NONSTANDARD REGULAR VARIATION OF IN-DEGREE AND OUT-DEGREE IN THE PREFERENTIAL ATTACHMENT MODEL

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1 NONSTANDARD REGULAR VARIATION OF IN-DEGREE AND OUT-DEGREE IN THE PREFERENTIAL ATTACHMENT MODEL GENNADY SAMORODNITSKY, SIDNEY RESNICK, DON TOWSLEY, RICHARD DAVIS, AMY WILLIS, AND PHYLLIS WAN Abstract. For the directed edge preferential attachment network growth model studied by Bollobás et al. 23) and Krapivsky and Redner 2), we prove that the joint distribution of in-degree and out-degree has jointly regularly varying tails. Typically the marginal tails of the in-degree distribution and the out-degree distribution have different regular variation indices and so the joint regular variation is non-standard. Only marginal regular variation has been previously established for this distribution in the cases where the marginal tail indices are different.. Introduction The directed edge preferential attachment model studied by Bollobás et al. 23) and Krapivsky and Redner 2) is a model for a growing directed random graph. The dynamics of the model are as follows. Choose as parameters nonnegative real numbers α, β, γ, δ in and δ out, such that α + β + γ =. To avoid degenerate situations we will assume that each of the numbers α, β, γ is strictly smaller than. At each step of the growth algorithm we obtain a new graph by adding one edge to an existing graph. We will enumerate the obtained graphs by the number of edges they contain. We start with an arbitrary initial finite directed graph, with at least one node and n edges, denoted Gn ). For n = n +, n + 2,..., Gn) will be a graph with n edges and a random number Nn) of nodes. If u is a node in Gn ), D in u) and D out u) denote the in and out degree of u respectively. The graph Gn) is obtained from Gn ) as follows. With probability α we append to Gn ) a new node v and an edge leading from v to an existing node w in Gn ) denoted v w). The existing node w in Gn ) is chosen with probability depending on its in-degree: pw is chosen) = D in w) + δ in n + δ in Nn ). With probability β we only append to Gn ) a directed edge v w between two existing nodes v and w of Gn ). The existing nodes v, w are chosen independently from the nodes of Gn ) with probabilities pv is chosen) = D out v) + δ out n + δ out Nn ), pw is chosen) = D in w) + δ in n + δ in Nn ). With probability γ we append to Gn ) a new node w and an edge v w leading from the existing node v in Gn ) to the new node w. The existing node v in Gn ) is 99 Mathematics Subject Classification. Primary 6G7, 5C8. Key words and phrases. multivariate heavy tails, preferential attachment model, scale free networks. The research of the authors was supported by MURI ARO Grant W9NF to Cornell University.

2 2 G. SAMORODNITSKY, S. RESNICK, D. TOWSLEY, R. DAVIS, A. WILLIS, AND P. WAN chosen with probability pv is chosen) = D out v) + δ out n + δ out Nn ). If either δ in =, or δ out =, we must have n > for the initial steps of the algorithm to make sense. For i, j =,, 2,... and n n, let N ij n) be the random) number of nodes in Gn) with in-degree i and out-degree j. Theorem 3.2 in Bollobás et al. 23) shows that there are nonrandom constants f ij ) such that N ij n).) lim = f ij a.s. for i, j =,, 2,.... n n Clearly, f =. Since we obviously have Nn) lim = β a.s., n n we see that the empirical joint in- and out-degree distribution in the sequence Gn)) of growing random graphs has as a nonrandom limit the probability distribution N ij n).2) lim n Nn) = f ij β =: p ij a.s. for i, j =,, 2,.... In Bollobás et al. 23) it was shown that the limiting degree distribution p ij ) has, marginally, regularly varying in fact, power-like) tails. Specifically, Theorem 3. ibid. shows that for some finite positive constants C in and C out we have.3) p i in) := p ij C in i α in as i, as long as αδ in + γ >, Here p j out) := j= p ij C out j αout as j, as long as γδ out + α >. i=.4) α in = + + δ inα + γ), α out = + + δ outα + γ). α + β γ + β We will prove that the limiting degree distribution p ij ) in.2) has jointly regularly varying tails and obtain the corresponding tail measure. This paper is organized as follows. We start with a summary of multivariate regular variation in Section 2. In Section 3 we show that the joint generating function of in-degree and out-degree satisfies a partial differential equation. We solve the differential equation and obtain an expression for the generating function. In Section 4 we represent the distribution corresponding to the generating function as a mixture of negative binomial random variables where the mixing distribution is Pareto. This allows direct computation of the tail measure of the non-standard regular variation of in- and out-degree without using transform methods. The tail measure is absolutely continuous with respect to two dimensional Lebesgue measure, and we exhibit its. We also present in Section 4. graphical evidence of the variety of dependence structures possible for the tail measure based on explicit formulae, simulation and iteration of the defining difference equation for limiting frequencies.

3 REGULAR VARIATION OF IN- AND OUT-DEGREES 3 Using the joint generating function of {p ij }, an alternate route for studing heavy tail behavior of in- and out-degree is to use transform methods and Tauberian theory. The multivariate Tauberian theory has been developed and we will report this elsewhere. 2. Multivariate regular variation We briefly review the basic concepts of multivariate regular variation Resnick, 27) which forms the mathematical framework for multivariate heavy tails. We restrict attention to two dimensions since this is the context for the rest of the paper. A random vector X, Y ) has a distribution that is non-standard regularly varying if there exist scaling functions ah) and bh) and a non-zero limit measure ν ) called the limit or tail measure such that as h, 2.) hp [ X/ah), Y/bh) ) ] v ν ) where v denotes vague convergence of measures in M + [, ] 2 \ {}) = M + E), the space of Radon measures on E. The scaling functions will be regularly varying and we assume their indices are positive and therefore, without loss of generality, we may suppose ah) and bh) are continuous and strictly increasing. The phrasing in 2.) implies the marginal distributions have regularly varying tails. In case ah) = bh), X, Y ) has a distribution with standard regularly varying tails Resnick, 27, Section 6.5.6). Given a vector with a distribution which is non-standard regularly varying, there are at least two methods Resnick, 27, Section 9.2.3) for standardizing the vector so that the transformed vector has standard regular variation. The simplest is the power method which is justified when the scaling functions are power functions: For instance with c = γ 2 /γ, ah) = h γ, bh) = h γ 2, γ i >, i =, ) hp [ X c /h γ 2, Y/h γ 2 ) ] v ν ), where if T x, y) = x c, y), then ν = ν T. Since the two scaling functions in 2.2) are the same, the regular variation is now standard. The measure ν will have a scaling property and for an appropriate change of coordinate system, the correspondingly transformed ν can be factored into a product; for example the polar coordinate transform is one such coordinate system change which factors ν into a product of a Pareto measure and an angular measure and this is one way to describe the asymptotic dependence structure of the standardized X, Y ) Resnick, 27, Section 6..4). Another suitable transformation is given in Section 4 based on ratios. 3. The joint generating function of in-degree and out-degree Define the joint generating function of the limit distribution {p ij } of in-degree and out-degree in.2) by 3.) ϕx, y) = i= j= x i y j p ij, x, y. The following lemma shows that the generating function satisfies a partial differential equation.

4 4 G. SAMORODNITSKY, S. RESNICK, D. TOWSLEY, R. DAVIS, A. WILLIS, AND P. WAN Lemma 3.. The function ϕ is continuous on the square [, ] 2 and is infinitely continuously differentiable in the interior of the square. In this interior it satisfies the equation [ 3.2) c δ in x) + c 2 δ out y) + ] ϕ + c x x) ϕ x + c 2y y) ϕ y = α α + γ y + γ α + γ x, where 3.3) c = α + β + δ in α + γ), c 2 = β + γ + δ out α + γ). Proof. Only the form of the partial differential equation in 3.2) requires justification. The following recursive relation connecting the limiting probabilities p ij ) was established in Bollobás et al. 23), 3.4) p ij =c i + δ in )p i,j c i + δ in )p ij + c 2 j + δ out )p i,j c 2 j + δ out ) + α i =, j = ) + γ i =, j = ) α + γ α + γ for i, j =,, 2,..., with the understanding that any p with a negative subscript is equal to zero. Rearranging the terms, multiplying both sides by x i y j and summing up, we see that for < x, y <, 3.5) Since i= j= c δ in +c 2 δ out + + c i + c 2 j)x i y j p ij = α α + γ y + i= j= γ α + γ x + c i + δ in )x i y j p i,j + c 2 j + δ out )x i y j p i,j. ϕ x x, y) = i= j= ix i y j p ij, we can rearrange the terms in 3.5) to obtain 3.2). ϕ y x, y) = i= j= i= j= jx i y j p ij, The next theorem gives an explicit formula for the joint generating function ϕ in 3.). Theorem. Let 3.6) a = c 2 /c, where c and c 2 are given in 3.3). Then for x, y, ϕx, y) = α α + γ c y z +/c ) x + x)z ) δ in y + y)z a ) δ 3.7) out+) dz + γ α + γ c x z +/c ) x + x)z ) δ in +) y + y)z a ) δ out dz. Proof. The partial differential equation in 3.2) is a linear equation of the form 2), p.6 in Jones 97), and to solve it we follow the procedure suggested ibid.. Specifically, we write the equation 3.2) in the form 3.8) ax, y) ϕ x + bx, y) ϕ y = cx, y)ϕ + dx, y), with ax, y) = c x x), bx, y) = c 2 y y),

5 REGULAR VARIATION OF IN- AND OUT-DEGREES 5 cx, y) = c δ in x + c 2 δ out y ρ, dx, y) = αα + γ) y + γα + γ) x, where ρ = c δ in + c 2 δ out +. Consider the family of characteristic curves for the differential equation 3.8) defined by the ordinary differential equation dy bx, y) = dx ax, y). It is elementary to check that the characteristic curves form a one-parameter family, {C θ, θ > }, with the curve C θ given by 3.9) y = + θx a x) a, < x <. Along each characteristic curve C θ the function ux) = ϕ x, yx) ), < x <, satisfies the ordinary differential equation 3.) where du cx, y)u + dx, y) = = uψ x) + ψ 2 x), dx ax, y) ψ x) = c δ in x + c 2 δ out + θx a x) a) ρ c x x) ψ 2 x) = γx + α + θx a x) a). α + γ)c x x) Let H be a function satisfying 3.) H x) = ψ x), < x <, and define It follows from 3.) that Ax) = ux)e Hx), < x <. 3.2) A x) = ψ 2 x)e Hx), < x <. We compute the function u by solving the differential equations 3.) and 3.2). To solve 3.), write it first in the form H x) = δ in x ρ/c x x) + c 2 δ out /c + θx a x) a x x). It is elementary to check by differentiation that + θx a x) a x x) dx = log x) + a log x a + θ x) a) + C with C R. Therefore, for < x <, 3.3) Hx) = c log x) ρc log x + δ out log x a + θ x) a) + C, implying that A x) =e C γx + α + θx a x) a) x) /c x ρ/c x a + θ x) a) δ out α + γ)c x x) = e C α + γ)c γ x) +/c ) x δ in+/c + θx a x) a) δ out,

6 6 G. SAMORODNITSKY, S. RESNICK, D. TOWSLEY, R. DAVIS, A. WILLIS, AND P. WAN We can now write 3.4) + e C α + γ)c x) +/c ) x δ in +/c + θx a x) a) +δ out). x Ax) =e C γ t) +/c) t δ in+/c + θt a t) a) δ out dt α + γ)c + e C α x t) +/c) t δ in +/c + θt a t) a) +δ out) dt + C2 α + γ)c with C 2 R. Using 3.3) and 3.4) we obtain the following expression for the the function ux) = ϕ x, yx) ), < x < along the characteristic curve C θ. ux) =Ax)e Hx) = x γ α + γ c x)/c x ρ/c + θx a x) a) δ out t) +/c ) t δ in+/c + θt a t) a) δ out dt + α α + γ c x)/c x ρ/c + θx a x) a) δ out x t) +/c ) t δ in +/c + θt a t) a) +δ out) dt + C 3 x) /c x ρ/c + θx a x) a) δ out with C 3 = C 3 θ) R. Multiply both sides of this equation by x aδout+ρ/c and let x. Using the fact that the generating function is bounded, we see that C 3 =. We can now obtain an expression for the joint generating function ϕ everywhere in, ) 2 by noticing that a point x, y), < x, y <, lies on the characteristic curve C θ with θ = y)/y ) a. x)/x We conclude that ϕx, y) = γ α + γ c x)/c x ρ/c + θx a x) a) δ out ) δout x t) +/c) t δ in+/c y)/y + ) a t a t) a dt x)/x + α α + γ c x)/c x ρ/c + θx a x) a) δ out ) +δout) t) +/c) t δ in +/c y)/y + ) a t a t) a dt. x)/x x Changing the variable in both integrals to z = x t) t x) and rearranging the terms, we obtain 3.7) for < x, y <. Now we can extend this formula for the joint generating function to the boundary of the square [, ] 2 by continuity.

7 REGULAR VARIATION OF IN- AND OUT-DEGREES 7 4. Joint regular variation of the distribution of in-degree and out-degree In this section we analyze the explicit form 3.7) of the joint generating function of the limiting distribution of in-degree and out-degree obtained in Theorem to prove the nonstandard joint regular variation of in-degree and out-degree. We also obtain an expression for the of the tail measure. We start by writing the joint generating function in 3.7) as 4.) ϕx, y) = γ α + γ xϕ x, y) + α α + γ yϕ 2x, y), with 4.2) 4.3) ϕ x, y) =c ϕ 2 x, y) =c z +/c ) x + x)z ) δ in +) y + y)z a ) δ out dz, z +/c ) x + x)z ) δ in y + y)z a ) δ out+) dz for x, y. Each of these functions ϕ i is a mixture of a product of negative binomial generating functions of possibly fractional order. On some probability space we can find nonnegative integervalued random variables X j, Y j, j =, 2 such that ϕ j x, y) = E x X j y Y j ), x, y, j =, 2. If I, O) is a random vector with generating function given in 4.), then we can represent in distribution I, O) as 4.4) I, ) d = B + X, Y ) + B)X 2, + Y 2 ), where B is a Bernoulli switching variable independent of X j, Y j, j =, 2 with P [B = ] = P [B = ] = γ α + γ. Theorem 2 below shows that each of the random vectors X j, Y j ), j =, 2, has a bivariate regularly varying distribution. The decomposition 4.) then gives the joint regular variation of in-degree and out-degree. Theorem 2. Let α in and α out be given by.4). Then for each j =, 2 there is a Radon measure V j on [, ] 2 \ {} such that 4.5) hp h /α in ) X j, h /αout ) Y j ) ) v Vj ), as h vaguely in [, ] 2 \ {}. Furthermore, V and V 2 concentrate on, ) 2 where they have Lebesgue densities given, respectively, by f x, y) =c Γδin + )Γδ out ) ) x δ in y δout z 2+/c +δ in +aδ 4.6) out) e x/z+y/za) dz and 4.7) f 2 x, y) =c Γδin )Γδ out + ) ) x δ in y δout z +a+/c +δ in +aδ out) e x/z+y/za) dz. Therefore, a random vector I, O) with the joint probabilities given by p ij ) in.2) satisfies h /α 4.8) hp in ) I, h /αout ) O ) ) v γ α + γ V ) + α α + γ V 2 ) as h vaguely in [, ] 2 \ {}.

8 8 G. SAMORODNITSKY, S. RESNICK, D. TOWSLEY, R. DAVIS, A. WILLIS, AND P. WAN Proof. It is enough to prove 4.5) and 4.6). We treat the case j =. The case j = 2 is analogous and is omitted. Let T δ p) be a negative binomial integer valued random variable with parameters δ > and p, ). We abbreviate this as NBδ, p). The generating function of T δ p) is Es T δp) = s + s)p ) δ. It is well known and elementary to prove by switching to Laplace transforms that as p, pt δ p) Γ δ where Γ δ is a Gamma random variable with distribution F δ x) and F δ x) = e x x δ, x >. Γδ) Now suppose {T δ p), p, )} and { T δ2 p), p, )} are two independent families of NB random variables. We can represent the mixture in 4.2) as X, Y ) = T δin +Z ), T δout Z a ) ), where Z is a Pareto random variable on [, ) with index c, independent of the NB random variables. To ease writing, we set δ = δ in + and δ 2 = δ out. Define the measure ν c on, ] by ν c x, ] = x c, x >. We now claim, as h, in M +, ] [, ] 2 ), [ Z 4.9) hp h c, Z T δ Z ), Z a Tδ2 Z a ) )) ] v νc P [Γ δ ] P [Γ δ2 ]. To prove this, suppose x > and let gu, v) be a function bounded and continuous on [, ] 2 and it suffices to show, 4.) he where Γ δ Γ δ2. Observe as p, [Z/h c >x] g Z T δ Z ), Z a T δ2 Z a ) )) x c E gγδ, Γ δ2 ) ) E g pt δ p), p a Tδ2 p a ) )) E gγ δ, Γ δ2 ) ) and so, given ɛ >, there exists η > such that 4.) sup E g pt δ p), p a Tδ2 p a ) )) E gγ δ, Γ δ2 ) ) < ɛ. p<η Bound the difference between the LHS and RHS of 4.) by he [Z/h c >x] g Z T δ Z ), Z a Tδ2 Z a ) )) he [Z/h c >x] E gγ δ, Γ δ2 ) ) + he [Z/h c >x] E gγ δ, Γ δ2 ) ) x c E gγδ, Γ δ2 ) ) = A + B, where B = o) and is henceforth neglected. Write E Z ) = E Z) for the conditional expectation and bound A by 4.2) E h [Z/h c >x] E Z g Z T δ Z ), Z a Tδ2 Z a ) ) E gγ δ, Γ δ2 ) ) ). As soon as h is large enough so that h c x < η, 4.2) bounded by E h [Z/h c >x] ) ɛ ɛx c.

9 REGULAR VARIATION OF IN- AND OUT-DEGREES 9 Let ɛ and we have verified 4.) and therefore 4.9). The next step is to apply a mapping to the convergence in 4.9). Define χ :, ] [, ] 2, ] [, ] 2 by χ x, y, y 2 ) ) = x, xy, x a y 2 ) ). This transformation satisfies the compactness condition in Resnick, 27, Proposition 5.5, page 4) or the bounded away condition in Lindskog et al., 23, Section 2.2). Following the product discussion of Example 3.3 in Lindskog et al. 23) or Maulik et al., 22, Corollary 2., page 682), we apply χ to the convergence in 4.9) which yields in M +, ] [, ] 2 ), as h, [ Z 4.3) hp h c, T δ Z ) h c, T δ2 Z a )) ) ] ) v h c ν 2 c P [Γ δ ] P [Γ δ2 ] χ ), where we used the fact that ac = c 2. We must extract from 4.3) the desired convergence in M + [, ] 2 \ {}), [ Tδ Z ) 4.4) hp, T δ2 Z a ) ) ] ) v ν c P [Γ δ ] P [Γ δ2 ] χ, ] )). h c h c 2 Assuming 4.4), we evaluate the convergence in 4.4) on a set of the form x, ] y, ] for x >, y > to get [ Tδ Z ) hp h c > x, T δ2 Z a ) ] h c > y ν 2 c du)f δ dv)f δ2 dw) = u,v,w):uv>x,u a w>y F δ x/u) F δ y/u a )ν c du). The right side is the limit measure of the distribution of X, Y ) evaluated on x, ] y, ] for x >, y >. Differentiating first with respect to x and then with respect to y yields after some algebra the limit measure s f x, y) in 4.6). To prove that 4.4) can be obtained from 4.3), we need the following result about negative binomial random variables whose proof is deferred. Suppose T δ p) is NBδ, p). For any δ >, k =, 2,... there is cδ, k), ) such that 4.5) E T δ p) ) k cδ, k)p k for all < p <. Suppose g : [, ] 2 \ {} [, ) is continuous, bounded by g with compact support in [, ɛ] [, ɛ]) c for some ɛ >. Using a Slutsky style argument, 4.3) implies 4.4) if = lim lim sup he [Z/h c x] g T δ Z )/h c, T δ2 Z a )/h c ) 2 heg T δ Z )/h c, T δ2 Z a )/h c ) 2 x h = lim lim sup he [Z/h c x] g T δ Z )/h c, T δ2 Z a )/h c ) 2. x h Keeping in mind the support of g, the previous expectation is bounded by g hp [ Z h c x, [T δ Z )/h c > ɛ] [T δ2 Z a )/h c 2 > ɛ] ]. Bounding the probability of the union by the sum of two probabilities, we show how to deal with the first since the second is analogous. Then neglecting the factor g we have hp [ [ ]) Z h c x,t δ Z )/h c > ɛ] = he [Z h c x] P T δ Z )/h c > ɛ Z and picking k > c and using 4.5) we get the bound he [Z h c x] cδ, k)z/h c ) k ɛ k

10 G. SAMORODNITSKY, S. RESNICK, D. TOWSLEY, R. DAVIS, A. WILLIS, AND P. WAN x =cδ, k)ɛ k u k hp [Z/h c du] and by Karamata s theorem or direct calculation, as h we get the limit =cδ, k)ɛ k c k c x k c which converges to as x as desired. Finally we verify 4.5). Begin with δ = so T p) is geometric with success probability p. It is enough to prove that for some constant Ck), ), k ) 4.6) E T p) j) j= Differentiating the generating function, we obtain, k ) 4.7) E T p) j) j= Ck)p k. = k! p) k p k k!p k. Next, for integer δ =, 2,..., and independent copies T, p), T,2 p),..., of T p) random variables, we have E T δ p) ) k =E T, p) + T,2 p) T,δ p) ) k and applying the c r inequality in Loève, 977, p. 77) gives Finally, for any δ >, δ k E T p) k) δ k Ck)p k. proving 4.5) and completing the proof. E T δ p) ) k E T δ p) ) k δ k Ck)p k, Remark 3. A change of variables in the integrals in 4.6) and 4.7) shows that the random vector I, O) is bivariate regular varying with marginal exponents α in and α out accordingly, and with tail measure having of the form fx, y) = c 4.8) +c for < x, y <. γ/α + γ) Γδ in + )Γδ out ) xδ in y δout α/α + γ) Γδ in )Γδ out + ) xδ in y δout t /c +δ in +aδ out e xt+yta) dt t a +/c +δ in +aδ out e xt+yta) dt The powers of h used in the scaling functions in 4.5) are, in general, not equal and thus the regular variation in 4.8) is non-standard. However, as the scaling functions are pure powers, the vector I a, O) is standard regularly varying. One can then transform to the familiar polar coordinates. We consider the alternative transformation I a, O) O/I a, I) which gives the immediate conclusion by Theorem 2 that out-degree is roughly proportional to a power of the in-degree when either degree is large. We calculate the limiting of ratio R := O/I a given I is large.

11 REGULAR VARIATION OF IN- AND OUT-DEGREES Corollary 4. As m, the conditional distribution of the ratio O/I a given that I > m converges to a distribution F R on, ) with 4.9) f R r) = θ r δout I r) + θ 2 r δout I 2 r), r >, where and with I r) = t /c +δ in +aδ out e t+rta) dt, I 2 r) = θ = γ Γδ in + )Γδ out )D, θ 2 = D = γ Γ/c + δ in + ) Γδ in + ) Proof. Let h m = m αin. Notice that for every λ >, ) P O/I a h m P h /α in ) m λ I > m = t a +/c +δ in +aδ out e t+rta) dt, α Γδ in )Γδ out + )D, + α Γ/c + δ in ) Γδ in ). I >, h /αout ) m O/ h /α in ) m I ) ) a λ h m P h /α in ) m I > ) γv + αv 2 ) { x, y) : x >, y/x a λ }) γv + αv 2 ) { x, y) : x > }) as m by Theorem 2. The numerator of this ratio can be rewritten as fx, y) dxdy, x>, y/x a λ and the same can be done to the denominator in this ratio. Using the f in 4.8) and performing an elementary change of variable shows that the ratio can be written in the form λ with f R as in 4.9). This completes the proof. f R r) dr, 4.. Plots, simulation, iteration. For fixed values of α in, α out ), we investigate how the dependence structure of I, O) in 4.4) depends on the remaining parameters. We generate plots of f R r) and the spectral for various values of the input parameters using the explicit formulae and compare such plots to histograms obtained by network simulation and iteration of 3.4) The distribution of R. We fix two values of α in, α out ), namely 7, 5) and 5, 7), and then plot f R r) for several values of the remaining parameters to see the variety of possible shapes. Since α + β + γ =, fixing values for α, γ) also determines β and because of.4), assuming values for α in, α out, α, γ determine values for δ in, δ out. The plots are in Figure. Additionally, we employ two numerical strategies based on the convergence of the conditional distribution of O/I a given I > m as m. Strategy simulates a network of 6 nodes using software provided by James Atwood University of Massachusetts, Amherst) and then computes the histogram of O/I a for nodes whose in-degree I exceeds some large threshold m. For the network simulation illustration, we chose m to be the 99.95% quantile of the in-degrees. Strategy 2 computes p ij on a grid i, j) using the recursion given in 3.4) and then estimates the of O/I a using only the grid points with i larger than m, the m chosen to be the same value as used for the network simulation.

12 2 G. SAMORODNITSKY, S. RESNICK, D. TOWSLEY, R. DAVIS, A. WILLIS, AND P. WAN Density of R at r for α in =7, α out =5 Density of R at r for α in =5, α out =7 Density of R=O/I^a at r α=., γ=.3 α=., γ=.2 α=., γ=.4 α=., γ=.6 α=.35, γ=.35 α=.5, γ=.3 Density of R=O/I^a at r α=., γ=. α=., γ=.2 α=., γ=.4 α=., γ=.6 α=.35, γ=.35 α=.5, γ= r r Figure. The f R r) for α in, α out ) = 7, 5) left) and α in, α out ) = 5, 7) right) for various values of α, γ. We observe from Figure that the mode of f R r) can drift away from the origin depending on parameter values. So we transform R using the arctan function which gives all plots the same compact support [, π/2], instead of an infinite domain as in Figure. We compare the of R with the histogram based on network simulation and the approximation provided by iteration across varying sets of parameter values. The of arctan R with the plots from the alternative strategies based on simulation and iteration are displayed in Figure 2 for various choices of δ in, δ out ), with α = β =.5 and γ =. For these parameter choices, the plots of the theoretical with those resulting from network simulation and probability iteration are in good agreement Density of the angular measure. A traditional way to describe the asymptotic dependence structure of a standardized heavy tailed vector is by using the angular measure. We transform the standardized vector I a, O) arctano/i a ), O 2 + I 2a) to polar coordinates and then the distribution of arctano/i a ) given O 2 + I 2a > m, converges as m to the distribution to a random variable Θ. The distribution of Θ is called the angular measure. The of Θ can be calculated from Theorem 2 in a similar fashion as in Corollory 4 and is given by f Θ θ) γ δ in cos θ) δin + α δ out cos θ) δin a + a sin θ) δout a sin θ) δout t c +δ in+aδ out e tcos θ) a t a sin θ dt t a +c +δ in+aδ out e tcos θ) a t a sin θ dt. Two approximations for the spectral using network simulation and numerical iteration of the p ij are obtained in the same way as in Section 4... Using the same sets of parameters values as in Figure 2, we overlay the approximations with the theoretical

13 REGULAR VARIATION OF IN- AND OUT-DEGREES 3 alpha_in= 4, alpha_out= 4 alpha_in= 4.5, alpha_out= arctanr) arctanr) alpha_in= 4, alpha_out= 5 alpha_in= 4.5, alpha_out= arctanr) arctanr) alpha_in= 4, alpha_out= 6 alpha_in= 4.5, alpha_out= arctanr) arctanr) simulated numerical theoretical Figure 2. Comparison of the true with the estimated densities of arctan R over various values of α in, α out ). in Figure 3. The truncation level was the 99.95% percentile of O 2 + I 2a. The agreement between the theoretical and estimated densities is quite good across the range of parameter values used. The main difference between Figures 2 and 3 is the choice of conditioning set. In the first, I a was conditioned to be large, while in the second the sum of squares of the in- and out-degrees I 2a + O 2 ) was conditioned to be large. Since the latter conditioning set is bigger and allows for the case that the in-degree is small relative to the out-degree, the function in a neighborhood will have less weight in Figure 3 than Figure 2. References B. Bollobás, C. Borgs, J. Chayes and O. Riordan 23): Directed scale-free graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms Baltimore, 23). ACM, New York, pp F. Jones 97): Partial Differential Equations. Springer-Verlag, New York.

14 4 G. SAMORODNITSKY, S. RESNICK, D. TOWSLEY, R. DAVIS, A. WILLIS, AND P. WAN alpha_in= 4, alpha_out= 4 alpha_in= 4.5, alpha_out= Theta Theta alpha_in= 4, alpha_out= 5 alpha_in= 4.5, alpha_out= Theta Theta alpha_in= 4, alpha_out= 6 alpha_in= 4.5, alpha_out= Theta Theta simulated numerical theoretical Figure 3. Comparison of the true angular with estimates for various values of α in, α out ). P. Krapivsky and S. Redner 2): Organization of growing random networks. Physical Review E 63:6623: 4. F. Lindskog, S. Resnick and J. Roy 23): Regularly Varying Measures on Metric Spaces: Hidden Regular Variation and Hidden Jumps. Technical report, School of ORIE, Cornell University. Preprint. Available at: M. Loève 977): Probability Theory, volume. Springer-Verlag, New York. K. Maulik, S. Resnick and H. Rootzen 22): Asymptotic independence and a network traffic model. Journal of Applied Probability 39: S. Resnick 27): Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.

15 REGULAR VARIATION OF IN- AND OUT-DEGREES 5 School of Operations Research and Information Engineering, and Department of Statistical Science, Cornell University, Ithaca, NY address: gs8@cornell.edu School of Operations Research and Information Engineering, Cornell University, Ithaca, NY address: sir@cornell.edu Department of Computer Science, University of Massachusetts, Amherst, MA 3 address: towsley@cs.umass.edu Department of Statistics, Columbia University, New York, NY 27 address: rdavis@stat.columbia.edu Department of Statistical Science, Cornell University, Ithaca, NY address: adw96@cornell.edu Department of Statistics, Columbia University, New York, NY 27 address: phyllis@stat.columbia.edu