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- 1. Return times of random walk on generalized random graphs Naoki Masuda (The University of Tokyo, Japan) Norio Konno (Yokohama Natl. Univ., Japan) Ref: PRE, 69, 066113, 2004.
- 2. Random walk • Describes many types of phenomena. • Also serves to understand infinite particle systems (e.g. math books by Durrett 1988; Liggett 1999; Schinazi 1999) – More return of branching RW to the origin ↔ more survival of contact process – More return of coalescing RW to the origin ↔ more survival of voter model
- 3. RW on regular lattices • Has a long history (e.g. Spitzer, 1976). • Simple random walk: – Z1, Z2 : recurrent – Zd (d ≥ 3) : transient • Asymmetric random walk is transient.
- 4. RW on heterogeneous networks • Finite scale-free networks – Stationary density on node vi ≈ ki (e.g. Noh et al., 2004) • In this study, infinite tree-like networks with a general degree distribution
- 5. RW on generalized random graphs (also called Galton-Watson trees) pk: degree distribution
- 6. Recursion relation valid for small mean degrees • pk: degree distribution • qn: prob. that a walker returns to the origin at time 2n for the first time.
- 7. Use of generating functions
- 8. Expression of the return probability using the ‘moment’ generating function of the graph zM Q ( z ) pk k −1 Q ( z ) = z∑ ∑ Q ( z) ÷ = , k Q ( z) k=1 k n=0 ∞ n ∞ ∞ M ( z ) ≡ ∑ mn z n , n=1 ∞ mn ≡ ∑ k=1 ( k −1) k n n−1 pk .
- 9. Lagrange’s inversion formula z = w / f ( w) [ ] z d n g [ w( z ) ] = g ( 0) + ∑n =1 n −1 g ′( u ) f ( u ) n! du u =0 ∞ n n −1 For our purpose, set w(z)=Q(z), f(w)=M(w)/w, g(w)=w.
- 10. partition of n number of parts
- 11. Partition of integers • Partitions of 5: (15), (132), (123), (122), (14), (23), (5) Young diagrams
- 12. Restricted partition of integers • All the partition of 5 with 3 parts are: (1 23) and (122).
- 13. Ex1) Cayley trees • pk=δk,d : homogeneous vertex degree 1 Q( z ) = d − (d − 1)Q( z ) (d − 1) n −1 qn = Dn −1 2 n −1 d → where d − d 2 − 4(d − 1) z Q( z ) = 2(d − 1) Cn Dn = n +1 2n : Catalan number • Consistent with well-known results (e.g. Spitzer, 1976).
- 14. Ex2) Erdös-Renyí random graph • pk=λk e-λ/k! • Can be calculated up to a larger n as compared to the brute-force method. • A bit slower exponential decay than the Cayley tree case. qn numerical (5 ∙107 runs) theory Cayley trees (theory) n (time) d = λ =7 d = λ =10
- 15. Ex3) scale-free networks • pk ∝ k −γ (k ≥ kmin) • Slower decay than the Cayley tree case. Effect of heterogeneity. qn numerical theory d=7 Cayley trees (theory) d=8 n (time) mean degree = 7.09
- 16. Relation to other results • Generally, more heterogeneous vertex degree → more percolation, easier survival of CP and voter models. In scale-free networks with pk ∝ k−γ , critical point even disappears when γ≤ 3 – Percolation (Albert et al., 2000; Cohen et al., 2000) – CP (Pastor-Satorras & Vespignani, 2001; Eguíluz & Klemm, 2002). – SIR (Moreno et al., 2002) • For branching RW on GW trees (Madras & Schinazi, 1992; Schinazi, 1993), critical point for – global survival: λ1=1 ， – local survival: λ2=1/(1−τ) where qn ~ exp(−τn) ． • Our numerical results connect these two theoretical results.
- 17. Conclusions • Explicit formula for return probability of RW on generalized random graphs. • Calculation up to a large time is possible as compared to the brute-force method. • Young diagrams appear. Relation between asymptotic distribution of Young diagrams and RW, CP, voter model, etc.?