The degree sequences and spectra of scale-free random graphs

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1 The degee sequeces ad specta of scale-fee ado gaphs Joatha Joda Uivesity of Sheffield 30th July 004 Abstact We ivestigate the degee sequeces of scale-fee ado gaphs. We obtai a foula fo the liitig popotio of vetices with degee d, cofiig o-igoous aguets of Doogovtsev et al [0]. We also coside a geealisatio of the odel with oe adoisatio, povig siila esults. Fially, we use ou esults o the degee sequece to show that fo cetai values of paaetes localised eigefuctios of the adjacecy atix ca be foud. Itoductio Thee has bee cosideable iteest ecetly i ado scale-fee gaphs as odels of vaious eal-wold pheoea. As defied i [], these ae gowig gaphs costucted by, at each stage, addig vetices which ae coected to vetices aleady peset i the gaph usig a ule of pefeetial attachet, so that whe a edge is added fo a ew vetex to a existig vetex, that existig vetex is chose with pobability popotioal to its degee. I the liteatue, icludig [, 3, ], it is suggested that this is a useful odel fo the gowth of etwos foud i vaious fields, icludig techological, social ad biological etwos. Oe popety that ay eal-wold gaphs show is a powe-law degee sequece, ad a ea field aguet is used i [] that suggests that the pefeetial attachet ule leads to ube of vetices with degee d followig a powe law with idex 3. Futhe aalysis of gowig etwos ad pefeetial attachet ca be foud i [, 3, 4, 3]. A suvey of igoous atheatical esults o scale-fee gaphs appeas as [5]. As poited out i [5, 6], the odel as descibed i [] does ot actually ae atheatical sese, ad a atheatically pecise vesio Depatet of Pobability ad Statistics, Uivesity of Sheffield, Hics Buildig, Sheffield S3 7RH, UK AMS 000 Subject Classificatio: Piay 05C80. Key wods ad phases: scale-fee ado gaphs, degee sequeces, eigevalues.

2 of the odel is defied i [6] whee, fo this atheatically pecise vesio, the esult that the ube of vetices with degee d follows a powe law with idex 3 is igoously poved. As a way of costuctig gaphs with degee sequeces followig powe laws with idices othe tha 3, oe possibility is, istead of choosig a vetex with pobability popotioal to its degee, to choose a vetex with pobability popotioal to its degee plus soe costat q. This is itoduced i [0], whee a oigoous aguet shows that the degee sequece follows a powe law with idex depedig o q ad o the othe paaetes of the odel. Fo soe values of the paaetes, this is ade igoous i [7], agai fo a atheatically pecise vesio of the odel of [0]. This is eough to give exaples of powe laws with all itege idices 3. See also the oe geeal odel of [8]. I this pape, we use a slightly diffeet atheatically pecise vesio of the odel fo those defied i [6, 7], descibed i sectio. Fo this odel, we igoously obtai a pecise foula fo the expected popotio of vetices with degee d, which coespods to that i [0], ad show that the popotio of vetices with degee d coveges to this i pobability. This gives exaples of powe laws with all eal idices ad cofis the clai of [0] o the degee sequece. I the odel descibed i [], ad also those cosideed i [6, 7], each vetex added to the etwo iitially has degee, whee is a costat. A atual geealisatio is to allow the iitial degee of ew vetices to be idepedet ad idetically distibuted ado vaiables, ad we coside this i sectio 4, showig that, ude faily ild coditios o the distibutio of the ado vaiables, a foula siila to that foud i sectio 3 gives the liitig popotio of vetices with degee d. I [], the specta of adjacecy atices of vaious types of ado gaph, icludig scale-fee gaphs, ae obtaied usig siulatio. I [], the spectu of a olecula biological etwo is aalysed, ad the spectal popeties appea to be siila to those of scale-fee gaphs. Soe futhe wo o specta of coplex etwos, icludig scale-fee gaphs, appeas i [9]. I sectio 5 we apply ou esults o the degee sequece to obtai esults o the specta of the adjacecy atices of the gaphs. Copaiso with the siulatio esults obtaied i [] suggests that iteestig diffeeces i the specta ca esult fo vayig the paaetes of the scale-fee odel. The basic costuctio To costuct ado scale-fee gaphs, we use the followig costuctio, based o that i []. We tae a itege paaete. Statig fo a iitial gaph G 0, we the costuct a sequece of gaphs G N. We let 0 be the ube of vetices of G 0, ad let e 0 be the ube of edges of G 0. To costuct G + fo G, we add a ew vetex v, ad the add edges betwee v ad vetices of G. We choose ado vetices W +,, W +,,..., W +, accodig to a pefeetial attachet ule. If

3 δ w, is the degee of w i G, the we let W +,i = w with pobability idepedetly fo each i {,,..., }. δ w, u V G δ u, Note that this allows the possibility that we choose the sae vetex oe tha oce, ad hece the gaphs ay have ultiple edges. As a geealisatio of this odel, we set a costat q ad, istead of choosig a vetex with pobability popotioal to d, we choose it with pobability popotioal to d + q. That is, if δ w, is the degee of w i G, the we let W +,i = w with pobability idepedetly fo each i {,,..., }. q + δ w, u V G q + δ u, This equies d + q > 0 fo all possible vetex degees d. If each vetex of G 0 has degee at least the all vetices i all G will also have degee at least, so, if we choose a appopiate G 0, we ca use ay q,. The oigial odel of [] coespods to the special case whee q = 0. The ube of vetices i G is obviously + 0. Because edges ae added at each stage, the total ube of edges i G is + e 0, ad hece We defie so PW +,i = w = We ote that li c = + q. q + δ w, u V G q + δ u, = q + δ w, + e 0 + q 0 +. c = + e 0 + q 0 +, PW +,i = w = q + δ w, c. This costuctio diffes fo those descibed i [5, 6, 7] i that a ew vetex caot be coected to itself, so that thee ae o loops, ad that the iitial gaph G 0 is tae to be a geeal coected gaph athe tha fo statig fo a specific gaph. We also ote that it is possible to coside the gaphs costucted as diected gaphs, with edges cosideed as beig fo the added vetex to the olde vetices, ad that this is how the gaphs ae cosideed i [7, 0]. We defie the σ-algeba F = σg ; 0. 3

4 3 The degee sequece We defie the ado vaiable A d, to be the ube of vetices of G with degee d, ad we set Ãd, to be the popotio of vetices of G with degee, Ã d, = A d, + 0, ad set α d, = Ãd,, the expected popotio of vetices with degee d. We fist pove a siple lea o covegece of sequeces, which we will use i ou poofs. Lea. Fo N, let x, y, η, be eal ubes such that ad x + x = η + y x + + y x as. η > 0, ad thee exists N 0 such that, fo > N 0, η <. = η = As, /η 0 The x x as. Poof. Fist, we ote that, as η 0, we ca assue = 0 fo all, eplacig y with y + η. So x + = x η + + η + y, so that, fo > N 0, x + is a weighted aveage of x ad y. Now, give ɛ > 0, coside N ɛ > N 0 such that, fo N ɛ, y x < ɛ. Hece, if x > x ɛ fo soe N ɛ, the x + > x ɛ, ad siilaly if x ɛ < x fo soe N ɛ, the x ɛ < x +. Fially, if x < x ɛ fo all N ɛ, the ɛ x + x η + =, =N ɛ =N ɛ ad so x, which is a cotadictio. Siilaly if x > x+ɛ fo all N ɛ, the =N ɛ x x + =, which agai gives a cotadictio. So fo lage eough x x ɛ. 4

5 Theoe. Fo each d, the expected popotio satisfies + q α d, Γ3 + q + + q Γ3 + q + q + d + q as. + q + Γ + q Γd + q I the q = 0 case, this liit is + dd+d+, as foud i [6]. As i [0], we ote that popeties of gaa fuctios show that, as d, the foula i Theoe is appoxiately a powe law with idex 3 + q. As we ca have ay value of q, with coditios o G 0, as descibed i sectio, if q < 0, this gives all idices i,. Poof. Fo 0, the pobability that W +,i = w fo exactly values of i, ad so the degee of vetex w i G + is δ,w +, is c δ w, q δw, + q. c c Hece, coditioal o the value of A d,, the expected ube of vetices with degee d i G + ad degee d i G is c d q + d + q. A d, c c At each stage, oe ew vetex is added, with degee. So, A d,+ F = c d q + d + q A d, + I {d=} c c ad hece the popotios satisfy Ãd,+ F = c d q + d + q à d, + c c I {d=}, so the diffeece Ãd,+ F Ãd, is [ c à d, + 0 c d q c + Ãd, + 0 d c d q + ] + à d+q, + 0 d + q c d q + + c I {d=}, = 5

6 which, collectig tes i powes of, is [ c c Ãd,d + q + I {d=} Ãd,d + q + c whee C d,j, is a costat. So Ãd,+ F Ãd, = c + c j= [ d + q + c d + q d + q + c à d, + + j= j j C d,j, à d+q, ], d + q + c I {d=} Ãd, j ] C d,j, à d, j ad so the expected popotios satisfy α d,+ α d, = c [ d + q + c d + q d + q + c α d, + d + q + c I {d=} α d, j ] + j C d,j, α d, j= Because each ew vetex has degee, ad, i the odel, degees of vetices ca oly icease, thee ca be at ost 0 vetices of degee d, fo d < the 0 iitial vetices. So α d, 0 as, fo all d <. 3 Now, = d + q + c ad because the C d,j, ae costats ad α d,, we have j =, 4 c j C d,j, α d, 0 as, 5 fo each d,, j, so, statig with the d = case, we ca apply Lea with η + = + q + c, c 6

7 which has liit to show that + = y = c j= j j C,j, α,, c + q + c + + q + q + c α,, li y + q = + q + q +, + q α, + q + q as. 6 + Fo d >, we poceed usig iductio, agai usig Lea, but with y = d+q d+q+ c α d, to fid that α d, + q d + q + q + l=+ so, siplifyig the poduct, fo d, + q α d, Γ3 + q + + q Γ3 + q + q + d + q + q + Γ + q Γd + q l + q + q as 7 + q + l as. Theoe 3. The popotios Ãd, covege i L, ad hece i pobability, to the liitig expectatio as. + dd+d+ Poof. Reaagig, squaig, ad goupig tes which ae O as, [ [Ãd,+ F ] à d, d + q + c ] d + q c d + q + c à d, à d, à d, < K d, 8 whee K d is a costat, fo d > ad [ [Ã,+ F ] Ã, + q + c ] c + q + c Ã, Ã, < K. 9 Now, A d,+ A d,, so Ãd,+ Ãd, + +, so VaÃd,+ F à d,+ F à d, c + + ad hece [ d + q + c d + q à d + q + c d, à d, d,] à < ˆK d, 0 7

8 whee ˆK d is a costat, fo d > ad [ Ã,+ F Ã, + q + c ] + + c + q + c Ã, Ã, < ˆK. Now, Ã, + q +q+ q +, so taig expectatios i iplies that we ca use Lea with x = Ã, + ad y = q +q+ q +Ã, to show that showig that VaÃ, 0 as. + q Ã, + q + q + We ow poceed by iductio, assuig as ou iductio hypothesis that VaÃd, 0 as. The Ãd,Ãd, Ãd,Ãd, = CovÃd,, Ãd, 0 as as Va Ãd,. So taig expectatios i 0 iplies that we ca use Lea agai, with x = Ã d, ad y = d + q so that d + q + c Ãd,Ãd, d + q y li d + q + c + q = Γ3 + q + + q + q + q + Γ + q Ãd,Ãd, Γ3 + q + d + q Γd + q ad hece VaÃd, 0 as. This ow gives the esult. Usig popeties of gaa fuctios, this gives a degee sequece decayig appoxiately as a powe law with idex 3 + q. As, with a appopiate iitial gaph G 0, we ca use ay q,, we ca have ay idex γ <. 4 A vaiatio with ado We ow coside a odel whee the ube of old vetices each ew vetex is coected to is a ado vaiable. We defie a sequece of idepedet idetically distibuted positive-itege-valued ado vaiables 8

9 M N, with M + idepedet of the σ-algeba F. We also defie D = δ w, so that w V G PW i = w = δ w, D. This ado vaiable D coespods to the costat c used i the deteiistic odel. Theoe 4. If the idepedet ad idetically distibuted ado vaiables M have a oet geeatig fuctio which exists i a eighbouhood of 0, the the expected popotio of vetices of G havig degee d coveges to dd + d + M M + I {M d} as. Poof. We have Ad,+ + F, M + = M + + A d, M+ D d + D M+ d + D + I {d=m + }. So Ad,+ + A d, F, M + = M + + D M + + = + Ad, M + = M+ + A d, [ Ad, + D M + D d M+ ] d D d + M+ + D M + I {d=m+ } M+ D d M + D d + M + = M+ Ad, = A d, I {d=m+} + M + d D + M + + M + = A d, A d, M+ M +D d + M+ M+ + I {d=m+} d D 3 4 d D. 5 9

10 Note that A d, = 0 if d, so Ad,+ + A d, F, M + = + A M d, M + +D d + + D A d, I {d=m+ } + M + ax{,d } M+ d D A d, d. 6 We ow eove the coditioig o M +, witig µ = M + ad q = PM + = : Ad,+ + A d, F = + q I {d=} + A d, d D = A d, D d + + D ax{,d } A d, d 7 = + q d + A d, µd d A d, + µd d + D M ax{,d } A d, d. 8 If µ is fiite, the the Stog Law of Lage Nubes shows that D µ as, alost suely, ad A d, +0. So, fo ay ɛ 0, µ we ca, alost suely, fid a ado N ɛ such that, if N ɛ, ax{,d } D M A d, d µ M ɛ + ɛ d µ ɛ M + ɛ d µ ɛ M + ɛ d ad so D M ax{,d } A d, d µ ɛ M + ɛ d M µ ɛ M = + ɛ d M µ ɛ = + ɛ M d 0

11 which will covege to 0 as if it is fiite fo soe. But M µ ɛ M M µ ɛ = M d d M µ ɛ M d =0 M = d + M ɛ which will exist fo lage eough as log as the oet geeatig fuctio of M exists i a eighbouhood of 0. So, i 8, the last te coveges to 0 as if the oet geeatig fuctio of M exists i a eighbouhood of 0, so we use Lea, with y = q d + A d, µd d µd. d + Fo d =, A 0, = 0. As, D µ alost suely, so A, µq µ + µ = 3 q. 9 Siilaly, fo d >, A d, = = = = q d + d d + d + li d + q d d + q l + l l= d =l+ A d, + d + q d ll + d + q l dd + d + d l= l= q l ll + dd + d + dd + d + M M + I {M d}. As the popotio of vetices of G with degee d is A d, + 0 = A d, + 0, this is eough to give the esult.

12 5 High ultiplicity of eigevalues I [], specta of adjacecy atices of scale-fee gaphs with = 5 ad q = 0 wee ivestigated, the esults suggestig that the cetal pat of the scaled spectal desity coveges to a tiagula shape. We show that, i cotast, scale-fee gaphs with cetai values of q ad, icludig = ad q = 0, have stictly localised eigefuctios with eigevalue 0, givig a delta fuctio at 0 i the liitig spectal desity. As discussed i [9], a siila delta fuctio, due to localised eigefuctios, occus i the spectal desity of gaphs with vetices of degee. Ou esults diffe i that the localised eigefuctios ca be associated with vetices of degees othe tha. We begi with a lea which gives a coditio ude which the adjacecy atix of a gaph will have localised eigefuctios. Lea 5. If we ca patitio the vetex set of a gaph which ay have ultiple edges G ito sets A, A,..., A, B ad C with the popeties that Thee ae o edges of G coectig vetices i A i ad A j fo i j. Thee ae o edges of G coectig vetices i A i with vetices i C fo ay i. ach A i gives the sae subgaph of G, i.e. the vetices i each A i ca be labelled a i, a i,..., a is such that the ube of edges i G betwee A ip ad A iq is the sae fo each i. > B the, fo each λ which is a eigevalue of the s s adjacecy atix of the subgaph give by the A i, we ca fid sets of B liealy idepedet eigevectos of the adjacecy atix of G with the sae eigevalue. Poof. We let the adjacecy atix of G be M ad let the adjacecy atix of the A i subgaphs be ˆM, ad coside the equatio Mx = λx. We tae a eigevecto ˆx of ˆM with eigevalue λ, ad we coside vectos x o the vetices of G satisfyig x w = 0 fo w B C x aip = z iˆx p, fo soe z, z,..., z. The coditio o the sets A i, x beig zeo o B ad C, ad ˆx beig a eigevecto of ˆM with eigevalue λ esue that vectos of this fo satisfy Mx v = 0 fo all v C Mx aip = z i λˆx p = λx aip.

13 So x is a eigevecto of M if ad oly if Mx w = λx w = 0 fo each w B. But fo w B the equatio Mx w = 0 educes to a liea equatio i the uows z, z,..., z ad hece the atix equatio educes to B siultaeous liea equatios i uows, givig the esult. I what follows Lea 5 will be used i the case whee each A i is a sigle vetex, ad hece the eigevalue ivolved is zeo. This also applies i the case of the dead-ed vetices i.e. vetices with degee discussed i [9]. Lea 6. Fo each d, thee alost suely exists a ado N d < such that, fo N d, a vetex w of degee d has at ost d eighbous of degee. Poof. Uless it was oe of the 0 iitial vetices, a vetex w was added to the gaph at stage w, ad has edges to vetices W w,,..., W w,. The each of these eighbous othe tha those which wee i the set of 0 iitial vetices has degee at least +. Fo oe of the 0 iitial vetices w, PW, = w /, so alost suely the degee of w evetually gows to at least d +. So we ca fid N d such that all 0 iitial vetices have degee at least d +. Hece fo N d a vetex of degee d has at least edges to vetices with degee at least +, givig the esult. Theoe 7. Fo the odel descibed i sectio, if q < the, if we defie γ to be the expected ultiplicity of the eigevalue 0 i the spectu of G, the so that the spectu has a sigulaity at 0. li if γ + > 0 Poof. If q <, the usig the foula of Theoe the the expected popotio of vetices with degee i G coveges to β >, ad so is at least +ɛ, fo soe ɛ > 0, fo lage eough. The Lea 6 iplies that fo lage eough we ca use Lea 5 o G, with each A i cosistig of a sigle vetex of degee which has adjacecy atix 0. This ow gives the esult. We ow loo at the case with = ad q = 0, whee q = exactly. We set N = axn, N 3. Lea 6 shows that fo N thee ae o edges betwee degee vetices. We ca also coside the ube of the vetices which have degee 3 ad have oe eighbou of degee Lea 6 shows that fo N they caot have oe tha oe. We will call this Z, ad let z = Z N, ad let ζ be the expected popotio of such vetices ζ = z + 0 coditioal o > N. 3

14 Lea 8. The expected popotio of vetices with degee 3 ad oe degee eighbou as. Z 4 + q4 + q q4 + 3q, Poof. Give a vetex w of degee 3 i G, ad its eighbou w of degee, PW i w ad W i w = c 5 + q c ad, give a vetex w of degee i G, w has degee 3 with a degee eighbou with pobability + q + q c. c So ad so So the diffeece c 5 + q + q Z + F, > N = Z + A, c c A, ζ + = ζ c 5 + q c α + q, c + q c + q. c ζ + ζ = ζ c 5 + q + 0 c α, qc + q c [ + qc α, 5 + qc + c = ζ + D α, + D ζ + D ] 3α, + D 4 ζ whee D, D, D 3, D 4 ae costats. We ow use Lea agai, with η = 5+qc +c ad y = +q 5+q+c α, 4+q4+q 8+3q4+3q, ad deduce ζ 4 + q4 + q 8 + 3q4 + 3q as. 0 Usig Lea 6 the esult follows. Theoe 9. I the case whee = ad q = 0 the, if we defie γ to be the expected ultiplicity of the eigevalue 0 i the spectu of G, so that the spectu has a sigulaity at 0. li if γ + 35 > 0 Poof. We use Lea 5 agai. The A i each cosist of a sigle vetex of degee, C is the set of vetices with degee 3 ad o degee eighbou, ad B cosists of the eaiig vetices. The expected value of B + 0 coveges to + 35 = 35 as, givig the esult. 4

15 Refeeces [] K. Austi ad G. J. Rodges. Gowig etwos with two vetex types. Physica A, 36: , 003. [] A.-L. Baabási, R. Albet, ad H. Jeog. Mea-field theoy fo scale-fee ado etwos. Physica A, 7:73 87, 999. [3] A.-L. Baabási, H. Jeog, Z. Néda,. Ravasz, A. Schubet, ad T. Vicse. volutio of the social etwo of scietific collaboatios. Physica A, 3:590 64, 00. [4] G. Biacoi ad A.-L. Baabási. Copetitio ad ultiscalig i evolvig etwos. uophys. Lett., 54:436 44, 00. [5] B. Bollobás ad O. Rioda. Matheatical esults o scale-fee ado gaphs. I S. Boholdt ad H.G. Schuste, editos, Hadboo of Gaphs ad Netwos, pages 34. Wiley, 003. [6] B. Bollobás, O. Rioda, J. Spece, ad G. Tusády. The degee sequece of a scale-fee ado gaph pocess. Rado Stuctues ad Algoiths, 8:79 90, 00. [7] P. G. Bucley ad D. Osthus. Populaity based ado gaph odels leadig to a scale-fee degee sequece. Disc. Maths., 8:53 68, 004. [8] C. Coope ad A. Fieze. A geeal odel of web gaphs. Rado Stuct. Algoiths, :3 335, 003. [9] S. N. Doogovtsev, A. V. Goltsev, J. F. F. Medes, ad A. N. Sauhi. Specta of coplex etwos. Phys. Rev., , 003. [0] S. N. Doogovtsev, J. F. F. Medes, ad A. N. Sauhi. Stuctue of gowig etwos with pefeetial liig. Phys. Rev. Lett., 85:4633, 000. [] I. Faas, I. Deéyi, H. Jeog, Z. Néda, Z. N. Oltvai,. Ravasz, A. Schubet, A.-L. Baabási, ad T. Vicse. Netwos i life: scalig popeties ad eigevalue specta. Physica A, 34:5 34, 00. [] I. J. Faas, I. Deéyi, A-L. Baabási, ad T. Vicse. Specta of eal-wold gaphs: Beyod the sei cicle law. Phys. Rev., 64:06704, 00. [3] H. Jeog, Z. Néda, ad A.-L. Baabási. Measuig pefeetial attachet i evolvig etwos. uophys. Lett., 6:567 57,