Coupling on-line and off-line analyses for random power law graphs

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1 Coupling on-line and off-line analyses for random power law graphs Fan Chung Linyuan Lu Abstract We develop a coupling technique for analyzing on-line models by using off-line models. This method is especially effective for a growth-deletion model which generalizes and includes the preferential attachment model for generating large complex networks that simulate numerous realistic networks. By coupling the on-line model with the off-line model for random power law graphs, we derive strong bounds for a number of graph properties including diameter, average distances, connected components and spectral bounds. For example, we prove that a power law graph generated by the growth-deletion model almost surely has diameter O(log n) and average distance O(log log n). Introduction In the past few years, it has been observed that a variety of information networks including Internet graphs, social networks and biological networks among others [,3,4,5,8,9,]havetheso-calledpower law degree distribution. A graph is called a power law graph if the fraction of vertices with degree k is proportional to k for some constant β>0. There are basically two different models for β random power law graphs. The first model is an on-line model that mimics the growth of a network. Starting from a vertex (or some small initial graph), a new node and/or new edge is added at each unit of time following the so-called preferential attachment scheme [3, 4, 9]. The endpoint of a new edge is chosen with the probability proportional to their (current) degrees. By using a combination of adding new nodes and new edges with given respective probabilities, one can generate large power law graphs with exponents β greater than (see [3, 7] for rigorous proofs). Since realistic networks encounter both growth and deletion of vertices and University of California, San Diego, fan@ucsd.edu Research supported in part by NSF Grants DMS and ITR University of California, San Diego, llu@math.ucsd.edu

2 edges, here we consider a growth-deletion online model that generalizes and includes the preferential attachment model. Detailed definitions will be given in Section 3. The second model is an off-line model of random graphs with given expected degrees. For a given sequence w of weights w v, a random graph in G(w) is formed by choosing the edge between u and v with probability proportional to the product of w u and w v. The Erdős-Rényi model G(n, p) can be viewed as a special case of G(w) with all w i s equal. Because of the independence in the choices of edges, the model G(w) is amenable to a rigorous analysis of various graph properties and structures. In a series of papers [0,,, ], various graph invariants have been examined and sharp bounds have been derived for diameter, average distance, connected components and spectra for random power law graphs and, in general, random graphs with given expected degrees. The on-line model is obviously much harder to analyze than the off-line model. There has been some recent work on the on-line model beyond showing the generated graph has a power law degree distribution. Bollobás and Riordan [7] have derived a number of graph properties for the on-line model by coupling with G(n, p), namely, identifying (almost regular) subgraphs whose behavior can be captured in a similar way as graphs from G(n, p) for some appropriate p. In this paper, our goal is to couple the on-line model with the off-line model of random graphs with a similar power law degree distribution so that we can apply the techniques from the off-line model to the on-line model. The basic idea is similar to the martingale method but with substantially differences. Although a martingale involves a sequence of functions with consecutive functions having small bounded differences, each function is defined on a fixed probability space Ω. For the on-line model, the probability space for the random graph generated at each time instance is different in general. We have a sequence of probability spaces where two consecutive ones have small differences. To analyze this, we need to examine the relationship of two distinct random graph models, each of which can be viewed as a probability space. In order to do so, we shall describe two basic methods that are not only useful for our proofs here but also interesting in their own right. Comparing two random graph models We define the dominance of one random graph model over another in Section 4. Several key lemmas for controlling the differences are also given there. A general Azuma inequality A concentration inequality is derived for martingales that are almost Lipschitz. A complete proof is given in Section

3 5. The main result of this paper is to show the following results for the random graph G generated by the on-line model G(p,p,p 3,p 4,m)withp >p 3,p > p 4,asdefinedinSection5:. Almost surely the degree sequence of the random graph generated by growth-deletion model G(p,p,p 3,p 4,m) follows the power law distribution with exponent β =+(p +p 3 )/(p +p p 3 p 4 ).. Suppose m>log +ɛ n. For p >p 3 +p 4,wehave<β<3. Almost surely a random graph in G(p,p,p 3,p 4,m) has diameter Θ(log n) and log log n average distance O( log(/(β ) ). We note that the average distance is defined to be the average over all distances among pairs of vertices in the same connected component. 3. Suppose m>log +ɛ n. For p <p 3 +p 4,wehaveβ>3. Almost surely a random graph in G(p,p,p 3,p 4,m) has diameter Θ(log n) and average distance O( log n log d )wheredis the average degree. 4. Suppose m>log +ɛ n. Almost surely a random graph in G(p,p,p 3,p 4,m) has Cheeger constant at least /+o(). 5. Suppose m>log +ɛ n. Almost surely a random graph in G(p,p,p 3,p 4,m) has spectral gap λ at least /8+o(). We note that the Cheeger constant h G of a graph G, which is sometimes called the conductance, is defined by E(A, h G = Ā) min{vol(a), vol(ā)} where vol(a) = x Adeg(x). The Cheeger constant is closely related to the spectral gap λ of the Laplacian of a graph by the Cheeger inequality h G λ h G /. Thus both h G and λ are key invariants for controlling the rate of convergence of random walks on G. Strong properties of off-line random power law graphs For random graphs with given expected degree sequences satisfying a power law distribution with exponent β, we may assume that the expected degrees are 3

4 w i = ci β for i satisfying i0 i<n+i 0. Here c depends on the average degree and i 0 depends on the maximum degree m, namely,c= β β dn β,i0 = n( d(β ) m(β ) )β. Average distance and diameter Fact ([]) For a power law random graph with exponent β > 3 and average degree d strictly greater than, almost surely the average distance is ( + o()) log n and the diameter is Θ(log n). log d Fact ([]) Suppose a power law random graph with exponent β has average degree d strictly greater than and maximum degree m satisfying log m log n/ log log n. If <β<3, almost surely the diameter is Θ(log n) and the log log n average distance is at most ( + o()) log(/(β )). For the case of β =3, the power law random graph has diameter almost surely Θ(log n) and has average distance Θ(log n/ log log n). Connected components Fact 3 ([0]) Suppose that G is a random graph in G(w) with given expected degree sequence w. If the expected average degree d is strictly greater than, then the following holds: () Almost surely G has a unique giant component. Furthermore, the volume of the giant component is at least ( de + o())vol(g) if d 4 e = , and is at least ( +log d d + o())vol(g) if d<. () The second largest component almost surely has size O( log n log d ). Spectra of the adjacency matrix and the Laplacian The spectra of the adjacency matrix and the Laplacian of a non-regular graph can have quite different distribution. The definition for the Laplacian can be found in [8]. Fact 4 ([]). The largest eigenvalue of the adjacency matrix of a random graph with a given expected degree sequence is determined by m, themax- imum degree, and d, the weighted average of the squares of the expected degrees. We show that the largest eigenvalue of the adjacency matrix is almost surely ( + o()) max{ d, m} provided some minor conditions are satisfied. In addition, suppose that the k th largest expected degree m k is significantly larger than d. Then the k th largest eigenvalue of the adjacency matrix is almost surely ( + o()) m k.. For a random power law graph with exponent β>.5, the largest eigenvalue of a random power law graph is almost surely (+o()) m where m 4

5 is the maximum degree. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power law distribution with exponent β if the maximum degree is sufficiently large and k is bounded above by a function depending on β,m and d, the average degree. When <β<.5, the largest eigenvalue is heavily concentrated at cm 3 β for some constant c depending on β and the average degree. 3. We will show that the eigenvalues of the Laplacian satisfy the semi-circle law under the condition that the minimum expected degree is relatively large ( the square root of the expected average degree). This condition contains the basic case when all degrees are equal (the Erdös-Rényi model). If we weaken the condition on the minimum expected degree, we can still have the following strong bound for the eigenvalues of the Laplacian which implies strong expansion rates for rapidly mixing, max λ i ( + o()) 4 + g(n)log n i 0 w w min where w is the expected average degree, w min is the minimum expected degree and g(n) is any slow growing function of n. 3 A growth-deletion model for generating random power law graphs One explanation for the ubiquitous occurrence of power laws is the simple growth rules that can result in a power law distribution (see [3, 4] ). Nevertheless, realistic networks usually encounter both the growth and deletion of vertices and edges. Here we consider a general on-line model that combine deletion steps with the preferential attachment model. Vertex-growth-step: Add a new vertex v and form a new edge from v to an existing vertex u chosen with probability proportional to d u. Edge-growth-step: Add a new edge with endpoints to be chosen among existing vertices with probability proportional to the degrees. If existing in the current graph, the generated edge is discarded. The edge-growthstep is repeated until a new edge is successfully added. Vertex-deletion-step: Delete a vertex randomly. Edge-deletion-step: Delete an edge randomly. 5

6 For non-negative values p,p,p 3,p 4 summing to, we consider the following growth-deletion model G(p,p,p 3,p 4 ): At each step, with probability p, take a vertex-growth step; With probability p, take an edge-growth step; With probability p 3, take a vertex-deletion step; With probability p 4 = p p p 3, take an edge-deletion step. Here we assume p 3 <p and p 4 <p so that the number of vertices and edge grows as t goes to infinity. If p 3 = p 4 = 0, the model is just the usual preferential attachment model which generates power law graphs with exponent β =+ p p +p. An extensive survey on the preferential attachment model is given in [5] and rigorous proofs can be found in [3, 3]. This growth-deletion model generates only simple graphs because the multiple edges are disallowed at the edge-growth-step. The drawback is that the edge-growth-step could runs in a loop. It only happens that the current graph is a completed graph. If this happens, we simply restart the whole procedure from the same initial graph. With high probability, the model generates sparse graphs so that we could omit the analysis of this extreme case. Previously, Bollobás considered edge deletion after the power law graph is generated [7]. Very recently, Cooper, Frieze and Vera [4] independently consider the growth-deletion model with vertex deletion only. We will show (see Section 6) the following: Suppose p 3 <p and p 4 <p. Then almost surely the degree sequence of the growth-deletion model G(p,p,p 3,p 4 ) follows the power law distribution with the exponent p +p 3 β =+. p +p p 3 p 4 We note that a random graph in G(p,p,p 3,p 4 ) almost surely has expected average degree (p + p p 4 )/(p + p 3 ). For of p i s in certain ranges, this value can be below and the random graph is not connected. To simulate graphs with specified degrees, we consider the following modified model G(p,p,p 3,p 4,m), for some integer m which generates random graphs with the expected degree m(p + p p 4 )/(p + p 3 ). At each step, with probability p, add a new vertex v and form m new edges from v to existing vertices u chosen with probability proportional to d u. With probability p,takemedge-growth steps; With probability p 3, take a vertex-deletion step; With probability p 4 = p p p 3,takemedge-deletion steps. Suppose p 3 <p and p 4 <p. Then almost surely the degree sequence of the growth-deletion model G(p,p,p 3,p 4,m) follows the power law distribution 6

7 with the exponent β the same as the exponent for the model G(p,p,p 3,p 4 ). p +p 3 β =+. p +p p 3 p 4 Many results for G(p,p,p 3,p 4,m) can be derived in the same fashion as for G(p,p,p 3,p 4 ). Indeed, G(p,p,p 3,p 4 ) = G(p,p,p 3,p 4,) is usually the hardest case because of the sparseness of the graphs. 4 Comparing random graphs In the early work of Erdős and Rényi on random graphs, they first used the model F (n, m) that each graph on n vertices and m edges are chosen randomly with equal probability, where n and m are given fixed numbers. This model is apparently different from the later model G(n, p), for which a random graph is formed by choosing independently each of the ( n ) pairs of vertices to be an edge with probability p. Because of the simplicity and ease to use, G(n, p) is the model for the seminar work of Erdős and Rényi. Since then, G(n, p) has been widely used and often been referred to as the Erdős-Rényi model. For m = p ( n ), the two models are apparently correlated in the sense that many graph properties that are satisfied by both random graph models. To precisely define the relationship of two random graph models, we need some definitions. A graph property P can be viewed as a set of graphs. We say a graph G satisfies property P if G is a member of P. A graph property is said to be monotone if whenever a graph H satisfies A, then any graph containing H must also satisfy A. For example, the property A of containing a specified subgraph, say, the Peterson graph, is a monotone property. A random graph G is a probability distribution Pr(G = ). Given two random graphs G and G on n vertices, we say G dominates G, if for any monotone graph property A, the probability that a random graph from G satisfies A is greater than or equal to the probability that a random graph from G satisfies A, i.e., Pr(G satisfies A) Pr(G satisfies A). In this case, we write G G and G G. For example, for any p p,we have G(n, p ) G(n, p ). For any ɛ>0, we say G dominates G with an error estimate ɛ, iffor any monotone graph property A, the probability that a random graph from G satisfies A is greater than or equal to the probability that a random graph from G satisfies A up to an ɛ error term, i.e., Pr(G satisfies A)+ɛ Pr(G satisfies A). 7

8 If G dominates G with an error estimate ɛ = ɛ n, which goes to zero as n approaches infinity, we say G is almost surely dominates G. In this case, we write almost surely G G and G G. For example, for any δ>0, we have almost surely G(n, ( δ) ( m n )) F (n, m) G(n, ( + δ) ( m n )). We can extend the definition of domination to graphs with different sizes in the following sense. Suppose that random graph G i has n i vertices for i =,, and n <n. By adding n n isolated vertices, the random graph G is extended to the random graph G with the same size as G. We say G dominates G if G dominates G. We consider random graphs that are constructed inductively by pivoting at one edge at a time. Here we assume the number of vertices is n. Edge-pivoting : For an edge e K n, a probability q (0 q ), and a random graph G, a new random graph G can be constructed in the following way. For any graph H, we define { ( q)pr(g=h) if e E(H), Pr(G = H) = Pr(G = H)+qPr(G = H \{e}) if e E(H). It is easy to check that Pr(G = ) is a probability distribution. We say G is constructed from G by pivoting at the edge e with probability q. For any graph property A, we define the set A e to be Further, we define the set Aē to be A e = {H {e} H A.} Aē = {H \{e} H A.} In other words, A e consists of the graphs obtained by adding the edge e to the graphs in A; Aē consists of the graphs obtained by deleting the edge e from the graphs in A. We have the following useful lemma. Lemma Suppose G is constructed from G by pivoting at the edge e with probability q. Then for any property A, we have Pr(G A) =Pr(G A)+q[Pr((A A e )ē) Pr(A Aē)]. In particular, if A is a monotone property, we have Thus, G dominates G. Pr(G A) Pr(G A). 8

9 Proof: The set associated with a property A can be partitioned into the following subsets. Let A = A A e be the graphs of A containing the edge e, and A =A Aē be the graphs of A not containing the edge e. Wehave Pr(G A) = Pr(G A ) + Pr(G A ) = Pr(G = H)+ Pr(G = H) H A H A = (Pr(G = H)+qPr(G = H \{e})) H A + ( q) Pr(G = H) H A = Pr(G A )+Pr(G A )+qpr(g (A )ē) q Pr(A ) = Pr(G A)+q[Pr((A A e )ē) Pr(A Aē)]. If A is monotone, we have A (A )ē. Thus, Pr(G A) Pr(G A). Lemmaisproved. Lemma Suppose G i is constructed from G i by pivoting the edge e with probability q i,fori=,. If q q and G dominates G,thenG dominates G. Proof: Following the definitions of A, and letting A and A be as in the proof of Lemma, we have Pr(G A) = Pr(G A)+q [Pr(G (A )ē) Pr(G A )] = Pr(G A)+q Pr(G ((A )ē \ A )) Pr(G A)+q Pr(G ((A )ē \ A )) = Pr(G A)+q [Pr(G (A )ē) Pr(G A )] = Pr(G A). The proof of Lemma 4. is complete. 9

10 Let G and G be the random graphs on n vertices. We define G G to be the random graph as follows: Pr(G G = H) = Pr(G = H )Pr(G =H ) H H =H where H,H range over all possible pairs of subgraphs that are not necessarily disjoint. The following Lemma is a generalization of Lemma. Lemma 3 If G dominates G 3 with an error estimate ɛ and G dominates G 4 with an error estimate ɛ,theng G dominates G 3 G 4 with an error estimate ɛ + ɛ. Proof: For any monotone property A and any graph H, we define the set f(a, H) tobe f(a, H) ={G G H A}. We observe that f(a, H) is also a monotone property. Therefore, Pr(G G A) = Pr(G = H )Pr(G =H ) H A H H =H = Pr(G = H )Pr(G f(a, H )) H Similarly, we have Thus, we get H Pr(G = H )(Pr(G 4 f(a, H )) ɛ ) Pr(G G 4 A) ɛ. Pr(G G 4 A) Pr(G 3 G 4 A) ɛ. Pr(G G A) Pr(G 3 G 4 A) (ɛ + ɛ ), as desired.. Suppose φ is a sequence of random graphs φ(g ),φ(g ),..., where the indices of φ range over all graphs on n vertices. Recall that a random graph G is a probability distribution Pr(G = ) over the space of all graphs on n vertices. For any random graph G, we define φ(g) to be the random graph defined as follows: Pr(φ(G) =H)= Pr(G = H )Pr(φ(H )=H ). H H =H 0

11 We have Lemma 4 Let φ and φ be two sequences of random graphs where the indices of φ and φ range over all graphs on n vertices. Let G be any random graph. If Pr(G {H φ (H)dominates φ (H)with an error estimate ɛ }) ɛ, then φ (G) dominates φ (G) with an error estimate ɛ + ɛ. Proof: For any monotone property A and any graph H, we have Pr(φ (G) A) = Pr(G = H )Pr(φ (H )=H ) H A H H =H = Pr(G = H )Pr(φ (H ) f(a, H )) H H Pr(G = H )Pr(φ (H ) f(a, H )) ɛ ɛ Pr(φ (G) A) (ɛ + ɛ ), as desired, since f(a, H) ={G G H A}is also a monotone property.. Let G and G be the random graphs on n vertices. We define G \ G to be the random graph as follows: Pr(G \ G = H) = H \H =H where H,H range over all pairs of graphs. Pr(G = H )Pr(G =H ) Lemma 5 If G dominates G 3 with an error estimate ɛ and G is dominated by G 4 with an error estimate ɛ,theng \G dominates G 3 \ G 4 with an error estimate ɛ + ɛ. Proof: For any monotone property A and any graph H, we define the set ψ(a, H) tobe ψ(a, H) ={G G\H A}.

12 We observe that ψ(a, H) is also a monotone property. Therefore, Pr(G \ G A) = Pr(G = H )Pr(G =H ) H A H \H =H = H Pr(G = H ) Pr(G ψ(a, H )) H Pr(G = H )(Pr(G 3 ψ(a, H )) ɛ ) Pr(G 3 \ G A) ɛ. Similarly, we define the set θ(a, H) tobe θ(a, H) ={G H\G A}. We observe that the complement of the set θ(a, H) is a monotone property. We have Pr(G 3 \ G A) = Pr(G 3 = H )Pr(G =H ) H A H \H =H = Pr(G 3 = H ) Pr(G θ(a, H )) H Thus, we get H Pr(G 3 = H )(Pr(G 4 θ(a, H )) ɛ ) Pr(G 3 \ G 4 A) ɛ. Pr(G G A) Pr(G 3 G 4 A) (ɛ + ɛ ), as desired. A random graph G is called edge-independent (or independent, for short) if there is an edge-weighted function p: E(K n ) [0, ] satisfying Pr(G = H) = p e p e ). e H e H( For example, a random graph with a given expected degree sequence is edgeindependent. Edge-independent random graphs have many nice properties, several of which we derive here.

13 Lemma 6 Suppose that G and G are independent random graph with edgeweighted functions p and p,theng G is edge-independent with the edgeweighted function p satisfying p e = p e + p e p ep e. Proof: For any graph H, wehave Pr(G G = H) = Pr(G = H )Pr(G =H ) = H H =H p e p e ( p e3 ) ( p e 4 ) H H =H e H e H e 3 H e 4 H = ( p e )( p e) e ( p e H e H(p e)+( p e )p e +p e p e) = e H p e ( p e ). e H Lemma 7 Suppose that G and G are independent random graph with edgeweighted functions p and p,theng\g is independent with the edge-weighted function p satisfying p e = p e( p e ). Proof: For any graph H, wehave Pr(G \ G = H) = Pr(G = H )Pr(G =H ) = H \H =H p e p e ( p e3 ) ( p e 4 ) H \H =H e H e H e 3 H e 4 H = (p e ( p e)) p e p e p e H e H( e) = e H p e ( p e ). e H Let {p e } e E(Kn) be a probability distribution over all pairs of vertices. Let G be the random graph of one edge, where a pair e of vertices is chosen with probability p e. Inductively, we can define the random graph G m by adding one more random edge to G m, where a pair e of vertices is chosen (as the new edge) with probability p e. (There is a small probability to have the same edges chosen more than once. In such case, we will keep on sampling until we have 3

14 exactly m different edges.) Hence, G m has exactly m edges. The probability that G m has edges e,...,e m is proportional to p e p e p em. The following lemma states that G m can be sandwiched by two independent random graphs with exponentially small errors if m is large enough. Lemma 8 Assume p e = o( m ) for all e E(K n). Let G be the independent random graph with edge-weighted function p e = ( δ)mp e. Let G be the independent random graph with edge-weighted function p e =(+δ)mp e. Then G m dominates G with error e δ m/4,andg m is also dominated by G within an error estimate e δ m/4. Proof: For any Graph H, we define f(h) = e Hp e For any graph property B, we define f(b) = H Bf(H). Let C k be the set of all graphs with exact k edges. Claim : For a graph monotone property A and an integer k, wehave f(a C k ) f(c k ) f(a C k+). f(c k+ ) Proof: Both f(a C k )f(c k+ )andf(a C k+ )f(c k ) are homogeneous polynomials on {p e } of degree k +. We compare the coefficients of a general monomial p e p e r p er+ p ek r+ in f(a C k )f(c k+ )andf(a C k+ )f(c k ). The coefficient c of the monomial in f(a C k )f(c k+ ) is the number of (k r)-subsets {e i,e i,,e ik r }of e r+,...,e k r+ satisfying that the graph with edges {e,...,e r,e i,e i,,e ik r } belongs to A k. The coefficient c of the monomial in f(a C k )f(c k+ )isthe number of (k r + )-subset {e i,e i,,e ik r+ } of e r+,...,e k r+ satisfying that the graph with edges {e,...,e r,e i,e i,,e ik r+ } belongs to A k+. Since A is monotone, if the graph with edges {e,...,e r,e i,e i,,e ik r } belongs to A k then the graph with edges {e,...,e r,e i,e i,,e ik r+ } must belong to A k+. Hence c is always less than or equal to c.thus,wehave f(a C k )f(c k+ ) f(a C k+ )f(c k ). 4

15 The claim is proved. Now let p e = ( δ)mp e. ( δ)mpe +( δ)mp e =(+o())( δ)mp e, or equivalently, 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) e E(K n) f(a C m) f(c m ) = f(a C m) f(c m ) n k=m+ Pr(G C k ) m ( p e ) (( δ)m) k f(a C k )+Pr(G has more than m edges) k=0 m ( p e ) (( δ)m) k f(c k ) f(a C m) +Pr(G has more than m edges) f(c m ) k=0 e E(K n) m ( p e ) (( δ)m) k f(c k )+Pr(G has more than m edges) k=0 m Pr(G C k )+Pr(G has more than m edges) k=0 f(a C m) +Pr(G has more than m edges) f(c m ) = Pr(G m A)+Pr(G has more than m edges) Now we estimate the probability that G has more than m edges. Let X e be the 0- random variable with Pr(X e =)=p e. Let X = e X e.thene(x)= ( + o())m( δ). Now we apply the following large deviation inequality. We have Pr(X E(X) >a)e a (E(X)+a/3). Pr(X >m) = Pr(X E(X) > ( + o())δm) e (+o()) δ m ( δ)m+δm/3 For the other direction, let p implies =(+δ)mp e. p e p e e δ m/. e = (+δ)mpe +(+δ)mp e =(+o())( + δ)mp e, which 5

16 Pr(G A) = n Pr(G A C k ) k=0 n Pr(G A C k ) k=m 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 haslessthanmedges) f(c m ) = Pr(G m A) Pr(G haslessthanmedges) Now we estimate the probability that G has less than m edges. Let X e be the 0- random variable with Pr(X e =)=p e. Let X = e X e.thene(x)= ( + o())m( + δ). Now we apply the following large deviation inequality. We have Pr(X E(X) <a)e a E(X), Pr(X <m) = Pr(X E(X) < ( + o())δm) e (+o()) δ m (+δ)m e δ m/3. The proof of Lemma is completed. 5 General martingale inequalities In this subsection, we will extend and generalize the Azuma inequality to a martingale which is not strictly Lipschitz but is nearly Lipschitz. Similar techniques have been introduced by Kim and Vu [0] in their important work on deriving concentration inequalities for multivariate polynomials. Here we use a rather general setting and we shall give a complete proof here. 6

17 Suppose that Ω is a probability space and F is a σ-field. X is a random variable that is F-measurable. (The reader is referred to [7] for the terminology on martingales). A filter F is an increasing chain of σ-subfields {0, Ω} = F 0 F F n =F, a martingale (obtained from) X associated with a filter F is a sequence of random variables X 0,X,...,X n with X i = E(X F i ) and, in particular, X 0 = E(X) andx n =X. For c =(c,c,...,c n ) a positive vector, the martingale X is said to be c-lipschitz if X i X i c i for i =,,...,n. A powerful tool for controlling martingales is the following: Azuma s inequality: If a martingale X is c-lipschitz, then a Pr( X E(X) <a)e È n c i= i, where c =(c,...,c n ). Here we are only interested in finite probability spaces and we use the following computational model. The random variable X can be evaluated by a sequence of decisions Y,Y,...,Y n. Each decision has no more than r outputs. The probability that an output is chosen depends on the previous history. We can describe the process by a decision tree T. T is a complete rooted r-tree with depth n. Each edge uv of T is associated with a probability p uv depending on the decision made from u to v. We allow p uv to be zero and thus include the case of having fewer than r outputs. Let Ω i denote the probability space obtained after first i decisions. Suppose Ω = Ω n and X is the random variable on Ω. Let π i :Ω Ω i be the projection mapping each point to its first i coordinates. Let F i be the σ-field generated by Y,Y,...,Y i. (In fact, F i = π ( Ωi ) is the full σ-field via the projection π i.) F i form a natural filter: {0, Ω} = F 0 F F n =F. Any vertex u of T is associated with a real value f(u). If u is a leaf, we define f(u) = X(u). For a general u, here are several equivalent definitions for f(u).. For any non-leaf node u, f(u) is the weighted average over the f-values of the children of u. r f(u) = p uvi f(v i ), i= where v,v,...,v r are the children of u. 7

18 . For a non-leaf node u, f(u) is the weighted average over all leaves in the sub-tree T u rooted at u. f(u) = p u (v)f(v) v leaf in T u where p u (v) denotes the product of edge-weights over edges in the unique path from u to v. 3. Let X i be a random variable of Ω, which for each node u of depth i, assumes the value f(u) for every leaf in the subtree T u.thenx 0,X,...,X n form a martingale, i.e., X i = E(X n F i ). In particular, X = X n is the restriction of f to leaves of T. We note that the Lipschitz condition X i X i c i is equivalent to f(u) f(v) c i for any edge uv from a vertex u with depth i toavertexvwith depth i. We say an edge uv is bad if f(u) f(v) >c i. We say a node u is good if the path from the root to u does not contain any node of a bad edge. The following theorem further generalizes the Azuma s Inequality. A similar but more restricted version can be found in [0]. Theorem For any c,c,...,c n,amartingalexsatisfies Pr( X E(X) <a)e È n c i= i +Pr(B), where B is the set of all bad leaves of the decision tree associated with X. Proof: We define a modified labeling f on T so that f (u) =f(u)ifuis a good node in T. For each bad node u,letxy be the first bad edge which intersects the path from the root to u at x. We define f (u) =f(x). Claim A: f (u) = r i= p uv i f (v i ), for any u with children v,...,v r. If u is a good vertex, we always have f (v i )=f(v i ) no matter whether v i is good or not. Since f(u) = r i= p uv i f(v i ), we have f (u) = r i= p uv i f (v i ). If u is a bad vertex, v,...,v r are all bad by the definition. 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 B: f is c-lipschitz. For any edge uv with u of depth i andvof depth i, Ifuis a good vertex, then uv is a good edge, and a f (u) f (v) c i. 8

19 If u is a bad vertex, we have f (u) =f (v) and thus f (u) f (v) c i. Let X be the random variable which is the restriction of f to the leaves. X is c-lipschitz. We can apply Azuma s Inequality to X.Namely, From the definition of f, wehave a Pr( X E(X ) <a)e È n c i= i. E(X )=E(X). Let B denote the set of bad leaves in the 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) e È a n c i= i +Pr(B). The proof of the theorem is complete. For some applications, even nearly Lipschitz condition is still not feasible. Here we consider an extension of Azuma s inequality. Our starting point is the following well known concentration inequality (see [3]): Theorem Let X be the martingale associated with a filter F satisfying. Var(X i F i ) σ i,forin;. X i X i M,forin. Then, we have a Pr(X E(X) a) e È ( n σ i= i +Ma/3). In this paper, we consider a strenghtened version of the above inequality where the variance Var(X i F i ) is instead upper bounded by a constant factor of X i. We first need some terminology. For a filter F: {0, Ω} = F 0 F F n =F, 9

20 a sequence of random variables X 0,X,...,X n is called a submartingale if X i is F i -measurable and E(X i F i )X i,forin. A sequence of random variables X 0,X,...,X n is said to be a supermartingale if X i is F i -measurable and E(X i F i ) X i,forin. We have the following theorem. Theorem Suppose that a submartingale X, associated with a filter F, satisfies Var(X i F i ) φ i X i and for i n. Then we have X i E(X i F i ) M Pr(X n >X 0 +a)e È ((X 0 +a)( n φ i= i )+Ma/3). Proof: for a positive λ (to be chosen later), 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 ) Let g(y) = y k k= g(y), for y<0. lim y 0 g(y) =. k=0 e λe(xi Fi )+È k= λk k! E((Xi E(Xi Fi )k ) F i ) k! = (ey y) y g(y) is monotone increasing, when y 0. When b<3, we have g(b) = Since X i E(X i F i ) M, wehave k= = a. We use the following facts: k= k= b k k! b k 3 k b/3. () λ k k! E((X i E(X i F i ) k ) F i ) g(λm) λ Var(X i F i ). 0

21 We define λ i 0for0<in, satisfying λ i = λ i + g(λ0m) φ i λ i, while λ 0 will be chosen later. Then λ n λ n λ 0, and E(e λixi F i ) e λie(xi Fi )+ g(λ i M) λ i Var(Xi Fi ) e λixi + g(λ 0 M) λ i φixi = e λi Xi. since g(y) is increasing for y>0. By Markov s inequality, we have Pr(X n >X 0 +a) e λn(x0+a) E(e λnxn ) = e λn(x0+a) E(E(e λnxn F n )) e λn(x0+a) E(e λn Xn ) e λn(x0+a) E(e λ0x0 ) = e λn(x0+a)+λ0x0 Note that Hence λ n = λ 0 n (λ i λ i ) i= n g(λ 0 M) = λ 0 φ i λ i i= λ 0 g(λ 0M) n λ 0 φ i. i= Pr(X n >X 0 +a) e λn(x0+a)+λ0x0 e (λ0 g(λ 0 M) λ 0 È n i= φi)(x0+a)+λ0x0 = e λ0a+ g(λ 0 M) λ 0 (X0+a) È n i= φi a Now we choose λ 0 = (X È n 0+a)( φi)+ma/3. Using the fact that λ 0M<3and i= inequality (), we have Pr(X n >X 0 +a) e λ0a+λ 0 (X0+a)È n i= φi ( λ 0 M/3) e È a ((X 0 +a)( n φ i= i )+Ma/3).

22 The proof of the theorem is finished. The condition of Theorem can be further relaxed using the same technique as in Theorem and we have the following theorem. The proof will be omitted. Theorem 3 For a filter F {0, Ω} = F 0 F F n =F, suppose a random variable X j is F i -measurable, for i n. LetB be the bad set in the decision tree associated with X s where at least one of the following conditions is violated: 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 È ((X 0 +a)( n φ i= i )+Ma/3) +Pr(B ). The theorem for supermartingale is slightly different due to the asymmetry of the condition on variance. Theorem 4 Suppose a supermartingale X, associated with a filter F, satisfies, for i n, Var(X i F i ) φ i X i a and Then we have for any a X 0. E(X i F i ) X i M. a Pr(X n <X 0 a)e È (X 0 ( n φ i= i )+Ma/3), Proof: The proof is similar to that of Theorem. The following inequality still holds. 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((E(X i F i ) X i ) k ) F i ) k=0 e λe(xi Fi )+È k= λk k! E((E(Xi Fi ) Xi)k ) F i ) g(λm) λie(xi Fi )+ e λ Var(X i F i ) g(λm) λixi + e λ φ ix i.

23 We now define λ i 0, for 0 i<nsatisfying λ i = λ i g(λn) φ i λ i. λ n will be defined later. Then we have and λ 0 λ λ n, E(e λixi F i ) e λie(xi Fi )+ g(λ i M) λ i Var(Xi Fi ) By Markov s inequality, we have g(λn M) λixi + e λ i φixi = e λi Xi. 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 λn(x0 a) E(e λn Xn ) e λn(x0 a) E(e λ0x0 ) = e λn(x0 a) λ0x0 We note Thus, we have λ 0 = λ n + n (λ i λ i ) i= n g(λ n M) = λ n φ i λ i i= λ n g(λ nm) n λ n φ i. i= We choose λ n = Pr(X n <X 0 a) e λn(x0 a) λ0x0 X 0( È n a e λn(x0 a) (λn g(λnm) λ nè n i= φi)x0 g(λnm) È λna+ = e λ n n X0 i= φi i= φi)+ma/3. Wehaveλ nm<3and Pr(X n <X 0 a) e λna+λ n X0 È n i= φi ( λn M/3) e È a (X 0 ( n φ i= i )+Ma/3). 3

24 It remains to verify that all λ i s are non-negative. Indeed, λ i λ 0 λ n g(λ nm) λ n n i= φ i λ n ( ( λ n M/3) λ n = λ n ( a ) X 0 0. n φ i ) i= The proof of the theorem is complete. Again, the above theorem can further relaxed as follows: Theorem 5 For a filter F {0, Ω} = F 0 F F n =F, suppose a random variable X j is F i -measurable, for i n. Let B be the bad set where at least one of the following conditions is violated: 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 È (X 0 ( n φ i= i )+Ma/3) +Pr(B ), 6 Main theorems for the growth deletion model We say a random graph G is almost surely edge-independent if there are two edge-independent random graphs G and G on the same vertex set satisfying. G dominates G.. G is dominated by G. 3. For any two vertices u and v, letp (i) uv be the probability of edge uv in G i for i =,. We have p () uv =( o())p() uv. a 4

25 We will prove the following: (p Theorem 6 Suppose p 3 <p,p 4 <p,andlog n m<t +p ). Then. Almost surely the degree sequence of the growth-deletion model G(p,p,p 3,p 4,m) follows the power law distribution with the exponent p (t) ij β =+ p +p 3 p +p p 3 p 4.. G(p,p,p 3,p 4,m) is almost surely edge-independent. It dominates and is dominated by an edge-independent graph with probability p (t) ij of having an edge between vertices i and j, i < j, attimet, satisfying: { p m l α p 4τ (p p 4) i α j ( + ( p4 α p )( j t ) τ +α ) if i α j α pmtα 4τ p 4 ( + o()) p4τ p m iα j α t α where α = p(p+p p3 p4) (p +p p 4)(p p 3) and τ = (p+p p4)(p p3) p +p 3. p if i α j α pmtα 4τ p 4 Without the assumption on m, we have the following genernal but weaker result: Theorem 7 In G(p,p,p 3,p 4,m) with p 3 <p,p 4 <p,letsbe the set of vertices with index i satisfying i m α t α. Let G S be the induced subgraph of G(p,p,p 3,p 4,m) on S. Then we have. G S dominates a random power law graph G, in which the expected degrees are given by p m t α d i p p 4 τ(p p 4 )( p p 3 α) i α.. G S is dominated by a random power law graph G, in which the expected degrees are given by m t α d i p p 4 τ( p p 3 α) i α. Theorem 8 In G(p,p,p 3,p 4,m) with p 3 <p,p 4 <p,lettbe the set of vertices with index i satisfying i m α t α. Then the induced subgraph G T of G(p,p,p 3,p 4,m)is almost a complete graph. Namely, G T dominates an edge-independent graph with p ij = o(). 5

26 Let n t (or τ t ) be the number of vertices (or edges) at time t. We assume the initial graph has n 0 vertices and τ 0 edges. When t is large enough, the graph at time t depend on the initial graph only in a mild manner. The number of vertices n 0 and edges τ 0 in the initial graph affect only a lower order term to random variables under consideration. We first establish the following lemmas on the number of vertices and the number of edges. Lemma 9 For any t and k>,ing(p,p,p 3,p 4 ) with an initial graph on n 0 vertices, the number n t of vertices at time t satisfies (p p 3 )t kt log t n t n 0 (p p 3 )t + kt log t. () with probability at least t k. Proof: The expected number of vertices n t satisfies the following recurrence relation: E(n t+ )=E(n t )+p p 3. Hence, E(n t+ )=n 0 +(p p 3 )t. Since we assume p 3 <p, the graph grows as time t increases. By Azuma s martingale inequality, we have Pr( n t E(n t ) >a)e a t. By choosing a = kt log t, with probability at least t k,wehave (p p 3 )t kt log t n t n 0 (p p 3 )t + kt log t. (3) Lemma 0 The number τ t of edges in G(p,p,p 3,p 4,m), with an initial graph on n 0 vertices and τ 0 edges, satisfies at time t where τ = (p+p p4)(p p3) p +p 3. E(τ t ) τ 0 τmt = O( tlog t) Proof: The expected number of edges satisfies E(τ t+ )=E(τ t )+mp + mp p 3 E( τ t ) mp 4. (4) n t Let C denote a large constant satisfying the following:. C> 8p3 (p p 3), 6

27 . C>4 s log s for some large constant s. We shall inductively prove the following inequality. When t = s, wehave E(τ t ) τ 0 mτt <Cm tlog t for t s. (5) E(τ s ) τ 0 mτs ms Cm slog s, by the definition of C. By the induction assumption, we assume that E(τ t ) τ 0 τmt C tlog t holds. Then we consider E(τ t+ ) τ 0 τm(t+) = E(τ t )+mp + mp p 3 E( τ t ) mp 4 τ 0 τm(t +) n t = E(τ t ) τ 0 τmt p 3 E( τ t mτ )+p 3 ) n t p p 3 p 3 = ( (p p 3 )t )(E(τ t) mτt τ 0 ) p 3 (E( τ t ) E(τ t) n t (p p 3 )t ) p 3 (p p 3 )t τ 0 ( p 3 (p p 3 )t )(E(τ t) mτt τ 0 ) +p 3 E( τ t n t ) E(τ t) (p p 3 )t + p 3 (p p 3 )t τ 0 E(τ t ) τ 0 τmt +p 3 E(τ t ( n t (p p 3 )t )) + O( t ) We wish to substitute n t by nt + n 0 + O( kt log t) if possible. However, E(τ t ( n t (p p 3)t )) can be large. We consider S, the event that n t n 0 (p + p 3 )t < 4 t log t. WehavePr(S)> t from Lemma 9. Let S be the indicator random variable for the event S and S denotes the complement event of S. We can derive an upper bound for E((τ t+ τ 0 τm(t +)) S ) in a similar argument as above and obtain E((τ t+ τ 0 τm(t+)) S ) E((τ t τ 0 τmt) S ) +p 3 E(τ t ( n t (p p 3 )t ) S) + O( ). (6) t We consider each term in the last inequality separately. p 3 E(τ t ( n t (p p 3 )t ) S) p 3 mt (p p 3 )t 4 t log t (p p 3 )t 8p3 log t (p p 3 ) + O( log t ). (7) t t 7

28 Since Pr( S) t and τ t τ 0 + mt, wehave E((τ t+ τ 0 τm(t+ ))) = E((τ t+ τ 0 τm(t+)) S ) + E((τ t+ τ 0 τm(t+)) S) E((τ t+ τ 0 τm(t+)) S ) +m(t+)pr( S) E((τ t+ τ 0 τm(t+)) S ) +m(t+) t By inequalities (6) and (7), we have E((τ t+ τ 0 τm(t + ))) 8p3 log t E((τ t τ 0 τmt) S ) +( (p p 3 ) t 8p3 log t E(τ t τ 0 τmt) + (p p 3 ) t Cm 8p3 log t tlog t +( (p p 3 ) + O( log t )) t t Cm (t + ) log(t +). + O( log t )) + mt t t + O( log t )+4m t t The proof of Lemma 0 is complete. To derive the concentration result on τ t, we will need to bound E(τ t )asthe initial number of edge τ 0 changes. Lemma We consider two random graphs G t and G t in G(p,p,p 3,p 4,m). Suppose that G t initially has τ 0 edges and n 0 vertices, and G t initially have τ 0 edges and n 0 vertices. Let τ t and τ t denote the number of edges in G t and G t, respectively. If n 0 n 0 = O(), then we have E(τ t ) E(τ t) τ 0 τ 0 +O(log t). Proof: From equation (4), we have E(τ t+ ) = E(τ t )+mp + mp p 3 E( τ t n t ) mp 4 E(τ t+ ) = E(τ t )+mp + mp p 3 E( τ t n ) mp 4 t Then, E(τ t+ τ t+) =E(τ t τ t) p 3 E( τ t τ t n t n ). (8) t Since both n t n 0 and n t n 0 follow the same distribution, we have Pr(n t = x)=pr(n t = x+n 0 n 0) for any x. 8

29 We can rewrite E( τt n t τ t n ) as follows. t E( τ t τ t n t n ) = t x = x x E(τ t n t = x)pr(n t = x) y x E(τ t n t = x)pr(n t = x) y E(τ t n t = y)pr(n t = y) x+n x 0 n E(τ t n t = x+n 0 n 0)Pr(n t = x+n 0 n 0) 0 = ( ) Pr(n t = x) x E(τ t n t = x) x+n x 0 n E(τ t n t = x+n 0 n 0 ) 0 = x Pr(n t = x)( x (E(τ t n t = x) E(τ t n t = x + n 0 n 0)) ( x x + n 0 n )E(τ t n t = x + n 0 n 0)) 0 From Lemma 9, with probability at least t,wehave n t n 0 (p p 3 )t tlog t. Let S denote the set of x satisfying x n 0 (p p 3 )t tlog t. The probability for x not in S is at most t. If this case happens, the contribution to E( τt n t τ t n )iso( t t ), which is a minor term. In addition, τ t is always upper bounded by τ 0 + mt. We can bound the second term as follows. ( x x + n x 0 n )E(τ t n t = x + n 0 n 0)Pr(n t = x) 0 = ( x x+n x S 0 n )E(τ t n t = x+n 0 n 0 )Pr(n t = x) +O( 0 t ) n 0 n 0 x (τ 0 +mt)pr(n t = x) (n 0 +(p p 3 )t tlog t)(n 0 +(p p 3 )t tlog t) + O( t ) n 0 n 0 (τ 0 + mt) (n 0 +(p p 3 )t tlog t)(n 0 +(p p 3 )t tlog t) + O( t ) = O( t ). 9

30 Hence, we obtain E( τ t τ t n t n ) = t x Pr(n t = x) x (E(τ t n t = x) E(τ t n t = x+n 0 n 0 )) + O( t ) = x S Pr(n t = x) x (E(τ t n t = x) E(τ t n t = x+n 0 n 0)) + O( t ) = n 0 +(p p 3 )t+o( tlog t) x S Pr(n t = x)(e(τ t n t = x) E(τ t n t = x + n 0 n 0 )) + O( t ) = n 0 +(p p 3 )t+o( tlog t) ( x S E(τ t n t = x)pr(n t = x) x E(τ t n t = x+n 0 n 0 )Pr(n t = x+n 0 n 0 )) + O( t ) = n 0 +(p p 3 )t+o( tlog t) (E(τ t) E(τ t )) + O( t ). Combine this with equation (8), we have E(τ t+ τ t+) =( Therefore we have p 3 n 0 +(p p 3 )t+o( tlog t) )E(τ t τ t)+o( t ). E(τ t+ τ t+ ) ( p 3 n 0 +(p p 3 )t+o( tlog t) )E(τ t τ t ) + O( t ) E(τ t τ t ) +O( t ) t τ 0 τ 0 + O( i ) i= = τ 0 τ 0 + O(log t). The proof of the lemma is complete. In order to prove the concentration result for the number of edges for G(p,p,p 3,p 4,m), we shall use the general martingale inequality. To establish the near Lipschitz coefficients, we will derive upper bounds for the degrees by considering the special case without deletion. For p = α, p = α, p 3 =p 4 =0,G(α, α, 0, 0,m) is just the preferential attachment model. The number of edge increases by m at a time. The total number of edges at time t is exactly mt + τ 0,whereτ 0 is the number of edge of the initial graph at t =0. We label the vertex u by i if u is generated at time i. Let d i (t) denotethe degree of the vertex i at time t. 30

31 Lemma For the preferential attachment model G(γ, γ,0,0,m), we have, with probability at least t k (any k>),thedegreeofvertexiat time t satisfies d i (t) mk( t i ) γ/ log t. Proof: For the preferential attachment model G(γ, γ,0,0,m), the total number of edge at time t is τ t = mt + τ 0. The recurrence for the expected value of d i (t) satisfies E(d i (t +) d i (t)) = m ( + γ τ t + j )d i(t) j= m( γ) = (+ + O( m τ t t ))d i(t). The term O( m t d i (t)) captures the difference from adding m edges sequentially to adding m edges simultaneously. We denote θ t =+ m( γ) τ t + O( m t ). Let X t be the scaled version of d i (t) defined as follows. X t = d i (t) t j=i+ θ. j We have E(X t+ X t )= E(d i(t+) d i (t)) t j=i+ θ j = θ td i (t) t j=i+ θ j = X t. Thus X t forms a martingale with E(X t )=X i+ = d i (i +)=m. We apply Theorem. First we compute Var(X t+ X t ) = = = t j=i+ θ j Var(d i (t +) d i (t)) t j=i+ θ j E((d i (t +) d i (t)) d i (t)) t m(γ d i(t) +( γ) d i(t) ) j=i+ θ τ j t τ t t (θ t )d i (t) j=i+ θ j θ t θ t t j=i+ θ X t. j 3

32 Let φ t = Éθ t θ t t.wehave j=i+ θj φ t = = θ t θ t t j=i+ θ j m( γ) (mt+τ 0) + O( m t ) ( + m( γ) (mt+τ + O( m 0) t )) t m( γ) j=i+ ( + (mj+τ 0) )+O(m t ) ( γ ) t e ( γ ) È t j=i+ ( γ ) t ( i t ) γ/ j+ τ 0 m ( γ ) i γ/ t γ/. In particular, we have t j=i+ φ j ( γ t ). j=i+ Concerning the last condition in Theorem, we have i γ/ j γ/ X t+ E(X t+ X t ) = t j=i+ θ (d i (t +) E(d i (t+) d i (t)) j t j=i+ θ (d i (t +) d i (t)) j t j=i+ θ j m m With M = m and t j=i+ φ j, Theorem gives Pr(X t <m+a)e a (m+a+ma/3). By choosing a = m(k log t ), with probability at least O(t k ), we have X t <m+m(klog t ) = mk log t. Hence, with probability at least O(t k ), d i (t) mk log t( t i ) γ/. Remark: In the above proof, d i (t +) d i (t) roughly follows the Poisson distribution with mean m( γ)d i (t) = O(i ( γ/) t γ/ )=O(t γ/ ). τ t 3

33 It follows with probability at least O(t k ), d i (t +) d i (t) is bounded by k γ. Applying Theorem 3 with M = k γ and t j=i+ φ j, we get Pr(X t <m+a)e a (m+a+ ka 3γ ) + O(t k ). When m log t, wecanchoosea= 3mk log t so that m dominates ka 3γ. In this case, we have Pr(X t <m+a)e a (m+a+ ka 3γ ) + O(t k )=O(t k ). With probability at least O(t k ), we have d i (t) (m+ 3mk log t)( t i ) γ/. Similarly arguments using Theorem 5 give the lower bound of the same order. If i survives at time t in the preferential attachment model G(γ, γ,0,0,m), then, with probability at least O(t k ), we have d i (t) (m 3mk log t)( t i ) γ/. The above bounds will be further generalized for model G(p,p,p 3,p 4,m) later in Lemma 5 with similar ideas. Lemma 3 For any k, i,andt, in graph G(p,p,p 3,p 4,m), the degree of i at time t satisfies: d i (t) Ckmlog t( t i ) p (p +p ) (9) with probability at least O( t k ), for some absolute constant C. Proof: We compare G(p,p,p 3,p 4,m) with the following preferential attachment model G(p,p,0,0,m) without deletion. At each step, with probability p, take a vertex-growth step and add m edges from the new vertex to the current graph; With probability p,takeanedge-growthstepandmedges are added into the current graph; With probability p p, do nothing. We wish to show that the degree d u (t) in the model G(p,p,p 3,p 4,m)(with deletion) is dominated by the degree sequence d u (t) in the model G(p,p,0,0,m). Basically it is a balls-and-bins argument, similar to the one given in [4]. The number of balls in the first bin (denoted by a ) represents the degree of u while the number of balls in the other bin (denoted by a ) represents the sum of degrees of the vertices other than u. When an edge incident to u is added to the 33

34 graph G(p,p,p 3,p 4,m), it increases both a and a by. When an edge not incident to u is added into the graph, a increases by while a remains the same. Without loss of generality, we can assume a is less than a in the initial graph. If an edge uv, which is incident to u, is deleted later, we delay adding this edge until the very moment the edge is to be deleted. At the moment of adding the edge uv, two bins have a and a balls respectively. When we delay adding the edge uv, the number of balls in two bins are still a and a comparing with a +anda + in the original random process. Since a <a,the random process with delay dominates the original random process. If an edge vw which is not incident to u is deleted, we also delay adding this edge until the very moment the edge is to be deleted. Equivalently, we compare the process of a and a balls in bins to the process with a and a + balls. The random process without delay dominates the one with delay. Therefore, for any u, the degrees of u in the model without deletion dominates the degrees in the model with deletion. It remains to derive an appropriate upper bound of d u (t)formodelg(p,p,0,0,m). If a vertex u is added at time i, welabelitbyi. Let us remove the idle steps and re-parameterize the time. For γ = p p +p,wehaveg(p,p,0,0,m)=g(γ, γ,0,0,m). We can use the upper bound for the degrees of G(γ, γ,0,0,m) as in Lemma. This completes the proof for Lemma 3. Lemma 0 can be further strengthened as follows: Lemma 4 In G(p,p,p 3,p 4,m) with intimal graph on n 0 vertices and τ 0 edges, the total number of edges at time t is τ t = τ 0 + τmt+o(kmt p 4(p +p ) log 3/ t) with probability at least O(t k ) where τ = (p+p p4)(p p3) p +p 3. Proof: For a fixed s, s t, we define τ s (t) =#{ij E(G t ) s i, j t}. We use Lemma 3 with the initial graph to be taken as the graph G s at time s. Then Lemma 3 implies that with probability at least O( t k ), we have τ t τ s (t) is d i (t) is C ( ) p t (p +p ) mk log t i ( ) p C t (p +p ) p mkt log t. (p +p ) s 34

35 By choosing s = t,wehave τ t =τ s (t)+o(t p 4(p +p ) mk log t). (0) We want to show that with probability at least O(t k/+ ), we have τ t E(τ t ) τ t τ s (t) + E(τ t ) E(τ s (t)) + τ s (t) E(τ s (t)) O(mkt p 4(p +p 4) log 3/ t) It suffices to show that τ s (t) E(τ s (t)) = O(mkt p 4(p +p 4) log 3/ t). We use the general martingale inequality as in Theorem as follows: Let c i = Ckm( i p s ) (p +p ) log t where C is the constant in equation (9). The nodes of the decision tree T are just graphs generated by graph model G(p,p,p 3,p 4,m). A path from root to a leaf in the decision tree T is associated with a chain of the graph evolution. The value f(i) ateachnodeg i (as defined in the proof of Theorem ) is the expected number of edges at time t with initial graph G i at time i. WenotethatX i might be different from the number of edges of G i, which is denoted by τ i. Let G i+ be any child node of G i in the decision T. We define f(i +)and τ i+ in a similar way. By Lemma, we have f(i +) f(i) τ i+ τ i + O(log t) ( + o())c i We say that an edge of the decision tree T is bad if and only if it deletes a vertex of degree greater than ( + o())c i at time i. A leaf of T is good if none of graphs in the chain contains a vertex with degree large than ( + o())c i at time i. Therefore, the probability for the set B consisting of bad leaves is at most t Pr(B) O(l k )=O(s k+ ). l=s By Theorem, we have Pr( τ s (t) E(τ s (t)) >a) e È a tl=s c l +Pr(B) e e a È p tl=s ( s l ) (p +p ) (Cmk log t) a C C t 3 p + p p +p s + O(s k+ ) p +p m k log t + O(s k+ ). We choose s = t and a = C Ct p 4(p +p ) mk log 3/ t. With probability at least O(t k/+ ), we have τ s (t) E(τ s (t)) = O(t p 4(p +p ) mk log 3/ t) 35