EconSocPhys_lecture_22Feb2016.keyNetworks: From the Small World
into the Real World
Sang Hoon Lee School of Physics, Korea Institute for Advanced
Study
http://newton.kias.re.kr/~lshlj82
Outline
• celebrated network models and properties for the past 18 years •
Watts-Strogatz “small-world” model: path length & clustering •
Barabási-Albert “scale-free” model: degree • community/modular and
other mesoscale structures
• more realistic approaches in this century • temporal networks,
spatial networks, multilayer networks • other mesoscale structures:
mesoscopic response function
(MRF) analysis, core-periphery structure and its relation to nested
structure
statistical physics: micro → interactions → macro
regular/random networks (interactions)
How about this? Something new but ubiquitous topology
ref) M. E. J. Newman, Phys. Rev. E 74, 036104 (2006).
Albert
Albert
Nakarado
Barabasi
Jeong
Aleksiejuk
Holyst
Stauffer
Allaria
Arecchi
Collaborations Between Network Scientists
This figure shows a network of collaborations between scientists
working on networks. It was compiled from the bibliographies of two
review articles, by M. Newman (SIAM Review 2003) and by S.
Boccaletti et al. (Physics Re- ports 2006). Vertices represent
scientists whose names appear as authors of papers in those bib-
liographies and an edge joins any two whose names appear on the
samepaper. A small num- ber of other references were added by hand
to bring the network up to date. This figure shows the largest
component of the resulting network, which contains 379 individuals.
Sizes of vertices are proportional to their so-called “community
centrality.” Colors represent ver- tex degrees with redder vertices
having higher degree.
a snapshot of “network of network scientists”
irregular, or “complex” (partially random) networks
How about this? Something new but ubiquitous topology
ref) M. E. J. Newman, Phys. Rev. E 74, 036104 (2006).
Albert
Albert
Nakarado
Barabasi
Jeong
Aleksiejuk
Holyst
Stauffer
Allaria
Arecchi
Collaborations Between Network Scientists
This figure shows a network of collaborations between scientists
working on networks. It was compiled from the bibliographies of two
review articles, by M. Newman (SIAM Review 2003) and by S.
Boccaletti et al. (Physics Re- ports 2006). Vertices represent
scientists whose names appear as authors of papers in those bib-
liographies and an edge joins any two whose names appear on the
samepaper. A small num- ber of other references were added by hand
to bring the network up to date. This figure shows the largest
component of the resulting network, which contains 379 individuals.
Sizes of vertices are proportional to their so-called “community
centrality.” Colors represent ver- tex degrees with redder vertices
having higher degree.
a snapshot of “network of network scientists”
(2004-2010)
How about this? Something new but ubiquitous topology
ref) M. E. J. Newman, Phys. Rev. E 74, 036104 (2006).
Albert
Albert
Nakarado
Barabasi
Jeong
Aleksiejuk
Holyst
Stauffer
Allaria
Arecchi
Collaborations Between Network Scientists
This figure shows a network of collaborations between scientists
working on networks. It was compiled from the bibliographies of two
review articles, by M. Newman (SIAM Review 2003) and by S.
Boccaletti et al. (Physics Re- ports 2006). Vertices represent
scientists whose names appear as authors of papers in those bib-
liographies and an edge joins any two whose names appear on the
samepaper. A small num- ber of other references were added by hand
to bring the network up to date. This figure shows the largest
component of the resulting network, which contains 379 individuals.
Sizes of vertices are proportional to their so-called “community
centrality.” Colors represent ver- tex degrees with redder vertices
having higher degree.
a snapshot of “network of network scientists”
(2004-2010)
(2010-2012)
How about this? Something new but ubiquitous topology
ref) M. E. J. Newman, Phys. Rev. E 74, 036104 (2006).
Albert
Albert
Nakarado
Barabasi
Jeong
Aleksiejuk
Holyst
Stauffer
Allaria
Arecchi
Collaborations Between Network Scientists
This figure shows a network of collaborations between scientists
working on networks. It was compiled from the bibliographies of two
review articles, by M. Newman (SIAM Review 2003) and by S.
Boccaletti et al. (Physics Re- ports 2006). Vertices represent
scientists whose names appear as authors of papers in those bib-
liographies and an edge joins any two whose names appear on the
samepaper. A small num- ber of other references were added by hand
to bring the network up to date. This figure shows the largest
component of the resulting network, which contains 379 individuals.
Sizes of vertices are proportional to their so-called “community
centrality.” Colors represent ver- tex degrees with redder vertices
having higher degree.
a snapshot of “network of network scientists”
(2004-2010)
(2010-2012)
(2012-2014)
How about this? Something new but ubiquitous topology
ref) M. E. J. Newman, Phys. Rev. E 74, 036104 (2006).
Albert
Albert
Nakarado
Barabasi
Jeong
Aleksiejuk
Holyst
Stauffer
Allaria
Arecchi
Collaborations Between Network Scientists
This figure shows a network of collaborations between scientists
working on networks. It was compiled from the bibliographies of two
review articles, by M. Newman (SIAM Review 2003) and by S.
Boccaletti et al. (Physics Re- ports 2006). Vertices represent
scientists whose names appear as authors of papers in those bib-
liographies and an edge joins any two whose names appear on the
samepaper. A small num- ber of other references were added by hand
to bring the network up to date. This figure shows the largest
component of the resulting network, which contains 379 individuals.
Sizes of vertices are proportional to their so-called “community
centrality.” Colors represent ver- tex degrees with redder vertices
having higher degree.
a snapshot of “network of network scientists”
(2004-2010)
(2010-2012)(2014-2015)
(2012-2014)
How about this? Something new but ubiquitous topology
ref) M. E. J. Newman, Phys. Rev. E 74, 036104 (2006).
Albert
Albert
Nakarado
Barabasi
Jeong
Aleksiejuk
Holyst
Stauffer
Allaria
Arecchi
Collaborations Between Network Scientists
This figure shows a network of collaborations between scientists
working on networks. It was compiled from the bibliographies of two
review articles, by M. Newman (SIAM Review 2003) and by S.
Boccaletti et al. (Physics Re- ports 2006). Vertices represent
scientists whose names appear as authors of papers in those bib-
liographies and an edge joins any two whose names appear on the
samepaper. A small num- ber of other references were added by hand
to bring the network up to date. This figure shows the largest
component of the resulting network, which contains 379 individuals.
Sizes of vertices are proportional to their so-called “community
centrality.” Colors represent ver- tex degrees with redder vertices
having higher degree.
a snapshot of “network of network scientists”
(2015-present)
(2004-2010)
(2010-2012)(2014-2015)
(2012-2014)
The most complicated system in the universe known to itself
microscale structure: neuron
Volume 12, Number 6, 2006 THE NEUROSCIENTIST 521
with its growth by creation of new nodes, which preferen- tially
form connections to existing hubs. One fMRI study has reported a
power law degree distribution for a func- tional network of
activated voxels (Eguíluz and others 2005). But the degree
distribution of whole-brain fMRI networks of cortical regions has
also been described as an exponentially truncated power law (Achard
and oth- ers 2006), meaning broadly that the probability of very
highly connected hubs is less in the brain than in the
WWW, but there is more probability of a hub in the brain than in a
random graph. The hubs of this network were predominantly regions
of the heteromodal and uni- modal association cortex.
Truncated power law degree distributions are wide- spread in
complex systems that are physically embedded or constrained, such
as transport or infrastructural net- works, and in systems in which
nodes have a finite life span, such as the social network of
collaborating Hollywood
Fig. 6. Small-world functional brain networks (Achard and others
2006). Anatomical map of a small-world human brain functional
network created by thresholding the scale 4 wavelet correlation
matrix representing functional connectivity in the frequency
interval 0.03 to 0.06 Hz. A, Four hundred five undirected edges,
~10% of the 4005 possible interregional connections, are shown in a
sagittal view of the right side of the brain. Nodes are located
according to the y and z coor- dinates of the regional centroids in
Talairach space. Edges representing connections between nodes
separated by a Euclidean distance <7.5 cm are red; edges
representing connections between nodes separated by Euclidean
distance >7.5 cm are blue. B, Degree distribution of a
small-world brain functional network. Plot of the log of the
cumulative prob- ability of degree, log(P(ki)), versus log of
degree, log(ki). The plus sign indicates observed data, the solid
line is the best- fitting exponentially truncated power law, the
dotted line is an exponential, and the dashed line is a power law.
C, Resilience of the human brain functional network (right column)
compared with random (left column) and scale-free (middle column)
networks. Size of the largest connected cluster in the network
(scaled to maximum; y axis) versus the proportion of total nodes
eliminated (x axis) by random error (dashed line) or targeted
attack (solid line). The size of the largest connected cluster in
the brain functional network is more resilient to targeted attack
and about equally resilient to random error compared with the
scale-free network. Reprinted from J Neurosci, 26(1), Achard S,
Salvador R, Whitcher B, Suckling J, Bullmore E, A resilient,
low-frequency, small-world human brain functional network with
highly connected association cortical hubs, 63-72, 2006, with
permission from the Society for Neuroscience.
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at Oxford University Libraries on March 14,
2014nro.sagepub.comDownloaded from
D. S. Bassett and E. Bullmore, “Small-World Brain Networks”, The
Neuroscientist 12, 512 (2006).
system-level approach! (including “mesoscale” structures)
The most complicated system in the universe known to itself
microscale structure: neuron
Ed BullmoreDanielle Bassett
Network terminology N = |V| = 7: number of nodes# of nodes
Network terminology
M = |E| = 13: number of edges# of edges N = |V| = 7: number of
nodes# of nodes
Network terminology
1
CCCCCCCCA
M = |E| = 13: number of edges# of edges N = |V| = 7: number of
nodes# of nodes
Network terminology
1
CCCCCCCCA
“microscale” structure
Network terminology
1
CCCCCCCCA
“microscale” structure
some kind of nontrivial “mesoscale structure”?
Network terminology
1
CCCCCCCCA
clustering coefficient: how well my neighbors are connected to each
other?
neighborsnearest its connecting edges theofnumber total theis and
neighborsnearest ofnumber theis where
)1( 2
where ki is the node i’s degree and
yi is the number of edges connecting its neighbors to each
other
i
clustering coefficient: how well my neighbors are connected to each
other?
neighborsnearest its connecting edges theofnumber total theis and
neighborsnearest ofnumber theis where
)1( 2
where ki is the node i’s degree and
yi is the number of edges connecting its neighbors to each
other
i ! C(i) =
2 2
4 3 =
1
3
clustering coefficient: how well my neighbors are connected to each
other?
neighborsnearest its connecting edges theofnumber total theis and
neighborsnearest ofnumber theis where
)1( 2
where ki is the node i’s degree and
yi is the number of edges connecting its neighbors to each
other
i ! C(i) =
2 2
4 3 =
i
Ci/N
clustering coefficient: how well my neighbors are connected to each
other?
neighborsnearest its connecting edges theofnumber total theis and
neighborsnearest ofnumber theis where
)1( 2
where ki is the node i’s degree and
yi is the number of edges connecting its neighbors to each
other
i ! C(i) =
2 2
4 3 =
average path length of a network
path length l between i and j: the number of edges in the shortest
path between i and j
1
2
3
4
5
6
7
average path length of a network
path length l between i and j: the number of edges in the shortest
path between i and j
1
2
3
4
5
6
7
average path length of a network
path length l between i and j: the number of edges in the shortest
path between i and j
1
2
3
4
5
6
7
average path length of a network
path length l between i and j: the number of edges in the shortest
path between i and j
1
2
3
4
5
6
7
average path length of a network
path length l between i and j: the number of edges in the shortest
path between i and j
1
2
3
4
5
6
7
l(1 ! 2) = 1 l(1 ! 7) = 2
average path length of a network
path length l between i and j: the number of edges in the shortest
path between i and j
1
2
3
4
5
6
7
l(1 ! 2) = 1 l(1 ! 7) = 2
average path length of a network
path length l between i and j: the number of edges in the shortest
path between i and j
1
2
3
4
5
6
7
average path length of a network
path length l between i and j: the number of edges in the shortest
path between i and j
1
2
3
4
5
6
7
average path length = l averaged over all of the node pairs
l(1 ! 2) = 1 l(1 ! 7) = 2 l(1 ! 6) = 4 . . .
average path length of a network
path length l between i and j: the number of edges in the shortest
path between i and j
1
2
3
4
5
6
7
average path length = l averaged over all of the node pairs real
networks: much smaller average path length than regular
networks!
l(1 ! 2) = 1 l(1 ! 7) = 2 l(1 ! 6) = 4 . . .
Watts-Strogatz “Small World” Network
D. J. Watts and S. H. Strogatz, Collective dynamics of
‘small-world’ networks, Nature 393, 440 (1998).
Duncan Watts Steven Strogatz
Watts-Strogatz “Small World” Network
D. J. Watts and S. H. Strogatz, Collective dynamics of
‘small-world’ networks, Nature 393, 440 (1998).
Duncan Watts Steven Strogatz
l / logN
ab = c ! b = loga c
p(k) / k
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
p( k)
p(k) / k
“hubs” with large degree
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
p( k)
p(k) / k
“hubs” with large degree
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
p( k)
ubiquitous topology, in fact!
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
(fully connected) initial seed nodes
attaching a new node to the existing node i with the
probability
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
k = 2 k = 2
attaching a new node to the existing node i with the
probability
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
k = 1
attaching a new node to the existing node i with the
probability
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
k = 1
attaching a new node to the existing node i with the
probability
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
k = 1
k = 1
attaching a new node to the existing node i with the
probability
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
k = 1
k = 1
attaching a new node to the existing node i with the
probability
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
k = 1k = 1
attaching a new node to the existing node i with the
probability
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
k = 1k = 1
attaching a new node to the existing node i with the
probability
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
k = 1k = 1
attaching a new node to the existing node i with the
probability
Barabási-Albert “Scale-Free” Network
A.-L. Barabási and R. Albert, Emergence of scaling in random
networks, Science 286, 509 (1999).
attaching a new node to existing ones with the probability (ki) =
ki/ P
j kj
k = 1k = 1
! p(k) / k3
k p( k )
attaching a new node to the existing node i with the
probability
Implication of the power-law degree distribution
p(k) / k
p(k) / k
Implication of the power-law degree distribution
p(k) / k
finite mean, diverging variance!
p(k) / k
finite mean, diverging variance!
kmin
p(k) / k
in general,
kmin
important for critical phenomena on networks!
mean-field critical behavior in the infinite size limit !Scalettar,
1991". The corresponding exact solution is given in Sec.
VI.A.1.a.
The conventional scaling relation between the critical exponents
takes place at !"3,
# + 2$ + ! = 2. !83"
Interestingly, the magnetic susceptibility % has universal critical
behavior with !=1 when #q2$&', i.e., at !"3. This agrees with
the scaling relation ! /(=2−) if we in- sert the standard
mean-field exponents: (= 1
2 and the Fisher exponent )=0; see Sec. IX.B. When 2&!*3, the
susceptibility % has a paramagnetic temperature depen- dence,
%+1/T, at temperatures T,J despite the system being in the ordered
state.
At T&Tc, the ferromagnetic state is strongly hetero- geneous
because the magnetic moment Mi fluctuates from vertex to vertex.
The ansatz !82" enables us to find an approximate distribution
function of Mi,
Y!M" % 1 N&
-!M − Mi" ' P(q!M")
$h!1 − M2" , !84"
where the function q!M" is a solution of an equation
M!q"=tanh!$hq". Near Tc, low-degree vertices have a small magnetic
moment, M!q"*q+Tc−T+1/2.1, while hubs with degree q"T / h/1 have
M!q"*1. The func- tion Y!M" is shown in Fig. 21. Note that the
distribution of magnetic moments in scale-free networks is more in-
homogeneous than in the Erdos-Rényi graphs. In the former case,
Y!M" diverges at M→1. A local magnetic moment depends on its
neighborhood. In particular, a magnetic moment of a spin
neighboring a hub may differ from a moment of a spin surrounded by
low-degree ver- tices !Giuraniuc et al., 2006".
At T=H=0, the exact distribution function 0!h" con- verges to a
function with two delta peaks,
0!h" = x-!h" + !1 − x"-!h − J" , !85"
where the parameter x is determined by an equation describing
percolation in networks; see Sec. III.B.1. Equation !85" tells us
that in the ground state, spins, which belong to a finite cluster,
have zero magnetic mo- ment while spins in a giant connected
component have magnetic moment 1. The average magnetic moment is
M=1−&qP!q"xq. This is exactly the size of the giant con- nected
component of the network.
3. Finite-size effects
When 2&!*3, a dependence of Tc on the size N is determined by
the finite-size cutoff qcut!N" of the degree distribution in Sec.
II.E.4. We obtain
Tc!N" ' , z1 ln N 4
at ! = 3 !86"
!! − 2"2z1qcut 3−!!N"
!3 − !"!! − 1" at 2 & ! & 3 !87"
!Bianconi, 2002; Dorogovtsev et al., 2002b; Leone et al., 2002".
These estimates agree with the numerical simula- tions of
Aleksiejuk et al. !2002" and Herrero !2004". No-
TABLE I. Critical behavior of the magnetization M, the spe- cific
heat -C, and the susceptibility % in the Ising model on networks
with a degree distribution P!q"*q−! for various val- ues of
exponent !. 1%1−T /Tc.
M -C!T&Tc" %
!=5 11/2 / !ln 1−1"1/2 1 / ln 1−1
3&!&5 11/!!−3" 1!5−!"/!!−3"
!=3 e−2T/#q$ T2e−4T/#q$ T−1
2&!&3 T−1/!3−!" T−!!−1"/!3−!"
2 < γ < 3
c
χ
C
M
T
b)
a)
M ,
C ,
χ
FIG. 20. Schematic representation of the critical behavior of the
magnetization M !dotted lines", the magnetic susceptibility %
!dashed lines", and the specific heat C !solid lines" for the
ferromagnetic Ising model on uncorrelated random networks with a
degree distribution P!q"*q−!. !a" !/1, the standard mean-field
critical behavior. A jump of C disappears when ! →5. !b" 4&!*5,
the ferromagnetic phase transition is of sec- ond order. !c"
3&!*4, the transition becomes of higher order. !d" 2&!*3,
the transition is of infinite order, and Tc→' as N→'.
0.001
0.01
10
1
0.1
) M
FIG. 21. Distribution function Y!M" of magnetic moments M in the
ferromagnetic Ising model on the Erdos-Rényi graph with mean degree
z1=5 !dashed line" and scale-free graphs with !=4 and 3.5 !solid
and dotted lines" at T close to Tc, $h =0.04.
1303Dorogovtsev, Goltsev, and Mendes: Critical phenomena in complex
networks
Rev. Mod. Phys., Vol. 80, No. 4, October–December 2008
review paper: S. N. Dorogovtsev, A. V. Goltsev, and J. F. F.
Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80,
1275 (2008).
Sergey Dorogovtsev José MendesAlexander Goltsev
Implication of the power-law degree distribution
p(k) / k
in general,
kmin
important for critical phenomena on networks!
mean-field critical behavior in the infinite size limit !Scalettar,
1991". The corresponding exact solution is given in Sec.
VI.A.1.a.
The conventional scaling relation between the critical exponents
takes place at !"3,
# + 2$ + ! = 2. !83"
Interestingly, the magnetic susceptibility % has universal critical
behavior with !=1 when #q2$&', i.e., at !"3. This agrees with
the scaling relation ! /(=2−) if we in- sert the standard
mean-field exponents: (= 1
2 and the Fisher exponent )=0; see Sec. IX.B. When 2&!*3, the
susceptibility % has a paramagnetic temperature depen- dence,
%+1/T, at temperatures T,J despite the system being in the ordered
state.
At T&Tc, the ferromagnetic state is strongly hetero- geneous
because the magnetic moment Mi fluctuates from vertex to vertex.
The ansatz !82" enables us to find an approximate distribution
function of Mi,
Y!M" % 1 N&
-!M − Mi" ' P(q!M")
$h!1 − M2" , !84"
where the function q!M" is a solution of an equation
M!q"=tanh!$hq". Near Tc, low-degree vertices have a small magnetic
moment, M!q"*q+Tc−T+1/2.1, while hubs with degree q"T / h/1 have
M!q"*1. The func- tion Y!M" is shown in Fig. 21. Note that the
distribution of magnetic moments in scale-free networks is more in-
homogeneous than in the Erdos-Rényi graphs. In the former case,
Y!M" diverges at M→1. A local magnetic moment depends on its
neighborhood. In particular, a magnetic moment of a spin
neighboring a hub may differ from a moment of a spin surrounded by
low-degree ver- tices !Giuraniuc et al., 2006".
At T=H=0, the exact distribution function 0!h" con- verges to a
function with two delta peaks,
0!h" = x-!h" + !1 − x"-!h − J" , !85"
where the parameter x is determined by an equation describing
percolation in networks; see Sec. III.B.1. Equation !85" tells us
that in the ground state, spins, which belong to a finite cluster,
have zero magnetic mo- ment while spins in a giant connected
component have magnetic moment 1. The average magnetic moment is
M=1−&qP!q"xq. This is exactly the size of the giant con- nected
component of the network.
3. Finite-size effects
When 2&!*3, a dependence of Tc on the size N is determined by
the finite-size cutoff qcut!N" of the degree distribution in Sec.
II.E.4. We obtain
Tc!N" ' , z1 ln N 4
at ! = 3 !86"
!! − 2"2z1qcut 3−!!N"
!3 − !"!! − 1" at 2 & ! & 3 !87"
!Bianconi, 2002; Dorogovtsev et al., 2002b; Leone et al., 2002".
These estimates agree with the numerical simula- tions of
Aleksiejuk et al. !2002" and Herrero !2004". No-
TABLE I. Critical behavior of the magnetization M, the spe- cific
heat -C, and the susceptibility % in the Ising model on networks
with a degree distribution P!q"*q−! for various val- ues of
exponent !. 1%1−T /Tc.
M -C!T&Tc" %
!=5 11/2 / !ln 1−1"1/2 1 / ln 1−1
3&!&5 11/!!−3" 1!5−!"/!!−3"
!=3 e−2T/#q$ T2e−4T/#q$ T−1
2&!&3 T−1/!3−!" T−!!−1"/!3−!"
2 < γ < 3
c
χ
C
M
T
b)
a)
M ,
C ,
χ
FIG. 20. Schematic representation of the critical behavior of the
magnetization M !dotted lines", the magnetic susceptibility %
!dashed lines", and the specific heat C !solid lines" for the
ferromagnetic Ising model on uncorrelated random networks with a
degree distribution P!q"*q−!. !a" !/1, the standard mean-field
critical behavior. A jump of C disappears when ! →5. !b" 4&!*5,
the ferromagnetic phase transition is of sec- ond order. !c"
3&!*4, the transition becomes of higher order. !d" 2&!*3,
the transition is of infinite order, and Tc→' as N→'.
0.001
0.01
10
1
0.1
) M
FIG. 21. Distribution function Y!M" of magnetic moments M in the
ferromagnetic Ising model on the Erdos-Rényi graph with mean degree
z1=5 !dashed line" and scale-free graphs with !=4 and 3.5 !solid
and dotted lines" at T close to Tc, $h =0.04.
1303Dorogovtsev, Goltsev, and Mendes: Critical phenomena in complex
networks
Rev. Mod. Phys., Vol. 80, No. 4, October–December 2008
review paper: S. N. Dorogovtsev, A. V. Goltsev, and J. F. F.
Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80,
1275 (2008). true mean-field
Sergey Dorogovtsev José MendesAlexander Goltsev
Implication of the power-law degree distribution
p(k) / k
in general,
kmin
important for critical phenomena on networks!
mean-field critical behavior in the infinite size limit !Scalettar,
1991". The corresponding exact solution is given in Sec.
VI.A.1.a.
The conventional scaling relation between the critical exponents
takes place at !"3,
# + 2$ + ! = 2. !83"
Interestingly, the magnetic susceptibility % has universal critical
behavior with !=1 when #q2$&', i.e., at !"3. This agrees with
the scaling relation ! /(=2−) if we in- sert the standard
mean-field exponents: (= 1
2 and the Fisher exponent )=0; see Sec. IX.B. When 2&!*3, the
susceptibility % has a paramagnetic temperature depen- dence,
%+1/T, at temperatures T,J despite the system being in the ordered
state.
At T&Tc, the ferromagnetic state is strongly hetero- geneous
because the magnetic moment Mi fluctuates from vertex to vertex.
The ansatz !82" enables us to find an approximate distribution
function of Mi,
Y!M" % 1 N&
-!M − Mi" ' P(q!M")
$h!1 − M2" , !84"
where the function q!M" is a solution of an equation
M!q"=tanh!$hq". Near Tc, low-degree vertices have a small magnetic
moment, M!q"*q+Tc−T+1/2.1, while hubs with degree q"T / h/1 have
M!q"*1. The func- tion Y!M" is shown in Fig. 21. Note that the
distribution of magnetic moments in scale-free networks is more in-
homogeneous than in the Erdos-Rényi graphs. In the former case,
Y!M" diverges at M→1. A local magnetic moment depends on its
neighborhood. In particular, a magnetic moment of a spin
neighboring a hub may differ from a moment of a spin surrounded by
low-degree ver- tices !Giuraniuc et al., 2006".
At T=H=0, the exact distribution function 0!h" con- verges to a
function with two delta peaks,
0!h" = x-!h" + !1 − x"-!h − J" , !85"
where the parameter x is determined by an equation describing
percolation in networks; see Sec. III.B.1. Equation !85" tells us
that in the ground state, spins, which belong to a finite cluster,
have zero magnetic mo- ment while spins in a giant connected
component have magnetic moment 1. The average magnetic moment is
M=1−&qP!q"xq. This is exactly the size of the giant con- nected
component of the network.
3. Finite-size effects
When 2&!*3, a dependence of Tc on the size N is determined by
the finite-size cutoff qcut!N" of the degree distribution in Sec.
II.E.4. We obtain
Tc!N" ' , z1 ln N 4
at ! = 3 !86"
!! − 2"2z1qcut 3−!!N"
!3 − !"!! − 1" at 2 & ! & 3 !87"
!Bianconi, 2002; Dorogovtsev et al., 2002b; Leone et al., 2002".
These estimates agree with the numerical simula- tions of
Aleksiejuk et al. !2002" and Herrero !2004". No-
TABLE I. Critical behavior of the magnetization M, the spe- cific
heat -C, and the susceptibility % in the Ising model on networks
with a degree distribution P!q"*q−! for various val- ues of
exponent !. 1%1−T /Tc.
M -C!T&Tc" %
!=5 11/2 / !ln 1−1"1/2 1 / ln 1−1
3&!&5 11/!!−3" 1!5−!"/!!−3"
!=3 e−2T/#q$ T2e−4T/#q$ T−1
2&!&3 T−1/!3−!" T−!!−1"/!3−!"
2 < γ < 3
c
χ
C
M
T
b)
a)
M ,
C ,
χ
FIG. 20. Schematic representation of the critical behavior of the
magnetization M !dotted lines", the magnetic susceptibility %
!dashed lines", and the specific heat C !solid lines" for the
ferromagnetic Ising model on uncorrelated random networks with a
degree distribution P!q"*q−!. !a" !/1, the standard mean-field
critical behavior. A jump of C disappears when ! →5. !b" 4&!*5,
the ferromagnetic phase transition is of sec- ond order. !c"
3&!*4, the transition becomes of higher order. !d" 2&!*3,
the transition is of infinite order, and Tc→' as N→'.
0.001
0.01
10
1
0.1
) M
FIG. 21. Distribution function Y!M" of magnetic moments M in the
ferromagnetic Ising model on the Erdos-Rényi graph with mean degree
z1=5 !dashed line" and scale-free graphs with !=4 and 3.5 !solid
and dotted lines" at T close to Tc, $h =0.04.
1303Dorogovtsev, Goltsev, and Mendes: Critical phenomena in complex
networks
Rev. Mod. Phys., Vol. 80, No. 4, October–December 2008
review paper: S. N. Dorogovtsev, A. V. Goltsev, and J. F. F.
Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80,
1275 (2008). true mean-field
“heterogeneous” mean-field Sergey Dorogovtsev José MendesAlexander
Goltsev
finite-size scaling
a scaling relation with the scale variable kN!1=! (not shown here)
[16]. For k > N1=!, the distribution P"k# becomes almost flat
for each realization of networks and the degree exponent ! loses
its identity. Therefore, vertices of such a high degree contribute
in a trivial way and the cutoff beyond this range (kc > N1=!)
should not be distin- guishable [15]. This argument is supported by
our numeri- cal results which cannot differentiate the FSS scaling
in the static and UCM networks.
Now we move to a typical model exhibiting a nonequi- librium phase
transition, namely, the directed percolation (DP) system [19]. It
is well known that most of the non- equilibrium models showing an
absorbing-type phase tran-
sition belong to the DP universality class. Among such models, we
here consider the contact process (CP) and the
susceptible-infected-susceptible (SIS) model [6].
The CP is an interacting particle model on a lattice. A particle
creates another particle in one of its neighboring sites with rate
p and a particle annihilates with rate 1. In the SIS model, the
particle creation is attempted in all neigh- boring sites. A
particle-particle interaction comes in through disallowance of
multiple occupancy at a site. As p increases, the system undergoes
a phase transition at pc from a quiescent vacuum (absorbing) phase
to a noisy many-particle (active) phase in the steady state. Near
the absorbing phase transition, the order parameter (particle
density) "$ #$, the fluctuations %0 % N"!"#2 $ #!&0 , the
susceptibility %$ j#j!&, the correlation length '$ j#j!(, the
relaxation time )$ j#j!(t , and the survival probability Ps $
#$
0 with the reduced coupling constant # %
"p! pc#=pc. It is known that $ % $0 due to the time- reversal
symmetry in the DP systems [19] and &0 ! & in general
nonequilibrium systems.
Consider the droplet (cluster) excitation starting from a localized
seed in the absorbing phase. The average space- time size S of a
cluster is estimated as
S$ )‘'dc $ j#j!*; (7)
where )‘ and 'c are the average lifetime and typical size of a
droplet, respectively. Usually )‘ diverges near the tran- sition as
)‘ $ j#j!(t&$
0 for (t > $0 [19], but )‘ is a O"1#
constant otherwise. In the MF regime, it is shown later that the
latter always applies. The droplet size diverges as 'c $ j#j!(T ,
which leads to * % d(T &maxf(t ! $0; 0g. It is well known that
the susceptibility is proportional to the cluster mass, which
yields & % *! $ [19]. Finally, we arrive at the generalized
exponent relation as
& % d(T ! $&maxf(t ! $0; 0g: (8)
The fluctuation exponent &0 satisfies the standard hyper-
scaling relation as &0 % d(T ! 2$.
In SF networks, we propose a phenomenological modi- fication of the
MF Langevin equation describing the DP models, similar to the free
energy modification of the Ising model in Eq. (5):
d dt ""t# % #"! b"2 ! d"+!1 & !!!!
" p
,"t#; (9)
where ""t# is the particle density at time t and ,"t# is a Gaussian
noise. Our modification to the standard MF the- ory comes in by the
third "+!1 term and it is straightfor- ward to show that the
exponent + % ! for the CP and + % !! 1 for the SIS.
By dropping the noise term, one may easily get the MF steady-state
solution for ". We find that $ % 1 for +> 3 and $ % 1="+! 2# for
2< +< 3. For +< 2, there is no phase transition at finite
p. The same result may be ob- tained from the well-established
k-dependent noiseless MF
TABLE I. Critical exponents of the Ising model on the static and
the UCM networks, and the CP on the UCM networks, compared with our
MF predictions.
Ising Network ! $= "( "( &0= "( MF !> 5 1=4 2 1=2
3< !< 5 1 !!1
!!1 !!3
!!3 !!1
Static 7.08 0.26(4) 2.0(2) 0.45(5) 4.45 0.28(2) 2.4(2) 0.45(3) 3.87
0.37(5) 3.5(3) 0.26(4)
UCM 6.50 0.24(4) 2.0(2) 0.51(5) 4.25 0.31(1) 2.5(1) 0.39(1) 3.75
0.38(6) 3.9(2) 0.24(3)
CP Network ! $= "( "( &0= "( MF !> 3 1=2 2 0
2< !< 3 1 !!1
!!1 !!2
!!3 !!1
UCM 4.0 0.49(1) 2.1(1) 0.00(5) 2.75 0.58(1) 2.4(1) !0:16"2# 2.25
0.78(1) 4.0(5) !0:55"5#
0
5
10
15
20
25
-10 -8 -6 -4 -2 0 2 4 6 8 10
m N
0. 37
128000 256000
χ’
N
FIG. 1 (color online). Data collapse of m for the static network
with ! % 3:87, using $= "( % 0:37 and "( % 3:5. Insets: Double
logarithmic plots of the critical decay ofm and %0 against N with
slopes $= "( % 0:37"5# and &0= "( % 0:26"4#.
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S
week ending 22 JUNE 2007
258701-3
a scaling relation with the scale variable kN!1=! (not shown here)
[16]. For k > N1=!, the distribution P"k# becomes almost flat
for each realization of networks and the degree exponent ! loses
its identity. Therefore, vertices of such a high degree contribute
in a trivial way and the cutoff beyond this range (kc > N1=!)
should not be distin- guishable [15]. This argument is supported by
our numeri- cal results which cannot differentiate the FSS scaling
in the static and UCM networks.
Now we move to a typical model exhibiting a nonequi- librium phase
transition, namely, the directed percolation (DP) system [19]. It
is well known that most of the non- equilibrium models showing an
absorbing-type phase tran-
sition belong to the DP universality class. Among such models, we
here consider the contact process (CP) and the
susceptible-infected-susceptible (SIS) model [6].
The CP is an interacting particle model on a lattice. A particle
creates another particle in one of its neighboring sites with rate
p and a particle annihilates with rate 1. In the SIS model, the
particle creation is attempted in all neigh- boring sites. A
particle-particle interaction comes in through disallowance of
multiple occupancy at a site. As p increases, the system undergoes
a phase transition at pc from a quiescent vacuum (absorbing) phase
to a noisy many-particle (active) phase in the steady state. Near
the absorbing phase transition, the order parameter (particle
density) "$ #$, the fluctuations %0 % N"!"#2 $ #!&0 , the
susceptibility %$ j#j!&, the correlation length '$ j#j!(, the
relaxation time )$ j#j!(t , and the survival probability Ps $
#$
0 with the reduced coupling constant # %
"p! pc#=pc. It is known that $ % $0 due to the time- reversal
symmetry in the DP systems [19] and &0 ! & in general
nonequilibrium systems.
Consider the droplet (cluster) excitation starting from a localized
seed in the absorbing phase. The average space- time size S of a
cluster is estimated as
S$ )‘'dc $ j#j!*; (7)
where )‘ and 'c are the average lifetime and typical size of a
droplet, respectively. Usually )‘ diverges near the tran- sition as
)‘ $ j#j!(t&$
0 for (t > $0 [19], but )‘ is a O"1#
constant otherwise. In the MF regime, it is shown later that the
latter always applies. The droplet size diverges as 'c $ j#j!(T ,
which leads to * % d(T &maxf(t ! $0; 0g. It is well known that
the susceptibility is proportional to the cluster mass, which
yields & % *! $ [19]. Finally, we arrive at the generalized
exponent relation as
& % d(T ! $&maxf(t ! $0; 0g: (8)
The fluctuation exponent &0 satisfies the standard hyper-
scaling relation as &0 % d(T ! 2$.
In SF networks, we propose a phenomenological modi- fication of the
MF Langevin equation describing the DP models, similar to the free
energy modification of the Ising model in Eq. (5):
d dt ""t# % #"! b"2 ! d"+!1 & !!!!
" p
,"t#; (9)
where ""t# is the particle density at time t and ,"t# is a Gaussian
noise. Our modification to the standard MF the- ory comes in by the
third "+!1 term and it is straightfor- ward to show that the
exponent + % ! for the CP and + % !! 1 for the SIS.
By dropping the noise term, one may easily get the MF steady-state
solution for ". We find that $ % 1 for +> 3 and $ % 1="+! 2# for
2< +< 3. For +< 2, there is no phase transition at finite
p. The same result may be ob- tained from the well-established
k-dependent noiseless MF
TABLE I. Critical exponents of the Ising model on the static and
the UCM networks, and the CP on the UCM networks, compared with our
MF predictions.
Ising Network ! $= "( "( &0= "( MF !> 5 1=4 2 1=2
3< !< 5 1 !!1
!!1 !!3
!!3 !!1
Static 7.08 0.26(4) 2.0(2) 0.45(5) 4.45 0.28(2) 2.4(2) 0.45(3) 3.87
0.37(5) 3.5(3) 0.26(4)
UCM 6.50 0.24(4) 2.0(2) 0.51(5) 4.25 0.31(1) 2.5(1) 0.39(1) 3.75
0.38(6) 3.9(2) 0.24(3)
CP Network ! $= "( "( &0= "( MF !> 3 1=2 2 0
2< !< 3 1 !!1
!!1 !!2
!!3 !!1
UCM 4.0 0.49(1) 2.1(1) 0.00(5) 2.75 0.58(1) 2.4(1) !0:16"2# 2.25
0.78(1) 4.0(5) !0:55"5#
0
5
10
15
20
25
-10 -8 -6 -4 -2 0 2 4 6 8 10
m N
0. 37
128000 256000
χ’
N
FIG. 1 (color online). Data collapse of m for the static network
with ! % 3:87, using $= "( % 0:37 and "( % 3:5. Insets: Double
logarithmic plots of the critical decay ofm and %0 against N with
slopes $= "( % 0:37"5# and &0= "( % 0:26"4#.
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S
week ending 22 JUNE 2007
258701-3
ref) H. Hong, M. Ha, and H. Park, Finite-size scaling in complex
networks, Phys. Rev. Lett. 98, 258701 (2007).
a typical size !T of a disordered droplet excitation out of the
uniformly ordered environment. As the free energy cost by the
droplet excitation is compensated by the thermal energy, !!f"!dT #
kBT, we find !T # "$#T with #T % 2=d % 1=yT . The Gaussian length
scale diverges as !G # "$#G with #G % 1=2.
For d > du % 4 (the upper critical dimension), !G domi- nates
over !T , which leads to the correlation length ex- ponent # % #G
and the MF theory is valid. However, the FSS variable LyT" becomes
!L=!T"yT , implying that the competing length scale is not the
dominant correlation length but the droplet size. Substituting the
linear size L by the volume N # Ld, Eq. (2) reads
m % N$$= "# !N1= "#""; (4)
where the FSS (droplet volume) exponent "# % d#T % 2 in the MF
regime. For the general %q MF theory (f % $"m2 & umq), we find
that "# % du#G with du % 2q=!q$ 2", which is consistent with the
earlier result by Botet et al. for models with infinite-range
interactions [12].
We are now ready to explore the FSS in networks. Networks have no
space dimensionality and may be con- sidered as a limiting case of
d! 1. So we expect that any model in networks displays a MF-type
critical behavior. In particular, the MF FSS exponent "# is
independent of d, which leads to the natural conjecture that Eq.
(4) also applies in networks. These predictions have been con-
firmed by numerical simulations for various models in random
networks, small-world networks, and complete graphs. Moreover, the
relation of "# % du#G has been ex- ploited to calculate the value
of du via simulations in networks for complex nonequilibrium models
[13].
In SF networks with the degree distribution P!k" # k$&, there
appears a nontrivial &-dependent MF critical scaling for
&< &u (highly heterogeneous networks) while the standard
MF theory applies for &> &u [1]. Naturally, we expect a
nontrivial FSS theory associated with the non- trivial MF scaling
for &< &u. Previous studies pay atten- tion to the MF
analysis in the thermodynamic limit and hardly discuss the FSS in
the general context. Recently, a few numerical efforts have been
attempted to confirm the MF predictions, but huge finite-size
effects and the lack of the FSS theory disallowed any decisive
conclusion for highly heterogeneous networks [4,7]. Most recently,
even a non-MF scaling has been claimed for the contact process
[14,15] and the question arises as to whether the cutoff in degree
k influences the FSS.
We start with the phenomenological MF free energy for the SF
networks proposed in [1,2]
f!m" % $"m2 & um4 & vjmj&$1 &O!m6"; (5)
where the &-dependent term originates from the singular
behavior of the higher moments of degree in SF networks. For
&> &u % 5, the &-dependent term is irrelevant and we
recover the usual %4 MF theory, yielding $ % 1=2 and
"# % 2. For 3< &< 5, the &-dependent term becomes
relevant and we find the %q MF theory with q % &$ 1. A simple
algebra leads to $ % 1=!&$ 3" and the free energy density in
the ordered phase is f#$"1&2$. One can estimate the typical
droplet volume NT # !!f"$1, yielding NT # "$ "# with "# % 1& 2$
% !&$ 1"=!&$ 3". By including the external field term hm in
Eq. (5), one can show ' % 1 for all &> 3.
!$;'; "#" % (
1 2 ; 1; 2 for &> 5:
(6)
For &< 3, no phase transitions occur at finite temperatures
and, at & % 5, a multiplicative logarithmic correction is
expected [1]. It is interesting to notice that a naive power
counting for the %q local theory with the Gaussian spatial
fluctuation term !rm"2 yields the same result for "# by using the
relation of "# % du#G [16]. Our conjecture for "# bears no
reference to the degree cutoff kc caused by the finite system size
N. We will argue later that the cutoff is irrelevant if it is not
too strong: kc > N1=& [15].
We check our conjecture via numerical simulations. Two typical SF
networks are considered, namely, the static model [17] and the
uncorrelated configuration model (UCM) [18]. As these networks have
different degree cut- offs (natural cutoff kc # N1=!&$1" versus
forced sharp cut- off kc # N1=2) in finite systems, one may look
for a possibility of the cutoff-dependent FSS behavior if any. It
turns out that both cutoffs are not strong enough to influ- ence
the FSS for &> 2.
We performed Monte Carlo simulations at various val- ues of &
up to N % 107. We measure the magnetization m, the fluctuation (0 %
N!!m"2, and the Binder cumulant B and average over#103 network
realizations. The transition temperature Tc is estimated by the
asymptotic limit of the crossing points of B for successive system
sizes as well as of the peak points of (0. At criticality, Eq. (4)
leads to m# N$$= "# and similarly (0 # N'0= "# with (0 # j"j$'0 in
the thermodynamic limit. This power-law behavior in N pro- vides an
alternative check for the criticality as well as the estimates for
the exponent ratios. In equilibrium systems, the
fluctuation-dissipation theorem guarantees '0 % '. By collapsing
the data over the range of temperatures, we estimate the value of
the FSS exponent "#. Our numerical data for m and (0 collapse very
well for all values of & in both static and UCM networks. In
Fig. 1, the data collapse is shown for & % 3:87 in static
networks. We summarize in Table I the numerical estimates for $=
"#, "#, and '0= "# at various values of & in static and UCM
networks. All data agree reasonably well with our
predictions.
We also measure the degree-dependent quantities like the
magnetization on vertices of degree k, mk, and its fluctuation
!!mk"2. These quantities are found to satisfy
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S
week ending 22 JUNE 2007
258701-2
a typical size !T of a disordered droplet excitation out of the
uniformly ordered environment. As the free energy cost by the
droplet excitation is compensated by the thermal energy, !!f"!dT #
kBT, we find !T # "$#T with #T % 2=d % 1=yT . The Gaussian length
scale diverges as !G # "$#G with #G % 1=2.
For d > du % 4 (the upper critical dimension), !G domi- nates
over !T , which leads to the correlation length ex- ponent # % #G
and the MF theory is valid. However, the FSS variable LyT" becomes
!L=!T"yT , implying that the competing length scale is not the
dominant correlation length but the droplet size. Substituting the
linear size L by the volume N # Ld, Eq. (2) reads
m % N$$= "# !N1= "#""; (4)
where the FSS (droplet volume) exponent "# % d#T % 2 in the MF
regime. For the general %q MF theory (f % $"m2 & umq), we find
that "# % du#G with du % 2q=!q$ 2", which is consistent with the
earlier result by Botet et al. for models with infinite-range
interactions [12].
We are now ready to explore the FSS in networks. Networks have no
space dimensionality and may be con- sidered as a limiting case of
d! 1. So we expect that any model in networks displays a MF-type
critical behavior. In particular, the MF FSS exponent "# is
independent of d, which leads to the natural conjecture that Eq.
(4) also applies in networks. These predictions have been con-
firmed by numerical simulations for various models in random
networks, small-world networks, and complete graphs. Moreover, the
relation of "# % du#G has been ex- ploited to calculate the value
of du via simulations in networks for complex nonequilibrium models
[13].
In SF networks with the degree distribution P!k" # k$&, there
appears a nontrivial &-dependent MF critical scaling for
&< &u (highly heterogeneous networks) while the standard
MF theory applies for &> &u [1]. Naturally, we expect a
nontrivial FSS theory associated with the non- trivial MF scaling
for &< &u. Previous studies pay atten- tion to the MF
analysis in the thermodynamic limit and hardly discuss the FSS in
the general context. Recently, a few numerical efforts have been
attempted to confirm the MF predictions, but huge finite-size
effects and the lack of the FSS theory disallowed any decisive
conclusion for highly heterogeneous networks [4,7]. Most recently,
even a non-MF scaling has been claimed for the contact process
[14,15] and the question arises as to whether the cutoff in degree
k influences the FSS.
We start with the phenomenological MF free energy for the SF
networks proposed in [1,2]
f!m" % $"m2 & um4 & vjmj&$1 &O!m6"; (5)
where the &-dependent term originates from the singular
behavior of the higher moments of degree in SF networks. For
&> &u % 5, the &-dependent term is irrelevant and we
recover the usual %4 MF theory, yielding $ % 1=2 and
"# % 2. For 3< &< 5, the &-dependent term becomes
relevant and we find the %q MF theory with q % &$ 1. A simple
algebra leads to $ % 1=!&$ 3" and the free energy density in
the ordered phase is f#$"1&2$. One can estimate the typical
droplet volume NT # !!f"$1, yielding NT # "$ "# with "# % 1& 2$
% !&$ 1"=!&$ 3". By including the external field term hm in
Eq. (5), one can show ' % 1 for all &> 3.
!$;'; "#" % (
1 2 ; 1; 2 for &> 5:
(6)
For &< 3, no phase transitions occur at finite temperatures
and, at & % 5, a multiplicative logarithmic correction is
expected [1]. It is interesting to notice that a naive power
counting for the %q local theory with the Gaussian spatial
fluctuation term !rm"2 yields the same result for "# by using the
relation of "# % du#G [16]. Our conjecture for "# bears no
reference to the degree cutoff kc caused by the finite system size
N. We will argue later that the cutoff is irrelevant if it is not
too strong: kc > N1=& [15].
We check our conjecture via numerical simulations. Two typical SF
networks are considered, namely, the static model [17] and the
uncorrelated configuration model (UCM) [18]. As these networks have
different degree cut- offs (natural cutoff kc # N1=!&$1" versus
forced sharp cut- off kc # N1=2) in finite systems, one may look
for a possibility of the cutoff-dependent FSS behavior if any. It
turns out that both cutoffs are not strong enough to influ- ence
the FSS for &> 2.
We performed Monte Carlo simulations at various val- ues of &
up to N % 107. We measure the magnetization m, the fluctuation (0 %
N!!m"2, and the Binder cumulant B and average over#103 network
realizations. The transition temperature Tc is estimated by the
asymptotic limit of the crossing points of B for successive system
sizes as well as of the peak points of (0. At criticality, Eq. (4)
leads to m# N$$= "# and similarly (0 # N'0= "# with (0 # j"j$'0 in
the thermodynamic limit. This power-law behavior in N pro- vides an
alternative check for the criticality as well as the estimates for
the exponent ratios. In equilibrium systems, the
fluctuation-dissipation theorem guarantees '0 % '. By collapsing
the data over the range of temperatures, we estimate the value of
the FSS exponent "#. Our numerical data for m and (0 collapse very
well for all values of & in both static and UCM networks. In
Fig. 1, the data collapse is shown for & % 3:87 in static
networks. We summarize in Table I the numerical estimates for $=
"#, "#, and '0= "# at various values of & in static and UCM
networks. All data agree reasonably well with our
predictions.
We also measure the degree-dependent quantities like the
magnetization on vertices of degree k, mk, and its fluctuation
!!mk"2. These quantities are found to satisfy
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S
week ending 22 JUNE 2007
258701-2
Landau free energy of the form
mean-field critical behavior in the infinite size limit !Scalettar,
1991". The corresponding exact solution is given in Sec.
VI.A.1.a.
The conventional scaling relation between the critical exponents
takes place at !"3,
# + 2$ + ! = 2. !83"
Interestingly, the magnetic susceptibility % has universal critical
behavior with !=1 when #q2$&', i.e., at !"3. This agrees with
the scaling relation ! /(=2−) if we in- sert the standard
mean-field exponents: (= 1
2 and the Fisher exponent )=0; see Sec. IX.B. When 2&!*3, the
susceptibility % has a paramagnetic temperature depen- dence,
%+1/T, at temperatures T,J despite the system being in the ordered
state.
At T&Tc, the ferromagnetic state is strongly hetero- geneous
because the magnetic moment Mi fluctuates from vertex to vertex.
The ansatz !82" enables us to find an approximate distribution
function of Mi,
Y!M" % 1 N&
-!M − Mi" ' P(q!M")
$h!1 − M2" , !84"
where the function q!M" is a solution of an equation
M!q"=tanh!$hq". Near Tc, low-degree vertices have a small magnetic
moment, M!q"*q+Tc−T+1/2.1, while hubs with degree q"T / h/1 have
M!q"*1. The func- tion Y!M" is shown in Fig. 21. Note that the
distribution of magnetic moments in scale-free networks is more in-
homogeneous than in the Erdos-Rényi graphs. In the former case,
Y!M" diverges at M→1. A local magnetic moment depends on its
neighborhood. In particular, a magnetic moment of a spin
neighboring a hub may differ from a moment of a spin surrounded by
low-degree ver- tices !Giuraniuc et al., 2006".
At T=H=0, the exact distribution function 0!h" con- verges to a
function with two delta peaks,
0!h" = x-!h" + !1 − x"-!h − J" , !85"
where the parameter x is determined by an equation describing
percolation in networks; see Sec. III.B.1. Equation !85" tells us
that in the ground state, spins, which belong to a finite cluster,
have zero magnetic mo- ment while spins in a giant connected
component have magnetic moment 1. The average magnetic moment is
M=1−&qP!q"xq. This is exactly the size of the giant con- nected
component of the network.
3. Finite-size effects
When 2&!*3, a dependence of Tc on the size N is determined by
the finite-size cutoff qcut!N" of the degree distribution in Sec.
II.E.4. We obtain
Tc!N" ' , z1 ln N 4
at ! = 3 !86"
!! − 2"2z1qcut 3−!!N"
!3 − !"!! − 1" at 2 & ! & 3 !87"
!Bianconi, 2002; Dorogovtsev et al., 2002b; Leone et al., 2002".
These estimates agree with the numerical simula- tions of
Aleksiejuk et al. !2002" and Herrero !2004". No-
TABLE I. Critical behavior of the magnetization M, the spe- cific
heat -C, and the susceptibility % in the Ising model on networks
with a degree distribution P!q"*q−! for various val- ues of
exponent !. 1%1−T /Tc.
M -C!T&Tc" %
!=5 11/2 / !ln 1−1"1/2 1 / ln 1−1
3&!&5 11/!!−3" 1!5−!"/!!−3"
!=3 e−2T/#q$ T2e−4T/#q$ T−1
2&!&3 T−1/!3−!" T−!!−1"/!3−!"
2 < γ < 3
c
χ
C
M
T
b)
a)
M ,
C ,
χ
FIG. 20. Schematic representation of the critical behavior of the
magnetization M !dotted lines", the magnetic susceptibility %
!dashed lines", and the specific heat C !solid lines" for the
ferromagnetic Ising model on uncorrelated random networks with a
degree distribution P!q"*q−!. !a" !/1, the standard mean-field
critical behavior. A jump of C disappears when ! →5. !b" 4&!*5,
the ferromagnetic phase transition is of sec- ond order. !c"
3&!*4, the transition becomes of higher order. !d" 2&!*3,
the transition is of infinite order, and Tc→' as N→'.
0.001
0.01
10
1
0.1
M
FIG. 21. Distribution function Y!M" of magnetic moments M in the
ferromagnetic Ising model on the Erdos-Rényi graph with mean degree
z1=5 !dashed line" and scale-free graphs with !=4 and 3.5 !solid
and dotted lines" at T close to Tc, $h =0.04.
1303Dorogovtsev, Goltsev, and Mendes: Critical phenomena in complex
networks
Rev. Mod. Phys., Vol. 80, No. 4, October–December 2008
review paper: S. N. Dorogovtsev, A. V. Goltsev, and J. F. F.
Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80,
1275 (2008).
true mean-field
“heterogeneous” mean-field
finite-size scaling
a scaling relation with the scale variable kN!1=! (not shown here)
[16]. For k > N1=!, the distribution P"k# becomes almost flat
for each realization of networks and the degree exponent ! loses
its identity. Therefore, vertices of such a high degree contribute
in a trivial way and the cutoff beyond this range (kc > N1=!)
should not be distin- guishable [15]. This argument is supported by
our numeri- cal results which cannot differentiate the FSS scaling
in the static and UCM networks.
Now we move to a typical model exhibiting a nonequi- librium phase
transition, namely, the directed percolation (DP) system [19]. It
is well known that most of the non- equilibrium models showing an
absorbing-type phase tran-
sition belong to the DP universality class. Among such models, we
here consider the contact process (CP) and the
susceptible-infected-susceptible (SIS) model [6].
The CP is an interacting particle model on a lattice. A particle
creates another particle in one of its neighboring sites with rate
p and a particle annihilates with rate 1. In the SIS model, the
particle creation is attempted in all neigh- boring sites. A
particle-particle interaction comes in through disallowance of
multiple occupancy at a site. As p increases, the system undergoes
a phase transition at pc from a quiescent vacuum (absorbing) phase
to a noisy many-particle (active) phase in the steady state. Near
the absorbing phase transition, the order parameter (particle
density) "$ #$, the fluctuations %0 % N"!"#2 $ #!&0 , the
susceptibility %$ j#j!&, the correlation length '$ j#j!(, the
relaxation time )$ j#j!(t , and the survival probability Ps $
#$
0 with the reduced coupling constant # %
"p! pc#=pc. It is known that $ % $0 due to the time- reversal
symmetry in the DP systems [19] and &0 ! & in general
nonequilibrium systems.
Consider the droplet (cluster) excitation starting from a localized
seed in the absorbing phase. The average space- time size S of a
cluster is estimated as
S$ )‘'dc $ j#j!*; (7)
where )‘ and 'c are the average lifetime and typical size of a
droplet, respectively. Usually )‘ diverges near the tran- sition as
)‘ $ j#j!(t&$
0 for (t > $0 [19], but )‘ is a O"1#
constant otherwise. In the MF regime, it is shown later that the
latter always applies. The droplet size diverges as 'c $ j#j!(T ,
which leads to * % d(T &maxf(t ! $0; 0g. It is well known that
the susceptibility is proportional to the cluster mass, which
yields & % *! $ [19]. Finally, we arrive at the generalized
exponent relation as
& % d(T ! $&maxf(t ! $0; 0g: (8)
The fluctuation exponent &0 satisfies the standard hyper-
scaling relation as &0 % d(T ! 2$.
In SF networks, we propose a phenomenological modi- fication of the
MF Langevin equation describing the DP models, similar to the free
energy modification of the Ising model in Eq. (5):
d dt ""t# % #"! b"2 ! d"+!1 & !!!!
" p
,"t#; (9)
where ""t# is the particle density at time t and ,"t# is a Gaussian
noise. Our modification to the standard MF the- ory comes in by the
third "+!1 term and it is straightfor- ward to show that the
exponent + % ! for the CP and + % !! 1 for the SIS.
By dropping the noise term, one may easily get the MF steady-state
solution for ". We find that $ % 1 for +> 3 and $ % 1="+! 2# for
2< +< 3. For +< 2, there is no phase transition at finite
p. The same result may be ob- tained from the well-established
k-dependent noiseless MF
TABLE I. Critical exponents of the Ising model on the static and
the UCM networks, and the CP on the UCM networks, compared with our
MF predictions.
Ising Network ! $= "( "( &0= "( MF !> 5 1=4 2 1=2
3< !< 5 1 !!1
!!1 !!3
!!3 !!1
Static 7.08 0.26(4) 2.0(2) 0.45(5) 4.45 0.28(2) 2.4(2) 0.45(3) 3.87
0.37(5) 3.5(3) 0.26(4)
UCM 6.50 0.24(4) 2.0(2) 0.51(5) 4.25 0.31(1) 2.5(1) 0.39(1) 3.75
0.38(6) 3.9(2) 0.24(3)
CP Network ! $= "( "( &0= "( MF !> 3 1=2 2 0
2< !< 3 1 !!1
!!1 !!2
!!3 !!1
UCM 4.0 0.49(1) 2.1(1) 0.00(5) 2.75 0.58(1) 2.4(1) !0:16"2# 2.25
0.78(1) 4.0(5) !0:55"5#
0
5
10
15
20
25
-10 -8 -6 -4 -2 0 2 4 6 8 10
m N
0. 37
128000 256000
χ’
N
FIG. 1 (color online). Data collapse of m for the static network
with ! % 3:87, using $= "( % 0:37 and "( % 3:5. Insets: Double
logarithmic plots of the critical decay ofm and %0 against N with
slopes $= "( % 0:37"5# and &0= "( % 0:26"4#.
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S
week ending 22 JUNE 2007
258701-3
a scaling relation with the scale variable kN!1=! (not shown here)
[16]. For k > N1=!, the distribution P"k# becomes almost flat
for each realization of networks and the degree exponent ! loses
its identity. Therefore, vertices of such a high degree contribute
in a trivial way and the cutoff beyond this range (kc > N1=!)
should not be distin- guishable [15]. This argument is supported by
our numeri- cal results which cannot differentiate the FSS scaling
in the static and UCM networks.
Now we move to a typical model exhibiting a nonequi- librium phase
transition, namely, the directed percolation (DP) system [19]. It
is well known that most of the non- equilibrium models showing an
absorbing-type phase tran-
sition belong to the DP universality class. Among such models, we
here consider the contact process (CP) and the
susceptible-infected-susceptible (SIS) model [6].
The CP is an interacting particle model on a lattice. A particle
creates another particle in one of its neighboring sites with rate
p and a particle annihilates with rate 1. In the SIS model, the
particle creation is attempted in all neigh- boring sites. A
particle-particle interaction comes in through disallowance of
multiple occupancy at a site. As p increases, the system undergoes
a phase transition at pc from a quiescent vacuum (absorbing) phase
to a noisy many-particle (active) phase in the steady state. Near
the absorbing phase transition, the order parameter (particle
density) "$ #$, the fluctuations %0 % N"!"#2 $ #!&0 , the
susceptibility %$ j#j!&, the correlation length '$ j#j!(, the
relaxation time )$ j#j!(t , and the survival probability Ps $
#$
0 with the reduced coupling constant # %
"p! pc#=pc. It is known that $ % $0 due to the time- reversal
symmetry in the DP systems [19] and &0 ! & in general
nonequilibrium systems.
Consider the droplet (cluster) excitation starting from a localized
seed in the absorbing phase. The average space- time size S of a
cluster is estimated as
S$ )‘'dc $ j#j!*; (7)
where )‘ and 'c are the average lifetime and typical size of a
droplet, respectively. Usually )‘ diverges near the tran- sition as
)‘ $ j#j!(t&$
0 for (t > $0 [19], but )‘ is a O"1#
constant otherwise. In the MF regime, it is shown later that the
latter always applies. The droplet size diverges as 'c $ j#j!(T ,
which leads to * % d(T &maxf(t ! $0; 0g. It is well known that
the susceptibility is proportional to the cluster mass, which
yields & % *! $ [19]. Finally, we arrive at the generalized
exponent relation as
& % d(T ! $&maxf(t ! $0; 0g: (8)
The fluctuation exponent &0 satisfies the standard hyper-
scaling relation as &0 % d(T ! 2$.
In SF networks, we propose a phenomenological modi- fication of the
MF Langevin equation describing the DP models, similar to the free
energy modification of the Ising model in Eq. (5):
d dt ""t# % #"! b"2 ! d"+!1 & !!!!
" p
,"t#; (9)
where ""t# is the particle density at time t and ,"t# is a Gaussian
noise. Our modification to the standard MF the- ory comes in by the
third "+!1 term and it is straightfor- ward to show that the
exponent + % ! for the CP and + % !! 1 for the SIS.
By dropping the noise term, one may easily get the MF steady-state
solution for ". We find that $ % 1 for +> 3 and $ % 1="+! 2# for
2< +< 3. For +< 2, there is no phase transition at finite
p. The same result may be ob- tained from the well-established
k-dependent noiseless MF
TABLE I. Critical exponents of the Ising model on the static and
the UCM networks, and the CP on the UCM networks, compared with our
MF predictions.
Ising Network ! $= "( "( &0= "( MF !> 5 1=4 2 1=2
3< !< 5 1 !!1
!!1 !!3
!!3 !!1
Static 7.08 0.26(4) 2.0(2) 0.45(5) 4.45 0.28(2) 2.4(2) 0.45(3) 3.87
0.37(5) 3.5(3) 0.26(4)
UCM 6.50 0.24(4) 2.0(2) 0.51(5) 4.25 0.31(1) 2.5(1) 0.39(1) 3.75
0.38(6) 3.9(2) 0.24(3)
CP Network ! $= "( "( &0= "( MF !> 3 1=2 2 0
2< !< 3 1 !!1
!!1 !!2
!!3 !!1
UCM 4.0 0.49(1) 2.1(1) 0.00(5) 2.75 0.58(1) 2.4(1) !0:16"2# 2.25
0.78(1) 4.0(5) !0:55"5#
0
5
10
15
20
25
-10 -8 -6 -4 -2 0 2 4 6 8 10
m N
0. 37
128000 256000
χ’
N
FIG. 1 (color online). Data collapse of m for the static network
with ! % 3:87, using $= "( % 0:37 and "( % 3:5. Insets: Double
logarithmic plots of the critical decay ofm and %0 against N with
slopes $= "( % 0:37"5# and &0= "( % 0:26"4#.
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S
week ending 22 JUNE 2007
258701-3
ref) H. Hong, M. Ha, and H. Park, Finite-size scaling in complex
networks, Phys. Rev. Lett. 98, 258701 (2007).
a typical size !T of a disordered droplet excitation out of the
uniformly ordered environment. As the free energy cost by the
droplet excitation is compensated by the thermal energy, !!f"!dT #
kBT, we find !T # "$#T with #T % 2=d % 1=yT . The Gaussian length
scale diverges as !G # "$#G with #G % 1=2.
For d > du % 4 (the upper critical dimension), !G domi- nates
over !T , which leads to the correlation length ex- ponent # % #G
and the MF theory is valid. However, the FSS variable LyT" becomes
!L=!T"yT , implying that the competing length scale is not the
dominant correlation length but the droplet size. Substituting the
linear size L by the volume N # Ld, Eq. (2) reads
m % N$$= "# !N1= "#""; (4)
where the FSS (droplet volume) exponent "# % d#T % 2 in the MF
regime. For the general %q MF theory (f % $"m2 & umq), we find
that "# % du#G with du % 2q=!q$ 2", which is consistent with the
earlier result by Botet et al. for models with infinite-range
interactions [12].
We are now ready to explore the FSS in networks. Networks have no
space dimensionality and may be con- sidered as a limiting case of
d! 1. So we expect that any model in networks displays a MF-type
critical behavior. In particular, the MF FSS exponent "# is
independent of d, which leads to the natural conjecture that Eq.
(4) also applies in networks. These predictions have been con-
firmed by numerical simulations for various models in random
networks, small-world networks, and complete graphs. Moreover, the
relation of "# % du#G has been ex- ploited to calculate the value
of du via simulations in networks for complex nonequilibrium models
[13].
In SF networks with the degree distribution P!k" # k$&, there
appears a nontrivial &-dependent MF critical scaling for
&< &u (highly heterogeneous networks) while the standard
MF theory applies for &> &u [1]. Naturally, we expect a
nontrivial FSS theory associated with the non- trivial MF scaling
for &< &u. Previous studies pay atten- tion to the MF
analysis in the thermodynamic limit and hardly discuss the FSS in
the general context. Recently, a few numerical efforts have been
attempted to confirm the MF predictions, but huge finite-size
effects and the lack of the FSS theory disallowed any decisive
conclusion for highly heterogeneous networks [4,7]. Most recently,
even a non-MF scaling has been claimed for the contact process
[14,15] and the question arises as to whether the cutoff in degree
k influences the FSS.
We start with the phenomenological MF free energy for the SF
networks proposed in [1,2]
f!m" % $"m2 & um4 & vjmj&$1 &O!m6"; (5)
where the &-dependent term originates from the singular
behavior of the higher moments of degree in SF networks. For
&> &u % 5, the &-dependent term is irrelevant and we
recover the usual %4 MF theory, yielding $ % 1=2 and
"# % 2. For 3< &< 5, the &-dependent term becomes
relevant and we find the %q MF theory with q % &$ 1. A simple
algebra leads to $ % 1=!&$ 3" and the free energy density in
the ordered phase is f#$"1&2$. One can estimate the typical
droplet volume NT # !!f"$1, yielding NT # "$ "# with "# % 1& 2$
% !&$ 1"=!&$ 3". By including the external field term hm in
Eq. (5), one can show ' % 1 for all &> 3.
!$;'; "#" % (
1 2 ; 1; 2 for &> 5:
(6)
For &< 3, no phase transitions occur at finite temperatures
and, at & % 5, a multiplicative logarithmic correction is
expected [1]. It is interesting to notice that a naive power
counting for the %q local theory with the Gaussian spatial
fluctuation term !rm"2 yields the same result for "# by using the
relation of "# % du#G [16]. Our conjecture for "# bears no
reference to the degree cutoff kc caused by the finite system size
N. We will argue later that the cutoff is irrelevant if it is not
too strong: kc > N1=& [15].
We check our conjecture via numerical simulations. Two typical SF
networks are considered, namely, the static model [17] and the
uncorrelated configuration model (UCM) [18]. As these networks have
different degree cut- offs (natural cutoff kc # N1=!&$1" versus
forced sharp cut- off kc # N1=2) in finite systems, one may look
for a possibility of the cutoff-dependent FSS behavior if any. It
turns out that both cutoffs are not strong enough to influ- ence
the FSS for &> 2.
We performed Monte Carlo simulations at various val- ues of &
up to N % 107. We measure the magnetization m, the fluctuation (0 %
N!!m"2, and the Binder cumulant B and average over#103 network
realizations. The transition temperature Tc is estimated by the
asymptotic limit of the crossing points of B for successive system
sizes as well as of the peak points of (0. At criticality, Eq. (4)
leads to m# N$$= "# and similarly (0 # N'0= "# with (0 # j"j$'0 in
the thermodynamic limit. This power-law behavior in N pro- vides an
alternative check for the criticality as well as the estimates for
the exponent ratios. In equilibrium systems, the
fluctuation-dissipation theorem guarantees '0 % '. By collapsing
the data over the range of temperatures, we estimate the value of
the FSS exponent "#. Our numerical data for m and (0 collapse very
well for all values of & in both static and UCM networks. In
Fig. 1, the data collapse is shown for & % 3:87 in static
networks. We summarize in Table I the numerical estimates for $=
"#, "#, and '0= "# at various values of & in static and UCM
networks. All data agree reasonably well with our
predictions.
We also measure the degree-dependent quantities like the
magnetization on vertices of degree k, mk, and its fluctuation
!!mk"2. These quantities are found to satisfy
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S
week ending 22 JUNE 2007
258701-2
a typical size !T of a disordered droplet excitation out of the
uniformly ordered environment. As the free energy cost by the
droplet excitation is compensated by the thermal energy, !!f"!dT #
kBT, we find !T # "$#T with #T % 2=d % 1=yT . The Gaussian length
scale diverges as !G # "$#G with #G % 1=2.
For d > du % 4 (the upper critical dimension), !G domi- nates
over !T , which leads to the correlation length ex- ponent # % #G
and the MF theory is valid. However, the FSS variable LyT" becomes
!L=!T"yT , implying that the competing length scale is not the
dominant correlation length but the droplet size. Substituting the
linear size L by the volume N # Ld, Eq. (2) reads
m % N$$= "# !N1= "#""; (4)
where the FSS (droplet volume) exponent "# % d#T % 2 in the MF
regime. For the general %q MF theory (f % $"m2 & umq), we find
that "# % du#G with du % 2q=!q$ 2", which is consistent with the
earlier result by Botet et al. for models with infinite-range
interactions [12].
We are now ready to explore the FSS in networks. Networks have no
space dimensionality and may be con- sidered as a limiting case of
d! 1. So we expect that any model in networks displays a MF-type
critical behavior. In particular, the MF FSS exponent "# is
independent of d, which leads to the natural conjecture that Eq.
(4) also applies in networks. These predictions have been con-
firmed by numerical simulations for various models in random
networks, small-world networks, and complete graphs. Moreover, the
relation of "# % du#G has been ex- ploited to calculate the value
of du via simulations in networks for complex nonequilibrium models
[13].
In SF networks with the degree distribution P!k" # k$&, there
appears a nontrivial &-dependent MF critical scaling for
&< &u (highly heterogeneous networks) while the standard
MF theory applies for &> &u [1]. Naturally, we expect a
nontrivial FSS theory associated with the non- trivial MF scaling
for &< &u. Previous studies pay atten- tion to the MF
analysis in the thermodynamic limit and hardly discuss the FSS in
the general context. Recently, a few numerical efforts have been
attempted to confirm the MF predictions, but huge finite-size
effects and the lack of the FSS theory disallowed any decisive
conclusion for highly heterogeneous networks [4,7]. Most recently,
even a non-MF scaling has been claimed for the contact process
[14,15] and the question arises as to whether the cutoff in degree
k influences the FSS.
We start with the phenomenological MF free energy for the SF
networks proposed in [1,2]
f!m" % $"m2 & um4 & vjmj&$1 &O!m6"; (5)
where the &-dependent term originates from the singular
behavior of the higher moments of degree in SF networks. For
&> &u % 5, the &-dependent term is irrelevant and we
recover the usual %4 MF theory, yielding $ % 1=2 and
"# % 2. For 3< &< 5, the &-dependent term becomes
relevant and we find the %q MF theory with q % &$ 1. A simple
algebra leads to $ % 1=!&$ 3" and the free energy density in
the ordered phase is f#$"1&2$. One can estimate the typical
droplet volume NT # !!f"$1, yielding NT # "$ "# with "# % 1& 2$
% !&$ 1"=!&$ 3". By including the external field term hm in
Eq. (5), one can show ' % 1 for all &> 3.
!$;'; "#" % (
1 2 ; 1; 2 for &> 5:
(6)
For &< 3, no phase transitions occur at finite temperatures
and, at & % 5, a multiplicative logarithmic correction is
expected [1]. It is interesting to notice that a naive power
counting for the %q local theory with the Gaussian spatial
fluctuation term !rm"2 yields the same result for "# by using the
relation of "# % du#G [16]. Our conjecture for "# bears no
reference to the degree cutoff kc caused by the finite system size
N. We will argue later that the cutoff is irrelevant if it is not
too strong: kc > N1=& [15].
We check our conjecture via numerical simulations. Two typical SF
networks are considered, namely, the static model [17] and the
uncorrelated configuration model (UCM) [18]. As these networks have
different degree cut- offs (natural cutoff kc # N1=!&$1" versus
forced sharp cut- off kc # N1=2) in finite systems, one may look
for a possibility of the cutoff-dependent FSS behavior if any. It
turns out that both cutoffs are not strong enough to influ- ence
the FSS for &> 2.
We performed Monte Carlo simulations at various val- ues of &
up to N % 107. We measure the magnetization m, the fluctuation (0 %
N!!m"2, and the Binder cumulant B and average over#103 network
realizations. The transition temperature Tc is estimated by the
asymptotic limit of the crossing points of B for successive system
sizes as well as of the peak points of (0. At criticality, Eq. (4)
leads to m# N$$= "# and similarly (0 # N'0= "# with (0 # j"j$'0 in
the thermodynamic limit. This power-law behavior in N pro- vides an
alternative check for the criticality as well as the estimates for
the exponent ratios. In equilibrium systems, the
fluctuation-dissipation theorem guarantees '0 % '. By collapsing
the data over the range of temperatures, we estimate the value of
the FSS exponent "#. Our numerical data for m and (0 collapse very
well for all values of & in both static and UCM networks. In
Fig. 1, the data collapse is shown for & % 3:87 in static
networks. We summarize in Table I the numerical estimates for $=
"#, "#, and '0= "# at various values of & in static and UCM
networks. All data agree reasonably well with our
predictions.
We also measure the degree-dependent quantities like the
magnetization on vertices of degree k, mk, and its fluctuation
!!mk"2. These quantities are found to satisfy
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S
week ending 22 JUNE 2007
258701-2
Landau free energy of the form
mean-field critical behavior in the infinite size limit !Scalettar,
1991". The corresponding exact solution is given in Sec.
VI.A.1.a.
The conventional scaling relation between the critical exponents
takes place at !"3,
# + 2$ + ! = 2. !83"
Interestingly, the magnetic susceptibility % has universal critical
behavior with !=1 when #q2$&', i.e., at !"3. This agrees with
the scaling relation ! /(=2−) if we in- sert the standard
mean-field exponents: (= 1
2 and the Fisher exponent )=0; see Sec. IX.B. When 2&!*3, the
susceptibility % has a paramagnetic temperature depen- dence,
%+1/T, at temperatures T,J despite the system being in the ordered
state.
At T&Tc, the ferromagnetic state is strongly hetero- geneous
because the magnetic moment Mi fluctuates from vertex to vertex.
The ansatz !82" enables us to find an approximate distribution
function of Mi,
Y!M" % 1 N&
-!M − Mi" ' P(q!M")
$h!1 − M2" , !84"
where the function q!M" is a solution of an equation
M!q"=tanh!$hq". Near Tc, low-degree vertices have a small magnetic
moment, M!q"*q+Tc−T+1/2.1, while hubs with degree q"T / h/1 have
M!q"*1. The func- tion Y!M" is shown in Fig. 21. Note that the
distribution of magnetic moments in scale-free networks is more in-
homogeneous than in the Erdos-Rényi graphs. In the former case,
Y!M" diverges at M→1. A local magnetic moment depends on its
neighborhood. In particular, a magnetic moment of a spin
neighboring a hub may differ from a moment of a spin surrounded by
low-degree ver- tices !Giuraniuc et al., 2006".
At T=H=0, the exact distribution function 0!h" con- verges to a
function with two delta peaks,
0!h" = x-!h" + !1 − x"-!h − J" , !85"
where the parameter x is determined by an equation describing
percolation in networks; see Sec. III.B.1. Equation !85" tells us
that in the ground state, spins, which belong to a finite cluster,
have zero magnetic mo- ment while spins in a giant connected
component have magnetic moment 1. The average magnetic moment is
M=1−&qP!q"xq. This is exactly the size of the giant con- nected
component of the network.
3. Finite-size effects
When 2&!*3, a dependence of Tc on the size N is determined by
the finite-size cutoff qcut!N" of the degree distribution in Sec.
II.E.4. We obtain
Tc!N" ' , z1 ln N 4
at ! = 3 !86"
!! − 2"2z1qcut 3−!!N"
!3 − !"!! − 1" at 2 & ! & 3 !87"
!Bianconi, 2002; Dorogovtsev et al., 2002b; Leone et al., 2002".
These estimates agree with the numerical simula- tions of
Aleksiejuk et al. !2002" and Herrero !2004". No-
TABLE I. Critical behavior of the magnetization M, the spe- cific
heat -C, and the susceptibility % in the Ising model on networks
with a degree distribution P!q"*q−! for various val- ues of
exponent !. 1%1−T /Tc.
M -C!T&Tc" %
!=5 11/2 / !ln 1−1"1/2 1 / ln 1−1
3&!&5 11/!!−3" 1!5−!"/!!−3"
!=3 e−2T/#q$ T2e−4T/#q$ T−1
2&!&3 T−1/!3−!" T−!!−1"/!3−!"
2 < γ < 3
c
χ
C
M
T
b)
a)
M ,
C ,
χ
FIG. 20. Schematic representation of the critical behavior of the
magnetization M !dotted lines", the magnetic susceptibility %
!dashed lines", and the specific heat C !solid lines" for the
ferromagnetic Ising model on uncorrelated random networks with a
degree distribution P!q"*q−!. !a" !/1, the standard mean-field
critical behavior. A jump of C disappears when ! →5. !b" 4&!*5,
the ferromagnetic phase transition is of sec- ond order. !c"
3&!*4, the transition becomes of higher order. !d" 2&!*3,
the transition is of infinite order, and Tc→' as N→'.
0.001
0.01
10
1
0.1
M
FIG. 21. Distribution function Y!M" of magnetic moments M in the
ferromagnetic Ising model on the Erdos-Rényi graph with mean degree
z1=5 !dashed line" and scale-free graphs with !=4 and 3.5 !solid
and dotted lines" at T close to Tc, $h =0.04.
1303Dorogovtsev, Goltsev, and Mendes: Critical phenomena in complex
networks
Rev. Mod. Phys., Vol. 80, No. 4, October–December 2008
review paper: S. N. Dorogovtsev, A. V. Goltsev, and J. F. F.
Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80,
1275 (2008).
true mean-field
“heterogeneous” mean-field
my contributions: SHL, H. Jeong, and J. D. Noh, Random field Ising
model on networks with inhomogeneous connections, Phys. Rev. E 74,
031118 (2006); SHL, M. Ha, H. Jeong, J. D. Noh, and H. Park,
Critical behavior of the Ising mod