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arXiv:0911.4001v2 [cond-mat.stat-mech] 26 May 2010

Cluster aggregation model for discontinuous percolation transition

Y.S. Cho, B. Kahng and D. Kim

Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea

(Dated: May 27, 2010)

The evolution of the Erd˝ os-R´ enyi (ER) network by adding edges is a basis model for irreversible

kinetic aggregation phenomena. Such ER processes can be described by a rate equation for the

evolution of the cluster-size distribution with the connection kernel Kij ∼ ij, where ij is the product

of the sizes of two merging clusters. Here we study that when the giant cluster is discouraged to

develop by a sub-linear kernel Kij ∼ (ij)ωwith 0 ≤ ω < 1/2, the percolation transition (PT) is

discontinuous. Such discontinuous PT can occur even when the ER dynamics evolves from proper

initial conditions. The obtained evolutionary properties of the simple model sheds light on the origin

of the discontinuous PT in other non-equilibrium kinetic systems.

PACS numbers: 64.60.ah,02.50.Ey,89.75.Hc

Irreversible cluster aggregations are widespread phe-

nomena occurring in a diverse range of fields, including

dust and colloid formation, aerosol growth, droplet nucle-

ation and growth, gelation transition, etc [1]. The Smolu-

chowski coagulation equation [2–4] can successfully de-

scribe such cluster aggregation processes. In linear poly-

merization, molecules with two reactive ends can react

to form long chains. In this case, the reaction kernel is

given as Kij= 1, where i and j are the masses of the two

reactants. For the aggregation of branched polymers, the

reaction kernel has the form Kij= (ai+b)(aj+b), where

a and b are constants. When clusters have a compact

shape, the reaction kernel has the form Kij∼ (ij)1−1/d,

where d is the spatial dimension. Intensive studies have

been carried out using the Smoluchowski coagulation

equation with such different kernel types [1, 5–8], and

it is known that sol-gel transitions can occur at either

finite or infinite transition points. They are continuous

transitions.

During the past decade, the evolution of complex net-

works has been of much interest to the science commu-

nities in multidisciplinary fields. To study percolation

transition (PT) during network evolution, the branch-

ing process approach [9, 10] and the Potts model formal-

ism [11] have been used. Such complex network evolution

can also be viewed as a cluster aggregation phenomenon,

and can be studied by the rate-equation approach [6].

For example, in the evolution of the classical random

network, called the Erd˝ os-R´ enyi (ER) model, an edge

is added at each time step, thereby either connecting

two separate clusters (inter-cluster edge) or increasing

the edge number in one cluster without changing clus-

ter numbers (intra-cluster edge). Fig. 1 shows that the

frequency of inter-cluster connections is dominant until

the percolation threshold. Thus, the cluster aggregation

picture of the ER network evolution comes in naturally.

In this paper, we extend the cluster aggregation dynam-

ics in networks to more general cases. Specifically, the

model is as follows: In a system composed of N vertices,

we perform the following tasks at each time step.

• Two clusters of sizes i and j are chosen with proba-

bilities qiand qj, respectively. The two clusters can

be the same. Probability qiis given as ki/?

where ki and ni are the weight and density of an

i-sized cluster, respectively.

sksns,

• Two vertices are selected randomly one each from

the selected clusters. If they are not yet connected,

then they are connected by an edge. If they are

already connected, we choose another pair of ver-

tices in the same manner until a link can be added.

Self-loop cases are excluded.

We repeat these simple steps until a given time t ≡ L/N,

where L, the number of edges added to the system, is

tuned. This model is called the cluster aggregation net-

work model hereafter. In this model, the two selected

clusters can be the same, and thus, the evolution can

proceed even after one giant cluster remains. The ER

network corresponds to the case ki= i. Here, we show

that when the weight is sub-linear, as ki = iωwith

0 ≤ ω < 1/2, a discontinuous PT occurs at a finite

transition point. Moreover, under certain initial con-

ditions, the ER dynamics also exhibits a discontinuous

PT. This observation is remarkable, because a discontin-

uous PT has rarely been discovered in irreversible kinetic

systems, except for recent observations in the ER [13]

and other networks [14–19] under the so-called Achliop-

tas process [13]. On the other hand, it is noteworthy

that the cluster aggregation network model evolves by

single-edge dynamics, as compared with the ER network

under the Achlioptas process, which involves a pair of

edges at each time step. Thus, this cluster aggregation

network model allows us to study the underlying mecha-

nism of the discontinuous PT analytically for some cases,

which is shown later.

The cluster aggregation processes in the model are de-

scribed via a rate equation for the cluster density, which

takes the following form in the thermodynamic limit:

dns(t)

dt

=

?

i+j=s

kini

c(t)

kjnj

c(t)− 2ksns

c(t),(1)

where c(t) =?

kikj/c2. The first term on the right hand side repre-

sents the aggregation of two clusters of sizes i and j with

sksns(t). The connection kernel Kij ≡

Page 2

2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

t ≡ L/N

0.8 1 1.2 1.4

fraction

ER

FIG. 1: (Color online) The fraction of each type of attached

edges, inter-cluster (•) or intra-cluster (▽) edges for the ER

model. Arrow indicates percolation threshold at pc = 1/2.

i + j = s and the second term represents a cluster of

size s merging with another cluster of any size.

rate equation differs from the Smoluchowski coagulation

equation in two aspects. First, the connection kernel is

time-dependent through c(t) when ω ?= 1. Second, the

second term on the right hand side of Eq. (1) includes the

process of merging with an infinite-size cluster. Hence,

Eq. (1) with ω = 1 and c = 1 describes the ER pro-

cess, while the conventional Smoluchowski coagulation

equation with ω = 1 does not, because only sol-sol reac-

tions are taken into account. However, the case including

the infinite-size cluster in the Smoluchowski coagulation

equation was also considered in Ref [6], which was called

the F-model. Owing to the presence of c(t), a PT occurs

at a finite transition point even when a PT does not occur

in the Smoluchowski coagulation equation, for example,

when ω = 0. Here, we study the cases ki= 1 (ω = 0),

ki= iωwith 0 < ω < 1, and ki= i (ω = 1), separately.

The case ω = 0: In this case, c(t) =?

the total density of the clusters, which decreases linearly

with time. The generating function of ns(t) is defined

as f(z,t) =?

range 0 < z < 1. Then, one can obtain the differential

equation for f(z,t) from Eq. (1) and solve in a closed

form as f(z,t) = (1 − t)2z/(1 − zt) for t < 1 and 0 for

t > 1 in the thermodynamic limit. Expanding f(z,t) as

a series in z, we obtain

The

sns becomes

sns(t)zs, where z is the fugacity in the

ns(t) = (1 − t)2ts−1

(2)

for t < 1. This formula shows that the cluster size dis-

tribution decays exponentially as s becomes large. Par-

ticularly, when δ ≡ 1 − t is small, ns(δ) ≈ δ2e−s/s∗with

s∗≈ 1/δ. The characteristic size s∗diverges as δ → 0.

As shown in the inset of Fig. 2(a), ns(t) is almost flat at

δ = 10−4for N = 106, indicating that large-size clusters

are relatively abundant. The merging of these clusters

causes a sudden jump in the giant cluster size, leading to

a first-order transition.

We find the giant cluster size G(t) by using the relation,

G(t) = 1−f′(1,t) ≡ 1−?′

excludes an infinite-size cluster. We find that

ssns(t), where the summation

G(t) =

?0 if

1 if

0 < t < 1,

t > 1,

(3)

10-6

10-5

10-4

10-3

10-2

10-1

100

5 10 15 20 25 30 35 40

ns

(a)

ki = 1

10-7

10-6

10-5

0 20000 40000

10-12

10-10

10-8

10-6

10-4

10-2

100

100

101

102

103

104

105

ns

(b)

ki = i0.4

10-12

10-10

10-8

10-6

10-4

10-2

100

100

101

102

103

104

105

ns

(c)

ki = i0.8

10-12

10-10

10-8

10-6

10-4

10-2

100

100

101

102

103

104

105

ns

s

(d)

ki = i

FIG. 2: (Color online) The cluster-size distributions of the

cluster aggregation network models with (a) ki = 1, (b) ki =

i0.4, (c) ki = i0.8, and (d) ki = 1 (ER model) with mono-

disperse initial condition. (a) is drawn in semi-logarithmic

scale, while the others are in double-logarithmic scales. Data

points in (a) are at t/tc = 0.3(•), 0.5(▽) and 0.7(◦), and

in (b), (c), and (d), they are obtained at t/tc = 0.50 (•),

0.95 (▽), 0.998 (◦)), and 1.003 (⋄)). Data points in the inset

of (a) are at δ = 1−t = 10−4. In (b), there exists a hump for

data points (◦). The system size N is taken as 106for (a) and

105for (b), (c), and (d). Solid lines in (b) and (c) represent

analytic formulae Eq. (4)

in the thermodynamic limit (Fig. 3(a)). Thus, the PT

is first-order at tc= 1. This result differs from what we

obtain from the Smoluchowski coagulation equation, in

which the transition point tc= ∞.

The case 0 < ω < 1: For this case, while exact solu-

tion for ns(t) is not obtained, ns(tc) is done under cer-

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3

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

G

t

(a)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

t

(b)

FIG. 3: (Color online) The density of the giant cluster size G

versus time t showing (a) discontinuous transitions in ki = 1

(◦), ki = i0.2(△), ki = i0.4(•), and (b) continuous transitions

in the cluster aggregation network model ki = i0.6(▽), ki =

i0.8(△), and the ER network (◦). N = 105in both (a) and

(b).

tain assumptions. To proceed, we define the generat-

ing function gω(µ,t) ≡?

presume that ns(tc) ∼ s−τ. Next, we use the assump-

tion made in Ref. [5, 7] for the Smoluchowski coagulation

equation that ns(t) = ns(tc)/(1 + b(t − tc)) near t = t+

where b is an s-independent constant. Then, comparing

the most singular terms in the series of the generating

functions f(eµ,tc) and g2

ssωns(t)eµs/c(t) (µ < 0), and

c,

ω(µ,tc) in µ, we find that

τ =

?1 + 2ω

3/2 + ω if

if0 < ω < 1/2,

1/2 < ω < 1.

(4)

This result is confirmed numerically in Fig 2. When t <

tc, ns(t) follows a power-law function with an exponential

cutoff for 1/2 < ω < 1, but it exhibits a hump in a large-

size region for 0 < ω < 1/2 (Fig. 2(b)).

We examine G(t) as a function of time for various ω

cases. G(t) exhibits a transition at finite tc, which is

continuous for 1/2 < ω ≤ 1, discontinuous for 0 ≤ ω <

1/2 (Fig. 3), and marginal for ω = 1/2. The first-order

transition is tested in Fig. 4 using the scaling approach

introduced in Ref. [13]. We define ∆ ≡ t1− t0, where t0

and t1are chosen as the times at which the value of G(t)

reaches 1/√N and 0.8 for the first time, respectively. We

find numerically that for 0 ≤ ω < 0.5, ∆ decays as N →

∞, while for 0.5 < ω ≤ 1, ∆ converges to a finite value.

This result suggests that the transition is discontinuous

(continuous) for 0 ≤ ω < 0.5 (0.5 < ω ≤ 1).

The case ω = 1: This case is exactly solvable as the

case of the Smoluchowski coagulation equation [6]. We

consider an arbitrary initial condition of ns(0). In this

case, c(t) =?

first moment M1(t) =?′

ing the largest cluster, is not. The generating function

g1(µ,t) =?

˙ g1= 2(g1− 1)g′

ssns(t) is conserved as c(t) = 1, but the

ssns(t), with the sum exclud-

ssns(t)exp(µs) satisfies the relation

1, (5)

where the dot (prime) is the derivative with respect to

time t (µ). Then, g1is the solution of

g1(µ,t) = 1 − H(−µ − 2t(g1(µ,t) − 1)), (6)

10-2

10-1

100

10-7

10-6

10-5

1/N

10-4

10-3

∆

ω = 0.8

= 0.6

= 0.5

= 0.4

= 0.2

FIG. 4: (Color online) Test of the discontinuous or continuous

PT for the cluster aggregation network models with various

ω values. When 0 < ω < 0.5, ∆ decays with increasing N;

however, when 0.5 < ω < 1, it converges to a finite value.

The straight lines are guidelines for eyes.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

G

t

(a)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

t

(b)

FIG. 5: (Color online) Plot of the giant cluster size G versus

time for the ER network with different initial conditions. In

(a), the cluster-size distribution ns(0) is flat with different

cutoff values sm = Nη, with η = 0, 0.1, 0.2, and 0.3 from right

to left. In (b), ns(0) decays according to a power law with

exponent τ = 3, 2.5, 2, and 1.5 from right to left. N = 107in

both (a) and (b).

where H(µ) ≡ 1 − g1(−µ,0) is fixed by the initial con-

ditions of ns(0). The giant cluster size G(t), defined as

G = 1 − g1(0−,t), can be solved by the self-consistent

equation G = H(2tG). The obtained G(t) has the form

near tcas

G(t) =2M2

2(0)

M3(0)

(2M2(0)t − 1), (7)

where Mn(0) =

ment. Also, one finds that the second moment M2(t) ≡

?′

M2(t) =

|1 − 2M2(0)t|

for t < tc= 1/(2M2(0)), and for t > tcas t → t+

These solutions for arbitrary initial conditions are used

to study the first-order transition in the ER network be-

low. For the cluster aggregation network model, the ini-

tial condition is ns(0) = δs,1. Then, M2(0) = M3(0) = 1,

?′

ssnns(0) is the initial n-th mo-

ss2ns(t), obtained from g′

1(0−,t), behaves as

M2(0)

(8)

c.

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TABLE I: When the number of cluster sizes at initial time obeys a power law ns(0) = As−τfor s = 1,...,sm, listed are the

amplitude A, the second moment at initial time M2(0), the critical point tc, and the critical behavior of the giant cluster size

G(t). Type of PT is specified for each case. The listed tc and G are the ones in the thermodynamic limit.

τAM2(0)

(2−τ)sm

3−τ

sm

ln sm

s3−τ

m

ζ(τ−1)(3−τ)

ln sm

ζ(2)

ζ(τ−2)

ζ(τ−1)

ζ(2)

ζ(3)

ζ(τ−2)

ζ(τ−1)

tc

0

G(t > tc)

1

type of PT

discontinuous

(i)0 ≤ τ < 2

2−τ

s2−τ

m

1

ln sm

1

ζ(τ−1)

1

ζ(τ−1)

1

ζ(τ−1)

1

ζ(τ−1)

1

ζ(τ−1)

(ii)τ = 2

01

τ−2

3−τ

discontinuous

(iii) 2 < τ < 3

0

∝ tcontinuous

(iv)τ = 3

0

∝

1

2te−ζ(2)/2t

∝ (t − tc)

continuous

(v)3 < τ < 4

ζ(τ−1)

2ζ(τ−2)

ζ(3)

2ζ(2)

ζ(τ−1)

2ζ(τ−2)

1

τ−3

continuous

(vi)τ = 4

∝

t−tc

ln(t−tc)

∝ t − tc

continuous

(vii) τ > 4

continuous

and consequently, tc= 1/2, which is the well-known ER

value. The giant cluster size exhibits a continuous tran-

sition at tcwith ns(tc) ∼ s−5/2.

It is often the case that starting from ns(0) = δs,1,

the cluster-size distribution ns(t) exhibits a power-law

behavior (or with hump) in s just before or at the tran-

sition point, even when the dynamics is different from

the ER (see Fig. 2). To see how such ns evolves un-

der the ER dynamics from then on, we consider here

two particular cases in which M2(0) and M3(0) depend

on N. First, we assume that ns(0) follows a flat distri-

bution, ns(0) = n0, in the range 0 < s < sm, where

sm, the size of the largest cluster at t = 0, depends

on N as sm = Nη.Then, n0 = 2N−2η, M2(0) ∝

Nη, and M3(0) ∝ N2η.

tc(N) = 1/2M2(0) ∝ N−η, and G(t) ∼ r(2M2(0)t − 1)

for t > tc(N) from Eq. (7), where r turns out to be in

O(1).

one can show that G(t′) is the solution of G =˜H(3t′G),

where˜H(x) = 2?∞

lar function qualitatively similar to H(x) = 1 − e−xof

the standard ER problem. Hence, G(t′) has a mean field

behavior similar to the original ER case. This scaling be-

havior implies that while δG(t) ≡ G(t1)−G(t0) increases

by O(1), ∆ ≡ t1− t0does so by ∼ O(N−η). Thus, we

have a first-order transition as N → ∞ (Fig. 5(a)).

Second, we suppose that the initial condition is given

as ns(0) = As−τin the range 0 < s < sm, where A is

the normalization constant determined by the condition

?

τ, together with the initial second moment for arbitrary

Then, a PT takes place at

Thus, if time t is scaled as t′= tM2(0), then

n=1(−1)n+1xn/(n!(n + 2)) is a regu-

ssns= 1. A is given in Table I for various ranges of

sm, the critical point, and the giant cluster size G(t),

obtained from Eq. (6)

In short, the PT occurs at t = 0 for τ ≤ 3, and at finite

tcfor τ > 3. This behavior is related to the divergence

of the second moment M2(0) since time is scaled in the

form t′= 2tM2(0). The transition is discontinuous when

τ < 2, but continuous when 2 < τ < 4. This difference

originates from the fact that the ratio M2

finite for the former, while it vanishes for the latter. For

τ > 4, both M2(0) and M3(0) are finite, resulting in the

classical percolation behavior at a finite tc.

In summary, we have introduced a cluster aggregation

network model,in which discontinuous percolation

transitions occur when the connection kernel is sub-

linear as Kij ∼ (ij)ωwith 0 ≤ ω < 1/2. Even for the

ER network, a discontinuous PT can also be obtained

by using initial conditions where M2(0) diverges and

M2

2(0)/M3(0) remains finite. The simple model mani-

fests explicitly the role of the abundance of large-size

clusters just before a transition point as a mechanism of

the discontinuous PT [17]. We expect that the cluster

aggregation network model can be used to study under-

lying dynamics of the explosive percolation transition

of random ER network under the Achlioptas process [13].

2(0)/M3(0) is

This work was supported by a KOSEF grant Acceler-

ation Research (CNRC) (Grant No.R17-2007-073-01001-

0) and by the NAP of KRCF.

Note added.–After the submission of this paper, we

became aware of a work [20], which starts from the same

motivation as ours.

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