LOGIN TO YOUR ACCOUNT

Username
Password
Remember Me
Or use your Academic/Social account:

CREATE AN ACCOUNT

Or use your Academic/Social account:

Congratulations!

You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.

Important!

Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message

CREATE AN ACCOUNT

Name:
Username:
Password:
Verify Password:
E-mail:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Radin, Charles; Sadun, Lorenzo (2013)
Languages: English
Types: Preprint
Subjects: Mathematics - Combinatorics, Mathematical Physics, Condensed Matter - Statistical Mechanics, 82B26, 05C35, 05C80
We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its analyticity in various regimes. In particular this implies the existence of a phase transition between distinct heterogeneous multipartite phases at low triangle density, and a phase transition between these phases and the disordered phase at high triangle density.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • [A] P.W. Anderson, Basic Notions of Condensed Matter Physics, Benjamin/Cummings, Menlo Park, 1984, Chapter 2.
    • [AR1] D. Aristoff and C. Radin, Rigidity in solids, J. Stat. Phys. 144 (2011) 1247-1255.
    • [AR2] D. Aristoff and C. Radin, Emergent structures in large networks, J. Appl. Probab. (to appear), arXiv:1110.1912 [BCLSV] C. Borgs, J. Chayes, L. Lov´asz, V.T. So´s and K. Vesztergombi, Convergent graph sequences I: subgraph frequencies, metric properties, and testing, Adv. Math. 219 (2008) 1801-1851.
    • [BFMH] P.G. Bolhuis, D. Frenkel, S.-C. Muse and D.A. Huse, Nature (London) 388 (1997) 235-236 .
    • [Br] S. G. Brush, Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager, Princeton University Press, Princeton, 1983, 277.
    • [CD] S. Chatterjee, and P. Diaconis, Estimating and understanding exponential random graph models, arXiv: 1102.2650v3.
    • [CV] S. Chatterjee and S.R.S. Varadhan, The large deviation principle for the Erdo¨s-R´enyi random graph, Eur. J. Comb. 32 (2011) 1000-1017 [GR] C.S. Gardner and C. Radin, The infinite volume ground state of the Lennard-Jones potential J. Stat. Phys., 20 (1979), 719-724.
    • [LMP] J.L. Lebowitz, A.E. Mazel and E. Presutti, Liquid-vapor phase transitions for systems with finiterange interactions, J. Stat. Phys. 94 (1999) 955-1025.
    • [Lov] L. Lov´asz, Large networks and graph limits, American Mathematical Society, Providence, 2012.
    • [LS1] L. Lov´asz and B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B 98 (2006) 933-957.
    • [LS2] L. Lov´asz and B. Szegedy, Szemer´edi's lemma for the analyst, GAFA 17 (2007) 252-270.
    • [LS3] L. Lov´asz and B. Szegedy, Finitely forcible graphons, J. Combin. Theory Ser. B 101 (2011) 269-301.
    • [Low] H. Lo¨wen, Fun with hard spheres, In: “Spatial Statistics and Statistical Physics”, edited by K. Mecke and D. Stoyan, Springer Lecture Notes in Physics, volume 554, pages 295-331, Berlin, 2000.
    • F. Nicolo` and C. Radin, A first-order phase transition between crystal phases in the shift model, J. Stat. Phys. 28 (1982) 473-478.
    • J. Park and M.E.J. Newman, Solution for the properties of a clustered network, Phys. Rev. E 72 (2005) 026136.
    • O. Pikhurko and A. Razborov, Asymptotic structure of graphs with the minimum number of triangles, arXiv:1203.4393 C. Radin and M. Yin, Phase transitions in exponential random graphs, Ann. Appl. Probab. (to appear), arXiv:1108.0649.
    • B. Simon, Fifteen problems in mathematical physics, in Perspectives in Mathematics: Anniversary of Oberwolfach 1984, Birkh¨auser Verlag, Basil-Berlin-Boston, 1984, 442.
    • D. Strauss, On a general class of models for interaction, SIAM Rev. 28 (1986) 513-527.
    • F. Theil, A proof of crystallization in two dimensions, Comm. Math. Phys. 262 (2006), 209 - 236 G.E. Uhlenbeck, in Fundamental Problems in Statistical Mechanics II, edited by E. G. D. Cohen, Wiley, New York, 1968, 16-17.
    • L.V. Woodcock, Nature (London) 385 (1997) 141-143.
    • Charles Radin, Department of Mathematics, The University of Texas at Austin, Austin, TX 78712 E-mail address: Lorenzo Sadun, Department of Mathematics, The University of Texas at Austin, Austin, TX 78712 E-mail address:
  • No related research data.
  • No similar publications.

Share - Bookmark

Funded by projects

Cite this article

Collected from