Reducing degeneracy in maximum entropy models of networks

Physical Review Letters

Volumn 114, Issue 15, 2015, Pages

  Horvát, Szabolcs ^a   Czabarka, Éva ^b   Toroczkai, Zoltán ^a,c  

^a UNIVERSITY OF NOTRE DAME   (United States)

^b UNIVERSITY OF SOUTH CAROLINA   (United States)

^c University of Notre Dame   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL TRANSFORMATIONS;

DEGENERACY PROBLEMS; DENSITY OF STATE; EMPIRICAL DATA; MAXIMUM ENTROPY MODELS; MAXIMUM ENTROPY PRINCIPLE; NETWORK PROPERTIES; NON-LINEAR RELATIONSHIPS; SPONTANEOUS SYMMETRY BREAKING;

ENTROPY;

EID: 84929590189     PISSN: 00319007     EISSN: 10797114     Source Type: Journal    
DOI: 10.1103/PhysRevLett.114.158701     Document Type: Article

References (54)

1. E.T. Jaynes, Phys. Rev. 106, 620 (1957). PHRVAO 0031-899X 10.1103/PhysRev.106.620
2. E.T. Jaynes, Phys. Rev. 108, 171 (1957). PHRVAO 0031-899X 10.1103/PhysRev.108.171
3. C.E. Shannon, Bell Syst. Tech. J. 27, 379 (1948). BSTJAN 0005-8580 10.1002/j.1538-7305.1948.tb01338.x
4. S. Pressé, K. Ghosh, J. Lee, and K.A. Dill, Rev. Mod. Phys. 85, 1115 (2013). RMPHAT 0034-6861 10.1103/RevModPhys.85.1115
5. A.K. Livesey and J.C. Brochon, Biophys. J. 52, 693 (1987). BIOJAU 0006-3495 10.1016/S0006-3495(87)83264-2
6. P.J. Steinbach, R. Ionescu, and C.R. Matthews, Biophys. J. 82, 2244 (2002). BIOJAU 0006-3495 10.1016/S0006-3495(02)75570-7
7. G. Yeo and C.B. Burge, J. Comput. Biol. 11, 377 (2004). JCOBEM 1066-5277 10.1089/1066527041410418
8. L.P. Watkins, H. Chang, and H. Yang, J. Phys. Chem. A 110, 5191 (2006). JPCAFH 1089-5639 10.1021/jp055886d
9. M. Vergassola, E. Villermaux, and B.I. Shraiman, Nature (London) 445, 406 (2007). NATUAS 0028-0836 10.1038/nature05464
10. Z.M. Saul and V. Filkov, Bioinformatics 23, 2604 (2007). BOINFP 1367-4803 10.1093/bioinformatics/btm370
11. A.M. Walczak, G. Tkačik, and W. Bialek, Phys. Rev. E 81, 041905 (2010). PLEEE8 1539-3755 10.1103/PhysRevE.81.041905
12. F. Morcos, N.P. Schafer, R.R. Cheng, J.N. Onuchic, and P.G. Wolynes, Proc. Natl. Acad. Sci. U.S.A. 111, 12408 (2014). PNASA6 0027-8424 10.1073/pnas.1413575111
13. J. Shlens, G.D. Field, J.L. Gauthier, M.I. Grivich, D. Petrusca, A. Sher, A.M. Litke, and E.J. Chichilnisky, J. Neurosci. 26, 8254 (2006). JNRSDS 0270-6474 10.1523/JNEUROSCI.1282-06.2006
14. E. Schneidman, M.J. Berry, R. Segev, and W. Bialek, Nature (London) 440, 1007 (2006). NATUAS 0028-0836 10.1038/nature04701
15. A. Tang, D. Jackson, J. Hobbs, W. Chen, J.L. Smith, H. Patel, A. Prieto, D. Petrusca, M.I. Grivich, A. Sher, P. Hottowy, W. Dabrowski, A.M. Litke, and J.M. Beggs, J. Neurosci. 28, 505 (2008). JNRSDS 0270-6474 10.1523/JNEUROSCI.3359-07.2008
16. O. Marre, S. El Boustani, Y. Frégnac, and A. Destexhe, Phys. Rev. Lett. 102, 138101 (2009). PRLTAO 0031-9007 10.1103/PhysRevLett.102.138101
17. F.-C. Yeh, A. Tang, J.P. Hobbs, P. Hottowy, W. Dabrowski, A. Sher, A. Litke, and J.M. Beggs, Entropy 12, 89 (2010). ENTRFG 1099-4300 10.3390/e12010089
18. G.J. Stephens, L.C. Osborne, and W. Bialek, Proc. Natl. Acad. Sci. U.S.A. 108, 15565 (2011). PNASA6 0027-8424 10.1073/pnas.1010868108
19. E. Ganmor, R. Segev, and E. Schneidman, Proc. Natl. Acad. Sci. U.S.A. 108, 9679 (2011). PNASA6 0027-8424 10.1073/pnas.1019641108
20. T. Watanabe, S. Hirose, H. Wada, Y. Imai, T. Machida, I. Shirouzu, S. Konishi, Y. Miyashita, and N. Masuda, Nat. Commun. 4, 1370 (2013). NCAOBW 2041-1723 10.1038/ncomms2388
21. M. Ercsey-Ravasz, N.T. Markov, C. Lamy, D.C. Van Essen, K. Knoblauch, Z. Toroczkai, and H. Kennedy, Neuron 80, 184 (2013). NERNET 0896-6273 10.1016/j.neuron.2013.07.036
22. S.J. Phillips, R.P. Anderson, and R.E. Schapire, Ecol. Model. 190, 231 (2006). ECMODT 0304-3800 10.1016/j.ecolmodel.2005.03.026
23. J. Harte, Maximum Entropy and Ecology (Oxford University Press, New York, 2011).
24. P. Fronczak, A. Fronczak, and J.A. Hołyst, Phys. Rev. E 75, 026103 (2007). PLEEE8 1539-3755 10.1103/PhysRevE.75.026103
25. A. Wimmer and K. Lewis, Am. J. Sociology 116, 583 (2010). 0002-9602 10.1086/653658
26. F.M. Bass, J. Market. Res. 11, 1 (1974). JMKRAE 0022-2437 10.2307/3150989
27. S. Agrawal, Z. Wang, and Y. Ye, in Internet and Network Economics, edited by C. Papadimitriou and S. Zhang, Lecture Notes in Computer Science Vol. 5385 (Springer-Verlag, Berlin, 2008), pp. 126-137.
28. S.F. Gull and G.J. Daniell, Nature (London) 272, 686 (1978). NATUAS 0028-0836 10.1038/272686a0
29. U. Skoglund, L.G. Öfverstedt, R.M. Burnett, and G. Bricogne, J. Struct. Biol. 117, 173 (1996). JSBIEM 1047-8477 10.1006/jsbi.1996.0081
30. R. Rosenfeld, Comput. Speech Lang. 10, 187 (1996). 0885-2308 10.1006/csla.1996.0011
31. P.W. Holland and S. Leinhardt, J. Am. Stat. Assoc. 76, 33 (1981). JSTNAL 0162-1459 10.1080/01621459.1981.10477598
32. D. Strauss, SIAM Rev. 28, 513 (1986). SIREAD 0036-1445 10.1137/1028156
33. J. Park and M.E.J. Newman, Phys. Rev. E 70, 066146 (2004). PLEEE8 1539-3755 10.1103/PhysRevE.70.066146
34. J. Park and M.E.J. Newman, Phys. Rev. E 70, 066117 (2004). PLEEE8 1539-3755 10.1103/PhysRevE.70.066117
35. J. Park and M.E.J. Newman, Phys. Rev. E 72, 026136 (2005). PLEEE8 1539-3755 10.1103/PhysRevE.72.026136
36. G.L. Robins, P. Pattison, Y. Kalish, and D. Lusher, Soc. Networks 29, 173 (2007). 0378-8733 10.1016/j.socnet.2006.08.002
37. P. Fronczak, A. Fronczak, and M. Bujok, Phys. Rev. E 88, 032810 (2013). PLEEE8 1539-3755 10.1103/PhysRevE.88.032810
38. T. House, EPL 105, 68006 (2014). EULEEJ 0295-5075 10.1209/0295-5075/105/68006
39. In the figures, however, we indicate the full range of the values.
40. B. Bollobás, Random Graphs, 2nd ed. (Cambridge University Press, Cambridge, England, 2001).
41. H. Kim, Z. Toroczkai, I. Miklós, P. Erdos, and L. Székely, J. Phys. A 42, 392001 (2009). JPAMB5 1751-8113 10.1088/1751-8113/42/39/392001
42. C.I. Del Genio, H. Kim, Z. Toroczkai, and K.E. Bassler, PLoS One 5, e10012 (2010). POLNCL 1932-6203 10.1371/journal.pone.0010012
43. É. Czabarka, A. Dutle, P.L. Erdos, and I. Miklós, Discrete Appl. Math. 181, 283 (2015). DAMADU 0166-218X 10.1016/j.dam.2014.10.012
44. M.R. Jerrum, L.G. Valiant, and V.V. Vazirani, Theor. Comput. Sci. 43, 169 (1986). TCSCDI 0304-3975 10.1016/0304-3975(86)90174-X
45. V. Vazirani, Approximation Algorithms (Springer, New York, 2003).
46. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.114.158701 for animated versions of Fig. 1, proof of the theorem in arbitrary dimensions, non-network examples of degenerate maximum entropy models, and for validation of the method using a larger network data set. It includes Refs. [27,47-49].
47. P. Ramond, Group Theory: A Physicist's Survey (Cambridge University Press, Cambridge, England, 2010).
48. M. Bóna, Combinatorics of Permutations (Discrete Mathematics and its Applications) (Chapman and Hall/CRC Press, London, 2004).
49. P.M. Gleiser and L. Danon, Adv. Compl. Syst. 06, 565 (2003). 0219-5259 10.1142/S0219525903001067
50. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, England, 2004).
51. Equivalent to the usual definition for strict concavity (Equation presented), (Equation presented) with (Equation presented).
52. T.A.B. Snijders, P.E. Pattison, G.L. Robins, and M.S. Handcock, Sociol. Methodol. 36, 99 (2006). 10.1111/j.1467-9531.2006.00176.x
53. Since (Equation presented) is bijective, the counts of states with (Equation presented) is the same as the counts of states with (Equation presented).
54. W. W. Zachary, Journal of Anthropological Research 33, 452 (1977). 0091-7710 JAPRCP