Presentation slides for Masuda and Konno. Physical Review E, 69, 066113 (2004).

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).

14. 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).

15. 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

16. 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

17. 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.

18. 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.?