메뉴 건너뛰기




Volumn 85, Issue 5, 2012, Pages

Entropy of stochastic blockmodel ensembles

Author keywords

[No Author keywords available]

Indexed keywords

BLOCK STRUCTURES; BLOCKMODEL; DEGREE CORRELATION; DEGREE SEQUENCE; GENERAL NATURE; GROUP MEMBERSHIPS; LOG-LIKELIHOOD FUNCTIONS; MULTIGRAPHS; NETWORK MODELS; OBSERVED DATA;

EID: 84862006668     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.85.056122     Document Type: Article
Times cited : (150)

References (77)
  • 3
    • 0040979748 scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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
  • 16
    • 84860550726 scopus 로고    scopus 로고
    • 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
  • 18
    • 38349157799 scopus 로고    scopus 로고
    • 1742-5689 10.1098/rsif.2007.1229
    • T. Gross and B. Blasius, J. R. Soc. Interface 1742-5689 10.1098/rsif.2007.1229 5, 259 (2008).
    • (2008) J. R. Soc. Interface , vol.5 , pp. 259
    • Gross, T.1    Blasius, B.2
  • 20
    • 0038752085 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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 scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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
  • 32
    • 37649029436 scopus 로고    scopus 로고
    • 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
  • 35
    • 35148824046 scopus 로고    scopus 로고
    • 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
  • 39
    • 0011216447 scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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
  • 43
  • 46
    • 33748286498 scopus 로고    scopus 로고
    • 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
  • 49
    • 84862009126 scopus 로고    scopus 로고
    • 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.
  • 51
    • 0035420732 scopus 로고    scopus 로고
    • PLEEE8 1539-3755 10.1103/PhysRevE.64.026118
    • M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.64.026118 64, 026118 (2001).
    • (2001) Phys. Rev. e , vol.64 , pp. 026118
    • Newman, M.E.J.1    Strogatz, S.H.2    Watts, D.J.3
  • 52
    • 0038718854 scopus 로고    scopus 로고
    • 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
  • 54
    • 84862012125 scopus 로고    scopus 로고
    • 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 scopus 로고
    • 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
  • 58
    • 0030491229 scopus 로고    scopus 로고
    • 0033-3123 10.1007/BF02294547
    • S. Wasserman and P. Pattison, Psychometrika 0033-3123 10.1007/BF02294547 61, 401 (1996).
    • (1996) Psychometrika , vol.61 , pp. 401
    • Wasserman, S.1    Pattison, P.2
  • 60
    • 84862009601 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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.
  • 63
    • 77956890234 scopus 로고
    • 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 scopus 로고
    • 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 scopus 로고
    • 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
  • 66
    • 70350422326 scopus 로고    scopus 로고
    • JSTPBS 0022-4715 10.1007/s10955-009-9821-2
    • A. C. C. Coolen, A. Martino, and A. Annibale, J. Stat. Phys. JSTPBS 0022-4715 10.1007/s10955-009-9821-2 136, 1035 (2009).
    • (2009) J. Stat. Phys. , vol.136 , pp. 1035
    • Coolen, A.C.C.1    Martino, A.2    Annibale, A.3
  • 67
    • 84862012130 scopus 로고    scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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 scopus 로고
    • 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 scopus 로고    scopus 로고
    • 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.
  • 76
    • 80054751596 scopus 로고    scopus 로고
    • 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


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.