LIMIT DISTRIBUTIONS OF VERTEX DEGREES IN A CONDITIONAL CONFIGURATION GRAPH

The configuration graph where vertex degrees are independent identically distributed random variables is often used for modeling of complex networks such as the Internet. We consider a random graph consisting of N vertices. The random variables η1, . . . , ηN are equal to the degrees of vertices with the numbers 1, . . . , N. The probability P{ηi = k}, i = 1, . . . , N, k = 1, 2, . . . , is equivalent to h(k)/k as k → ∞, where h(x) is a slowly varying function integrable in any finite interval, τ > 1. We obtain the limit distributions of the maximum vertex degree and the number of vertices with a given degree under the condition that the sum of degrees is equal to n and N,n→∞.


Introduction
The study of random graphs has been causing growing interest in connection with the wide use of these models for the description of complex networks (see, e. g. [3,6,11].Such models can be used to adequate by describe the topology of transport, electricity, social, telecommunication networks and global Internet.Observations on real networks showed that their topology can be described by random graphs with vertex degrees being independent identically distributed random variables with power-law distribution.In [3] it was suggested that for large k the number of vertices with the degree k is proportional to k −τ , where τ > 1.That is why in [11] it was suggested that the distribution of the vertex degree η is P{η k} = h(k)k −τ +1 , k = 1, 2, . . ., (1) where h(k) is a slowly varying function.
We consider a random graph consisting of N +1 vertices.Let random variables η 1 , . . ., η N be equal to the degrees of vertices with the numbers 1, . . ., N. Each vertex is assigned a certain degree in accordance with the distribution (1).The vertex degree is the number of stubs (or semiedges) that are numbered in an arbitrary order.Stubs are vertex edges for which adjacent nodes are not yet determined.The vertex 0 is auxiliary and has degree 0 if the sum of all other vertices is even, else the degree is 1.It is clear that we need to use the auxiliary vertex 0 for the sum of degrees to be even.The graph is constructed by joining all the stubs pairwise equiprobably to form links.
There are many papers where the results describing the limit behaviour of different random graph characteristics were obtained.The authors of [11] were sure (without proof) that the function h(k) in (1) does not influence the limit results, and that to study the configuration graph one can replace h(k) with the constant 1.In our work we will show that the role of the slowly varying function h(k) is more complicated.
We consider the subset of random graphs under the condition η 1 + • • • + η N = n.Such conditional graphs can be useful for modeling of networks for which we can estimate the number of communications.They are useful also for studying networks without conditions on the number of links by averaging the results of conditional graphs with respect to the distribution of the sum of degrees.Conditional random graphs were first analyzed in [9], where h(k) ≡ 1. Obviosly, the limit behaviour of a random graph depends on the degree structure.In [9] the limit distributions were obtained for the maximum vertex degree and the number of vertices of a given degree as N and n tend to infinity in such a way that 1 < n/N < ζ(τ ), where ζ(τ ) is the value of the Rimman's zeta-function at the point τ.For other zones of parameters analogous results were obtained in papers [7,8,10].
Here we extend the results on the maximum vertex degree and the number of vertices of a given degree to the configuration graphs with degree distribution (1), where h(k) is not constant.In the following section the main results are formulated, then auxiliary statements are proved.And the last section contains proofs of the main results.

Main Results
In the paper we assume that the distributions of node degrees are where i = 1, . . ., N, k = 1, 2, . . ., τ > 1, h(k) is a slowly varying function integrable in any finite interval and We denote also by ξ 1 , . . ., ξ N the auxiliary independent identically distributed random variables such that where i = 1, . . ., N, k = 1, 2, . . .and the parameter λ = λ(n, N ) belongs to the interval (0, 1).From ( 2)-( 4) we obtain Let the parameter λ = λ(n, N ) of the distribution (4) be determined by the relation We denote by η (N ) and µ r the maximum vertex degree and the number of vertices with the degree r, respectively.We get the following results.
where γ is a nonnegative constant.Then We introduce the conditions: where ε is some sufficiently small positive constant.
where γ is a positive constant.Then Theorem 3. Let n, N → ∞ and one of the following conditions is fulfilled Then for a nonnegative integer k uniformly with respect to u = (k − N p r (λ))/(σ rr √ N ) lies in any fixed finite interval where uniformly with respect to (k −N p r (λ))/ N p r (λ) lies in any fixed finite interval.This assertion remains true for r → ∞ if 1 < n/N < Σ(1, τ − 1)/Σ(1, τ ) under one of the following conditions: Remark.In [2], a case of these theorems under the condition

Auxiliary results
We prove some auxiliary statements (Lemmas 1-6), and use them to prove Theorems 1-5.The technique of obtaining these theorems is based on the generalized allocation scheme suggested by V. F. Kolchin [5].It is readily seen that for our subset of random graphs Therefore, the conditions of the generalized allocation scheme are valid (see [5]).Let ξ N be two sets of independent identically distributed random variables such that P{ ξ(r) From ( 2)-( 6) we can deduce the next lemma.
Lemma 1.Let N, n → ∞.The next assertions are true: Let us consider the limit behaviour of ζ N .
Lemma 2. Under the conditions of Theorems 1-4 uniformly with respect to integers k such that (k − n)/(σ √ N ) lies in any fixed finite interval.
Proof.Let ϕ(t) be the characteristic function of the random variable ξ 1 .Then Further we will need an explicit form of the third derivative of ln ϕ(t).From (4) it is not hard to get that Let n/N → 1.From ( 2)-( 4) it is easy to obtain that Let ϕ N (t) be the characteristic function of the random variable Then from Lemma 1, (12) and (13) follows relation Let n/N Σ(1, τ − 1)/Σ(1, τ ).It is well known (see e.g.[4]) that the slowly varying function integrable in any finite interval has the following properties: Using the properties (15) and Lemma 1 we can deduce that for j = 0, 1, 2, 3 |Σ(e it λ, τ where Φ(x, s, a) is the Lerch transcendent function: It is well known (see e.g.[1]) that for the Lerch transcendent function the following properties are valid: From ( 3), ( 5), ( 11), ( 16)-( 19) it is not hard to get that The next expression is valid for a sufficiently small t: where Using ( 13), ( 20)-( 23) and (A1)-(A5) we get (14).
According to the inversion formula we represent the probability P{ζ N = k} as the integral the difference can be rewritten as the sum of four integrals: R = I 1 + I 2 + I 3 + I 4 , where the positive constants A and a will be chosen later.Lemma 2 will be proved if we show that by choosing sufficiently large n, N the difference R can be made arbitrarily small.From (14) we get that I 1 → 0.Moreover, and the integral I 4 is as small as desired, provided that A is large enough.
Let us estimate the integral I 2 .From ( 23) and ( 12) we obtain that for sufficiently small A<|t| e −C11t 2 dt, and the integral I 2 is small for large enough A. From (13), ( 17) and ( 20) we obtain the same estimate where From (13) we get that where In the integration domains of the integral I 2 t/(σ √ N ) → 0, then from Lemma 1, (17), ( 20) and ( 27) we obtain: It follows that for small enough a |ϕ N (t)| exp{−C 12 t 2 }.Therefore and the integral I 2 is as small as desired, provided that A is large enough.To estimate the integral I 2 we expand the function Σ(λz, τ ), where z = e it/(σ √ N ) in the Taylor series near the point z = 1.Then Using (27), the conditions (A1) -(A5) and the inequality we can show that Let us consider the integral I 3 .For ε |t| π the inequality is valid.Then under the condition that n/N → 1 it can be shown that From this and Lemma 1 we get that for ε Therefore using relations (12) and (25) it is not hard to see that Let n/N → Σ(1, τ − 1)/Σ(1, τ ).From the conditions (A1) -(A5), ( 21) and (28) we get that Thus Lemma 2 is proved.
Let ϕ r (t) be the characteristic function of the random variable (ζ Then uniformly with respect to t in any fixed finite interval the next conclusions are true where γ is a positive constant, parameters τ, N, n are determined by the conditions (A1) -(A5) then ϕ r (t) → e −t 2 /2 .
Proof.From ( 7) and (10) it is easy to see that ϕ r (t) = (30) It is not hard to get that
. We divide I 2 into integrals I 2 and I 2 , where the integration domains are {t : A < |t| aB(λ, τ )σ