Random walk and Pair-Annihilation Processes on Scale-Free Networks

Abstract

We investigate the dynamic scaling properties of stochastic particle systems on a non-deterministic scale-free network. It has been known that the dynamic scaling behavior depends on the degree distribution exponent of the underlying scale-free network. Our study shows that it also depends on the global structure of the underlying network. In random walks on the tree structure scale-free network, we find that the relaxation time follows a power-law scaling $\tau\sim N$ with the network size $N$. And the random walker return probability decays algebraically with the decay exponent which varies from node to node. On the other hand, in random walks on the looped scale-free network, they do not show the power-law scaling. We also study a pair-annihilation process on the scale-free network with the tree and the looped structure, respectively. We find that the particle density decays algebraically in time both cases, but with the different exponent.

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arXiv:cond-mat/0509564v1 [cond-mat.stat-mech] 22 Sep 2005
Random walk and Pair-Annihilation Processes on Scale-Free Networks
Jae Dong Noh and Sang-Woo Kim
Department of Physics, Chungnam National University, Daejeon 305-764, Korea
(Received February 2, 2008)
We investigate the dynamic scaling properties of stochastic particle systems on a non-deterministic
scale-free network. It has been known that the dynamic scaling behavior depends on the degree
distribution exponent of the underlying scale-free network. Our study shows that it also depends
on the global structure of the underlying network. In random walks on the tree structure scale-free
network, we find that the relaxation time follows a power-law scaling τ ∼ N with the network size
N. And the random walker return probability decays algebraically with the decay exponent which
varies from node to node. On the other hand, in random walks on the looped scale-free network,
they do not show the power-law scaling. We also study a pair-annihilation process on the scale-free
network with the tree and the looped structure, respectively. We find that the particle density
decays algebraically in time both cases, but with the different exponent.
I. INTRODUCTION
Networks composed of nodes and edges have been at-
tracting a lot of research interest recently [1]. Unlike
conventional networks such as a periodic regular network
and a random network, many real-world networks have
complex structure. Many of them belong to the class of
so-called scale-free (SF) networks. Denoting the degree
k of a node as the number of edges attached to it, a SF
is characterized with the power-law degree distribution
Pdeg(k) ∼ k−γ. (1)
Here γ is called the degree distribution exponent.
The power-law degree distribution implies that the SF
network has an inhomogeneous structure. On the one
hand, it is a challenging problem to characterize and un-
derstand the organization principle of the SF network.
On the other hand, it is also interesting to study thermo-
dynamic or dynamic systems on such an inhomogeneous
structure. Various physical problems have been studied.
Examples include the ferromagnetic phase transitions in
the Ising model [2], the non-equilibrium phase transition
in the epidemic spreading model [3], the random walk
process [4, 5, 6, 7, 8, 9], the dynamic scaling in pair-
annihilation process [10, 11], and so on. In those studies,
the γ-dependent scaling properties have been studied.
In the present work, we address the question how the
global structure of the underlying SF network influences
the dynamic scaling behavior of stochastic particle sys-
tems. For that purpose we study the random walk pro-
cess and the pair-annihilation process, and investigate
their dynamic scaling property. Both systems are studied
on the tree structure SF (TSF) networks and the looped
structure SF (LSF) networks, respectively.
To be specific, we study the stochastic systems on
the Dorogovtsev-Mendes-Samukhin (DMS) network [12],
which is a generalization of the Barab´ asi-Albert (BA)
network [13]. It is a non-deterministic model for a grow-
ing network: Each time step, a new node is added, and
linked with m nodes which are selected among existing
nodes with the probability given by Πi∝ (ki+ a). Here
the parameter a is called an initial attractiveness. The
BA model corresponds to the a = 0 case of the DMS net-
work. The resulting network is scale-free and the degree
distribution exponent is given by
γ = 3 +a
m.
(2)
With the parameters m and a, one can vary the value
of the degree exponent. At the same time, one can also
generate a TSF network with m = 1 or a LSF network
with m ?= 1. Hence, we can study the effect of the degree
distribution exponent and the global network structure
systematically.
This paper is organized as follows: In Sec. II, we
present the results on the random walk process. Sec-
tion III is devoted to the scaling property of the pair-
annihilation process. Summary will be given in Sec. IV.
II. RANDOM WALK PROCESS
As a basic and fundamental stochastic process, the ran-
dom walk process [14] on complex networks has been at-
tracting a lot of research interest [4, 5, 6, 7, 8, 9]. In
this work we concentrate on relaxation dynamics on SF
networks.
On a network with N nodes, a random walker is as-
signed to a starting node denoted by s at time t = 0.
Then, at each unit time step ∆t = 1, it hops to one of
the neighboring nodes selected randomly with the equal
probability. Defining P(i,t;s,0) as the probability to find
the walker at node i at time t, one finds that it evolves
in time as
P(i,t + 1;s,0) =
N
?
j=1
Aij
kj
P(j,t;s,0)(3)
with the initial condition P(i,0;s,0) = δi,s. Here A =
{Aij} is the adjacency matrix whose elements are Aij=
1 (0) if two nodes i and j are connected (disconnected).
The relaxation dynamics is studied with the so-called
return probability Rs(t) ≡ P(s,t;s,0). For a given net-
work, one can solve numerically the master equation

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100
101
102
103
104
105
t
10-5
10-4
10-3
10-2
10-1
100
R
N = 1000
N = 4000
N = 16000
N = 64000
100
101
t
102
10-6
10-5
10-4
10-3
10-2
10-1
100
R
( a )
( b )
FIG. 1:
RP (filled symbols) in the DMS networks with γ = 3 and
with m = 1 in (a) and m = 2 in (b).
The return probability RH (open symbols) and
Eq. (3) iteratively to obtain Rs(t). We then average it
over different realizations of the networks.
On a network with loops, the probability distribution
converges to the stationary one Pstat.(i) = ki/(?
verges to Rs(t = ∞) = ks/(?
lating in time, so is the return probability. The random
walker cannot return to a starting node in odd time steps,
which means that Rs(t) = 0 at odd t. In this case, we
only measure the return probability at even time steps,
which converges to stationary value 2ks/(?
broadly that one can define an exponent qsfor each node
s describing the degree scaling with the network size:
jkj) in
the t → ∞ limit [6]. Hence, the return probability con-
jkj). On the other hand,
on a tree network, the probability distribution is oscil-
jkj).
In a SF network, the node degree is distributed so
ks∼ Nqs.(4)
For instance, in the DMS network with the degree distri-
bution exponent γ, a peripheral node with the minimum
degree has q = 0, while the hub with the maximum de-
gree has q = 1/(γ − 1). Then, the return probability in
the stationary state scales with the network size as
Rs(t → ∞) ∼ N−(1−qs).(5)
Due to the broad degree distribution, the return prob-
ability Rs(t) may have a different scaling property at
different starting node s with different values of qs. So,
we measured the return probability RH(t) for the hub
and RP(t) for a peripheral node on the DMS networks.
We present the numerical data for the return probability
RH and RP in Fig. 1. We compare the data obtained
from the TSF networks and the LSF networks. The data
show that the return probability behaves distinctly de-
pending on the global structure of the network, which
will be discussed in detail in the following.
A.Random walks on tree networks
We perform the quantitative analysis of the scaling
property in the TSF. From Fig. 1, one finds that the
103
104
N
105
101
102
103
104
105
τ
γ = 4.0
γ = 3.0
γ = 2.5
FIG. 2: Relaxation time τ for RH(t) (open symbols) and
RP(t) (filled symbols) on the tree structure SF networks. The
solid line has the slope 1.
return probability relaxes much slower in the TSF net-
work with m = 1. We estimated the relaxation time
τ using the condition Rs(t = τ) = cRs(t = ∞) with
a constant c = 2. The relaxation time is plotted as a
function of the network size N in Fig. 2. It shows that
the relaxation time estimated from RH and RP follows
a power-law scaling
τ ∼ Nz
(6)
with the same dynamic exponent
z = 1.0 . (7)
We provide a theoretical argument for the numeri-
cal result from the analysis of the mean first passage
time (MFPT). The MFPT problem has been studied on
complex networks recently [6, 7, 9], and some rigorous
results are known [6, 7]. Consider the MFPT, denoted
by Tj,i, from an arbitrary node i (degree ki) to one of
its neighboring node j. Let {n1,n2,···,nki= j} denote
the neighbors of the node i. Then, following Ref. [7], the
MFPT on tree networks satisfies the recursion relation
Tj,i= ki+
?
l?=ki
Ti,nl. (8)
Applying the recursion relation repeatedly until one ar-
rives at dangling ends, one can find the explicit solution
for Tj,i. Without the link between i and j, the tree net-
work would be decomposed into two parts. Denoting the
number of nodes in the i side by Ni, the MFPT is simply
given by
Tj,i= 2Ni− 1 .(9)
Hence, for a typical adjacent nodes i and j, one has that
Tj,i∼ N1. For non-adjacent nodes i and j, the MFPT is
given by the sum of the MFPT’s given by Eq. (9) along
the path between them. Hence, the MFPT between a
typical node pair is given by T ∼ DNN, where DN is
the mean diameter of the networks. For SF networks,
the mean diameter scales at most logarithmically with

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10-4
10-2
t / N
100
102
10-1
100
101
102
103
104
105
R N1 − q
( a )
N = 1000
N = 4000
N = 16000
N = 64000
10-4
10-2
t / N
100
102
10-1
100
101
102
103
104
105
R N1 − q
( b )
FIG. 3: Scaling analysis of RH (open symbols) and RP (filled
symbols) in the TSF networks with γ = 3 (a) and γ = 4 (b).
the network size [15]. Therefore, we conclude that the
relaxation time follows the power-law scaling τ ∼ Nz
with the dynamic exponent z = 1.
The power-law scaling of the relaxation time suggests
that the return probability decays algebraically as
Rs(t) ∼ t−δs
(10)
with the decay exponent δs. For a node s with the degree
scaling ks ∼ Nqs, the decay exponent can be deduced
from the finite-size-scaling ansatz
Rs(t) = N−(1−qs)f(t/Nz)(11)
with the dynamic exponent z = 1. For large t ≫ Nz,
the return probability should converge to the stationary
value Rs(t = ∞) ∼ N−(1−qs)(see Eq. (5)). It requires
that the scaling function should behave as f(x ≫ 1) =
constant. The power-law scaling for t ≪ Nzrequires that
the scaling function should behave as f(x ≪ 1) ∼ x−δs,
which yields that Rs∼ t−δsNδsz−(1−qs). Therefore, the
finite-size-scaling ansatz predicts that the decay expo-
nent is given by
δs= 1 − qs. (12)
It is interesting to note that the decay exponent varies
with the degree scaling exponent qsof the starting node
s. We confirm the scaling behavior with the scaling plot
of RsN1−qsversus t/N for the hub and the peripheral
node in Fig. 3. For γ = 3, qH = 1/(γ − 1) = 1/2 for
the hub and qP= 0 for the peripheral node, which yields
that δH = 1/2 and δP = 1. Similarly, for γ = 4, one
expects that δH = 2/3 and δP = 1. One finds that all
data from different network sizes collapse very well in the
scaling plot, which supports the result in Eq. (12).
It is also interesting that the return probability decays
as Rs(t) ∼ t−δswith the exponent δs≤ 1 at all nodes.
It indicates that the random walks are recurrent in the
N → ∞ limit [14].
00.0001
1 / N
0.6
0.65
0.7
0.75
0.8
θ
γ = 2.5
γ = 3
γ = 4
00.10.2
1 / ln N
0.5
0.6
0.7
0.8
0.9
1
θ
( a )
( b )
FIG. 4: Plots of θ obtained from RP(t) on the DMS networks
with γ = 2.5,3,4 and with m = 2 (open symbols) and m =
4 (filled symbols). θ is plotted against 1/N in (a) and 1/lnN
in (b).
B. Random walks on looped networks
From Fig. 1, one can see that the return probability
decays much faster in the LSF networks. The downward
curvature in the log-log plot implies that the decay is
faster than a power-law decay. The relaxation time mea-
surement also indicates a faster decay. We found that
the relaxation time, measured using the condition that
Rs(τ) = 2Rs(∞), grows at most logarithmically with
the network size N. This is in contrast to the power-
law growth in the TSF networks. In this subsection, we
address a question how the return probability decays in
time in looped SF networks.
In the random network, it is known that the return
probability follows a stretched exponential decay as
Rs(t) − Rs(∞) ∼ e−atθ
(13)
with a constant a and the exponent θ = 1/3 [16]. It was
reported that the return probability in the small-world
network follows the stretched exponential decay, too [17].
It might suggest that the looped SF network follow the
stretched exponential decay, too.
On the DMS networks with m ≥ 2, we measured Rs(t)
numerically and fitted the data to the form in Eq. (13)
to estimate θ. If the return probability follows an expo-
nential decay, one would have θ = 1. On the other hand,
one would have θ < 1, if it follows the stretched expo-
nential decay. In Fig. 4, we present the data for θ(N) for
RP(t) obtained on the DMS networks with various values
of γ = 2.5,3,4 and m = 2,4 up to sizes N ≃ 106. Similar
behaviors are observed in θ(N) for RH(t). At small val-
ues of N, the exponent seems to depend on m and to be
smaller than 1. One may interpret that as an evidence
of the stretched exponential decay with a non-universal
exponent θ.
However, we observe that there is a very strong finite
size effect. The plot of θ versus 1/lnN in Fig. 4 (b) shows
that the finite size effect is significant even for N ≃ 106.
With this strong finite size effect, one can not exclude the
possibility of the exponential decay with θ = 1. We sug-

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4
gest that an analytic approach be necessary to conclude
whether the return probability follows the exponential or
the stretched exponential decay.
Before closing this section, we remark on the previ-
ous studies on the random walks on deterministic SF
networks [7, 9]. Unlike the non-deterministic LSF net-
works studied in this work, the deterministic LSF net-
works [18, 19, 20] behave similarly as the TSF networks.
For example, in the hierarchical network [20] which has a
looped structure, the relaxation time scales algebraically
as τ ∼ N [7]. The reason why the hierarchical network
behaves as TSF networks is clear. The network has a
symmetry, due to which the random walks on it can be
mapped to the walks on TSF networks [7]. Therefore,
our general conclusion should not be applied to the de-
terministic LSF networks with high symmetry.
III.PAIR-ANNIHILATION PROCESS
The pair-annihilation process is a diffusion-limited
reaction-diffusion process. In this process, each node in
a given network may be occupied by a particle (denoted
by A) or empty (denoted by ∅). The particles perform
random walks on the network, and annihilate pairwise
whenever they meet at a same node (A + A → ∅).
The pair-annihilation process on SF networks was
studied numerically by Gallos and Argyrakis [10]. They
found that the particle density decays algebraically
ρ(t) ∼ t−αwith the γ-dependent decay exponent α =
α(γ) ≥ 1. This is contrasted with the d-dimensional pe-
riodic lattice case where α = α(d) ≤ 1. Namely, the
particle density decays faster in SF networks.
Later on, Catanzaro et al. [11] developed a mean field
theory for the pair-annihilation process, which will be
reviewed briefly hereafter. Let us define ρk as the av-
erage particle density at nodes with degree k. It is re-
lated to the total density through the relation ρ(t) =
?
dρk(t)
dt
kPdeg(k)ρk(t). In a mean field level, one can write
down the rate equation for ρkas
= −ρk(t) +
k
?k?[1 − 2ρk(t)]ρ(t) ,(14)
where ?k? is the mean degree. Multiplying Pdeg(k) and
summing over k, one finds that
dρ(t)
dt
= −2ρ(t)
?
1
?k?
?
k
kPdeg(k)ρk(t)
?
. (15)
Then, Catanzaro et al. made a quasistatic approximation
neglecting the time derivative in Eq. (14). The approx-
imation assumes that the particles adjust themselves so
efficiently that their distribution ρk remains close to a
stationary one at a given value of ρ(t). It leads to the
relation
ρk(t) =
kρ(t)/?k?
1 + 2kρ(t)/?k?.(16)
100
101
102
t
10-5
10-4
10-3
10-2
10-1
100
ρ(t)
m = 1
m = 4
234
5
γ
-1
0
1
2
3
α , β
( a )
( b )
FIG. 5: (a) Density decay in the DMS networks with m = 1
and m = 4. The degree distribution exponent is γ = 2.5. (b)
Density decay exponent α (circles) and β (squares) defined in
Eq. (17) for the DMS networks with m = 1 (open symbols)
and m = 4 (filled symbols).
Substituting it in Eq. (15) and solving the resulting equa-
tion, Catanzaro et al. obtained that the density decay
follows a power law in the N → ∞ limit as
ρ(t) ∼ t−α(lnt)−β. (17)
The decay exponent α is given by
α(γ) =
?1/(γ − 2) , 2 < γ < 3
1, 3 ≤ γ
(18)
and the exponent for the logarithmic correction is given
by β = 1 for γ = 3 and β = 0 for γ ?= 3. This result
is qualitatively consistent with the simulation results of
Refs. [10, 11].
In the previous works [10, 11], only the γ-dependent
scaling behaviors have been studied. However, the study
on the random walks in Sec. II suggests that the scaling
behavior may also depend on the global structure of the
underlying SF network. In this section, we present the
results of our numerical works on the pair-annihilation
process on the DMS networks with m = 1 (tree structure)
and m ?= 1 (looped structure). Comparing the two differ-
ent cases, we will show that the global structure does also
matter for the scaling behavior of the pair-annihilation
process.
We have performed the Monte Carlo simulations on the
DMS networks of size N = 1024000 with various values of
γ and m. The fully-occupied state is taken as the initial
configuration. In Fig. 5 (a), the numerical data from the
TSF networks (m = 1) and the LSF networks (m = 4)
are compared. The data show that the particle density
decay follows the power law in both cases but with a
different exponent.
We estimate the decay exponent by fitting the data to
the form in Eq. (17), and plot α and β as a function of
γ in Fig. 5 (b). At m = 4, our result is qualitatively
consistent with the previous works; α ≃ 1 for γ ≥ 1, and
α varies with γ for γ < 3. And the logarithmic correction
is prominent at γ = 3. However, at m = 1, we obtain a

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100
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k
10-3
10-2
10-1
100
ρ k
100
101
102
103
104
105
k
10-3
10-2
10-1
100
ρ k
( a )
( b )
FIG. 6: ρkvs. k at different time steps t =20(top),···,28(bot-
tom).The data are taken from the DMS network with
m = 4 (a) and m = 1 (b). The network size is N = 1024000
and the degree distribution exponent is γ = 2.5 in both plots.
The dashed lines have the slope 1.
completely different result; α ≃ 1.0 at all values of γ and
the logarithmic correction is present at all values of γ.
These numerical results show that the global structure
of the underling SF networks affects the scaling behavior
of the pair-annihilation process.
We speculate the origin for the different scaling behav-
iors. On the LSF networks, the scaling behavior seems
to be consistent with the analytic mean field result in
Eq. (18). In the analytic approach one adopts the qua-
sistatic approximation [11] leading to the particle distri-
bution given by Eq. (16). It can be rewritten as
ρk(t) ≃
?kρ(t)/?k? , fork ≪ ?k?/ρ(t) ,
k ≫ ?k?/ρ(t) .1/2, for
(19)
Note that ρk∝ k for small k.
We investigate the particle distribution numerically. In
Fig. 6 (a), we plot ρk(t) against k for the DMS network
with the looped structure (m = 4). We find that the par-
ticle distribution ρk(t) is fully consistent with Eq. (19).
This supports that the analytic approach is appropriate
on the LSF networks.
However,
Eq. (19) on the TSF network. Figure 6 (b) shows that
there are three different regimes: ρk ∼ k for k < k1,
ρk∼ kηfor k1< k < k2, and ρk≃ 1/2 for k2< k. The
scaling exponent η in the intermediate regime is found
to be less than 1 and to vary with γ. This feature is
inconsistent with the quasistatic approximation.
The quasistatic approximation assumes that particles
rearrange themselves quickly upon the change in the to-
tal particle density via diffusion.
diffusion should be a fast process. In the previous sec-
tion, we have shown that the diffusion is a slow process
in the LSF networks. This explains why the quasistatic
approximation is invalid in the TSF network whereas it
is valid in the LSF network.
IV.SUMMARY
the particle distribution deviates from
It requires that the
In summary, we have investigated the scaling proper-
ties of the random walk and the pair-annihilation pro-
cesses on non-deterministic SF networks with the tree
structure and the looped structure, respectively. In the
random walk process on TSF networks, we find that the
relaxation time scales as τ ∼ N with the network size N
and that the return probability decay follows the power
law with the node-dependent exponent. The LSF net-
work does not display the power-law scalings.
pair-annihilation process, we find that the exponent de-
scribing the particle density decay is different in TSF and
LSF networks. Our results show that the global struc-
ture of the SF network, as well as the degree distribution
exponent, is the important ingredient in understanding
the dynamic scaling behaviors.
In the
Acknowledgments
This work was supported by Chungnam National Uni-
versity through the Internal Research Grant in 2004.
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