Skip to main content

Contraction of Measures on Graphs

Abstract

Given a finitely supported probability measure μ on a connected graph G, we construct a family of probability measures interpolating the Dirac measure at some given point oG and μ. Inspired by Sturm-Lott-Villani theory of Ricci curvature bounds on measured length spaces, we then study the convexity of the entropy functional along such interpolations. Explicit results are given in three canonical cases, when the graph G is either ℤn, a cube or a tree.

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

References

  1. Gozlan, N., Roberto, C., Samson, P.M., Tetali, P.: Displacement convexity of entropy and related inequalities on graphs. Probab. Theory Relat. Fields (2014)

  2. Harremoës, P., Johnson, O., Kontoyiannis, I.: Thinning and the law of small numbers. In: IEEE International Symposium on Information Theory, 2007. ISIT 2007, pp. 1491–1495. IEEE (2007)

  3. Hillion, E., Johnson, O., Yu, Y.: Translations of probability measures on ℤ

  4. Léonard, C.: From the Schrödinger problem to the Monge–Kantorovich problem. J. Funct. Anal. (2011)

  5. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991

  6. Ohta, S.: On the measure contraction property of metric measure spaces. Commentarii Mathematici Helvetici 82(4), 805–828 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  7. Rényi, A.: A characterization of poisson processes. Magyar Tud. Akad. Mat. Kutató. Int. Közl 1, 519–527 (1956)

    Google Scholar 

  8. Sturm, K.T.: On the geometry of metric measure spaces I. Acta Math. 196(1), 65–131 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  9. Sturm, K.T.: On the geometry of metric measure spaces II. Acta Math. 196(1), 133–177 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  10. Villani, C.: Topics in optimal transportation, vol. 58. Amer Mathematical Society (2003)

  11. Villani, C.: Optimal transport: old and new, vol. 338. Springer, Berlin (2009)

    Book  Google Scholar 

  12. Yu, Y., Johnson, O.: Concavity of entropy under thinning. In: IEEE International Symposium on Information Theory, 2009. ISIT 2009, pp. 144–148. IEEE (2009)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Erwan Hillion.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hillion, E. Contraction of Measures on Graphs. Potential Anal 41, 679–698 (2014). https://doi.org/10.1007/s11118-014-9388-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-014-9388-7

Keywords

  • Ricci curvature
  • Sturm-Lott-Villani theory
  • Convexity of entropy
  • Measure contraction property

Mathematic Subject Classifications (2010)

  • 60B99
  • 05C99