o weigh ed g aphs in he ollowing way. 9 and he spec al decomposi ion o i s no malized modula i y ma ix uns in polynomial ime in he educed numbe o he e ices. Unde he e ex- and clus e -balance condi ions his me hod can gi e qui e good app oxima ions o he mul iway cu s and helps us o ind he numbe o clus e s and iden i y he clus e s uc u e. In addi ion, aking in o accoun bo h he posi i e and nega i e, la ge absolu e alue eigen alues oge he wi h eigen ec o s, egula cu s can also be de ec ed, as he in es iga ed spec al cha ac e is ics gi e good es ima es o he olume egula i y’s cons an o he clus e pai s by Theo- em 3. Such egula cu s a e o impo ance in social o biological ne wo ks, e.g., i we wan o ind equally unc ioning synapses o he b ain. Acknowledgemen s We hank he anonymous e e ee o his/he cons uc i e commen s. Re e ences [1] Alon, N., 1986 Eigen alues and expande s, Combina o ica 6(1986), 83-96. 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