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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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 $= "( % 0:37"5# and &0= "( % 0:26"4#.
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending 22 JUNE 2007
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a scaling relation with the scale variable kN!1=! (not shown here) [16]. For k > N1=!, the distribution P"k# becomes almost flat for each realization of networks and the degree exponent ! loses its identity. Therefore, vertices of such a high degree contribute in a trivial way and the cutoff beyond this range (kc > N1=!) should not be distin- guishable [15]. This argument is supported by our numeri- cal results which cannot differentiate the FSS scaling in the static and UCM networks.
Now we move to a typical model exhibiting a nonequi- librium phase transition, namely, the directed percolation (DP) system [19]. It is well known that most of the non- equilibrium models showing an absorbing-type phase tran-
sition belong to the DP universality class. Among such models, we here consider the contact process (CP) and the susceptible-infected-susceptible (SIS) model [6].
The CP is an interacting particle model on a lattice. A particle creates another particle in one of its neighboring sites with rate p and a particle annihilates with rate 1. In the SIS model, the particle creation is attempted in all neigh- boring sites. A particle-particle interaction comes in through disallowance of multiple occupancy at a site. As p increases, the system undergoes a phase transition at pc from a quiescent vacuum (absorbing) phase to a noisy many-particle (active) phase in the steady state. Near the absorbing phase transition, the order parameter (particle density) "$ #$, 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 $= "( % 0:37"5# and &0= "( % 0:26"4#.
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending 22 JUNE 2007
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ref) H. Hong, M. Ha, and H. Park, Finite-size scaling in complex networks, Phys. Rev. Lett. 98, 258701 (2007).
a typical size !T of a disordered droplet excitation out of the uniformly ordered environment. As the free energy cost by the droplet excitation is compensated by the thermal energy, !!f"!dT # kBT, we find !T # "$#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 $= "#, "#, and '0= "# at various values of & in static and UCM networks. All data agree reasonably well with our predictions.
We also measure the degree-dependent quantities like the magnetization on vertices of degree k, mk, and its fluctuation !!mk"2. These quantities are found to satisfy
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending 22 JUNE 2007
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a typical size !T of a disordered droplet excitation out of the uniformly ordered environment. As the free energy cost by the droplet excitation is compensated by the thermal energy, !!f"!dT # kBT, we find !T # "$#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 $= "#, "#, and '0= "# at various values of & in static and UCM networks. All data agree reasonably well with our predictions.
We also measure the degree-dependent quantities like the magnetization on vertices of degree k, mk, and its fluctuation !!mk"2. These quantities are found to satisfy
PRL 98, 258701 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending 22 JUNE 2007
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Landau free energy of the form
mean-field critical behavior in the infinite size limit !Scalettar, 1991". The corresponding exact solution is given in Sec. VI.A.1.a.
The conventional scaling relation between the critical exponents takes place at !"3,
# + 2$ + ! = 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