Skip to main content

Statistical Physics and Network Optimization Problems

  • Chapter

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 2141)

Abstract

The scope of these lecture notes is to provide an introduction to modern statistical physics mean-field methods for the study of phase transitions and optimization problems over random structures. We first give a brief introduction to the field using as tutorial example the percolation problem in random graphs. Next we describe the so called cavity method and the related message-passing algorithms (Belief Propagation and variants) which can be used to analyze and solve optimization problems over random structures.

Keywords

  • Partition Function
  • Random Graph
  • Constraint Satisfaction Problem
  • Soft Constraint
  • Hard Constraint

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Throughout this chapter, we will omit for simplicity the Boltzmann’s constant k; this is always possible by choosing appropriate measurement units, such that k = 1.

  2. 2.

    When the graph does not form a single connected component it is often called a forest, but this distinction is moot four our purposes.

  3. 3.

    Using the most biased variable is a simple and reasonable heuristic which works well in practice, but other strategies may be considered.

  4. 4.

    One step may correspond to one update of all messages (synchronous update scheme), or more often to the update of the messages associated to one randomly chosen variable or function node (asynchronous update scheme).

  5. 5.

    This requires to extend the BP equations to models with continuous variables, which was omitted here for simplicity, but is rather straightforward.

References

  1. R. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965)

    Google Scholar 

  2. S. Ma, Statistical Mechanics (World Scientific, Singapore, 1985)

    CrossRef  MATH  Google Scholar 

  3. K. Huang, Statistical Mechanics (Wiley, New York, 1967)

    Google Scholar 

  4. R.S. Ellis, Entropy, Large Deviations, and Statistical Mechanics (Springer, New York, 1985)

    CrossRef  MATH  Google Scholar 

  5. P. Erdős, A. Rényi, On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960)

    Google Scholar 

  6. B. Bollobás, Random Graphs (Academic, New York, 1985)

    MATH  Google Scholar 

  7. R. Potts, Proc. Camb. Philos. Soc. 48, 106 (1952)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. P. Kasteleyn, C. Fortuin, J. Phys. Soc. Jpn. Suppl. 26, 1114 (1969)

    Google Scholar 

  9. F. Wu, The potts model. Rev. Mod. Phys. 54, 235 (1982)

    CrossRef  Google Scholar 

  10. A. Engel, R. Monasson, A.K. Hartmann, On large-deviations properties of Erdős-Rényi random graphs. J. Stat. Phys. 117, 387 (2004)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. O. Martin, R. Monasson, R. Zecchina, Statistical mechanics methods and phase transitions in optimization problems. Theor. Comput. Sci. 265, 3 (2001)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. M. Mézard, G. Parisi, M.A. Virasoro, Spin-Glass Theory and Beyond. Lecture Notes in Physics, vol. 9 (World Scientific, Singapore, 1987)

    Google Scholar 

  13. M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, Oxford, 2009)

    CrossRef  MATH  Google Scholar 

  14. M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. de L’IHÉS 81(1), 73–205 (1995)

    CrossRef  MATH  Google Scholar 

  15. M. Mézard, R. Zecchina, Random K-satisfiability: from an analytic solution to a new efficient algorithm. Phys. Rev. E 66, 056126 (2002)

    CrossRef  Google Scholar 

  16. A. Braunstein, M. Mézard, R. Zecchina, Survey propagation: an algorithm for satisfiability. Random Struct. Algorithm 27, 201–226 (2005)

    CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carlo Baldassi .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Baldassi, C., Braunstein, A., Ramezanpour, A., Zecchina, R. (2015). Statistical Physics and Network Optimization Problems. In: Fagnani, F., Fosson, S., Ravazzi, C. (eds) Mathematical Foundations of Complex Networked Information Systems. Lecture Notes in Mathematics(), vol 2141. Springer, Cham. https://doi.org/10.1007/978-3-319-16967-5_2

Download citation