Explosive Growth in Biased Dynamic Percolation on Two-Dimensional Regular Lattice Networks
Robert M. Ziff*
Center for the Study of Complex Systems and Department of Chemical Engineering, University of Michigan,
Ann Arbor, Michigan 48109-2136, USA
(Received 23 April 2009; revised manuscript received 18 June 2009; published 22 July 2009)
The growth of two-dimensional lattice bond percolation clusters through a cooperative Achlioptas type
of process, where the choice of which bond to occupy next depends upon the masses of the clusters it
connects, is shown to go through an explosive, first-order kinetic phase transition with a sharp jump in the
mass of the largest cluster as the number of bonds is increased. The critical behavior of this growth model
is shown to be of a different universality class than standard percolation.
DOI: 10.1103/PhysRevLett.103.045701 PACS numbers: 64.60.ah, 68.43.Jk, 89.75.Da, 89.75.Hc
Percolation concerns the formation of long-range con-
nectivity in systems [1] and has applications to numerous
practical problems including conductivity in composite
materials, flow through porous media, and polymerization
[2]. In the usual random (Bernouilli) percolation, bonds are
formed randomly and independently throughout the sys-
tem, and, at a critical concentration pc of occupied bonds, a
finite fraction of the sites (vertices) are connected together
and percolation takes place through a continuous or
second-order transition. By the universality hypothesis,
all large-scale properties near the transition point (i.e.,
the fractal dimension, critical exponents) are the same in
a given dimensionality, irrespective of the microscopic
details of the model [1].
Recently, Achlioptas, D’Souza, and Spencer [3] modi-
fied the growth of percolation clusters to produce a first-
order kinetic transition on the mean-field-like random
graph, through a procedure that is known as an
Achlioptas process. In this optimization method, intro-
duced to study problems in graph theory, two alternate
choices of adding a bond are considered, and a specific
strategy to favor a desired result (such as delaying or
accelerating the appearance of a giant component or the
formation of a Hamiltonian cycle) is followed [4,5]. In
Ref. [3], the authors considered a process of growth of
clusters in the Erdo˝s-Re´nyi random graph network model
in which two unoccupied edges between clusters are
chosen at random, and the one that minimizes the product
of the two connecting cluster masses is preferentially
chosen as the next occupied bond. This process led to a
sharp jump in the growth dynamics, which the authors
labeled as explosive percolative growth, in contrast to the
normal percolation phase transition, on both regular and
random lattices, where the transition is second-order and
continuous.
Erdo˝s-Re´nyi random networks are formed by randomly
linking pairs of points, irrespective of their distance apart.
These networks are essentially mean-field and infinite-
dimensional, and, for this problem, only the mass of the
clusters needs to be kept track of. There is a close connec-
tion between the percolation on Erdo˝s-Re´nyi networks and
percolation on the infinite-dimensional Bethe lattice and
polymerization of nonlooping branched polymers, whose
percolation or gelation theory goes back to Flory [6]. When
bonds are added randomly between sites in the random
graph, the net effective probability that a cluster of mass s
1
and a cluster of mass s
2
are joined is proportional to the
product of their masses s
1
s
2
. This leads to the product
kernel of the polymer growth process as described by the
Smoluchowski equation, and this growth process is equiva-
lent to percolation on the Bethe lattice [7]. Choosing the
bond with the minimum product leads to the minority
product rule (PR) of Ref. [3].
While random and scale-free networks have recently
received a great deal of attention [8], many actual networks
of interaction, whether physical or social, are restricted
spatially, and bonds can form only between close neigh-
bors. The two-dimensional percolation model is the stron-
gest form of such a restriction and has been the subject in
intense study for over 50 years [1]. Random percolation
can be viewed as the result of a growth process where
bonds are added to a single cluster (as in the Leath algo-
rithm) or between different clusters, as considered in
Ref. [9]. Both of these processes lead to second-order
transitions, as do their more dynamic cousins, the
directed-percolation and contact processes. Here we con-
sider the question of whether the PR rule produces a
discontinuous transition in the two-dimensional percola-
tion model also. Indeed, we find that the transition is very
sharp and explosive and find strong indication that the
transition is indeed first-order.
We consider bond percolation on L
L square lattices
with periodic boundary conditions in both directions, with
n ¼ L2 sites. Bonds are added randomly and one at a time,
keeping track of the current cluster structure of the system
(the so-called Newman-Ziff process) [9], modified by the
consideration of two unoccupied bonds that could connect
distinct clusters. The bond that minimizes the product of
the masses of the two clusters it joins is preferentially
chosen to become occupied, and the clusters are merged
into one. Time t represents the number of successful bonds
added, and each new bond, which always connects differ-
PRL 103, 045701 (2009) P HY S I CA L R EV I EW LE T T E R S
week ending
24 JULY 2009
0031-9007=09=103(4)=045701(4) 045701-1
2009 The American Physical Society

ent clusters, reduces the number of clusters by one. Sites
with no bonds are considered to be clusters of one site, so
initially we have n clusters. After t bonds are added, the
number of clusters N in the system equals n t. We
characterize the mass of a cluster by the number of sites
s. Because we add bonds only between sites of different
clusters, the bonds themselves form ‘‘minimum spanning
trees’’ over each cluster, yet for the conventional growth
process, where bonds are added one at a time, standard
percolation clusters (characterized by the sites that are
connected) are created.
We have carried out simulations of the cluster growth
using both the regular and PR processes, like in Ref. [3],
measuring the maximum cluster size C as a function of t.
The main plot of Fig. 1 shows the results of these simula-
tions for single realizations on a 1024
1024 lattice. In the
PR model, the transition is quite sharp. An expansion of
this curve shows that the jump substantially occurs over a
small number of time steps, as shown in the inset for
different system sizes (here with the axes scaled by n). In
contrast, for regular percolation, the transition is much
smoother. The simulations on different size lattices show
that the first-order transition in the PR model is robust.
For regular percolation, we can locate the transition
point exactly. While we do not know the effective overall
bond occupation fraction p, which includes bonds between
sites in a cluster in addition to the spanning tree, and thus
do not know when the square-lattice threshold pc ¼ 1=2 is
reached, we can identify the transition by finding the point
where the number of clusters per site reaches its critical
value [10,11]
Nc
n

3
ffiffiffi
3
p
 5
2
 0:098 076; (1)
which is known to have small finite-size corrections for a
periodic system [11]. This density will be reached when
t=n ¼ 1 Nc=n ¼ ð7 3
ffiffiffi
3
p
Þ=2  0:901 924. Of course,
for a standard system at pc, there will normally be some
fluctuation in Nc, but for large systems these fluctuations
will be small, so finding where the density exactly equals
1 Nc=n gives an excellent estimate of the critical point.
We have verified that, at this point, the average C for
different size systems scales as LD with D  1:8953, con-
sistent with the theoretical value 91=48  1:8958. On the
other hand, for the PR rule, we have no a priori knowledge
on where the transition point should be.
Achlioptas, D’Souza, and Spencer characterized the
transitions by two times, the time t
0
where C equals
ffiffiffi
n
p
and the time t
1
where C equals n=2, and then considered

¼ t
1
 t
0
. For unbiased growth, they argued that

should be proportional to n, while for the PR growth they
found
 n2=3. We have measured the same quantities for
regular and PR growth on square lattices with L ¼
32; 64; . . . ; 8192, with the number of realizations ranging
from 100 000 for the smaller sizes to 100 for the largest.
The results for
are plotted in Fig. 2.
This figure provides clear evidence that the two transi-
tions are of a quite different nature. The regular percolation
model does not correspond to
=n going to a constant as in
the Erdo˝s-Re´nyi case but instead decreases as
=n
L0:383 ¼ n0:192. In fact, t
1
converges rapidly to tc (be-
cause, for finite lattices of this size, at the critical pointC=n
is close to 1=2, which is the condition used to calculate t
1
).
The main variation in
is due to the variation in t
0
. By
usual scaling arguments, we expect the typical cluster mass
s to scale as s  jp pcj1=
, with
¼ 36=91 
0:3956 in 2d [1], so setting s ¼
ffiffiffi
n
p
¼ L to correspond
to the point t
0
, and using the fact that p
0
 pc is propor-
tional to ðt
0
 tcÞ=n  
=n for small p0  pc, we de-
duce that

=n n
=2 ¼ L36=91; (2)
which agrees fairly well with our numerical observations.
Actually, the maximum cluster size C differs from the
FIG. 1 (color online). Plot of regular percolation (blue) and the
PR (red) for bond percolation on a lattice of size 1024
1024,
showing the delayed and explosive growth in the PR model.
Points are every 1024 time steps. The inset shows the PR model
on an expanded scale, on systems of sizes L ¼ 256, 1024, 2048,
and 8192, with the scaled axes C=n vs t=n with n ¼ L2.
FIG. 2 (color online). Scaling of
=n ¼ ðt
0
 t
1
Þ=n for regu-
lar percolation (upper curve) and the PR model (lower curve)
with system size L.
PRL 103, 045701 (2009) P HY S I CA L R EV I EW LE T T E R S
week ending
24 JULY 2009
045701-2