Scale free random trees

Transcription

1 Scale free random trees Tamás F. Móri Department of Probability Theory and Statistics, Eötvös Loránd University, 7 Budapest, Pázmány Péter s. /C moritamas@ludens.elte.hu Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T-2962

2 . Model (Barabási Albert) Erdős Rényi: n vertices, each of the ( n 2) possible edges is included with the same probability p, and independently of each other. Degree distribution: approximately Poisson light tail. Real life networks (e.g. the Internet) much heavier tail behaviour, power law degree distributions. Evolution of random trees Step : a single edge with endpoints labelled 0 and. Further steps: new vertices and edges one by one. Choose one of the existing edges at random, then one of its endpoints at random, and draw an edge from it to a new vertex (labelled n at step n). Each existing vertex is selected with probability proportional to its degree. Barabási, Albert (999) Generalization β > parameter, weight of a vertex = degree + β. Each existing vertex is selected with probability proportional to its weight. Probability of being selected at step n + = degree + β, where = (2 + β)n + β (total weight after step n). Problems Degree distribution (limit as n of the proportion of vertices with degree k) Maximal degree Profile, height, width Mean distance, Wiener index

3 2. Degree distribution SLLN a n,i number of vertices of degree i after step n. Bollobás, Riordan, Spencer, Tusnády (200): β = 0 Instead of the proportions a n,i /n we deal with the relative weights a n,i (i + β)/. They stabilize around q i = i j= j+β j+2+2β Γ(2β+3) Γ(β+) e β i β+2 i. In the B A model (β = 0) q i = 2 (i+)(i+2). b n,i = a n,i (i + β) q i centered variables ( ) Γ n + β i+β c[n, i] = ( ) n normalizing constants Γ n i Theorem 2.. For every i =, 2,... the sequence Z[n, i] = c[n, i] i j= ( ) i j ( i+β i j ) b n,j, n i is a martingale. Theorem 2.2. With probability lim n a n,i (i+β) = q i, i =, 2,...

4 3. Degree distribution CLT b n,i = a n,i (i + β) q i centered variables t, t 2,..., t i fixed real numbers Theorem 3.. n i d ( ) t j b n,j N 0, σ 2 i, j= as n. In addition, the variance of the left-hand side also converges to the asympotic variance σ 2 i. Corollary. The distribution of the random vector n (b n,, b n,2,..., b n,i ) converges to an i-variate multinormal law. The variables b n,i can be expressed in terms of the martingales Z[n, i] with the help of combinatorial inversion: b n,i = i j= ( ) i+β Z[n, j] i j c[n, j]. Writing every martingale Z[n, j] as the sum of its differences we can express n /2 (t b n, + + t i b n,i ) as rowwise sums of a certain martingale difference array, to which we can apply standard martingale CLTs.

5 4. Maximal degree basic martingales X[n, j] weight of vertex j after the n-th step Normalizing constants: for k =, 2,... c[n, k] = ( Γ n+ β ( Γ n+ k+β ) ) n k, n. Theorem 4.. For j = 0,,... and k =, 2,... ( ) X[n, j]+k Z[n; j, k] = c[n, k], n max{j, } k is a (positive) martingale. X[n, j] can be considered as the number of white balls in a generalized Pólya Eggenberger urn after n j draws: in the beginning (2 + β)j black and + β white balls, draw white add white and + β black balls, draw black add 2 + β black and no white balls. Theorem 4.2. For j = 0,,... n X[n, j] ζ j a.s., and in L p, p, as n. The limits ζ j are positive and their joint distribution is absolutely continuous. In addition, Eζ k j = (+β)() (k+β)c[j, k].

6 5. Maximal degree SLLN M n maximal degree after n steps M[n] = max{z[n; j, ] : 0 j n} = c[n, ](M n + β) normalized max. weight µ = sup j 0 ζ j it can be proved that µ = max j 0 ζ j ; it is finite, positive, and attained uniquely. Theorem 5.. With probability lim n n M n = lim n The convergence also holds in L p, p. M[n] = µ. Being the maximum of martingales, M[n] is a submartingale; and it is bounded in L p, p : if k > 2 + β, EM[n] k n EZ[n; j, ] k j=0 (k+β) k j=0 j=0 c[j, k] <. Eζ k j Remark. λ n = min{j : 0 j n, Z[n; j, ] = M[n]} label of the vertex with maximal degree. λ n does not change if n is large enough, thus it has a proper limit distribution.

7 6. Maximal degree CLT Theorem 6.. As n, (i) n 2() (n d M n µ) µ N (0, ), ( (ii) n 2() µ /2 n d M n µ) N (0, ). Here n M n can be replaced by M[n]. Doob Meyer decomposition of M[n] into a martingale and an increasing predictable process: M[n] = Y n + A n, where Y n Y n = M[n] E ( M[n] Fn ), A n A n = E ( M[n] Fn ) M[n ]. Both Y n and A n converge. Y n obeys the CLT, while A n turns out to be negligible. Q n = # {0 j n : M[n] = Z[n; j, ]} multiplicity of the maximal degree. Q n = eventually, hence A n A n = M[n ] Q n + = 0 for all sufficiently large n. In the martingale difference array { } n 2() (Y i Y i ), i > n, n =, 2,... the row sums are just n 2() (Y n Y ), the row sums of conditional variances µ, the Lindeberg condition holds by the uniform boundedness of differences.

8 7. Profile lower layers Rooted tree : root = vertex 0, every new vertex is the offspring of the node it is bound to. layer 0 = {root} layer k = offsprings of nodes in layer k, L[n, k] = size of layer k after step n {L[n, k] : k n} profile of the tree W n = max{l[n, k] : k n} width of the tree H n = max{k : L[n, k] 0} height of the tree Drmota, Gittenberger (997) Profile of simply generated (Galton Watson) trees. W n n ξ. Chauvin, Drmota, Jabbour-Hattab (200) Profile of binary search trees. W n n/ π log n. Bollobás, Riordan (2003) Diameter of scale free (Barabási) random graphs. Theorem 7.. For every fixed k, with probability, ζ ( +β ) k L[n, k] (k )! log n n ( +β ) = n ζ Poi k log n where ζ is a positive random variable, and Poi i (λ) denotes the i-th term of the Poisson distribution of parameter λ. Method: recursion, using the Doob Meyer decomposition of the submartingale c[n, ]L(n, k). Expectation: widest layers must be around k +β and W n = O ( n/ log n ) log n,

9 8. Profile width, height W [n, 0] = L[n, ] + β, W [n, k] = L[n, k + ] + βl[n, k] total weight of nodes in layer k, g n (z) = n EW [n, k]z k, z. Then by recursion g n (z) = k=0 n ( +β ) ( z), S i i= this is just the pgf of the sum η n = ξ + + ξ n, where the ξ i are independent Bernoulli variables, Eξ i = ( + β)/s i. By the local version of the CLT we obtain Theorem 8.. Let µ(n) = +β (i) EL [ n, µ(n) + t µ(n) ] log n, then n 2πµ(n) e t2 /2, t R (ii) max{el[n, k] : k n} ( ) /2 n. +β 2π log n The same can be proved for the L[n, k] themselves, by following Chauvin, Drmota, Jabbour-Hattab (200), i.e. W n ( ) /2 n. +β 2π log n By applying large deviation results to η n we get: Theorem 8.2. With probability +β lim inf n H n log n, lim sup n H n log n min z>0 + +β z log( + z).

10 9. Mean distance, Wiener index d(i, j) distance of vertices i and j X[n, i] weight of vertex i after step n n n i=0 j=0 D n = Sn 2 X[n, i]x[n, j] d(i, j) mean distance of two independent random vertices (weighted average) D n = n 2 n n i=0 j=0 d(i, j) mean distance I n = 0 i<j n d(i, j) = n2 D n/2 Wiener index Let again µ(n) = +β log n. Theorem 9.. D n 2µ(n) converges with probability as n, and the same holds for D n. Consequently I n = +β n2 log n + O(n 2 ). Family of a vertex = itself + children + grandchildren, etc. V [n, k] total weight of the family of vertex k after step n. Then D n = 2 n k= n V [n, k] ( + β) k= V [n, k] 2 S 2 n n k=0 S k n V [n, k] 2. k= martingale, bounded in L 2. n ( V [n, k] ) 2 increasing sum of squared martingales, hence submartingale. k= Bounded in L, hence convergent. D n D n 2 with probability.

11 References. Barabási, A.-L., and Albert, R., Emergence of scaling in random networks, Science 286 (999), Bollobás, B., Riordan, O., Spencer, J., and Tusnády, G., The degree sequence of a scale-free random graph process, Random Structures Algorithms 8 (200), Bollobás, B., and Riordan, O., The diameter of a scalefree random graph, Combinatorica (2003) (to appear). 4. Chauvin, B., Drmota, M., and Jabbour-Hattab, J., The profile of binary search trees, Ann. Appl. Prob. (200), Drmota, M., and Gittenberger, B., On the profile of random trees, Random Structures Algorithms 0 (997), Gouet, R., Martingale functional central limit theorems for a generalized Pólya urn, Ann. Probab. 2 (993), Móri, T. F., On random trees, Studia Sci. Math. Hungar. 39 (2002), Móri, T. F., The maximum degree of the Barabási random tree, Combinatorics, Probability & Computing (to appear). 9. Pittel, B., Note on the heights of random recursive trees and random m-ary search trees, Random Structures Algorithms 5 (994),