-
3
-
-
0040979748
-
-
0378-8733 10.1016/0378-8733(92)90013-W
-
K. Faust and S. Wasserman, Social Networks 0378-8733 10.1016/0378- 8733(92)90013-W 14, 5 (1992).
-
(1992)
Social Networks
, vol.14
, pp. 5
-
-
Faust, K.1
Wasserman, S.2
-
7
-
-
74049087026
-
-
PRPLCM 0370-1573 10.1016/j.physrep.2009.11.002
-
S. Fortunato, Phys. Rep. PRPLCM 0370-1573 10.1016/j.physrep.2009.11.002 486, 75 (2010).
-
(2010)
Phys. Rep.
, vol.486
, pp. 75
-
-
Fortunato, S.1
-
9
-
-
37249033857
-
Role models for complex networks
-
DOI 10.1140/epjb/e2007-00340-y
-
J. Reichardt and D. R. White, Eur. Phys. J. B EPJBFY 1434-6028 10.1140/epjb/e2007-00340-y 60, 217 (2007). (Pubitemid 350277664)
-
(2007)
European Physical Journal B
, vol.60
, Issue.2
, pp. 217-224
-
-
Reichardt, J.1
White, D.R.2
-
12
-
-
79951710564
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.83.016107
-
B. Karrer and M. E. J. Newman, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.83.016107 83, 016107 (2011).
-
(2011)
Phys. Rev. e
, vol.83
, pp. 016107
-
-
Karrer, B.1
Newman, M.E.J.2
-
13
-
-
80053018115
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.84.036103
-
B. Ball, B. Karrer, and M. E. J. Newman, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.84.036103 84, 036103 (2011).
-
(2011)
Phys. Rev. e
, vol.84
, pp. 036103
-
-
Ball, B.1
Karrer, B.2
Newman, M.E.J.3
-
14
-
-
79960953976
-
-
1932-6203 10.1371/journal.pone.0021282
-
J. Reichardt, R. Alamino, and D. Saad, PLoS ONE 1932-6203 10.1371/journal.pone.0021282 6, e21282 (2011).
-
(2011)
PLoS ONE
, vol.6
, pp. 21282
-
-
Reichardt, J.1
Alamino, R.2
Saad, D.3
-
15
-
-
84555195640
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.84.066106
-
A. Decelle, F. Krzakala, C. Moore, and L. Zdeborová, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.84.066106 84, 066106 (2011).
-
(2011)
Phys. Rev. e
, vol.84
, pp. 066106
-
-
Decelle, A.1
Krzakala, F.2
Moore, C.3
Zdeborová, L.4
-
16
-
-
84860550726
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.85.041908
-
T. P. Peixoto, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.85.041908 85, 041908 (2012).
-
(2012)
Phys. Rev. e
, vol.85
, pp. 041908
-
-
Peixoto, T.P.1
-
20
-
-
0038752085
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.67.026126
-
M. E. J. Newman, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.67.026126 67, 026126 (2003).
-
(2003)
Phys. Rev. e
, vol.67
, pp. 026126
-
-
Newman, M.E.J.1
-
22
-
-
70350239119
-
-
EULEEJ 0295-5075 10.1209/0295-5075/81/28005
-
G. Bianconi, EPL (Europhysics Letters) EULEEJ 0295-5075 10.1209/0295-5075/81/28005 81, 28005 (2008).
-
(2008)
EPL (Europhysics Letters)
, vol.81
, pp. 28005
-
-
Bianconi, G.1
-
23
-
-
70350247586
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.80.045102
-
K. Anand and G. Bianconi, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.045102 80, 045102 (2009).
-
(2009)
Phys. Rev. e
, vol.80
, pp. 045102
-
-
Anand, K.1
Bianconi, G.2
-
24
-
-
65249124591
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.79.036114
-
G. Bianconi, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79.036114 79, 036114 (2009).
-
(2009)
Phys. Rev. e
, vol.79
, pp. 036114
-
-
Bianconi, G.1
-
25
-
-
77954823947
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.82.011116
-
K. Anand and G. Bianconi, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.82.011116 82, 011116 (2010).
-
(2010)
Phys. Rev. e
, vol.82
, pp. 011116
-
-
Anand, K.1
Bianconi, G.2
-
27
-
-
0022904506
-
-
SIREAD 0036-1445 10.1137/1028156
-
D. Strauss, SIAM Review SIREAD 0036-1445 10.1137/1028156 28, 513 (1986).
-
(1986)
SIAM Review
, vol.28
, pp. 513
-
-
Strauss, D.1
-
28
-
-
0035475792
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.64.046118
-
Z. Burda, J. D. Correia, and A. Krzywicki, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.64.046118 64, 046118 (2001).
-
(2001)
Phys. Rev. e
, vol.64
, pp. 046118
-
-
Burda, Z.1
Correia, J.D.2
Krzywicki, A.3
-
31
-
-
37649033050
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.69.046117
-
G. Palla, I. Derényi, I. Farkas, and T. Vicsek, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.69.046117 69, 046117 (2004).
-
(2004)
Phys. Rev. e
, vol.69
, pp. 046117
-
-
Palla, G.1
Derényi, I.2
Farkas, I.3
Vicsek, T.4
-
32
-
-
37649029436
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.69.026106
-
Z. Burda, J. Jurkiewicz, and A. Krzywicki, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.69.026106 69, 026106 (2004).
-
(2004)
Phys. Rev. e
, vol.69
, pp. 026106
-
-
Burda, Z.1
Jurkiewicz, J.2
Krzywicki, A.3
-
34
-
-
32844466371
-
Fluctuation-dissipation relations in complex networks
-
DOI 10.1103/PhysRevE.73.016108
-
A. Fronczak, P. Fronczak, and J. A. Hołyst, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.73.016108 73, 016108 (2006). (Pubitemid 43254395)
-
(2006)
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
, vol.73
, Issue.1
, pp. 016108
-
-
Fronczak, A.1
Fronczak, P.2
Holyst, J.A.3
-
35
-
-
35148824046
-
Phase transitions in social networks
-
DOI 10.1140/epjb/e2007-00270-8
-
P. Fronczak, A. Fronczak, and J. A. Hołyst, Eur. Phys. J. B EPJBFY 1434-6028 10.1140/epjb/e2007-00270-8 59, 133 (2007). (Pubitemid 47536706)
-
(2007)
European Physical Journal B
, vol.59
, Issue.1
, pp. 133-139
-
-
Fronczak, P.1
Fronczak, A.2
Holyst, J.A.3
-
37
-
-
77951709870
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.81.046115
-
D. Foster, J. Foster, M. Paczuski, and P. Grassberger, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.81.046115 81, 046115 (2010).
-
(2010)
Phys. Rev. e
, vol.81
, pp. 046115
-
-
Foster, D.1
Foster, J.2
Paczuski, M.3
Grassberger, P.4
-
39
-
-
0011216447
-
-
DSMHA4 0012-365X 10.1016/0012-365X(74)90118-6
-
E. A. Bender, Discrete Mathematics DSMHA4 0012-365X 10.1016/0012-365X(74) 90118-6 10, 217 (1974).
-
(1974)
Discrete Mathematics
, vol.10
, pp. 217
-
-
Bender, E.A.1
-
41
-
-
0000537111
-
-
in edited by J. D. Lamb and D. A. Preece, Vol. 276 (Cambridge University Press, Cambridge
-
N. Wormald, in Surveys in Combinatorics, London Mathematical Society Lecture Note Series, edited by, J. D. Lamb, and, D. A. Preece, Vol. 276 (Cambridge University Press, Cambridge, 1999), pp. 239-298.
-
(1999)
Surveys in Combinatorics
, pp. 239-298
-
-
Wormald, N.1
-
46
-
-
33748286498
-
-
PLEEE8 1539-3755 10.1103/PhysRevE.68.026112
-
J. Park and M. E. J. Newman, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.68.026112 68, 026112 (2003).
-
(2003)
Phys. Rev. e
, vol.68
, pp. 026112
-
-
Park, J.1
Newman, M.E.J.2
-
47
-
-
77949347416
-
-
PRLTAO 0031-9007 10.1103/PhysRevLett.104.108702
-
S. Johnson, J. J. Torres, J. Marro, and M. A. Muñoz, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.104.108702 104, 108702 (2010).
-
(2010)
Phys. Rev. Lett.
, vol.104
, pp. 108702
-
-
Johnson, S.1
Torres, J.J.2
Marro, J.3
Muñoz, M.A.4
-
49
-
-
84862009126
-
-
Note that not all degree sequences are allowed in the first place, since they must be graphical. Imposing a block structure further complicates things, since the graphical condition needs to be generalized to blockmodels. We will not pursue this here, as we consider only the sufficiently sparse situation, where this issue can be neglected.
-
Note that not all degree sequences are allowed in the first place, since they must be graphical. Imposing a block structure further complicates things, since the graphical condition needs to be generalized to blockmodels. We will not pursue this here, as we consider only the sufficiently sparse situation, where this issue can be neglected.
-
-
-
-
52
-
-
0038718854
-
-
SIREAD 0036-1445 10.1137/S003614450342480
-
M. E. J. Newman, SIAM Rev. SIREAD 0036-1445 10.1137/S003614450342480 45, 167 (2003).
-
(2003)
SIAM Rev.
, vol.45
, pp. 167
-
-
Newman, M.E.J.1
-
53
-
-
41349096849
-
Comment on subgraphs in random networks
-
DOI 10.1103/PhysRevE.70.058101, 058101
-
O. D. King, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.70.058101 70, 058101 (2004). (Pubitemid 40086459)
-
(2004)
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
, vol.70
, Issue.52
, pp. 0581011-0581022
-
-
King, O.D.1
Itzkovitz, S.2
Milo, R.3
Kashtan, N.4
Newman, M.E.J.5
Alon, U.6
-
54
-
-
84862012125
-
-
The difference between ensembles with "hard" and "soft" degree constraints is analyzed in detail in Ref. for the case without block structures.
-
The difference between ensembles with "hard" and "soft" degree constraints is analyzed in detail in Ref. for the case without block structures.
-
-
-
-
56
-
-
0010746686
-
-
JCBTB8 0095-8956 10.1016/S0095-8956(81)80022-6
-
N. Wormald, J. Comb. Theory, Ser. B JCBTB8 0095-8956 10.1016/S0095- 8956(81)80022-6 31, 168 (1981).
-
(1981)
J. Comb. Theory, Ser. B
, vol.31
, pp. 168
-
-
Wormald, N.1
-
60
-
-
84862009601
-
-
We could easily use any of the other entropy expressions derived previously, to accommodate the diverse variants of the ensemble, which could be directed, multigraphs, etc. However, the use of the expressions derived for the "hard" degree constraints have a more limited validity, since it assumes stronger sparsity conditions. We focus therefore on ensembles with soft degree constraints, since they are more generally applicable.
-
We could easily use any of the other entropy expressions derived previously, to accommodate the diverse variants of the ensemble, which could be directed, multigraphs, etc. However, the use of the expressions derived for the "hard" degree constraints have a more limited validity, since it assumes stronger sparsity conditions. We focus therefore on ensembles with soft degree constraints, since they are more generally applicable.
-
-
-
-
61
-
-
84862012128
-
-
This simple method can be very inefficient in certain cases, especially if the network is very large, since one may always finish in local maxima which are far away from the global optimum. We have also used the better variant know as the Kernighan-Lin algorithm, adapted to the blockmodel problem in Ref., which can escape such local solutions. However, for the simple examples considered here, we found almost no difference in the results.
-
This simple method can be very inefficient in certain cases, especially if the network is very large, since one may always finish in local maxima which are far away from the global optimum. We have also used the better variant know as the Kernighan-Lin algorithm, adapted to the blockmodel problem in Ref., which can escape such local solutions. However, for the simple examples considered here, we found almost no difference in the results.
-
-
-
-
62
-
-
5744249209
-
-
JCPSA6 0021-9606 10.1063/1.1699114
-
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.1699114 21, 1087 (1953).
-
(1953)
J. Chem. Phys.
, vol.21
, pp. 1087
-
-
Metropolis, N.1
Rosenbluth, A.W.2
Rosenbluth, M.N.3
Teller, A.H.4
Teller, E.5
-
63
-
-
77956890234
-
-
BIOKAX 0006-3444 10.1093/biomet/57.1.97
-
W. K. Hastings, Biometrika BIOKAX 0006-3444 10.1093/biomet/57.1.97 57, 97 (1970).
-
(1970)
Biometrika
, vol.57
, pp. 97
-
-
Hastings, W.K.1
-
64
-
-
0042926219
-
-
in edited by K. L. McAvaney, Vol. 884 (Springer, Berlin/Heidelberg
-
R. Taylor, in Combinatorial Mathematics VIII, edited by, K. L. McAvaney, Vol. 884 (Springer, Berlin/Heidelberg, 1981), pp. 314-336.
-
(1981)
Combinatorial Mathematics VIII
, pp. 314-336
-
-
Taylor, R.1
-
65
-
-
85046697891
-
-
in edited by K. L. McAvaney, Vol. 884 (Springer, Berlin/Heidelberg
-
R. B. Eggleton and D. A. Holton, in Combinatorial Mathematics VIII, edited by, K. L. McAvaney, Vol. 884 (Springer, Berlin/Heidelberg, 1981), pp. 155-172.
-
(1981)
Combinatorial Mathematics VIII
, pp. 155-172
-
-
Eggleton, R.B.1
Holton, D.A.2
-
67
-
-
84862012130
-
-
rs. However, as mentioned in Sec. , this ensemble is equivalent to the microcanonical version for sufficiently large samples.
-
r s. However, as mentioned in Sec., this ensemble is equivalent to the microcanonical version for sufficiently large samples.
-
-
-
-
68
-
-
84862024285
-
-
An alternative which circumvents this problem is the so-called maximum a posteriori (MAP) approach, which uses parameter distributions, instead of a single set of parameters when maximizing the log-likelihood. Instead of the log-likelihood increasing monotonically with B, the parameter distributions become broader. This approach has been applied to the degree-corrected stochastic blockmodel in Ref., using belief propagation. This method, however, has the disadvantage of being numerically less efficient for large networks.
-
An alternative which circumvents this problem is the so-called maximum a posteriori (MAP) approach, which uses parameter distributions, instead of a single set of parameters when maximizing the log-likelihood. Instead of the log-likelihood increasing monotonically with B, the parameter distributions become broader. This approach has been applied to the degree-corrected stochastic blockmodel in Ref., using belief propagation. This method, however, has the disadvantage of being numerically less efficient for large networks.
-
-
-
-
69
-
-
0001065513
-
-
NEUCEB 0899-7667 10.1162/neco.1995.7.2.399
-
A. Treves and S. Panzeri, Neural Computation NEUCEB 0899-7667 10.1162/neco.1995.7.2.399 7, 399 (1995).
-
(1995)
Neural Computation
, vol.7
, pp. 399
-
-
Treves, A.1
Panzeri, S.2
-
70
-
-
84862011704
-
-
We note that Eq. should be understood only as a rule of thumb which gives a lower bound on the bias of L, since it is obtained only from the first term of Eq. , and assumes that the number of blocks in each partition fluctuates independently, which is not likely to hold in general since the block partition is a result of an optimization algorithm.
-
We note that Eq. should be understood only as a rule of thumb which gives a lower bound on the bias of L, since it is obtained only from the first term of Eq., and assumes that the number of blocks in each partition fluctuates independently, which is not likely to hold in general since the block partition is a result of an optimization algorithm.
-
-
-
-
74
-
-
79955707583
-
-
1932-6203 10.1371/journal.pone.0018961
-
A. Lancichinetti, F. Radicchi, J. J. Ramasco, and S. Fortunato, PLoS ONE 1932-6203 10.1371/journal.pone.0018961 6, e18961 (2011).
-
(2011)
PLoS ONE
, vol.6
, pp. 18961
-
-
Lancichinetti, A.1
Radicchi, F.2
Ramasco, J.J.3
Fortunato, S.4
-
76
-
-
80054751596
-
-
PRLTAO 0031-9007 10.1103/PhysRevLett.107.178701
-
C. I. Del Genio, T. Gross, and K. E. Bassler, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.107.178701 107, 178701 (2011).
-
(2011)
Phys. Rev. Lett.
, vol.107
, pp. 178701
-
-
Del Genio, C.I.1
Gross, T.2
Bassler, K.E.3
|