Eötvös Loránd University, Budapest. 13 May 2005

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1 A NEW CLASS OF SCALE FREE RANDOM GRAPHS Zsolt Katona and Tamás F Móri Eötvös Loránd University, Budapest 13 May 005 Consider the following modification of the Barabási Albert random graph At every step a new vertex is added to the graph It is connected to the old vertices randomly, with probabilities proportional to the degree of the other vertex, and independently of each other We show that the proportion of vertices of degree decreases at the rate of 3 Furthermore, we prove a strong law of large numbers for the maximum degree 1 The model The examination of complex networs, such as the World Wide Web, led to the emerging role of random networs Barabási and Albert [1] pointed out that many complex real world networs cannot be adequately described by the classical Erdős- Rényi random graph model, where the possible edges are included independently, with the same probability p In this model the degree distribution is approximately Poisson with parameter np, while in real networs for example the WWW power law degree distributions have been observed That is, the ratio of vertices with degree is P c γ with some constant c and the γ exponent around 3 Here the parameter is independent of n Barabási and Albert [1] proposed a random graph model where they start with a single edge and at every step we add a new vertex and connect it to m of the old vertices by m edges The old vertices are chosen randomly with probability proportional to their degrees It was shown by Bollobás, Riordan, Spencer and Tusnády [] that the proportion of vertices of degree decreases at the rate 3 as found in the empirical studies In the most general model Cooper and Frieze [4] allow edges to be established between old vertices and the number of edges established in a step is determined randomly by a given distribution They show that the degree distribution is power-law with exponent γ > However, they do not consider the case when edges from a new vertex are added independently Consider the following modification of the Barabási Albert random graph At every step a new vertex is added to the graph It is connected to the old 1991 Mathematics Subject Classification 05C80, 60C05 Key words and phrases random graph, scale-free disribution Research supported by the Hungarian National Foundation for Scientific Research, Grant No T Typeset by AMS-TEX

2 ZSOLT KATONA AND TAMÁS F MÓRI vertices randomly, with probabilities proportional to the degree of the other vertex, and independently of each other Since we are interested in asymptotic analysis, the initial configuration can be arbitrary, but, for the sae of simplicity, let us start from the very simple graph consisting of two points and the edge between them Let us number the vertices in the order of their creation; thus the vertex set of the graph after n steps is {0, 1,, n} Let X[n, ] and Y [n, ] denote the number of vertices of degree exactly and at least, resp, after n steps Thus, X[n, 0] + X[n, 1] + n+1 Let 1 X[n, ] 1 Y [n, ], the sum of degrees, or equivalently, the double of the number of edges At the nth step an old vertex of degree is connected to the new one with probability λ/s This quantity remains below 1, provided the proportionality coefficient λ is less than, which will therefore be assumed in the sequel Let F n denote the σ-field generated by the first n steps Let [n, ] be the number of new edges into the set of old vertices of degree at the nth step, and let n 1 [n, ] be the total number of new edges Obviously, the conditional distribution of [n+1, ] with respect to F n is binomial with parameters X[n, ] and λ, hence E n+1 Fn λ The aim of the present paper is to study some asymptotic properties of this random graph as the number of vertices tends to infinity In Section we prove a strong law of large numbers for the maximum degree, and in Section 3 it will be shown that the proportion of vertices of degree converges as to a constant, which, as a function of, decreases in the order of 3 as The maximum degree First we deal with the asymptotics of Theorem 1 λn + o n 1/+ε, ε > 0 Proof With 1 1 define ζ n n j1 j λ / nλ Then ζ n, F n is a square integrable martingale, and the increasing process associated with ζ n by the Doob decomposition is A n Var j F j 1 j j 1 X[j, ] λ S j 1 λ S j nλ It is well nown [7], Proposition VII--4 that ζ n o A 1/+ε n ae on the event A n This completes the proof Let us turn to M n max{ : X[n, ] > 0}, the maximum degree after n steps Theorem We have lim n M n / n µ with probability 1, where the limit µ differs from zero with positive probability Proof We follow the same lines as in the proof of Theorem 31 of [6] Let W [n, j] denote the degree of vertex j after the nth step, with the initial values W [n, j] 0 for n < j, W [1, 0] W [1, 1] 1, W [j, j] j Then M n

3 A NEW CLASS OF SCALE FREE RANDOM GRAPHS 3 max{w [n, j] : j 0} Let us introduce the normalizing terms c[n, ] For n, with probability 1 we have c[n, ] exp λ S i, n 1, 1 S i +λ 1 + λ S i 1 + o1 Si 1 Since o i 1/+ε, we obtain that the exponent in 1 differs from S i λi log n only by a term converging with probability 1 Thus c[n, ] γ n /, with an appropriate positive random variable γ We clearly have that E W [n+1, j] F n W [n, j] + λ W [n, j] W [n, j] +λ Hence Z[n; j, 1] c[n, 1] W [n, j], Fn, n max{j, 1} is either a positive martingale or constant zero, thus it converges as to some ζ j To estimate the moments of ζ j, consider W [n, j]+ 1 Z[n; j, ] c[n, ] Since W [n+1, j] W [n, j] is equal to either 1 or 0, we can write W [n+1, j]+ 1 W [n, j]+ 1 + W [n+1, j] W [n, j] W [n, j]+ 1 1 W [n, j]+ 1 W [n+1, j] W [n, j] 1 + W [n, j] By taing conditional expectation with respect to F n we can see that W [n+1, j]+ 1 W [n, j]+ 1 E F n 1 + λ, hence Z[n; j, ], Fn, n max{j, 1} is also a convergent martingale Since c[n, 1] c[n, ], we can majorize Z[n; j, 1] by! Z[n; j, ]

4 4 ZSOLT KATONA AND TAMÁS F MÓRI Now, c[n, 1]M n max{z[n; j, 1] : 0 j n}, being the maximum of an increasing number of nonnegative martingales, is a submartingale The proof can be completed by showing that this submartingale is bounded in L, for some 1 Let us start from the estimation E c[n, 1]M n E max{z[n; j, 1] : 0 j n}!!ez[1; 0, ] +! EZ[j; j, ]! +! E c[j, ] j1 j1 EZ[n; j, ] j0 j + 1 Here j + 1 E c[j, ] [ j + 1 E E c[j, ] F j 1] [ j + 1 E c[j, ]E F j 1] 3 Next we show that, independently of j, j + 1 π+ 1 E F j 1 E, 4 where π stands for a Poissonλ random variable Remembering that j [j, 0]+ + [j, j 1], we can write j + 1 l 0+ +l j [j, 0] [j, j 1] 1 5 l 0 The binomial coefficients on the right-hand side are conditionally independent The conditional distribution of each [j, i] is binomial Let ξ be a Binomialn, p random variable and η a Poisson one, with the same expectation Then E ξ l n p l npl E l l! l j 1 η l Thus, if all random variables [j, i] on the right-hand side of 5 are replaced by conditionally independent Poisson variables, the conditional expectation cannot decrease Hence 4 follows By and 3, for the L -boundedness of the submartingale c[n, 1]M n it is sufficient to chec that j1 Ec[j, ] < The convergence of j1 c[j, ] when > is clear from the asymptotics obtained for c[n, ], but the integrability does not follow immediately Let 8 and N max{n : > 4λn} Then for j > N we have j 1 c[j, 8] 1 8λ S i +8λ j 1 in+1 1 n+ l j N +1N +, jj +1

5 A NEW CLASS OF SCALE FREE RANDOM GRAPHS 5 but this is obviously true even for j N Thus, for the finiteness of j1 Ec[j, 8] it is sufficient to prove that EN < By the usual large deviation arguments we have P N n P > 4λn P / > 4 λn E / 4 λn 6 Thus, we have to estimate the moment generating function of With 1 1 we can write / n i, and E Sn/ E E Sn/ F E S/ E n F E S / 1 + λj X[,j] S E S / exp { j1 j1 λjx[, j] S } e λ E S / Therefore E Sn/ e λn, which, combined with 6, implies that P N n e/4 λn Thus EN <, indeed 3 Degree distribution In this section we prove that the degree distribution of our graph stabilizes almost surely, as n, around a power law with exponent 3 First we show that stochastic process of new vertices approaches a stationary regime More precisely, the random variables n are asymptotically independent and asymptotically Poissonλ distributed By LeCam s theorem on Poisson approximation [5] we have P n+1 λ Fn 0! e λ n λ X[n, ] λ M n o Similarly, the conditional distribution of [n+1, ] with respect to F n is also X[n, ] asymptotically Poisson with parameter λ Theorem 31 For every 0, 1, The proportion of vertices of degree converges as as n : X[n, ] P lim n n+1 x 1, where x 0 p 0, x +1+ ii+1p i, p λ! e λ Remar x λ+λ 3 as

6 6 ZSOLT KATONA AND TAMÁS F MÓRI Proof First we will show by induction over that Y [n, ] P lim n n+1 y 1, where the constants y satisfy the following recursion: y 0 1, y 1 +1 y 1 + q +1, q p + p The sequence X[n, 0] obeys the strong law of large numbers, because [4], Corollary VII--6 implies that X[n, 0] I i 0 P i 0 Fi 1 ne λ Hence we obtain that Y [n, 1] P lim n n+1 1 e λ 1 For the induction step suppose our assertion holds true for 1 This time we introduce the normalizing factors d[n, ] 1 I S i 1 1 1λ S i Their asymptotic behaviour can be treated similarly to what we have done in 1 Thus, with probability 1, we can write d[n, ] exp 1λ I S i λ S i 1+o1 Si By Theorem 1 we have o i 1/+ε, thus the exponent differs S i λi from 1 log n only by an as convergent term Therefore d[n, ] δ n 1/, with some random variable δ > 0 Since Y [n+1, ] Y [n, ] + [n+1, 1] + I n+1, it is easy to see that where E d[n+1, ] Y [n+1, ] F n d[n, ] Y [n, ] + b[n, ], Y [n, 1] b[n, ] d[n+1, ] 1λ + d[n+1, ] P n+1 Fn d[n+1, ]λ + 1

7 In addition, we have A NEW CLASS OF SCALE FREE RANDOM GRAPHS 7 Var d[n+1, ] Y [n+1, ] F n d[n+1, ] Var [n+1, 1] + I n+1 d[n+1, ] Var [n+1, 1] + Var I n+1 d[n+1, ]b[n, ] O n 1 33 Let us introduce a square integrable martingale by its differences ξ n d[n, ] Y [n, ] d[, ] Y [, ] b[, ] The increasing process associated with the square of this martingale is A n E ξi F i 1 which is of order On by 33 Hence ξ i d[n, ] Y [n, ] From all these we obtain that n Var d[i, ]Y [i, ] F i 1, b[i 1, ] o n /+ε Y [n, ] n+1 1 n+1d[n, ] b[i 1, ] + o1 Now, by 31 and the induction hypothesis, 1 b[i, ] d[i+1, ] y 1 + q By applying the asymptotics we obtained for b[i, ] we arrive at the formula which was to be proved Y [n, ] n y 1 + q +1 + o1, Next we show that the solution of the recursion 3 is y +1 iq i Let r q + q +1 + and z y + y +1 +, then from the recursion for y one can derive that z 1z 1 + r From that we have z r i, which easily yields the above mentioned explicite form of y, and finally, the desired expression for x

8 8 ZSOLT KATONA AND TAMÁS F MÓRI References 1 Barabási, A-L, and Albert, R, Emergence of scaling in random networs, Science , Bollobás, B, Riordan, O, Spencer, J, and Tusnády, G, The degree sequence of a scale-free random graph process, Random Structures and Algorithms , Chow, Y S, and Teicher, H, Probability Theory: Independence, Interchangeability, Martingales, Springer, New Yor, Cooper, C, and Frieze, A, A general model of web graphs, Random Structures and Algorithms 003, LeCam, L, An approximation theorem for the Poisson binomial distribution, Pacific J Math , Móri, T F, The maximum degree of the Barabási Albert random tree, Combin Probab Comput , Neveu, J, Discrete-Parameter Martingales, North-Holland, Amsterdam, 1975 Department of Probability Theory and Statistics, Eötvös Loránd University, Pázmány Péter s 1/C, Budapest, Hungary H address: zsatona@csbmehu, moritamas@ludenseltehu