Coupling Online and Offline Analyses for Random Power Law Graphs

Transcription

1 Inerne Mahemaics Vol, No 4: Coupling Online and Offline Analyses for Random Power Law Graphs FanChungandLinyuanLu Absrac We develop a coupling echnique for analyzing online models by using offline models This mehod is especially effecive for a growh-deleion model ha generalizes and includes he preferenial aachmen model for generaing large complex neworks which simulae numerous realisic neworks By coupling he online model wih he offline model for random power law graphs, we derive srong bounds for a number of graph properies including diameer, average disances, conneced componens, and specral bounds For example, we prove ha a power law graph generaed by he growh-deleion model almos surely has diameer O(log n) and average disance O(log log n) Inroducion In he pas few years, i has been observed ha a variey of informaion neworks, including Inerne graphs, social neworks, and biological neworks among ohers [Aiello e al 00, Aiello e al 02, Barabási and Alber 99, Barabási e al 00, Jeong e al 00, Kleinberg e al 99, Lu 0], have he so-called power law degree disribuion A graph is called a power law graph if he fracion of verices wih degree k is proporional o for some consan β > 0 There are basically wo k β differen models for random power law graphs The firs model is an online model ha mimics he growh of a nework Saring from a verex (or some small iniial graph), a new node and/or new edge is added a each uni of ime following he so-called preferenial aachmen A K Peers, Ld /04 $050 per page 409

2 40 Inerne Mahemaics scheme [Aiello e al 02, Barabási and Alber 99, Kleinberg e al 99] The endpoin of a new edge is chosen wih he probabiliy proporional o heir (curren) degrees By using a combinaion of adding new nodes and new edges wih given respecive probabiliies, one can generae large power law graphs wih exponens β greaer han 2 (see [Aiello e al 02, Bollabás and Riordan 03] for rigorous proofs) Since realisic neworks encouner boh growh and deleion of verices and edges, we consider a growh-deleion online model ha generalizes and includes he preferenial aachmen model Deailed definiions will be giveninsecion3 The second model is an offline model of random graphs wih given expeced degrees For a given sequence w of weighs w i, a random graph in G(w) is formed by choosing he edge beween u and v wih probabiliy proporional o he produc of w u and w v The Erdős-Rényi model G(n, p) can be viewed as a special case of G(w) wihallw i equal Because of he independence in he choices of edges, he model G(w) is amenable o a rigorous analysis of various graph properies and srucures In a series of papers [Chung and Lu 02a, Chung and Lu 02b, Chung e al 03, Lu 0], various graph invarians have been examined and sharp bounds have been derived for diameer, average disance, conneced componens, and specra for random power law graphs and, in general, random graphs wih given expeced degrees The online model is obviously much harder o analyze han he offline model There has been some recen work on he online model beyond showing ha he generaed graph has a power law degree disribuion Bollobás and Riordan [Bollabás and Riordan 03] have derived a number of graph properies for he online model by coupling wih G(n, p), namely, idenifying (almos regular) subgraphs whose behavior can be capured in a similar way as graphs from G(n, p) for some appropriae p In his paper, our goal is o couple he online model wih he offline model of random graphs wih a similar power law degree disribuion so ha we can apply he echniques from he offlinemodeloheonlinemodelthebasicidea is similar o he maringale mehod bu wih subsanial differences Alhough a maringale involves a sequence of funcions wih consecuive funcions having small bounded differences, each funcion is defined on a fixed probabiliy space Ω For he online model, he probabiliy space for he random graph generaed a each ime insance is differen in general We have a sequence of probabiliy spaces where wo consecuive ones have small differences To analyze his, we need o examine he relaionship of wo disinc random graph models, each of which can be viewed as a probabiliy space In order o do so, we shall describe wo basic mehods ha are no only useful for our proofs here bu also ineresing in heir own righ

3 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 4 Comparing wo random graph models We define he dominance of one random graph model over anoher in Secion 4 Several key lemmas for conrolling he differences are also given here A general Azuma inequaliy A concenraion inequaliy is derived for maringales ha are almos Lipschiz A complee proof is given in Secion 5 The main goal of his paper is o show he following resuls for he random graph G generaed by he online model G(p,p 2,p 3,p 4,m)wihp >p 3,p 2 >p 4, as definedinsecion5: Almos surely he degree sequence of he random graph generaed by growh-deleion model G(p,p 2,p 3,p 4,m) follows he power law disribuion wih exponen β =2+(p + p 3 )/(p +2p 2 p 3 2p 4 ) 2 Suppose m>log + nforp 2 <p 3 +p 4,wehave2< β < 3 Almos surely a random graph in G(p,p 2,p 3,p 4,m)hasdiameerΘ(log n) andaverage log log n disance O( ) We noe ha he average disance is defined o be log(/(β 2) he average over all disances among pairs of verices in he same conneced componen 3 Suppose m>log + n For p 2 p 3 + p 4,wehaveβ > 3 Almos surely a random graph in G(p,p 2,p 3,p 4,m)hasdiameerΘ(log n) andaverage disance O( log n )whered is he average degree log d 4 Suppose m>log + n Almos surely a random graph in G(p,p 2,p 3,p 4,m) has Cheeger consan a leas /2 + o() 5 Suppose m>log + n Almos surely a random graph in G(p,p 2,p 3,p 4,m) has specral gap λ a leas /8+o() We noe ha he Cheeger consan h G of a graph G, which is someimes called he conducance, isdefined by E(A, h G = Ā) min{vol(a), vol(ā)}, where vol(a) = x A deg(x) The Cheeger consan is closely relaed o he specral gap λ of he Laplacian of a graph by he Cheeger inequaliy 2h G λ h 2 G/2 Thus, boh h G and λ are key invarians for conrolling he rae of convergence of random walks on G

4 42 Inerne Mahemaics 2 Srong Properies of Offline Random Power Law Graphs For random graphs wih given expeced degree sequences saisfying a power law disribuion wih exponen β, we may assume ha he expeced degrees are w i = ci β for i saisfying i0 i<n+ i 0 Here c depends on he average degree, and i 0 depends on he maximum degree m, namely,c = β 2 β dn β and 2 Average Disance and Diameer β d(β 2) i 0 = n m(β ) Fac 2 ([Chung and Lu 02b]) Forapowerlawrandomgraphwihexponenβ > 3 and average degree d sricly greaer han, almos surely he average disance is ( + o()) log n log d and he diameer is Θ(log n) Fac 22 ([ChungandLu02b])Suppose a power law random graph wih exponen β has average degree d sricly greaer han and maximum degree m saisfying log m log n/ log log n If 2 < β < 3, almos surely he diameer is Θ(log n) log log n and he average disance is a mos (2 + o()) log(/(β 2)) For he case of β =3, he power law random graph has diameer almos surely Θ(log n) and has average disance Θ(log n/ log log n) 22 Conneced Componens Fac 23 ([Chung and Lu 02a]) Suppose ha G is a random graph in G(w) wih given expeced degree sequence w If he expeced average degree d is sricly greaer han, hen he following hold: Almos surely G has a unique gian componen Furhermore, he volume of hegiancomponenisaleas( 2 de +o())vol(g) if d 4 =475 e and is a leas ( +log d + o())vol(g) if d<2 d 2 The second larges componen almos surely has size O( log n log d ) 23 Specra of he Adjacency Marix and he Laplacian The specra of he adjacency marix and he Laplacian of a non-regular graph can have quie differen disribuion The definiion for he Laplacian can be found in [Chung 97]

5 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 43 Fac 24 ([Chung e al 03]) The larges eigenvalue of he adjacency marix of a random graph wih a given expeced degree sequence is deermined by m, he maximum degree, and d, he weighed average of he squares of he expeced degrees We show ha he larges eigenvalue of he adjacency marix is almos surely ( + o()) max{ d, m} provided ha some minor condiions are saisfied In addiion, suppose ha he kh larges expeced degree m k is significanly larger han d 2 Then he kh larges eigenvalue of he adjacency marix is almos surely ( + o()) m k 2 For a random power law graph wih exponen β > 25, he larges eigenvalue of a random power law graph is almos surely (+o()) m,wherem is he maximum degree Moreover, he k larges eigenvalues of a random power law graph wih exponen β have power law disribuion wih exponen 2β if he maximum degree is sufficienly large and k is bounded above by a funcion depending on β, m,andd, he average degree When 2 < β < 25, he larges eigenvalue is heavily concenraed a cm 3 β for some consan c depending on β and he average degree 3 We will show ha he eigenvalues of he Laplacian saisfy he semicircle law under he condiion ha he minimum expeced degree is relaively large ( he square roo of he expeced average degree) This condiion conains he basic case when all degrees are equal (he Erdös-Rényi model) If we weaken he condiion on he minimum expeced degree, we can sill have he following srong bound for he eigenvalues of he Laplacian which implies srong expansion raes for rapidly mixing: max λ i ( + o()) 4 + g(n)log2 n, i=0 w w min where w is he expeced average degree, w min is he minimum expeced degree, and g(n) is any slow growing funcion of n 3 A Growh-Deleion Model for Generaing Random Power Law Graphs One explanaion for he ubiquious occurrence of power laws is he simple growh rules ha can resul in a power law disribuion (see [Aiello e al 02, Barabási and Alber 99]) Neverheless, realisic neworks usually encouner boh he growh and deleion of verices and edges Here we consider a general online model ha combine deleion seps wih he preferenial aachmen model

6 44 Inerne Mahemaics Verex-growh sep Add a new verex v and form a new edge from v o an exising verex u chosen wih probabiliy proporional o d u Edge-growh sep Add a new edge wih endpoins o be chosen among exising verices wih probabiliy proporional o he degrees If exising in he curren graph, he generaed edge is discarded The edge-growh sep is repeaed unil a new edge is successfully added Verex-deleion sep Delee a verex randomly Edge-deleion sep Delee an edge randomly For nonnegaive values p,p 2,p 3,p 4 summing o, we consider he following growh-deleion model G(p,p 2,p 3,p 4 ): A each sep, wih probabiliy p, ake a verex-growh sep; wih probabiliy p 2, ake an edge-growh sep; wih probabiliy p 3, ake a verex-deleion sep; wih probabiliy p 4 = p p 2 p 3, ake an edge-deleion sep Here we assume ha p 3 <p and p 4 <p 2 so ha he number of verices and edge grows as goes o infiniy If p 3 = p 4 = 0, he model is jus he usual preferenial aachmen model ha generaes power law graphs wih exponen β = 2 + p p +2p 2 An exensive survey on he preferenial aachmen model is given in [Mizenmacher 05] and rigorous proofs can be found in [Aiello e al 02, Cooper and Frieze 03] This growh-deleion model generaes only simple graphs because he muliple edges are disallowed a he edge-growh sep The drawback is ha he edgegrowh sep could run in a loop I only happens if he curren graph is a compleed graph If his happens, we simply resar he whole procedure from he same iniial graph Wih high probabiliy, he model generaes sparse graphs so ha we could omi he analysis of his exreme case Previously, Bollobás considered edge deleion afer he power law graph is generaed [Bollabás and Riordan 03] Very recenly, Cooper, Frieze, and Vera [Cooper e al 04] independenly consider he growh-deleion model wih verex deleion only We will show (see Secion 6) he following Suppose ha p 3 <p and p 4 <p 2 Then almos surely he degree sequence of he growh-deleion model G(p,p 2,p 3,p 4 ) follows he power law disribuion wih he exponen β =2+ p + p 3 p +2p 2 p 3 2p 4

7 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 45 We noe ha a random graph in G(p,p 2,p 3,p 4 ) almos surely has expeced average degree (p + p 2 p 4 )/(p + p 3 ) For of p i s in cerain ranges, his value can be below and he random graph is no conneced To simulae graphs wih specified degrees, we consider he following modified model G(p,p 2,p 3,p 4,m), for some ineger m ha generaes random graphs wih he expeced degree m(p + p 2 p 4 )/(p + p 3 ): A each sep, wih probabiliy p, add a new verex v and form m new edges from v o exising verices u chosen wih probabiliy proporional o d u ; wih probabiliy p 2,akem edge-growh seps; wih probabiliy p 3, ake a verex-deleion sep; wih probabiliy p 4 = p p 2 p 3,akem edge-deleion seps Suppose ha p 3 <p and p 4 <p 2 Then almos surely he degree sequence of he growh-deleion model G(p,p 2,p 3,p 4,m) follows he power law disribuion wih he exponen β he same as he exponen for he model G(p,p 2,p 3,p 4 ): p + p 3 β =2+ p +2p 2 p 3 2p 4 Many resuls for G(p,p 2,p 3,p 4,m) can be derived in he same fashion as for G(p,p 2,p 3,p 4 ) Indeed, G(p,p 2,p 3,p 4 ) = G(p,p 2,p 3,p 4, ) is usually he hardes case because of he sparseness of he graphs 4 Comparing Random Graphs In he early work of Erdős and Rényi on random graphs, hey firs used he model F (n, m) haeachgraphonn verices and m edgesischosenrandomly wih equal probabiliy, where n and m are given fixed numbers This model is apparenly differen from he laer model G(n, p), for which a random graph is formed by choosing independenly each of he n 2 pairs of verices o be an edge wih probabiliy p Because of he simpliciy and ease o use, G(n, p) is he model for he seminar work of Erdős and Rényi Since hen, G(n, p) has been widely used and ofen been referred o as he Erdős-Rényi model For m = p n 2,he wo models are apparenly correlaed in he sense ha many graph properies are saisfied by boh random graph models To precisely define he relaionship of wo random graph models, we need some definiions AgraphproperyP can be viewed as a se of graphs We say ha a graph G saisfies propery P if G is a member of P A graph propery is said o be monoone if whenever a graph H saisfies A, hen any graph conaining H mus also saisfy A For example, he propery A of conaining a specified subgraph, say, he Peerson graph, is a monoone propery A random graph G

8 46 Inerne Mahemaics is a probabiliy disribuion Pr(G = ) Given wo random graphs G and G 2 on n verices, we say ha G dominaes G 2 if, for any monoone graph propery A, he probabiliy ha a random graph from G saisfies A isgreaerhanorequal o he probabiliy ha a random graph from G 2 saisfies A, ie, Pr(G saisfies A) Pr(G 2 saisfies A) In his case, we wrie G G 2 and G 2 G For example, for any p p 2,we have G(n, p ) G(n, p 2 ) For any > 0, we say ha G dominaes G 2 wih an error esimae if, for any monoone graph propery A, he probabiliy ha a random graph from G saisfies A is greaer han or equal o he probabiliy ha a random graph from G 2 saisfies A up o an error erm, ie, Pr(G saisfies A)+ Pr(G 2 saisfies A) If G dominaes G 2 wih an error esimae = n, which goes o zero as n approaches infiniy, we say ha G almos surely dominaes G 2 In his case, we wrie almos surely G G 2 and G 2 G For example, for any δ > 0, we have almos surely G n, ( δ) m n F (n, m) G n, ( + δ) 2 m n 2 We can exend he definiion of dominaion o graphs wih differen sizes in he following sense Suppose ha he random graph G i has n i verices for i =, 2, and n <n 2 By adding n 2 n isolaed verices, he random graph G is exended o he random graph G wih he same size as G 2 We say ha G 2 dominaes G if G 2 dominaes G We consider random graphs ha are consruced inducively by pivoing a one edge a a ime Here we assume he number of verices is n Edge-pivoing For an edge e K n, a probabiliy q (0 q ), and a random graph G, a new random graph G can be consruced in he following way For any graph H, wedefine Pr(G = H) = ( q)pr(g = H) Pr(G = H)+q Pr(G = H \{e}) if e E(H), if e E(H) I is easy o check ha Pr(G = ) is a probabiliy disribuion We say ha G is consruced from G by pivoing a he edge e wih probabiliy q

9 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 47 For any graph propery A, wedefine he se A e o be Furher, we define he se Aē o be A e = {H {e} H A} Aē = {H \{e} H A} In oher words, A e consiss of he graphs obained by adding he edge e o he graphs in A; Aē consiss of he graphs obained by deleing he edge e from he graphs in A We have he following useful lemma Lemma 4 Suppose ha G is consruced from G by pivoing a he edge e wih probabiliy q Then for any propery A, we have Pr(G A) =Pr(G A)+q[Pr((A A e )ē) Pr(A Aē)] In paricular, if A is a monoone propery, we have Thus, G dominaes G Pr(G A) Pr(G A) Proof The se associaed wih a propery A can be pariioned ino he following subses Le A = A A e be he graphs of A conaining he edge e, andle A 2 = A Aē be he graphs of A no conaining he edge e Wehave Pr(G A) = Pr(G A )+Pr(G A 2 ) = Pr(G = H)+ Pr(G = H) H A H A 2 = (Pr(G = H)+q Pr(G = H \{e})) H A + ( q)pr(g = H) H A 2 = Pr(G A )+Pr(G A 2 )+q Pr(G (A )ē) q Pr(A 2 ) = Pr(G A)+q[Pr((A A e )ē) Pr(A Aē)] If A is monoone, we have A 2 (A )ē Thus, Lemma 4 is proved Pr(G A) Pr(G A)

10 48 Inerne Mahemaics Lemma 42 Suppose ha G i is consruced from G i by pivoing he edge e wih probabiliy q i,fori =, 2 If q q 2 and G dominaes G 2,henG dominaes G 2 Proof Following he definiions of A, and leing A and A 2 be as in he proof of Lemma 4, we have Pr(G 2 A) = Pr(G 2 A)+q 2 [Pr(G 2 (A )ē) Pr(G 2 A 2 )] = Pr(G 2 A)+q 2 Pr(G 2 ((A )ē \ A 2 )) Pr(G A)+q Pr(G ((A )ē \ A 2 )) = Pr(G A)+q [Pr(G (A )ē) Pr(G A 2 )] = Pr(G A) The proof of Lemma 42 is complee Le G and G 2 be he random graphs on n verices We define G G 2 o be he random graph as follows: Pr(G G 2 = H) = Pr(G = H )Pr(G 2 = H 2 ) H H 2=H where H,H 2 range over all possible pairs of subgraphs ha are no necessarily disjoin The following lemma is a generalizaion of Lemma 42 Lemma 43 If G dominaes G 3 wih an error esimae and G 2 dominaes G 4 wih an error esimae 2,henG G 2 dominaes G 3 G 4 wih an error esimae + 2 Proof For any monoone propery A and any graph H, wedefine he se f(a, H) o be f(a, H) ={G G H A} We observe ha f(a, H) is also a monoone propery Therefore, Pr(G G 2 A) = Pr(G = H )Pr(G 2 = H 2 ) H A H H 2 =H = Pr(G = H )Pr(G 2 f(a, H )) H H Pr(G = H )(Pr(G 4 f(a, H )) 2 ) Pr(G G 4 A) 2

11 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 49 Similarly, we have Thus, we ge as desired Pr(G G 4 A) Pr(G 3 G 4 A) Pr(G G 2 A) Pr(G 3 G 4 A) ( + 2 ), Suppose ha φ is a sequence of random graphs φ(g ), φ(g 2 ),, where he indices of φ range over all graphs on n verices Recall ha a random graph G is a probabiliy disribuion Pr(G = ) over he space of all graphs on n verices For any random graph G, wedefine φ(g) o be he random graph defined as follows: Pr(φ(G) =H) = Pr(G = H )Pr(φ(H )=H 2 ) We have he following lemmas H H 2=H Lemma 44 Le φ and φ 2 be wo sequences of random graphs where he indices of φ and φ 2 range over all graphs on n verices Le G be any random graph If Pr(G {H φ (H)dominaes φ 2 (H)wih an erroresimae }) 2, hen φ (G) dominaes φ 2 (G) wih an error esimae + 2 Proof For any monoone propery A and any graph H, wehave Pr(φ (G) A) = Pr(G = H )Pr(φ (H )=H 2 ) H A H H 2 =H = Pr(G = H )Pr(φ (H ) f(a, H )) H Pr(G = H )Pr(φ 2 (H ) f(a, H )) 2 H Pr(φ 2 (G) A) ( + 2 ), as desired, since f(a, H) ={G G H A} is also a monoone propery Le G and G 2 be he random graphs on n verices We define G \ G 2 o be he random graph as follows: Pr(G \ G 2 = H) = Pr(G = H )Pr(G 2 = H 2 ), H \H 2=H where H and H 2 range over all pairs of graphs

12 420 Inerne Mahemaics Lemma 45 If G dominaes G 3 wih an error esimae and G 2 is dominaed by G 4 wih an error esimae 2,henG \ G 2 dominaes G 3 \ G 4 wih an error esimae + 2 Proof For any monoone propery A and any graph H, wedefine he se ψ(a, H) o be ψ(a, H) ={G G \ H A} We observe ha ψ(a, H) is also a monoone propery Therefore, Pr(G \ G 2 A) = Pr(G = H )Pr(G 2 = H 2 ) H A H \H 2=H = Pr(G 2 = H 2 )Pr(G ψ(a, H 2 )) H 2 Pr(G 2 = H 2 )(Pr(G 3 ψ(a, H 2 )) ) H 2 Pr(G 3 \ G 2 A) Similarly, we define he se θ(a, H) obe θ(a, H) ={G H \ G A} We observe ha he complemen of he se θ(a, H) is a monoone propery We have Pr(G 3 \ G 2 A) = Pr(G 3 = H )Pr(G 2 = H 2 ) H A H \H 2 =H = H Pr(G 3 = H )Pr(G 2 θ(a, H )) Thus, we ge as desired H Pr(G 3 = H )(Pr(G 4 θ(a, H )) 2 ) Pr(G 3 \ G 4 A) 2 Pr(G G 2 A) Pr(G 3 G 4 A) ( + 2 ), A random graph G is called edge-independen (or independen, for shor) if here is an edge-weighed funcion p: E(K n ) [0, ] saisfying Pr(G = H) = p e p e ) e H e H(

13 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 42 For example, a random graph wih a given expeced degree sequence is edgeindependen Edge-independen random graphs have many nice properies, several of which we derive here Lemma 46 Suppose ha G and G are independen random graph wih edgeweighed funcions p and p ; hen, G G is edge-independen wih he edgeweighed funcion p saisfying p e = p e + p e p e p e Proof For any graph H, wehave Pr(G G = H) = Pr(G = H )Pr(G = H 2 ) = H H 2 =H p e p e 2 ( p e3 ) ( p e 4 ) H H 2=H e H e 2 H 2 e 3 H e 4 H 2 = ( p e )( p e ) e ( p e H e H(p e )+( p e)p e + p ep e ) = e H p e ( p e ) e H Lemma 47 Suppose ha G and G are independen random graph wih edgeweighed funcions p and p ;hen,g \ G is independen wih he edge-weighed funcion p saisfying p e = p e ( p e) Proof For any graph H, wehave Pr(G \ G = H) = Pr(G = H )Pr(G = H 2 ) = H \H 2 =H p e p e 2 ( p e3 ) ( p e 4 ) H \H 2=H e H e 2 H 2 e 3 H e 4 H 2 = (p e ( p e)) p e p e p e H e H( e) = e H p e ( p e ) e H Le {p e } e E(Kn) be a probabiliy disribuion over all pairs of verices Le G be he random graph of one edge, where a pair e of verices is chosen wih probabiliy p e Inducively, we can define he random graph G m by adding

14 422 Inerne Mahemaics onemorerandomedgeog m,whereapaire of verices is chosen (as he new edge) wih probabiliy p e (There is a small probabiliy of having he same edges chosen more han once In such cases, we will keep on sampling unil we have exacly m differen edges) Hence, G m has exacly m edges The probabiliy ha G m has edges e,,e m is proporional o p e p e2 p em The following lemma saes ha G m can be sandwiched by wo independen random graphs wih exponenially small errors if m is large enough Lemma 48 Assume ha p e = o( m ) for all e E(K n) LeG be he independen random graph wih edge-weighed funcion p e = ( δ)mp e Le G be he independen random graph wih edge-weighed funcion p e =(+δ)mp ethen, G m dominaes G wih error e δ2m/4,andg m is also dominaed by G wihin an error esimae e δ2m/4 Proof For any Graph H, wedefine f(h) = e H p e For any graph propery B, we define f(b) = H B f(h) Le C k be he se of all graphs wih exac k edges Claim 49 For a graph monoone propery A and an ineger k, we have f(a C k ) f(c k ) f(a C k+) f(c k+ ) Proof of Claim 49 Boh f(a C k )f(c k+ )andf(a C k+ )f(c k ) are homogeneous polynomials on {p e } of degree 2k + Wecomparehecoefficiens of a general monomial p 2 e p 2 e r p er+ p e2k r+ in f(a C k )f(c k+ )andf(a C k+ )f(c k ) The coefficien c of he monomial in f(a C k )f(c k+ )ishenumberof(k r)-subses {e i,e i2,,e ik r } of e r+,,e 2k r+ saisfying ha he graph wih edges {e,,e r,e i,e i2,,e ik r } belongs o A k The coefficien c 2 of he monomial in f(a C k )f(c k+ )ishe number of (k r + )-subse {e i,e i2,,e ik r+ } of e r+,,e 2k r+ saisfying

15 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 423 ha he graph wih edges {e,,e r,e i,e i2,,e ik r+ } belongs o A k+ Since A is monoone, if he graph wih edges {e,,e r,e i,e i2,,e ik r } belongs o A k, hen he graph wih edges {e,,e r,e i,e i2,,e ik r+ } mus belong o A k+ Hence, c is always less han or equal o c 2 Thus,wehave The claim is proved f(a C k )f(c k+ ) f(a C k+ )f(c k ) Now le p e = ( δ)mpe +( δ)mp e ( δ)mp e =(+o())( δ)mp e,orequivalenly, p e p e = Pr(G A) = = n Pr(G A C k ) k=0 m Pr(G A C k )+ k=0 e E(K n) n k=m+ Pr(G C k ) m ( p e ) (( δ)m) k f(a C k ) k=0 +Pr(G has more han m edges) m ( p e ) (( δ)m) k f(c k ) f(a C m) f(c m ) e E(K n ) k=0 +Pr(G has more han m edges) f(a C m) m ( p f(c m ) e) (( δ)m) k f(c k ) e E(K n) k=0 +Pr(G has more han m edges) = f(a C m) m Pr(G C k )+Pr(G has more han m edges) f(c m ) k=0 f(a C m) +Pr(G has more han m edges) f(c m ) = Pr(G m A)+Pr(G has more han m edges) NowweesimaeheprobabiliyhaG has more han m edges Le X e be he 0- random variable wih Pr(X e =)=p e Le X = e X e Then, E(X) =(+o())m( δ) Now we apply he following large deviaion inequaliy: Pr(X E(X) >a) e a 2 2(E(X)+a/3)

16 424 Inerne Mahemaics We have Pr(X >m) = Pr(X E(X) > ( + o())δm) e (+o()) δ 2 m 2 2( δ)m+2δm/3 e δ2 m/2 For he oher direcion, le p e = (+δ)mpe implies ha p e p e Pr(G A) = =(+δ)mp e +(+δ)mp e n Pr(G A C k ) k=0 n Pr(G A C k ) k=m =(+o())( + δ)mp e,which = n ( p e ) (( + δ)m) k f(a C k ) e k=m ( p e ) n (( + δ)m) k f(c k ) f(a C m) f(c e m ) k=m f(a C m) n ( p f(c m ) e) (( + δ)m) k f(c k ) e k=m = f(a C m m) Pr(G C k ) f(c m ) k=0 f(a C m) Pr(G haslesshanm edges) f(c m ) = Pr(G m A) Pr(G has less han m edges) Now we esimae he probabiliy ha G haslesshanm edges Le X e be he 0- random variable wih Pr(X e =)=p e Le X = e X e Then E(X) =(+o())m(+δ) Now we apply he following large deviaion inequaliy: We have Pr(X E(X) <a) e a2 2E(X) Pr(X <m) = Pr(X E(X) < ( + o())δm) e (+o()) δ2 m 2 2(+δ)m e δ2 m/3 The proof of Lemma 48 is compleed

17 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs General Maringale Inequaliies In his subsecion, we will exend and generalize he Azuma inequaliy o a maringale ha is no sricly Lipschiz bu is nearly Lipschiz Similar echniques have been inroduced by Kim and Vu [Kim and Vu 00] in heir imporan work on deriving concenraion inequaliies for mulivariae polynomials Here we use a raher general seing, and we shall give a complee proof Suppose ha Ω is a probabiliy space and F is a σ-field; X is a random variable ha is F-measurable (The reader is referred o [Janson e al 00] for he erminology on maringales) A filer F is an increasing chain of σ-subfields {0, Ω} = F 0 F F n = F A maringale (obained from) X associaed wih a filer F is a sequence of random variables X 0,X,,X n wih X i = E(X F i ) and, in paricular, X 0 = E(X) andx n = X For c =(c,c 2,,c n ) a posiive vecor, he maringale X is said o be c- Lipschiz if X i X i c i for i =, 2,,n A powerful ool for conrolling maringales is he following: Azuma s inequaliy If a maringale X is c-lipschiz, hen where c =(c,,c n ) Pr( X E(X) <a) 2e a2 2 n c 2 i= i, Here we are only ineresed in finie probabiliy spaces, and we use he following compuaional model The random variable X can be evaluaed by a sequence of decisions Y,Y 2,,Y n Each decision has no more han r oupus The probabiliy ha an oupu is chosen depends on he previous hisory We can describe he process by a decision ree T ; T is a complee rooed r-ree wih deph n Eachedgeuv of T is associaed wih a probabiliy p uv depending on he decision made from u o v We allow p uv o be zero and hus include he case of having fewer han r oupus Le Ω i denoe he probabiliy space obained afer he firs i decisions Suppose ha Ω = Ω n and X is he random variable on Ω Le π i : Ω Ω i be he projecion mapping each poin o is firs i coordinaes Le F i be he σ-field generaed by Y,Y 2,,Y i (Infac,F i = π (2 Ωi )ishe full σ-field via he projecion π i ) F i forms a naural filer: {0, Ω} = F 0 F F n = F Any verex u of T is associaed wih a real value f(u) If u is a leaf, we define f(u) =X(u) For a general u, hereareseveralequivalendefiniions for f(u)

18 426 Inerne Mahemaics For any non-leaf node u, f(u) is he weighed average over he f-values of he children of u: r f(u) = p uvi f(v i ), i= where v,v 2,,v r are he children of u 2 For a non-leaf node u, f(u) is he weighed average over all leaves in he sub-ree T u rooed a u: f(u) = p u (v)f(v), v leaf in T u where p u (v) denoes he produc of edge-weighs over edges in he unique pah from u o v 3 Le X i be a random variable of Ω, whichforeachnodeu of deph i assumes he value f(u) for every leaf in he subree T u Then, X 0,X,,X n form a maringale, ie, X i = E(X n F i ) In paricular, X = X n is he resricion of f o leaves of T We noe ha he Lipschiz condiion X i X i c i is equivalen o f(u) f(v) c i for any edge uv from a verex u wih deph i oaverexv wih deph i We say an edge uv is bad if f(u) f(v) >c i Wesayanodeu is good if he pah from he roo o u does no conain any node of a bad edge The following heorem furher generalizes he Azuma s Inequaliy A similar bu more resriced version can be found in [Kim and Vu 00] Theorem 5 For any c,c 2,,c n,amaringalex saisfies Pr( X E(X) <a) 2e a 2 2 n i= c 2 i +Pr(B), where B is he se of all bad leaves of he decision ree associaed wih X Proof We define a modified labeling f on T so ha f (u) =f(u) ifu is a good node in T Foreachbadnodeu, lexy be he firs bad edge ha inersecs he pah from he roo o u a x Wedefine f (u) =f(x) Claim 52 f (u) = r i= p uv i f (v i ), for any u wih children v,,v r If u is a good verex, we always have f (v i )=f(v i )wheherv i is good or no Since f(u) = r i= p uv i f(v i ), we have f (u) = r i= p uv i f (v i )

19 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 427 If u is a bad verex, v,,v r are all bad by he definiion We have f (u) = f (v )= = f (v r ) Hence, r i= p uv i f (v i )=f(u) r i= p uv i = f(u) Claim 53 f is c-lipschiz For any edge uv wih u of deph i andv of deph i, ifu is a good verex, hen uv is a good edge, and f (u) f (v) c i If u is a bad verex, we have f (u) =f (v), and hus, f (u) f (v) c i Le X be he random variable ha is he resricion of f o he leaves; X is c-lipschiz We can apply Azuma s Inequaliy o X Namely, From he definiion of f, wehave Pr( X E(X ) <a) 2e E(X )=E(X) a 2 2 n i= c 2 i Le B denoe he se of bad leaves in he decision T of X Clearly, we have Therefore, we have Pr(u : X(u) = X (u)) Pr(B) Pr( X E(X) <a) Pr(X = X )+Pr( X E(X ) <a) 2e a2 2 n i= c 2 i +Pr(B) The proof of he heorem is complee For some applicaions, even nearly Lipschiz condiion is sill no feasible Here we consider an exension of Azuma s inequaliy Our saring poin is he following well-known concenraion inequaliy (see [McDiarmid 98]) Theorem 54 Le X be he maringale associaed wih a filer F saisfying Var(X i F i ) σi 2,for i n; 2 X i X i M, for i n

20 428 Inerne Mahemaics Then, we have Pr(X E(X) a) e a 2 2( n i= σ i 2+Ma/3) In his paper, we consider a srenghened version of he above inequaliy where he variance Var(X i F i ) is insead upper bounded by a consan facor of X i We firs need some erminology For a filer F, {0, Ω} = F 0 F F n = F A sequence of random variables X 0,X,,X n is called a submaringale if X i is F i -measurable and E(X i F i ) X i,for i n A sequence of random variables X 0,X,,X n is said o be a supermaringale if X i is F i -measurable and E(X i F i ) X i,for i n We have he following heorem Theorem 55 Suppose ha a submaringale X, associaed wih a filer F, saisfies and for i n Then, we have Var(X i F i ) φ i X i X i E(X i F i ) M Pr(X n >X 0 + a) e a 2 2((X 0 +a)( n φ i= i )+Ma/3) Proof For a posiive λ (o be chosen laer), we consider E(e λxi F i ) = e λe(xi Fi ) E(e λ(xi E(Xi Fi )) F i ) = e λe(xi Fi ) λ k k! E((X i E(X i F i ) k ) F i ) Le g(y) =2 y k 2 k=2 g(y), for y<0 lim y 0 g(y) = k=0 e λe(x i F i )+ k=2 λk k! E((X i E(X i F i ) k ) F i ) k! = 2(ey y) y 2 g(y) is monoone increasing, when y 0 When b<3, we have We use he following facs:

21 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 429 g(b) =2 k=2 b k 2 k! k=2 b k 2 3 k 2 = b/3 (5) Since X i E(X i F i ) M, wehave λ k k! E((X i E(X i F i ) k ) F i ) g(λm) λ 2 Var(X i F i ) 2 k=2 We define λ i 0for0<i n, saisfyingλ i = λ i + g(λ0m) 2 φ i λ 2 i, while λ 0 will be chosen laer Then, λ n λ n λ 0, and E(e λixi F i ) e λie(xi Fi )+ g(λ i M) 2 λ 2 i Var(Xi Fi ) since g(y) is increasing for y>0 By Markov s inequaliy, we have Noe ha Hence, e λ ix i + g(λ 0 M) 2 λ 2 i φ ix i = e λi Xi, Pr(X n >X 0 + a) e λn(x0+a) E(e λnxn ) = e λn(x0+a) E(E(e λnxn F n )) e λ n(x 0 +a) E(e λ n X n ) λ n = λ 0 e λn(x0+a) E(e λ0x0 ) = e λn(x0+a)+λ0x0 n (λ i λ i ) i= n g(λ 0 M) = λ 0 φ i λ 2 i 2 i= λ 0 g(λ 0M) n λ 2 0 φ i 2 Pr(X n >X 0 + a) e λ n(x 0 +a)+λ 0 X 0 i= e (λ0 g(λ 0 M) 2 λ 2 n 0 i= φi)(x0+a)+λ0x0 = e λ0a+ g(λ 0 M) 2 λ 2 0 (X0+a) n i= φi

22 430 Inerne Mahemaics a Now we choose λ 0 = (X 0 +a)( n i= φ i )+Ma/3 Using he fac ha λ 0M<3and Inequaliy (5), we have Pr(X n >X 0 + a) e λ0a+λ2 0 (X0+a) n i= φi 2( λ 0 M/3) The proof of he heorem is finished e a2 2((X 0 +a)( n i= φ i )+Ma/3) ThecondiionofTheorem55canbefurherrelaxedusinghesameechnique as in Theorem 5, and we have he following heorem The proof will be omied Theorem 56 For a filer F {0, Ω} = F 0 F F n = F, suppose ha a random variable X j is F i -measurable, for i n Le B be he bad se in he decision ree associaed wih Xs where a leas one of he following condiions is violaed: Then, we have E(X i F i ) X i, Var(X i F i ) φ i X i, X i E(X i F i ) M Pr(X n >X 0 + a) e 2((X 0 +a)( n i= φ i )+Ma/3) +Pr(B ) a 2 The heorem for supermaringale is slighly differen due o he asymmery of he condiion on variance Theorem 57 Suppose ha a supermaringale X, associaed wih a filer F, saisfies, for i n, Var(X i F i ) φ i X i and Then, we have for any a X 0 E(X i F i ) X i M Pr(X n <X 0 a) e a 2 2(X 0 ( n φ i= i )+Ma/3),

23 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 43 Proof The proof is similar o ha of Theorem 55 The following inequaliy sill holds: E(e λx i F i ) = e λe(x i F i ) E(e λ(x i E(X i F i )) F i ) = e λe(xi Fi ) λ k k! E((E(X i F i ) X i ) k ) F i ) k=0 e λe(xi Fi )+ k=2 λk k! E((E(Xi Fi ) Xi)k ) F i ) g(λm) λie(xi Fi )+ e 2 λ 2 Var(X i F i ) g(λm) λixi + e 2 λ 2 φ ix i We now define λ i 0, for 0 i<nsaisfying λ i = λ i g(λn) 2 φ i λ 2 i ; λ n will be defined laer Then, we have λ 0 λ λ n, and E(e λ ix i F i ) e λ ie(x i F i )+ g(λ i M) 2 λ 2 i Var(X i F i ) g(λnm) λixi + e 2 λ 2 i φixi = e λi Xi By Markov s inequaliy, we have Pr(X n <X 0 a) = Pr( λ n X n > λ n (X n a)) e λn(x0 a) E(e λnxn ) = e λn(x0 a) E(E(e λnxn F n )) e λ n(x 0 a) E(e λ n X n ) e λn(x0 a) E(e λ0x0 ) = e λn(x0 a) λ0x0 We noe ha λ 0 = λ n + n (λ i λ i ) i= n g(λ n M) = λ n φ i λ 2 i 2 i= λ n g(λ nm) n λ 2 n φ i 2 i=

24 432 Inerne Mahemaics Thus, we have We choose λ n = Pr(X n <X 0 a) e λ n(x 0 a) λ 0 X 0 g(λnm) λn(x0 a) (λn e 2 λ 2 n n i= φi)x0 = e λna+ g(λ nm) 2 λ 2 n X0 n i= φi a X 0 ( n i= φ i )+Ma/3 Wehaveλ nm<3and Pr(X n <X 0 a) e λna+λ2 n X0 n i= φi 2( λnm/3) e a 2 2(X 0 ( n i= φ i )+Ma/3) I remains o verify ha all λ i are nonnegaive Indeed, λ i λ 0 λ n g(λ nm) λ 2 n 2 λ n = λ n ( a 2X 0 ) 0 The proof of he heorem is complee n i= φ i 2( λ n M/3) λ n n φ i i= Again, he above heorem can furher relaxed as follows Theorem 58 For a filer F {0, Ω} = F 0 F F n = F, suppose ha a random variable X j is F i -measurable, for i n Le B 2 be he bad se where a leas one of he following condiions is violaed: Then, we have for any a X 0 E(X i F i ) X i, Var(X i F i ) φ i X i, E(X i F i ) X i M Pr(X n <X 0 a) e 2(X 0 ( n φ i= i )+Ma/3) +Pr(B 2 ), a2

25 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs Main Theorems for he Growh-Deleion Model We say ha a random graph G is almos surely edge-independen if here are wo edge-independen random graphs G and G 2 on he same verex se saisfying he following: G dominaes G 2 G is dominaed by G 2 3 For any wo verices u and v, lep (i) uv be he probabiliy of edge uv in G i for i =, 2 We have We will prove he following: p () uv =( o())p(2) uv Theorem 6 Suppose ha p 3 <p, p 4 <p 2,andlog n m< Almos surely he degree sequence of he growh-deleion model G(p,p 2,p 3,p 4,m) follows he power law disribuion wih he exponen β =2+ p + p 3 p +2p 2 p 3 2p 4 p 2(p +p 2) Then, 2 G(p,p 2,p 3,p 4,m) is almos surely edge-independen I dominaes and is dominaed by an edge-independen graph wih probabiliy p () ij of having an edge beween verices i and j, i < j, aime, saisfying: p 2m p () l 2α ij 2p 4 τ(2p 2 p 4 ) i α j + p4 j 2τ +2α α p 2 if i α j α p2m2α 4τ 2 p 4 ( + o()) 2p4τ p 2m iα j α 2α where α = p (p +2p 2 p 3 2p 4 ) 2(p +p 2 p 4 )(p p 3 ) and τ = (p +p 2 p 4 )(p p 3 ) p +p 3 if i α j α p2m2α 4τ 2 p 4 Wihou he assumpion on m, we have he following general bu weaker resul: Theorem 62 In G(p,p 2,p 3,p 4,m) wih p 3 <p and p 4 <p 2,leS be he se of verices wih index i saisfying i m α 2α Le G S be he induced subgraph of G(p,p 2,p 3,p 4,m) on S Then,

26 434 Inerne Mahemaics G S dominaes a random power law graph G, in which he expeced degrees are given by p 2 m α d i p 2p 4 τ(2p 2 p 4 )( p p 3 α) i α 2 G S is dominaed by a random power law graph G 2, in which he expeced degrees are given by d i m α p 2p 4 τ( p p 3 α) i α Theorem 63 In G(p,p 2,p 3,p 4,m) wih p 3 <p and p 4 <p 2,leT be he se of verices wih index i saisfying i m α 2α Then, he induced subgraph G T of G(p,p 2,p 3,p 4,m) is almos a complee graph Namely, G T dominaes an edge-independen graph wih p ij = o() Le n (or τ ) be he number of verices (or edges) a ime We assume ha he iniial graph has n 0 verices and τ 0 edges When is large enough, he graph a ime depends on he iniial graph only in a mild manner The number of verices n 0 and edges τ 0 in he iniial graph affec only a lower order erm o random variables under consideraion We firs esablish he following lemmas on he number of verices and he number of edges Lemma 64 For any and k>, ing(p,p 2,p 3,p 4,m) wih an iniial graph on n 0 verices, he number n of verices a ime saisfies (p p 3 ) 2klog n n 0 (p p 3 ) + 2k log, (6) wih probabiliy a leas 2 k Proof The expeced number of verices n saisfies he following recurrence relaion: E(n + )=E(n )+p p 3 Hence, E(n + )=n 0 +(p p 3 ) Since we assume ha p 3 <p, he graph grows as ime increases By Azuma s maringale inequaliy, we have Pr( n E(n ) >a) 2e a2 2

27 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 435 By choosing a = 2klog, wih probabiliy a leas 2 k,wehave (p p 3 ) 2klog n n 0 (p p 3 ) + 2k log (62) Lemma 65 The number τ of edges in G(p,p 2,p 3,p 4,m), wihaniniialgraphon n 0 verices and τ 0 edges, saisfies a ime where τ = (p +p 2 p 4 )(p p 3 ) p +p 3 E(τ ) τ 0 τm = O( log ), Proof The expeced number of edges saisfies E(τ + )=E(τ )+mp + mp 2 p 3 E( 2τ n ) mp 4 (63) Le C denoe a large consan saisfying he following: C> 8p3 (p p 3) 2 2 C>4 s log s for some large consan s We shall inducively prove he following inequaliy: When = s, wehave E(τ ) τ 0 mτ <Cm log for s (64) E(τ s ) τ 0 mτs 2ms Cm s log s, by he definiion of C By he inducion assumpion, we assume ha E(τ ) τ 0 τm C log holds Then, we consider E(τ + ) τ 0 τm( +) = E(τ 2τ )+mp + mp 2 p 3 E mp 4 τ 0 τm( +) n = E(τ τ mτ ) τ 0 τm 2p 3 E +2p 3 n p p 3

28 436 Inerne Mahemaics = 2p 3 (E(τ ) mτ τ 0 ) (p p 3 ) τ 2p 3 E E(τ ) 2p 3 n (p p 3 ) (p p 3 ) τ 0 2p 3 (E(τ ) mτ τ 0 ) (p p 3 ) τ +2p 3 E E(τ ) n (p p 3 ) + 2p 3 (p p 3 ) τ 0 E(τ ) τ 0 τm +2p 3 E τ n (p p 3 ) + O We wish o subsiue n by n + n 0 + O( 2k log ) ifpossiblehowever, E τ n (p p 3 ) can be large We consider S, he even ha n n 0 (p + p 3 ) < 4 log We have Pr(S) > from Lemma 64 Le 2 S be he indicaor random variable for he even S, and S denoes he complemen even of S We can derive an upper bound for E((τ + τ 0 τm( +)) S ) in a similar argumen as above and obain E((τ + τ 0 τm( +)) S ) E((τ τ 0 τm) S ) +2p 3 E τ n (p p 3 ) S + O (65) We consider each erm in he las inequaliy separaely 2p 3 E τ S n (p p 3 ) 2p 3 m (p p 3 ) 4 log (p p 3 ) 8p3 log log (p p 3 ) 2 + O (66) Since Pr( S) and τ 2 τ 0 + m, wehave E((τ + τ 0 τm( +))) = E((τ + τ 0 τm( +)) S ) + E((τ + τ 0 τm( +)) S) E((τ + τ 0 τm( +)) S ) +2m( +)Pr( S) E((τ + τ 0 τm( +)) S ) +2m( +) 2

29 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 437 By Inequaliies (65) and (66), we have E((τ + τ 0 τm( + ))) 8p3 log log E((τ τ 0 τm) S ) + (p p 3 ) 2 + O +2m 2 8p3 log log E(τ τ 0 τm) + + O +4m (p p 3 ) 2 Cm 8p3 log log log + (p p 3 ) 2 + O Cm ( +)log( +) The proof of Lemma 65 is complee To derive he concenraion resul on τ, we will need o bound E(τ )ashe iniial number of edge τ 0 changes Lemma 66 We consider wo random graphs G and G in G(p,p 2,p 3,p 4,m) Suppose ha G iniially has τ 0 edges and n 0 verices, and G iniially have τ0 edges and n 0 verices Le τ and τ denoe he number of edges in G and G, respecively If n 0 n 0 = O(), henwehave E(τ ) E(τ ) τ 0 τ 0 + O(log ) Proof From Equaion (63), we have τ E(τ + ) = E(τ )+mp + mp 2 2p 3 E mp 4, n τ E(τ+ ) = E(τ )+mp + mp 2 2p 3 E mp 4 n Then, E(τ + τ+) =E(τ τ) τ 2p 3 E τ n n (67) Since boh n n 0 and n n 0 follow he same disribuion, we have We can rewrie E τ n Pr(n = x) =Pr(n = x + n 0 n 0 ) for any x as follows: τ n

30 438 Inerne Mahemaics τ E τ n = x n x E(τ n = x)pr(n = x) y x E(τ n = x)pr(n = x) y E(τ n = y)pr(n = y) = x x + n x 0 n E(τ n = x + n 0 n 0)Pr(n = x + n 0 n 0) 0 = Pr(n = x) x E(τ n = x) x + n x 0 n 0 = Pr(n = x) x (E(τ n = x) E(τ n = x + n 0 n 0)) x x x + n 0 n E(τ n = x + n 0 n 0 ) 0 E(τ n = x + n 0 n 0 ) From Lemma 64, wih probabiliy a leas 2 2,wehave n n 0 (p p 3 ) 2 log Le S denoe he se of x saisfying x n 0 (p p 3 ) 2 log The probabiliy for x no in S is a mos 2 If his case happens, he conribuion 2 o E( τ n τ n )iso( ), which is a minor erm In addiion, τ is always upper bounded by τ 0 + m Wecanboundhesecond erm as follows x x + n x 0 n E(τ n = x + n 0 n 0 )Pr(n = x) 0 = x x + n x S 0 n E(τ n = x + n 0 n 0)Pr(n = x) 0 + O n 0 n 0 x (τ 0 + m)pr(n = x) (n 0 +(p p 3 ) 2 log )(n 0 +(p p 3 ) 2 log ) + O n 0 n 0 (τ0 + m) (n 0 +(p p 3 ) 2 log )(n 0 +(p p 3 ) 2 log ) + O = O

31 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 439 Hence, we obain τ E τ n n = Pr(n = x) x (E(τ n = x) E(τ n = x + n 0 n 0 )) + O x = Pr(n = x) x (E(τ n = x) E(τ n = x + n 0 n 0 )) + O x S = = = n 0 +(p p 3 ) + O( log ) x S Pr(n = x)(e(τ n = x) E(τ n = x + n 0 n 0)) + O n 0 +(p p 3 ) + O( E(τ n = x)pr(n = x) log ) x S E(τ n = x + n 0 n 0)Pr(n = x + n 0 n 0) x n 0 +(p p 3 ) + O( log ) (E(τ ) E(τ )) + O Combine his wih Equaion (67), and we have E(τ + τ +) = Therefore, we have + O 2p 3 n 0 +(p p 3 ) + O( E(τ τ log ) )+O E(τ + τ+) 2p 3 n 0 +(p p 3 ) + O( E(τ τ log ) ) + O E(τ τ ) + O τ 0 τ0 + O i i= = τ 0 τ0 + O(log ) The proof of he lemma is complee

32 440 Inerne Mahemaics In order o prove he concenraion resul for he number of edges for G(p,p 2,p 3,p 4,m), we shall use he general maringale inequaliy To esablish he near Lipschiz coefficiens, we will derive upper bounds for he degrees by considering he special case wihou deleion For p = α, p 2 = α, and p 3 = p 4 =0,G(α, α, 0, 0,m) is jus he preferenial aachmen model The number of edge increases by m a a ime The oal number of edges a ime is exacly m + τ 0,whereτ 0 is he number of edge of he iniial graph a =0 We label he verex u by i if u is generaed a ime i Le d i () denoehe degree of he verex i a ime Lemma 67 For he preferenial aachmen model G(γ, γ, 0, 0,m), wehave, wih probabiliy a leas k (any k>),hedegreeofverexi a ime saisfies γ/2 d i () mk log i Proof For he preferenial aachmen model G(γ, γ, 0, 0,m), he oal number of edge a ime is τ = m + τ 0 The recurrence for he expeced value of d i () saisfies E(d i ( +) d i ()) = d i ()+mγ d i() 2τ + m( γ) d i() = + τ m(2 γ) d i () 2τ We denoe θ =+ m(2 γ) 2τ Le X be he scaled version of d i () defined as follows: X = d i () j=i+ θ j We have E(X + X )= E(d i( +) d i ()) j=i+ θ j = θ d i () j=i+ θ j = X Thus, X forms a maringale wih E(X )=X i+ = d i (i +)=m We apply Theorem 55 Firs, we compue

33 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 44 Var(X + X ) = = = j=i+ θ2 j Var(d i ( +) d i ()) j=i+ θ2 j E((d i ( +) d i ()) 2 d i ()) m γ d i() +( γ) d i() j=i+ θ2 2τ j τ (θ )d i () j=i+ θ2 j θ θ j=i+ θ X j Le φ = θ θ j=i+ θ j Wehave φ = = In paricular, we have θ θ j=i+ θ j j=i+ + m(2 γ) 2(m+τ 0) γ 2 γ 2 γ 2 φ j m(2 γ) 2(m+τ 0) j=i+ e ( γ 2 ) j=i+ i i γ/2 2 γ/2 γ/2 + m(2 γ) 2(mj+τ 0) j+ τ 0 m γ i γ/2 2 j 2 γ/2 j=i+ Concerning he las condiion in Theorem 55, we have X + E(X + X ) = j=i+ θ (d i ( +) E(d i ( +) d i ())) j j=i+ θ (d i ( +) d i ()) j j=i+ θ j m m

34 442 Inerne Mahemaics Wih M = m and j=i+ φ j, Theorem 55 gives Pr(X <m+ a) e a 2 2(m+a+ma/3) By choosing a = m(k log ), wih probabiliy a leas O( k ), we have X <m+ m(k log ) = mk log Hence, wih probabiliy a leas O( k ), d i () mk log ( i ) γ/2 Remark 68 In he above proof, d i ( +) d i () roughly follows he Poisson disribuion wih mean m(2 γ)d i () = O i ( γ/2) γ/2 = O γ/2 2τ I follows wih probabiliy a leas O( k )had i ( +) d i () is bounded by 2k 2k γ Applying Theorem 56 wih M = γ and j=i+ φ j, we ge Pr(X <m+ a) e a 2 2(m+a+ 2ka 3γ ) + O( k ) When m log, wecanchoosea = 3mk log so ha m dominaes 2ka 3γ In his case, we have Pr(X <m+ a) e a 2 2 ( m+a+ 2ka 3γ ) + O( k )=O( k ) Wih probabiliy a leas O( k ), we have d i () (m + 3mk log )( i ) γ/2 Similarly argumens using Theorem 58 give he lower bound of he same order If i survives a ime in he preferenial aachmen model G(γ, γ, 0, 0,m), hen, wih probabiliy a leas O( k ), we have d i () m γ/2 3mk log i The above bounds will be furher generalized for model G(p,p 2,p 3,p 4,m) laer in Lemma 6 wih similar ideas Lemma 69 For any k, i, and in graph G(p,p 2,p 3,p 4,m), he degree of i a ime saisfies p 2(p +p 2 ) d i () Ckmlog (68) i wih probabiliy a leas O( k ), for some absolue consan C

35 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 443 Proof We compare G(p,p 2,p 3,p 4,m) wih he following preferenial aachmen model G(p,p 2, 0, 0,m) wihou deleion: A each sep, wih probabiliy p, ake a verex-growh sep and add m edges from he new verex o he curren graph; wih probabiliy p 2, ake an edge-growh sep and m edges are added ino he curren graph; wih probabiliy p p 2, do nohing We wish o show ha he degree d u () inhemodelg(p,p 2,p 3,p 4,m)(wih deleion) is dominaed by he degree sequence d u () in he model G(p,p 2, 0, 0,m) Basically, i is a balls-and-bins argumen, similar o he one given in [Cooper e al 04] The number of balls in he firs bin (denoed by a ) represens he degree of u while he number of balls in he oher bin (denoed by a 2 )represenshe sum of degrees of he verices oher han u When an edge inciden o u is added o he graph G(p,p 2,p 3,p 4,m), i increases boh a and a 2 by When an edge no inciden o u is added ino he graph, a 2 increases by 2 while a remains he same Wihou loss of generaliy, we can assume ha a is less han a 2 in he iniial graph If an edge uv, which is inciden o u, is deleed laer, we delay adding his edge unil he very momen ha he edge is o be deleed A he momen of adding he edge uv, hewobinshavea and a 2 balls, respecively When we delay adding he edge uv, he number of balls in each bin is sill a and a 2, respecively, compared wih a +and a 2 + in he original random process Since a <a 2, he random process wih delay dominaes he original random process If an edge vw, which is no inciden o u, is deleed, we also delay adding his edge unil he very momen ha he edge is o be deleed Equivalenly, we compare he process wih a and a 2 balls in he bins o he process wih a and a balls The random process wihou delay dominaes he one wih delay Therefore, for any u, he degrees of u in he model wihou deleion dominaes he degrees in he model wih deleion I remains o derive an appropriae upper bound of d u () formodel G(p,p 2, 0, 0,m) If a verex u is added a ime i, welabelibyi Le us remove he idle seps and re-parameerize he ime For γ = p p +p 2,wehaveG(p,p 2, 0, 0,m)=G(γ, γ, 0, 0,m) We can use he upper bound for he degrees of G(γ, γ, 0, 0,m) as in Lemma 67 This complees he proof for Lemma 69 Lemma 65 can be furher srenghened as follows:

36 444 Inerne Mahemaics Lemma 60 In G(p,p 2,p 3,p 4,m) wih iniial graph on n 0 verices and τ 0 edges, he oal number of edges a ime is τ = τ 0 + τm + O km p 4(p +p 2) log 3/2 wih probabiliy a leas O( k ) where τ = (p+p2 p4)(p p3) p +p 3 Proof For a fixed s wih s, wedefine τ s () =#{ij E(G ) s i, j } We use Lemma 69 wih he iniial graph o be aken as he graph G s a ime s Then, Lemma 69 implies ha, wih probabiliy a leas O( ), we have k τ τ s () is d i () is C p 2(p +p 2 ) mk log i p C 2(p +p 2 ) p mk log 2(p +p 2) s By choosing s =,wehave τ = τ s ()+O p 4(p +p 2) mk log (69) We wan o show ha wih probabiliy a leas O( k/2+ ), we have τ E(τ ) τ τ s () + E(τ ) E(τ s ()) + τ s () E(τ s ()) O mk p 4(p +p 4) log 3/2 I suffices o show ha τ s () E(τ s ()) = O(mk p 4(p +p 4) log 3/2 ) We use he general maringale inequaliy as in Theorem 5 as follows: le c i = Ckm( i p s ) 2(p +p 2) log where C is he consan in Equaion (68) The nodes of he decision ree T are jus graphs generaed by graph model G(p,p 2,p 3,p 4,m) A pah from he roo o a leaf in he decision ree T is associaed wih a chain of he graph evoluion The value f(i) aeachnodeg i (as defined in he proof of Theorem 5) is he expeced number of edges a ime wih iniial graph G i a ime i We noe ha X i migh be differen from he number of edges of G i, which is denoed by τ i Le G i+ be any child node of G i in he decision T We define f(i +)and τ i+ in a similar way By Lemma 66, we have f(i +) f(i) τ i+ τ i + O(log ) ( + o())c i

37 Chung and Lu: Coupling Online and Offline Analyses for Random Power Law Graphs 445 We say ha an edge of he decision ree T is bad if and only if i delees a verex of degree greaer han ( + o())c i a ime i A leaf of T is good if none of he graphs in he chain conains a verex wih degree larger han ( + o())c i a ime i Therefore, he probabiliy for he se B consising of bad leaves is a mos Pr(B) O l k )=O(s k+ By Theorem 5, we have l=s Pr( τ s () E(τ s ()) >a) 2e a 2 l=s c 2 l 2e 2e +Pr(B) a 2 p l=s ( s l )2 (p +p 2 ) (Cmk log ) 2 a 2 C C 2 3 p 2+ p p +p 2 s + O(s k+ ) p +p 2 m 2 k 2 log 2 + O(s k+ ) We choose s = and a = C C p 4(p +p 2) mk log 3/2 Wih probabiliy a leas O( k/2+ ), we have τ s () E(τ s ()) = O p 4(p +p 2) mk log 3/2, as desired Lemma 6 For he model G(p,p 2,p 3,p 4,m), le α = p (p +2p 2 p 3 2p 4 ) 2(p +p 2 p 4 )(p p 3 ) and γ = p p +p 2 Iflog m γ/2, we have he following: For p 3 > 0 and > 0, wih probabiliy a leas, no verex born before p 3 p survives a ime 2 If he verex i survives a ime, hen, wih probabiliy a leas O( k ), he degree d i () in a graph G of he model G(p,p 2,p 3,p 4 ) saisfies d i () (m C α mk log )( Ci γ/4 log 3/2 i), i d i () (m + C α mk log )( + Ci γ/4 log 3/2 i), i for some consan C depending on p,p 2,p 3, and p 4 Proof For a fixed and i, lez i denoe he number of verices lef a ime i wih indices less han 0 = p 3 p (ie, born before 0 ) Clearly, Z 0 0 For