ArticlePDF Available

Abstract and Figures

We describe the effect of power-law initial distributions of clusters on ordinary percolation and its generalizations, specifically, models of explosive percolation processes based on local optimization. These aggregation processes were shown to exhibit continuous phase transitions if the evolution starts from a set of disconnected nodes. Since the critical exponents of the order parameter in explosive percolation transitions turned out to be very small, these transitions were first believed to be discontinuous. In this article we analyze the evolution starting from clusters of nodes whose sizes are distributed according to a power law. We show that these initial distributions change dramatically the position and order of the phase transitions in these problems. We find a particular initial power-law distribution producing a peculiar effect on explosive percolation, namely before the emergence of the percolation cluster, the system is in a "critical phase" with an infinite generalized susceptibility. This critical phase is absent in ordinary percolation models with any power-law initial conditions. The transition from the critical phase is an infinite order phase transition, which resembles the scenario of the Berezinskii-Kosterlitz-Thouless phase transition. We obtain the critical singularity of susceptibility at this peculiar infinite-order transition in explosive percolation. It turns out that the susceptibility in this situation does not obey the Curie-Weiss law.
Content may be subject to copyright.
arXiv:1501.01312v1 [cond-mat.dis-nn] 6 Jan 2015
Solution of the explosive percolation quest. II. Infinite-order transition produced by
initial distributions of clusters
R. A. da Costa,1S. N. Dorogovtsev,1, 2 A. V. Goltsev,1, 2 and J. F. F. Mendes1
1Departamento de F´ısica da Universidade de Aveiro &I3N,
Campus Universit´ario de Santiago, 3810-193 Aveiro, Portugal
2A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
We describe the effect of power-law initial distributions of clusters on ordinary percolation and its
generalizations, specifically, models of explosive percolation processes based on local optimization.
These aggregation processes were shown to exhibit continuous phase transitions if the evolution
starts from a set of disconnected nodes. Since the critical exponents of the order parameter in
explosive percolation transitions turned out to be very small, these transitions were first believed
to be discontinuous. In this article we analyze the evolution starting from clusters of nodes whose
sizes are distributed according to a power law. We show that these initial distributions change
dramatically the position and order of the phase transitions in these problems. We find a particular
initial power-law distribution producing a peculiar effect on explosive percolation, namely before
the emergence of the percolation cluster, the system is in a “critical phase” with an infinite general-
ized susceptibility. This critical phase is absent in ordinary percolation models with any power-law
initial conditions. The transition from the critical phase is an infinite order phase transition, which
resembles the scenario of the Berezinskii–Kosterlitz–Thouless phase transition. We obtain the crit-
ical singularity of susceptibility at this peculiar infinite-order transition in explosive percolation. It
turns out that susceptibility in this situation does not obey the Curie-Weiss law.
PACS numbers: 64.60.ah, 05.40.-a, 64.60.F-
I. INTRODUCTION
The percolation transition is the key phase transition
occurring in disordered systems including disordered lat-
tices and random networks [1–4]. The gradual increase
of the number of links in a network or lattice leads to
the growth of clusters of connected nodes and eventu-
ally to the formation of a percolation cluster (giant con-
nected component) at the percolation threshold. This
phase transition was studied in detail, and understood
to be continuous in all disordered systems which were
explored. Recently a new class of irreversible percolation
processes, so-called “explosive percolation”, was intro-
duced [5], where the new links are added to the sys-
tem using metropolis-like algorithms. Although these
processes directly generalize ordinary percolation, they
demonstrate a set of features remarkably distinct from or-
dinary percolation. The unusual properties of this kind
of percolation led to the initial reports based on sim-
ulations [5–13] that these processes show discontinuous
transitions. Solving the problem analytically, we have
shown that the explosive percolation transitions are ac-
tually continuous [14]. These transitions have a so small
critical exponent of the percolation cluster size, that in
simulations of finite systems they can be easily perceived
as discontinuous [14, 15]. This conclusion was supported
by subsequent works of physicists [16–19] and mathe-
maticians [20].
In our previous work [21] we developed the scaling the-
ory of explosive percolation phase transitions for a wide
range of models, explaining the continuous nature of the
transitions and their unusual features. We obtained the
full set of relevant critical exponents and scaling functions
in the typical situation, in which the evolution starts from
isolated nodes or clusters with sufficiently rapidly decay-
ing size distribution. In our papers [14, 15, 21] we employ
the following model of explosive percolation, at which the
number of nodes Nis fixed and links are added one by
one. At each step we choose two sets of mrandom nodes,
from each set we select the node that is in the smallest
of mclusters, and then we add a new link between these
two nodes.
In the present article we show that the explosive perco-
lation transition, as well as ordinary percolation, strongly
depends on the initial conditions of the process. In par-
ticular, slowly decaying initial cluster size distributions
can change crucially the nature of these transitions. The
effects are interesting and add much to understanding of
explosive percolation and other generalizations of ordi-
nary percolation. So in the present article we explore in
detail the effect of initial conditions on the percolation
transitions in systems including ordinary and explosive
percolation models. We consider power-law initial clus-
ter size distributions with exponent ˜τ, and for different
values of the exponent find a spectrum of distinct criti-
cal behaviors. Here we introduce the initial cluster size
distribution exponent ˜τin contrast to the critical clus-
ter size distribution exponent traditionally denoted by τ.
Because of the power-law critical distribution we expect
that power-law initial conditions produce interesting ef-
fects. We will indicate the range of ˜τwhere the transition
point coincides with the initial moment of the process,
tc= 0. In particular, for ordinary percolation, if ˜τ= 3,
the percolation cluster emerges as Sexp(const/t),
where Sis the relative size of this cluster. In contrast,
for explosive percolation, we find that there exists a value
2
˜τ < 2 + 1/(2m1) ˜τ= 2 + 1/(2m1) ˜τ > 2 + 1/(2m1)
tc= 0 tc= 0 tc>0
P(s, 0) s1˜τP(s, 0) s1˜τP(s, tc)s3/2
m= 1 StβSexp(-const/t)S(ttc)β
β= (˜τ2)/(3 ˜τ) — β= 1
χt1χt2χ∼ |ttc|1
tc= 0 tc>0tc>0
P(s, 0) s1˜τP(s, tc)s1˜τlnλs P (s, tc)s1τ
StβSexp[-const/(ttc)µ]S(ttc)β
m > 1β=˜τ2
1(2m1)(˜τ2) µ=1
λ(2m1) 1τ2βe1.43m[21]
χt1χ(ttc)1µif t > tcχ∼ |ttc|1
χ=if t < tc
TABLE I. Summary of results. The initial distribution P(s, t = 0) s1˜τ, ˜τ > 2, Sis the relative size of the percolation
cluster, and χis the susceptibility. At ˜τ= 2 + 1/(2m1), we show only the most singular factor of S.
of ˜τat which the phase transition turns out to be infinite-
order and occurs at tc>0. In this situation, the system
at t < tcis in the “critical phase” with divergent sus-
ceptibility. We also find susceptibility at t > tcfor any
mand show that its critical exponent is nontraditional,
differing from the Curie–Weiss law. The main results of
the paper are presented in Table I.
The paper is organized in the following way. Section II
outlines our results and methods. In Sec. III we consider
effect of power-law initial conditions on ordinary perco-
lation (m= 1), which is the simplest particular case of
the more general model analyzed in this work. In Sec. IV
we study the effect of initial conditions on the explosive
percolation model (m2), which turns out to be prin-
cipally different from the case of m= 1.
II. RESULTS
To help the reader let us outline the main results of
this article. We use the following set of models. At each
time step a new link connecting two nodes is added to the
network of Nnodes. At each step sample two times: (i)
choose m1 nodes uniformly at random and compare
the clusters to which these nodes belong; select the node
within the smallest of these clusters; (ii) similarly choose
the second sampling of mnodes and, again, as in (i),
select the node belonging to the smallest of the mclus-
ters; (iii) add a link between the two selected nodes thus
merging the two smallest clusters. The resulting process
is described by the time dependent probability P(s, t)
that a randomly chosen node belongs to a finite cluster
of size s, where the time t=L/N , where Lis the num-
ber of added links (number of steps of the process). We
assume that Nis infinite. Then this aggregation process
is described by the evolution equation
∂P (s, t)
∂t =s
s1
X
u=1
Q(u, t)Q(su, t)2sQ(s, t),(1)
where Q(s, t) is the probability that a cluster chosen to
merge is of size s. This probability is expressed in terms
of P(s, t). In particular, when m= 1, the distribution
Q(s, t) coincides with P(s, t), and the model is reduced
to ordinary percolation.
We focus on the effect of slowly decaying initial distri-
butions P(s, t = 0), namely power laws P(s, 0)
=a0s1˜τ,
which can produce tc= 0 or, in the case of explo-
sive percolation, a critical phase. Using the evolution
equation (1) we analyze the Taylor expansion P(s, t) =
P(s, 0) + A1(s)t+A2t2+... and obtain the scaling form
of the distribution P(s, t)
=s1˜τf(st1 ). We obtain the
relation between the scaling function f(x) and the size
of the percolation cluster Stβ. We demonstrate that
when the exponent βis integer, there are two contribu-
tions to S. The first is the well known contribution deter-
mined by the scaling part of P(s, t) (see, for example, the
book [2]). However, there is a second, analytic, contri-
bution that was not considered in Ref. [2]. In particular,
the exponent βis 1 for ordinary percolation with rapidly
decaying P(s, 0). In this standard case, it is the combina-
tion of these two contributions S(t)
=const(|ttc|+ttc)
that produces S(t < tc) = 0 and the proper dependence
S(t > tc).
In Table I, we present a summary of the main results
of this article. The first row of the table shows the criti-
cal behaviors of the ordinary percolation model, m= 1,
for different ˜τ > 2. The values ˜τ > 3 result in a perco-
lation phase transition at tc>0 with standard critical
exponents. On the other hand, at 2 <˜τ3 the per-
colation cluster emerges at t= 0 with critical exponents
3
given in terms of ˜τ. In the marginal case of ˜τ= 3 we ob-
serve an infinite-order phase transition with a singularity
Sexp(const/t). This set of the critical singularities
of Sfor m= 1 agree with those obtained in Ref. [22].
Interestingly, the susceptibility χ=hsiPPssP (s)
obeys the Curie-Weiss law χ(ttc)1in all the con-
sidered situations except at ˜τ= 3, when χt2(note
that ˜τ3 gives tc= 0).
The second row of the table presents our results for ex-
plosive percolation m > 1. For these m, an infinite order
phase transition occurs when ˜τ= 2+1/(2m1) at tc>0.
Before tcthe system is in the critical phase, in which
the susceptibility [21] diverges, while in the percolation
phase the critical behavior of susceptibility χ(t > tc) dif-
fers from the Curie–Weiss law. We find that the size
distribution of clusters at the point of the infinite-order
phase transition is P(s, tc)s1˜τlnλs. We obtain the
exponent λclose to 1 solving the evolution equation nu-
merically.
Finally, we obtain a general relation between the sus-
ceptibility and the size of the percolation cluster close to
the critical point,
χ
=m
2
ln S
∂t .(2)
This simple relation is valid for all models and initial
conditions considered in this work. The detailed deriva-
tions of these analytical results are given in subsequent
sections.
III. EFFECT OF INITIAL CONDITIONS IN
ORDINARY PERCOLATION (m= 1)
In the case of m= 1, our process is actually ordi-
nary percolation, that is at each step we chose uniformly
at random two nodes and interconnect them. This can
be treated as an aggregation process in which at each
step two clusters, chosen with probability proportional
to their sizes, merge together. This process is described
by the probability distribution P(s, t) that a randomly
selected node belongs to a cluster of size sat moment
t(each step increases time tby 1/N) . In the infinite
system, the evolution of this distribution is described by
the following Smoluchowski equation [23, 24]:
∂P (s, t)
∂t =s
s1
X
u=1
P(u, t)P(su, t)2sP (s, t) (3)
for a given initial distribution P(s, t = 0). Defining the
generating function ρ(z, t) as
ρ(z, t) =
X
s=1
P(s, t)zs,(4)
we can rewrite Eq. (3) in terms of ρ, and analyze the
resulting partial differential equation:
∂ρ(z , t)
∂t =2[1 ρ(z , t)] ρ(z, t)
ln z,(5)
whose solution ρ(z, t) can be obtained from
ln z= 2t(1 ρ) + g(ρ),(6)
where the function g(ρ) is determined by initial condi-
tions.
Let us find this function. Our results will be completely
determined by the asymptotic of the initial distribution.
An initial cluster size distribution with a power-law tail,
i.e., P(s)
=a0s1˜τfor large s, leads to the following
singularity of the generating function
ρ(1) ρ(z)=
X
s=1
P(s)(1 zs) = a0X
s
s1˜τ(1 zs)
=a0(ln z)˜τ2Z
0
dx x1˜τ(1 ex)
=a0Γ(2 ˜τ) (ln z)˜τ2,(7)
at z= 1.
We find the function g(ρ), replacing the left-hand side
of Eq. (6) by Eq. (7) and putting ρ(1) = 1, since S=
1ρ(1) = 0 at t= 0. Then Eq. (6) becomes
ln z= 2t(1 ρ)1ρ
a0Γ(2 ˜τ)1/τ2)
,(8)
which is valid when z1. Putting z= 1 and 1 ρ=S
we get
2tS =S
a0Γ(2 ˜τ)1/τ2)
.(9)
The last equation has two solutions, the trivial one S= 0
and a nontrivial solution
S= (2t)τ2)/(3˜τ)[a0Γ(2 ˜τ)]1/(3˜τ),(10)
If ˜τ > 3 then Γ(2 ˜τ)>0, and Eq. (9) has only one
real solution S= 0, showing that the transition does not
occur at tc= 0 for this range of ˜τ. In this case, to find a
real solution S > 0 for t > tc>0 we must also consider
the analytic terms omitted in Eq. (7). For the range
2<˜τ < 3, we have Γ(2 ˜τ)<0 and the solution (10) is
real and positive, that is
S
=Bt( ˜τ2)/(3˜τ),(11)
for small t, and B= 2( ˜τ2)/(3˜τ)[a0Γ(2 ˜τ)]1/(3˜τ).
Therefore, if ˜τ < 3, the transition occurs at the initial
moment. Note that for ˜τ < 3, the first moment of the ini-
tial distribution diverges. For ordinary percolation this
moment has the meaning of susceptibility [2], and its di-
vergence indicated that the poit t= 0 is indeed the crit-
ical point. We will consider the case of ˜τ= 3 separately
in Sec. III B.
It is easy to see that the distribution P(s, t) is an ana-
lytic function of t. (i) The initial distribution P(s, 0) has
4
no divergencies at any s. (ii) Let us, for a moment as-
sume that P(s, t) at has singularity tφwith non-integer
φat t= 0. Then the lowest non-integer power on the
left-hand side of Eq. (3) is tφ1, while on the right-hand
side the lowest non-integer power is tφ, which shows that
the assumption was not correct. Then we can write the
Taylor expansion of the function P(s, t) around t= 0:
P(s, t) = A0(s) + A1(s)t+A2(s)t2+A3(s)t3+. . . . (12)
The first term in expansion (12) is the initial distribu-
tion, A0(s) = P(s, 0). The coefficient of the second term,
A1(s), is the first derivative tP(s, t = 0), A2(s) is the
second derivative (1/2)2
tP(s, t = 0), and so on. Then,
given an initial distribution P(s, 0), we can find the co-
efficients Ai(s) sequentially differentiating both sides of
Eq. (3). We analyze these coefficients in different ranges
of ˜τ.
The remainder of this section is organized in the fol-
lowing way. In Sec. III A we consider the case of ˜τ < 3.
In this region, we derive the scaling of the distribution
P(s, t) containing a scaling function. We obtain the rela-
tion between the analytical features of this scaling func-
tion and the singularity of the relative size Sof the perco-
lation cluster at t= 0. In Sec. III B we consider the case
of ˜τ= 3. We show that in this situation all the deriva-
tives of S(t) with respect to tare zero at t= 0 and derive
the respective scaling function. In Sec. I II C we analyze
the singularity of the susceptibility of this problem, and
its relation with the percolation cluster size S.
A. The case of ˜τ < 3
Let us first consider the case of ˜τ < 3, derive the scaling
of the distribution P(s, t), and the critical singularity of
the percolation cluster size.
1. Scaling of P(s, t)
We consider initial configurations without a percola-
tion cluster, so PsA0(s) = 1. We find the coefficients
An(s), n>0, using the identity (n)
tP(s, t = 0) An(s)n!
and Eq. (3). For the second coefficient we have
A1(s) = s
s1
X
u=1
A0(u)A0(su)2sA0(s).(13)
The sum Pu<s A0(u)A0(su) cannot be directly re-
duced to the integral a2
0s3τR1
0dx [x(1x)]1˜τfor large
s, because it diverges at both limits 0 and 1. So, following
our work [21], we rewrite Eq. (13) as
A1(s) =s
s1
X
u=1
[A0(u)A0(s)] [A0(su)A0(s)]
s(s1)[A0(s)]2+ 2A0(s)
s1
X
u=1
A0(u)2sA0(s)
=s
s1
X
u=1
[A0(u)A0(s)] [A0(su)A0(s)]
s(s1)[A0(s)]22sA0(s)
X
u=s
A0(u),(14)
where we took into account the normalization condition
PsA0(s) = 1. For large s, the sums in last equation can
be already reduced to integrals, which converge at both
limits, and A0(s) can be replaced by a0s1˜τ. Then we
get
A1(s)
=a2
0s4τZ1
0
dx x1˜τ1(1 x)1˜τ1
a2
0s4τ2a2
0s4τZ
1
dx x1˜τ
=a2
0
Γ(2 ˜τ)2
Γ(4 τ)s42 ˜τa1s42˜τ,(15)
where we have introduced the coefficient a1.
The case of ˜τ= 5/2 is special. For this ˜τthe coefficient
a1in Eq. (15) is zero, and A1decays faster than s4τ. To
find the asymptotics of A1in this situation, we must take
into account the higher-order terms that were neglected
when passing from the sums in Eq. (14) to the integrals
of Eq. (15). In general, we estimate the difference of the
respective integral and sum with arbitrary exponents ψ
and φ
sZs
0
du [uψsψ][(su)φsφ]
s
s1
X
u=1
[uψsψ][(su)φsφ] = O(smin(ψ,φ)),
(16)
where we assume that 0 < ψ , φ < 2. In particular, for
A1, we have ψ=φ= ˜τ1. Then, when ˜τ= 5/2, the
large sasymptotics of A1is
A1(s)∝ −s1˜τ∝ −s3/2if ˜τ= 5/2.(17)
In Appendix A we calculate the asymptotics of A2(s):
A2(s)
=a3
0
2Γ(2˜τ)3
3Γ(6τ)s73 ˜τa2s73˜τ.(18)
For the particular values ˜τ= 7/3 and 8/3 the coefficient
a2= 0. To find the asymptotic behavior of A2in these
5
cases it is necessary to consider higher-order terms that
were neglected when passing from Eq. (A2) to Eq. (A3),
similarly to how we treated A1. This analysis gives
A2(s)
s2˜τ∝ −s1/3if ˜τ= 7/3,
s4τ∝ −s4/3if ˜τ= 8/3.
(19)
These calculations can be repeated for the next terms
of expansion (12), and in general we find that the n-th
term
An(s)
=ans1˜τ+n(3˜τ)(20)
for large s. In Appendix B, we obtain the general expres-
sion
an=[2a0Γ(2˜τ)]n+1
2Γ[(n+1)(2˜τ)](n+1)!.(21)
Notice that this expression generalizes the results for a1
and a2, Eqs. (15) and (18), respectively. The coefficients
anare expressed in terms of a0and ˜τand become zero
for ˜τ= 3 1/(n+1). For this ˜τ, the coefficient An(s)
decays as s11/(n+1).
Equation (20) enables us to write the Taylor expansion
of P(s, t) in the form:
P(s, t)
=s1˜τ
X
n=0
ans3˜τtn=s1˜τf(st1/(3˜τ)).(22)
The function f(x) is the scaling function of the prob-
lem. This function is only analytic at zero for ˜τ= 5/2
when coefficients anare zero for odd n. In general, for
2<˜τ < 3, f(x) is represented as the series
f(x) = a0+a1xσ+a2x2σ+a3x3σ..., (23)
where
σ= 3 ˜τ . (24)
Let us estimate the radius of convergence xrc of this se-
ries. Expansion (12) is a convergent series for t < r,
r= lim
n→∞
An(s)
An+1(s)
s1˜τ+n(3˜τ)
s1˜τ+(n+1)(3˜τ)s˜τ3.(25)
Then series (23) is convergent for st1/(3˜τ)=x < xrc =
sr1/(3˜τ)1.
Let us consider, for example, the particular case ˜τ=
5/2. In this case the scaling function, after substituting
Eq. (21) into Eq. (23), reproduces the known result for
the mean-field percolation transition at tc>0 [1]:
f(x) = a0
X
n=0
(4πa2
0x)n
n!=a0e4πa2
0x.(26)
Note that when ˜τ= 5/2 the coefficients anin se-
ries of Eq. (23) with odd nare zero. If we set a0
to 1/2πin Eq. (26), we arrive at the known form
f(x) = exp(2x)/2πfor the percolation process start-
ing from isolated nodes [1]. Figure 1 shows scaling func-
tions for several values of ˜τ.
00.5 11.5 2
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
f(x)
00.5 11.5
xσ
0
0.2
0.4
0.6
f(x)
τ=2.7
τ=2.6
τ=2.5
τ=2.4
τ=2.2
FIG. 1. Scaling function f(x) calculated using the first 1000
terms of the series (23) for a0= 1/2πand different values of
˜τ. In inset, f(x) is shown as a function of xσs3˜τt, where
σ= 3 ˜τ. In this representation the function fis analytic
at the origin, where its derivative at the origin changes signs
when ˜τcrosses 5/2.
2. Relation between the singularity of Sand the scaling
function f(x)
The relative size of the percolation cluster Snear tcis
independent of the details of distribution P(s, tc) in the
region of small s(non-scaling region). This enables us to
use the scaling form of distribution P(s, t), Eq. (22), to
recover expression (11) for S(t) near tc. To this end we
start from the definition of S
S1X
s
P(s, t),
= 1 X
sA0(s) + A1(s)t+A2(s)t2+....(27)
At t= 0 the relative size of the percolation cluster S= 0,
which implies PsA0(s) = 1, and so we can write
S=X
n1
tnBn,(28)
which is the Taylor expansion of Sat zero with the coef-
ficients
BnX
s
An(s).(29)
In Eq. (28), the coefficients BnPss1˜τ+n(3˜τ). They
diverge if ˜τ < 31/(n+ 1) and converge if ˜τ
31/(n+ 1). (Recall that at ˜τ= 31/(n+1) the coef-
ficient anbecomes zero and asymptotics of An(s) decays
as s11/(n+1).) For any ˜τ < 3 the coefficients Bnare
infinite for n > β, where
β=˜τ2
3˜τ.(30)
6
The divergence of these coefficients in the Taylor se-
ries (28) indicates the singularity of Sat t= 0. We
extract this singularity from series (28) in the follow-
ing way. We divide the sum Pn1Bntninto two parts,
PnnBntn+Pn>nBntn. Here
n=β(31)
denotes the largest integer smaller or equal to β. In
the second part we replace Anby their asymptotic form,
namely
S=X
1nn
BntnX
n>nX
s
ans1˜τ+ tn
X
n>n
tnX
sAn(s)ans1˜τ+
=X
1nn
BntnX
s
s1˜τX
n>n
an(st1)
X
s
s2X
n>n
O[(st1)]
=X
1nn
BntnX
s
s1˜τf(st1 ).(32)
Here we have used the fact, following from Eq. (16),
that the deviations from the asymptotics An(s)
ans1˜τ+n(3˜τ)are of the order of s1˜τ+(n1)(3˜τ)for
large s. The function f(x) is a new scaling function
obtained from f(x) by subtracting the first n+ 1 terms
of its expansion over xσ,
f(x) = f(x)a0a1xσ... anxnσ,(33)
σ= 3 ˜τ. In particular, if ˜τ= 5/2, then β= 1, a1= 0,
and so f(x) = f(x)a0.
In Appendix B we find PsAn(s)Bnemploying the
generating functions approach:
Bn=
0 if n < β,
2n[a0Γ(2 ˜τ)]n+1
n+ 1 if n=β,
if n > β,
(34)
Therefore the first sum on the right-hand side of Eq. (32)
is zero, except when βis integer, equal to n, and the
sum is Bntβ. Thus, the singularity of the size of the
percolation cluster is
S
=X
s
s1˜τf(st1 )Bntβ
=tτ2)Z
0
dx x1˜τf(x) + Bn.(35)
Note that the integral R
0dx x1˜τf(x) converges at
the lower limit because we have subtracted from f(x)
all terms leading to divergence, see Eq. (33). Recall-
ing that σ= 3 ˜τ, we arrive at the same singularity
S=Bt( ˜τ2)/(3˜τ)as in Eqs. (11),
B= 2τ2)/(3˜τ)[a0Γ(2 ˜τ)]1/(3˜τ)
=Z
0
dx x1˜τf(x)Bn.(36)
The first term on the right-hand side of Eq. (35) is the
singular contribution from the scaling behavior P(s, t)
=
s1˜τf(st1 ). The second term on the right-hand side
of Eq. (35) is nonzero only when β= (˜τ2)/(3 ˜τ) is
integer and equal to n. The contribution of this term
comes from the finite sregion, BnPss11/(n+1) ,
so it is not included in the scaling function f(x). This
term is an analytic contribution to Sat ˜τ= 3 1/(n+ 1),
n= [1..], including ˜τ= 5/2.
Let us consider briefly the ordinary percolation model
(m= 1) with an initial distribution P(s, t) decaying
faster than s2. Then the transition takes place at
tc>0, and the critical distribution P(s, tc)
=a0s1τ
with τ= 5/2. In this situation, the Taylor expansion of
P(s, t) around t=tchas coefficients angiven by Eq. (21)
with 5/2 substituted for ˜τ. As a result, the scaling form
of the distribution P(s, t) is
P(s, t)
=a0s3/2e4πa2
0s(ttc)2(37)
for tapproaching tcfrom above and below, where the
scaling function is f(x) = a0exp(4πa2
0x) in both
phases. To obtain this result we simply replace ˜τby
5/2 and tby ttcin Eqs. (22) and (26). Making the
same replacements in Eq. (35), we get
S
=a0X
s
s3/2e4πa2
0s(ttc)21(ttc)B1
=−|ttc|a0Z
0
dx x3/2e4πa2
0x1(ttc)B1
=4πa2
0(|ttc|+ttc).(38)
Note that according to Eq. (34), for ˜τ= 5/2, the co-
efficient B1=R
0dx x3/2(e4πa2
0x1) = 4πa2
0. This
equation describes Sin both phases, giving S= 0 for
t < tc, and S= 8a2
0π(ttc) for t > tc.
When β= (τ2)is non-integer, only the scaling
region contributes to the singularity of the percolation
cluster size
S
=−|ttc|βZ
0
dx x1τf(x).
In this situation, as is noted in Ref. [2], the scaling func-
tion f(x) must be different below and above tc. Namely,
in the phase t > tcthe scaling function integral
Z
0
dx x1τf(x) = B,
7
while in the phase t < tcthe integral
Z
0
dx x1τf(x) = 0
to comply with S= 0. Consequently the scaling function
f(x) = f(x) + a0+... +anxnσmust have a maximum
f(xmax>0) > f (0) in the phase t < tc[2]. This asym-
metry of the scaling function is observed, for example,
for ordinary percolation at dimension d < 6 [25], and
in explosive percolation [6, 14, 21]. In these examples,
0< β < 1. On the other hand, when βis integer, the
non-scaling region additionally contributes to the singu-
larity of S. This contribution, which is important, in
particular, for ordinary percolation above the upper crit-
ical dimension 6 where β= 1, was not considered in
the book [2]. Thanks to the non-scaling contribution
B1(ttc) to S,f(x) is the same monotonically decreas-
ing function in both phases in this situation, see Eqs. (37)
and (38).
B. The case of ˜τ= 3
Now we show that in the case of τ= 3 all derivatives
of S(t) with respect to tare zero at t= 0 and then we
find the respective scaling function. In this situation the
singularity of the generating function ρ(z , 0) at z= 1
differs from that in Eq. (7), namely
1ρ(z, 0)
=a0(1 z) ln(1 z).(39)
The inverse function is
ln z
=z1
=1ρ
a0ln(1 ρ).(40)
The right-hand side of this expression gives the function
g(ρ) in Eq. (6). Using Eqs. (6) and (40) we write the
equation for the generating function ρ(z, t) near z= 1 as
follows
ln z= 2t(1 ρ) + 1ρ
a0ln(1 ρ).(41)
At z= 1 this equation gives the relative size of the per-
colation cluster for small t:
Se1/(2a0t).(42)
Therefore, if ˜τ= 3, the transition occurs at the initial
moment, and all the derivatives of Sare zero at t= 0,
so the percolation transition is infinite-order.
1. Scaling of P(s, t)
In the case of ˜τ= 3, instead of the Eqs. (20) and (21)
for ˜τ < 3, we derive the general asymptotic expression
for the coefficients An(s):
An(s)
=2n(n+1)an+1
0s2(ln s)n(43)
00.5 1
t lns
0
5
10
15
P(s,t)s2
0 0.1 0.2 0.3
t lns
0.6
1
1.4
P(s,t)s2
105
104
103
102
FIG. 2. Numerical solution of evolution equation (3) for
s105with the initial condition P(s, 0) = a0s2, where a0=
ζ(2)1. Solid lines, curves P(s, t)s2vs. tln sfor the cluster
sizes sindicated by the numbers in the plot. Dashed line,
scaling function from Eq. (45), f(tln s) = a0/(1 2a0tln s).
Inset, highlight of the small tln sregion, in which the curves
here depicted approach f(tln s).
(see Appendix C for the derivation). Within the radius
of convergence of the series (12), this expression gives the
following scaling form of the distribution P(s, t):
P(s, t)
=s2f(tln s),(44)
where
f(x) = a0(1 2a0x)2.(45)
The function f(x) plays the role of a scaling function,
which is represented by the Taylor series (43) for tup to
the radius of convergence, t < r = 1/(2a0ln s). Surpris-
ingly, in contrast to the case of ˜τ < 3, the scaling function
for ˜τ= 3 diverges approaching xrc = 1/(2a0) from below.
This divergence requires interpretation, since the distri-
bution P(s, t) itself cannot be divergent at any cluster
size, including s= exp[(2a0t)1]. In general, for t < r,
lim
s→∞ P(s, t)s˜τ1f(x),(46)
where xis a scaling variable such as |ttc|sσor |ttc|ln s.
If the scaling function converges everywhere in the range
of xbetween 0 and infinity, then the curves P(s, t)s˜τ1
vs. xwill collapse into f(x) at sufficiently large s. In the
case of ˜τ= 3 the curves P(s, t)s˜τ1vs. xtend to f(x)
when s→ ∞ in the region x < xr c, see Fig. 2. These
curves have a maximum near x= 1/(2a0). As is shown
in Fig. 2, the height of this maximum increases, and the
difference between P(s, t)s2and f(tln s) decreases, as s
grows. Note that the curves in Fig. 2 tend to f(tln s)
only for tln s < 1/(2a0), above this point they tend to
zero.
In the case of ˜τ= 3 we cannot formulate a relation
similar to Eq. (35) between the scaling function f(x) and
8
the singularity of S(t) at 0 . All the coefficients Bnof
the series S(t) = Pn1Bntnare 0, which indicates an
infinitely smooth singularity of S(t) at 0.
C. The singularity of susceptibility
Let us find the singularity of the susceptibility and its
relation with the percolation cluster size S. In mean-
field models, near the critical point, the susceptibility
typically follows the Curie-Weiss law χ∼ |ttc|1. It
is easy to see that this is the case for the ordinary per-
colation model if the exponent of the initial cluster size
distribution ˜τ6= 3. In ordinary percolation the suscep-
tibility is the average cluster size to which a uniformly
randomly chosen node belongs [2], namely
χ=hsiP=X
s
sP (s).(47)
For m= 1 the singularities of χand Scan be related in
the following way. Summing over sboth sides of Eq. (3)
gives
∂S
∂t = 2ShsiP= 2S χ. (48)
In the case of ˜τ < 3 the percolation cluster size Shas a
power-law singularity with exponent β= (˜τ2)/(3 ˜τ),
Eq. (11), then
χ=1
2
ln S
∂t
=˜τ2
2(3 ˜τ)t1(49)
for small t.
At ˜τ= 3, we obtain an anomalous singularity of the
susceptibility which diverges at t= 0, χ(t=0) Pss1.
Substituting Se1/(2a0t)[Eq. (42)] into Eq. (48) we
find
χ=1
2
ln S
∂t
=1
4a0
t2,(50)
differing from the Curie-Weiss law. Note that when
˜τ > 3, the initial susceptibility does not diverge, and
the transition occurs at tc>0 with standard exponents.
When ˜τ < 3, the initial susceptibility diverges, the transi-
tion occurs at tc= 0, and the susceptibility demonstrates
the standard behavior, χt1.
IV. EFFECT OF INITIAL CONDITIONS ON
EXPLOSIVE PERCOLATION (m > 1)
In this section we extend the analysis made in the pre-
vious section for m= 1 to m > 1. We consider the
following explosive percolation model [14, 15, 21]. The
evolution starts from a given distribution of clusters. At
each step, we choose at random two set of mnodes. Then
the node in the smallest cluster of each set is selected,
and these two nodes are interconnected. The evolution
equation for an arbitrary mtakes the form:
∂P (s, t)
∂t =s
s1
X
u=1
Q(u, t)Q(su, t)2sQ(s, t),(51)
where Q(s, t) is the probability that a chosen node seats
in a cluster of size s. The relation between Q(s) and P(s)
is
Q(s) = "1
s1
X
u=1
P(u)#m
"1
s
X
u=1
P(u)#m
=mP (s)"1
s
X
u=1
P(u)#m1
,(52)
where the last approximate equality takes place at large
s[21]. In the phase with the percolation cluster, this
relation between distributions P(s, t) and Q(s, t) is sim-
plified:
Q(s)
=mSm1P(s).(53)
Then we can write the partial differential equation for
the generating function ρ(z , t) defined in Eq. (4) for any
m:
∂ρ
∂t =2[S(t)]2(m1) m2(1ρ)m(m1)S(t)ρ
ln z.
(54)
This partial differential equation can be solved by ap-
plying the hodograph transformation [21], which leads to
the following equation
Zt
0
dtln z
∂t= ln z
=Zt
0
dt2[S(t)]2(m1) m2(1ρ)m(m1)S(t)
+g(ρ),(55)
where the function g(ρ) is determined by initial condi-
tions. Similarly to Eq. (8), we have
g(ρ) = 1ρ
a0Γ(2 ˜τ)1/τ2)
.(56)
At z= 1, the generating function ρ(1, t) = 1 S(t), so
0 = 2m2S(t)Zt
0
dt[S(t)]2(m1)
2(m2m)Zt
0
dt[S(t)]2m1S(t)
a0Γ(2˜τ)1/τ2)
.
(57)
One can see that
S=Btβ(58)
9
is a non-trivial solution of Eq. (57) with
β=˜τ2
1(2m1)(˜τ2) ,(59)
and
B= [a0Γ(2˜τ)]β/( ˜τ2)2m(˜τ2)[1+(m1)(˜τ2)]
(3 ˜τ)ββ
.
(60)
In Ref. [21] we introduced the generalized susceptibility
for these explosive percolation models. The susceptibil-
ity χis introduced in terms of the probability c2that
two nodes selected by our algorithm belong to the same
cluster
c2=χ
N+S2m=1
NX
s
sQ2(s)
P(s)+S2m.(61)
In particular, when m= 1, this susceptibility is reduced
to the standard one for ordinary percolation χ=hsiP[2].
In [21] we showed that the susceptibility is divergent at
t= 0 if the exponent of the initial cluster size distribution
˜τ2+1/(2m1). For ˜τ > 2 + 1/(2m1) the transition
occurs at tc>0, with the critical exponents and scaling
functions calculated in [15, 21]. If ˜τ < 2 + 1/(2m1)
the size of the percolation cluster follows the power-law
S
=Btβ, with βand Bgiven by Eqs. (59) and (60)
respectively. Here the case of ˜τ= 2 + 1/(2m1) for
m > 1 distinguishes itself significantly from that for m=
1. In the next subsection we show that an infinite-order
percolation transition takes place at tcexceeding zero
when ˜τ= 2 + 1/(2m1) and m > 1.
The remainder of this section is organized in the fol-
lowing way. In Sec. IV A we consider the region ˜τ <
2 + 1/(2m1). For this range of τwe derive the scal-
ing form of the distribution P(s, t) and relate the scaling
function and the singularity of S. In Sec. IV B we con-
sider the case of ˜τ= 2 + 1/(2m1) and show that in
this situation we have an infinite-order phase transition.
Next we derive scaling functions for this problem and the
singularity of the relative size of the percolation cluster
S. Finally in Sec. IV C we find the critical singularity of
susceptibility.
A. The case of ˜τ < 2 + 1/(2m1)
Let us first consider the region ˜τ < 2 + 1/(2m1),
derive the scaling form of the distribution P(s, t), and
obtain the critical singularity of the percolation cluster
size.
1. Scaling of P(s, t)
For m > 1, the coefficients An(s) of the Taylor expan-
sion of P(s, t), Eq. (12), can be calculated in a similar
way to the case of m= 1. Using Eq. (51) at t= 0 we
write
A1(s) =s
s1
X
u=1
Q(u, 0)Q(su, 0) 2sQ(s, 0)
=s
s1
X
u=1
[Q(u, 0) Q(s, 0)] [Q(su, 0) Q(s, 0)]
s(s1)Q(s, 0)22sQ(s, 0)
X
u=s
Q(u, 0),(62)
where we have used the normalization condition
PsQ(s, 0) = 1. The asymptotics of Q(s, 0) are obtained
substituting the power-law initial distribution P(s, 0) =
A0(s)
=a0s1˜τinto Eq. (52):
Q(s, 0)
=mA0(s)"X
u>s
A0(u)#m1
=mam
0
sm(2˜τ)1
τ2)m1,
(63)
Then the asymptotics of A1is
A1(s)
=m2a2m
0
s2m(2˜τ)
τ2)2m2"12Z
1
dx xm(2˜τ)1
+Z1
0
dx [xm(2˜τ)11][(1x)m(2˜τ)11]#
=a1s2m(2˜τ),(64)
where
a1=a2m
0m2Γ[mτ2)]2
τ2)2m2Γ[2m(˜τ2)] .(65)
For ˜τ= 2 + 1/(2m) the coefficient a1becomes zero, and
we need to find the next term. This term can be obtained
by taking into account the next-to-leading order term in
the expansion of the right-hand side of Eq. (62) in powers
of s. For this sake we use Eq. (16). In this way, when
a1= 0, we get the following asymptotics of A1(s)
sm(2˜τ)1=s3/2.
Repeating the procedure for the next coefficients An(s)
we find that the n-th coefficient has the power-law
asymptotics:
An(s)
=ans1˜τ+ ,(66)
where the exponent
σ= 1 (2m1)(˜τ2),(67)
for any m. We express the prefactors anin terms of m,
˜τand a0, similar to a1. For instance, for a2and a3we
10
find:
a2=2a4m1
0m2Γ[1mτ2)]3Γ[2(3m1)(˜τ2)]
τ2)4m2Γ[22m(˜τ2)]Γ[1(4m1)(˜τ2)] ,(68)
a3=4a6m2
0m4Γ[1mτ2)]4Γ[3(5m2)(˜τ2)]
3(˜τ2)6m4Γ[2(6m2)(˜τ2)]Γ[22mτ2)]2
×"Γ[2(3m1)(˜τ2)] Γ[2(3m1)(˜τ2)]
Γ[3(5m2)(˜τ2)]
[1(4m1)(˜τ2)]Γ[22m(˜τ2)]
mτ2)Γ[3(4m1)(˜τ2)] !
(m1)(˜τ2)Γ[m(˜τ2)]#.(69)
When m > 1, we have to derive expressions for the co-
efficients anindividually, unlike the general expression
(21) for all anin the case of m= 1. Each coefficient
an>0becomes zero when ˜τ= 2 + n/[n(2m1)+1]. In
this case, the resulting asymptotics of An(s) decay as
s1[n(m1)+1]/[n(2m1)+1].
The sums PsAn1(s)Bnin the equations for
An2(s), are obtained by the generating function ap-
proach explained in Appendix B in detail for m= 1.
We find the singularity
ρn(z)
=anΓ[2˜τ+](1z)˜τ2 ,(70)
by differentiating the evolution equation (51) and rela-
tion (52). The value of Bn=ρn(1) diverges or converges
depending on ˜τ, similarly to the case of m= 1:
Bn=
0 if n < β
anΓ[2˜τ+n{1(2m1)(˜τ2)}] if n=β
if n > β
(71)
where β= (˜τ2)/[1 (2m1)(˜τ2)].
The function P(s, t) can be written in the scaling form:
P(s, δ)
=s1˜τ
X
n=0
an(sσδ)n=s1˜τf(1 ).(72)
Here we used the asymptotic behavior of the coefficients
An(s). In the vicinity of x= 0, the expansion of the
function f(x) is
f(x) = a0+a1xσ+a2x2σ+a3x3σ..., (73)
where σ= 1(2m1)(˜τ2) and the coefficients anare
given by Eqs. (65), (68), (69), etc. The scaling function
and Sare interrelated in the same way as in Eq. (35)
from Sec. III A,
S
=δτ2)Z
0
dx x1˜τf(x) + Bn,(74)
where σ= 1 (2m1)(˜τ2), β= (˜τ2), and
n=β. The function f(x) = f(x)Pnnanxas
in Eq. (33). The constant
Bn=X
s
An(s) = ρn(1) = anΓ (2 ˜τ+nσ)
in nonzero only when βis integer, β=β⌋ ≡ n, see
Eq. (71).
B. The case of ˜τ= 2 + 1/(2m1)
Let us consider the case of ˜τ= 2+1/(2m1) and show
that in this situation we have an infinite-order phase tran-
sition. When m > 1, the coefficients Anin the case of
˜τ= 2+1/(2m1) have exactly the same form (66) as for
˜τ < 2 +1/(2m1). This coincidence is due to the conver-
gent convolution integral in Eq. (64) for A1and similar
integrals for An>1. This is in contrast to m= 1, ˜τ= 3,
where we have the coefficients Ans2(ln s)nwith log-
arithmic factors unlike Ans1˜τ+for m= 1, ˜τ < 3,
see Sec. III B. Recall that these logarithms emerged due
to the divergent convolution integral in Eq. (15) for A1
at ˜τ= 3 and similar integrals for An>1. In this respect,
there is a principle difference between m= 1 and m > 1.
Below we show that for m > 1 and ˜τ= 2 + 1/(2m1)
an infinite-order percolation transition takes place at a
tc>0.
1. Expansion of P(s, t)at t= 0
When ˜τ= 2+1/(2m1), we have the exponent σ= 0,
so the coefficients An(s)s1˜τfor all n, see Eq. (66).
Consequently the asymptotics of the cluster size distri-
bution is
P(s, t)
=s2m/(2m1)
X
n=0
antn=s2m/(2m1)f(t).
(75)
Substituting ˜τ2 = 1/(2m1) into Eqs. (65), (68), (69),
etc., we find the general expression for the coefficients an
an=a0Γ[n+1/(2m1)]
n! Γ[1/(2m1)]
×2m[a0(2m1)]2m1Γ[(m1)/(2m1)]2
Γ[2(m1)/(2m1)] n
,
(76)
and so
f(t) =
X
n=0
antn=a01t
r1/(12m)
,(77)
where
r=(2m1)12mΓ[2(m1)/(2m1)]
2a2m1
0mΓ[(m1)/(2m1)]2.(78)
11
00.05 0.1 0.15 0.2
t
0
0.2
0.4
0.6
P(s,t)s4/3
102
105103
104
FIG. 3. Numerical solution of the evolution equation (51),
for m= 2 and s105with the initial condition P(s, 0) =
a0s4/3, where a0=ζ(4/3)1. Solid lines, curves P(s, t)s4/3
vs. tfor the cluster sizes sindicated by the numbers in the
plot. Dashed line, function f(t) in Eq. (77).
The function f(t) diverges at t=r. Fig. 3 demonstrates
how the curves P(s, t)s2m/(2m1) vs. tapproach f(t) in
the region t < r as sapproaches infinity.
Substituting Eq. (75) into Eq. (52), we obtain the
asymptotics of the distribution Q(s, t):
Q(s, t)
=m(2m1)m1f(t)ms1m/(2m1).
The first moment of this distribution
hsiQ=X
s
sQ(s, t)f(t)mX
s
sm/(2m1)
is divergent in the interval 0 tr. Note that also
hsiP=f(t)Pss1/(2m1) and χf(t)2m1Pss1are
divergent in this interval. Summing over sboth sides of
Eq. (51) we get
∂S
∂t = 2SmhsiQ.(79)
Due to the divergence of hsiQ, the only solution of this
equation in the interval tris S= 0. Thus we have a
phase without percolation and with divergent susceptibil-
ity, which enables us to call this phase “critical”. Below
we show that the percolation threshold tcof this transi-
tion is exactly r. For t > r, we show in Appendix F that
the moments hsiQand hsiPare finite.
2. Scaling of P(s, t)at tc
For t < tcthe function f(t) is convergent, and for large
sthe curves P(s, t)s2m/(2m1) collapse into the func-
tion f(t), see Fig. 3. The value of P(s, tc)s2m/(2m1)
approaches infinity as s→ ∞. Our numerical results,
Fig. 4, indicate that P(s, tc)s2m/(2m1) grows linearly
100101102103104105
s
0.2
0.3
0.4
0.5
P(s,tc)s4/3
0.23+0.024 ln s
(a) m=2
100101102103104105
s
0.15
0.2
0.25
P(s,tc)s6/5
0.151+ 0.0088ln s
(b) m=3
FIG. 4. Logarithmic contribution to the asymptotics
of P(s, tc) for (a) m= 2 and (b) m= 3. Solid lines,
P(s, tc)s2m/(2m1) for sfrom 1 to 105found numerically from
the evolution equations with an initial condition P(s, 0) =
a0s2m/(2m1), where a0=ζ[2m/(2m1)]1. Dashed lines,
straight lines presented for reference.
with ln sfor sufficiently large s. In the range s[1..105],
however, it may be difficult to distinguish different slowly
varying functions, such as powers of logarithm. There-
fore, we cannot exclude the possibility that the solid line
asymptotics follows a law (ln s)λwith exponent λclose
to but different from 1.
In Appendix D we obtain the following scaling form of
P(s, t) near tc:
P(s, t)
=C(ln s)λs2m/(2m1)fhC(ln s)λ(2m1) (ttc)i,
(80)
where Cis a constant, λ1/(2m1), and f(x) is the
function defined in Eq. (77). Then, the critical distribu-
tion P(s, t =tc) behaves as
P(s, tc)
=a0C(ln s)λs2m/(2m1) (81)
for large s. Recall that P(s, 0)
=a0s2m/(2m1).
12
3. Critical behavior of S
In Appendix E we derive the critical singularity of the
percolation cluster size S:
Sexp(µ),(82)
where δ=ttc>0,
µ=1
λ(2m1) 1,(83)
and dis a constant
d= m(3m2)[λ1/(2m1)]1λ(2m1)
(m1){−a0CΓ[1/(2m1)]}12m!1/[1λ(2m1)]
.
(84)
Note that λcannot be smaller than 1/(2m1).
C. The singularity of susceptibility
Let us we find the critical singularity of susceptibility.
The critical behavior of χis determined by large s. In
the region in which S > 0, i.e. t > tc, we can substitute
Q(s)
=mSm1P(s), Eq. (53), into Eq. (61), which gives
χ
=m2S2m2X
s
sP (s, t) = m2S2m2hsiP(85)
close to tc. Summing Eq. (51) over sgives
∂S
∂t = 2SmX
s
sQ(s, t)
=2mS2m1hsiP.(86)
Combining the last two equations we finally get
χ
=m
2S
∂S
∂t =m
2
ln S
∂t .(87)
This remarkably general formula relates the critical sin-
gularities of the susceptibility and the percolation cluster
size in all situations shown in Table I. These situations in-
clude the finite- and infinite-order continuous phase tran-
sitions.
For ˜τ < 2 + 1/(2m1), when S
=Btβ, this equation
ensures that the susceptibility has the following singular-
ity:
χ
=
2t1.(88)
For m > 1 and ˜τ= 2 + 1/(2m1) the susceptibility
is divergent below tc, while above tcit has a power-law
singularity
χ
=mdµ
2δµ1,(89)
which we obtained by substituting Eq. (82) into Eq. (87).
10-1 100
t
10-1
100
101
χ
m=1 (a)
10-1 100
t-tc
10-3
10-2
10-1
100
101
102
χ
m=2 (b)
10-1 100
t-tc
10-3
10-2
10-1
100
101
102
χ
m=3 (c)
FIG. 5. Critical behavior of the susceptibility χfor (a) m=
1, (b) m= 2, and (c) m= 3. Solid lines, numerical solution
of 105evolution equations with initial condition P(s, 0) =
ζ[2m/(2m1)]1s2m/(2m1). Dashed lines, (a) power-law
in Eq. (50); (b) and (c) power-law in Eq. (89) assuming µ=
1/(2m2), and C= 0.024 and 0.0088 for m= 2 and 3,
respectively.
Figure 5 presents the susceptibility of the infinite-order
percolation transitions for ˜τ= 2 + 1/(2m1). We ob-
tain the solid curves in this figure inserting the numeri-
cal solution of 105evolution equations into the expression
χ=PssQ(s, t)2/P (s, t). The dashed lines are the power
laws in Eq. (50) and Eq. (89) for m= 1 and m > 1,
respectively. For m= 2 and 3 we used the value of
µ= 1/(2m2) corresponding to the asymptotic distri-
13
bution P(s, tc)s2m/(2m1) ln s, which is observed in
Fig. 4.
V. CONCLUSIONS
In this article we have explored the impact of the initial
cluster size distribution in a set of models generalizing or-
dinary percolation. Specifically, we focused on explosive
percolation models, but our approach could be applied
to a much wider range of generalized percolation mod-
els that can be reduced to various aggregation processes.
In particular, we considered initial cluster size distribu-
tions, for which the percolation phase transition turned
out to be remarkably different from that for the evolution
started from isolated nodes [14, 15, 21]. Our results are
summarized in Table I. We have found the special values
of the exponent ˜τof the initial cluster size distribution
s˜τ, namely ˜τ= 2 + 1/(2m1), for which the perco-
lation cluster emerges continuously, with all derivatives
zero. For ordinary percolation, m= 1, this singularity is
at t= 0, and the susceptibility diverges as t2in contrast
to the Curie–Weiss law typical for mean-field theories in-
cluding various percolation problems, in which the evo-
lution starts from isolated nodes. We have found that for
explosive percolation, m > 1, the situation is even more
interesting. When ˜τ= 2+1/(2m1), (i) the phase tran-
sition occurs at tc>0; (ii) the transition is continuous,
of infinite order; (iii) the phase 0 ttcis critical in
the sense that the generalized susceptibility diverges at
any ttc, and the size distribution of clusters has the
same asymptotics s˜τin the entire critical phase; (iv) the
susceptibility diverges above the transition with a criti-
cal exponent different from 1; (v) the size distribution of
clusters at tcis the same power law but with additional
logarithmic factor, see Table I. Finally, in the special case
of ˜τ= 2 + 1/(2m1), we have obtained unusual scaling
both for ordinary and explosive percolation. Note that,
counterintuitively, in the case of explosive percolation,
for this special value of ˜τthe power-law critical singular-
ity of the generalized susceptibility is accompanied by a
strong divergence of the moments hsiPand hsiQat the
critical point, hsiP∼ hsi2
Qexp[const(ttc)µ], see
Appendix F. We studied the range ˜τ > 2, in which the
initial distribution P(s, 0) s1˜τis normalizable and
the average cluster size is finite. In the case of ˜τ2 and
m= 1 it was found that S(t > 0) = 1 [22].
The infinite order singularity for the percolation clus-
ter at t= 0 was found in Ref. [22, 26] in aggrega-
tion processes with power-law kernels [23, 27] instead of
our power-law initial distribution of clusters for ordinary
percolation. This singularity at zero was also observed
in epidemic models and percolation problems on equi-
librium scale-free networks with the degree distribution
P(q)q3[3, 28, 29].
The critical phase and the infinite-order phase transi-
tion resemble the Berezinskii-Kosterlitz-Thouless transi-
tion [30, 31], which was observed in numerous systems
at a lower critical dimension. In addition, these singu-
larities were observed in heterogeneous one-dimensional
systems with long-range interactions [32, 33], in various
growing networks [34–39], and in percolation on specific
hierarchical graphs [40]. Interestingly, in these growing
networks, the critical distributions of finite cluster sizes
also had factors with powers of logarithms [35, 36].
In conclusion, we have found that initial conditions
can have dramatic effect on the scenario of percolation
transition and its various generalizations. For particu-
lar initial distributions of clusters in the case of explo-
sive percolation, we revealed a continuous phase tran-
sition of infinite-order singularity, which resembles the
Berezinskii-Kosterlitz-Thouless transition. We suggest
that our findings are valid for a wide range of gener-
alizations of percolation, in particular, for explosive per-
colation models with power-law kernels.
ACKNOWLEDGMENTS
This work was partially supported by the FET proac-
tive IP project MULTIPLEX 317532, by the FCT project
EXPL/FIS-NAN/1275/2013, and by the project “New
Strategies Applied to Neuropathological Disorders,” co-
funded by QREN and EU.
Appendix A: Calculation of the coefficient A2(s)for
m= 1 and ˜τ < 3
Let us derive expression (18) for the coefficient A2(s)
of the expansion (12) of P(s, t) in the case of ordinary
percolation. We differentiate both sides of Eq. (3) with
respect to tand replace (i)
tP(s, t)|t=0 with Ai(s)i!. In
this way we get
A2(s) = s
s1
X
u=1
A1(u)A0(su)sA1(s).(A1)
Rearranging this equation in order to cancel divergencies,
we write
A2(s) = s
s1
X
u=1
[A1(u)A1(s)] [A0(su)A0(s)]
s(s1)A0(s)A1(s)sA1(s)
X
u=s
A0(u)
+sA0(s)
s1
X
u=1
A1(u).(A2)
Similarly to A1, let us replace the sums over uin Eq. (A2)
by convergent integrals. The first and second sums on the
right-hand site of this relation can be directly replaced
by the respective integrals. So we have to analyze only
the third sum. For this sum, there are three possibilities:
14
(i) if ˜τ < 5/2, then the asymptotics of A1(u) have the
exponent 4 τ > 1, (ii) if ˜τ > 5/2, then the expo-
nent 4 2˜τ < 1, and (iii) if ˜τ= 5/2, then the coeffi-
cient a1from Eq. (15) is zero, and A1(s) decays as s3/2,
see Eq. (17). In the the first case the sum Pu<s A1(u)
can be directly replaced by an integral convergent at the
lower limit. In the second and third cases this sum must
be first replaced with P
u=1 A1(u)PusA1(u). The
sum P
u=1 A1(u) is a finite constant and PusA1(u) can
be replaced by a convergent integral. Substituting the
asymptotics of A0and A1, we obtain
A2(s)
=a0a1s7τZ1
0
dx x4τ1(1x)1˜τ1
a0a1s7τa0a1s7τZ
1
dx x1˜τ
+θ(5/2˜τ)a0a1s7τZ1
0
dx x4τ
+θτ5/2)a0s2˜τ"
X
u=1
A1(u)a1s5τZ
1
dx x4τ#,
(A3)
where the step function θ(x) is defined here as θ(x<0)=0
and θ(x0)=1. When ˜τ>5/2, the fourth term on the
right-hand side of this equation is zero, and in the last
term, the sum PuA1(u) = 0, see Appendix B. When
˜τ=5/2, the coefficient a1=0, and so the asymptotics of
A2(s) is given a0s2˜τPuA1(u). As we show in Ap-
pendix B, in this special case the sum PuA1(u) is finite,
see Eq. (B7). As a result, for any ˜τ < 3, we have
A2(s)
=a3
0
2Γ(2˜τ)3
3Γ(6τ)s73 ˜τ.(A4)
Appendix B: Generating functions approach for
ordinary percolation (m= 1)
Using generating functions, we will obtain PsAn(s)
for different exponents ˜τand the general expression for
the coefficients an. Let us consider the generating func-
tions of the coefficients An(s) in the series (12), that is
ρn(z) = X
s
zsAn(s).(B1)
We obtain the generating function ρ1(z) in terms
of ρ0(z) multiplying both sides of the evolution equa-
tion (13) by zsand summing over s,
ρ1(z)= 2 X
s"s1
X
u=1
zuuA0(u)zsuA0(su)zssA0(s)#
= 2 X
s
zssA0(s)"X
u
zuA0(u)1#
=ln z(ρ0(z)1)2.(B2)
Here we used PsPu<s f(u)f(su) = Psf(s)Puf(u).
The function ρ0(z) is the generating function of the
initial distribution P(s, 0) A0(s). For a power-law
P(s, 0)
=a0s1˜τthe singular behavior of ρ0(z) at z= 1
is given by Eq. (7), which we reproduce here for the sake
of clarity,
1ρ0(z)
=a0Γ(2 ˜τ)(ln z)˜τ2.(B3)
Inserting this result into Eq. (B2) we find the singularity
of ρ1(z) at z= 1:
ρ1(z)
=2a2
0Γ(2 ˜τ)2τ2)(1 z)2 ˜τ5.(B4)
Differentiating both sides of the evolution equation with
respect to t, and combining with Eq. (B2) and (B3), we
express the function ρ2(z) as
ρ2(z) =X
s
zsA0(s)X
u
zuuA1(u)
+X
s
zsA1(s)X
u
zuuA0(u)X
s
zssA1(s)
=2
3ln zln z(ρ0(z)1)3
=2a3
0Γ(2 ˜τ)3τ2)(3˜τ7)(1 z)τ8.(B5)
In a similar way, for a general n, we obtain
ρn(z)
=2nan+1
0Γ(2 ˜τ)n+1Γ[(n+ 1)(3 ˜τ)1]
(n+ 1)! Γ[(n+ 1)(2 ˜τ)]
×(1 z)(n+1)(˜τ3)+1.(B6)
The sums BnPsAn(s) are equal to the value of the
generating function ρn(z) at z= 1. Then according to
the last equation
Bn=
0 if n < β,
2n[a0Γ(2 ˜τ)]n+1
n+ 1 if n=β,
if n > β,
(B7)
where β= (˜τ2)/(3 ˜τ) is defined by the condition
1˜τ+β(3 ˜τ) = 1.
Equation (B6) can be used to find the general form
of the prefactors anin the asymptotics of An(s). We
obtain the singular contribution to ρn(z) near z= 1 by
inserting the respective power-law An(s)
=ans1˜τ+
into Eq. (B1):
ρn(z)
=anΓ[(n+ 1)(3 ˜τ)1](1 z)(n+1)(˜τ3)+1.(B8)
Comparing Eqs. (B8) and (B6) we readily get:
an=[2a0Γ(2 ˜τ)]n+1
2Γ[(n+ 1)(2 ˜τ)](n+ 1)!.(B9)