Intrinsic Degree: An Estimator of the Local Growth Rate in Graphs

  • Lorenzo von RitterEmail author
  • Michael E. Houle
  • Stephan Günnemann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11223)


The neighborhood size of a query node in a graph often grows exponentially with the distance to the node, making a neighborhood search prohibitively expensive even for small distances. Estimating the growth rate of the neighborhood size is therefore an important task in order to determine an appropriate distance for which the number of traversed nodes during the search will be feasible. In this work, we present the intrinsic degree model, which captures the growth rate of exponential functions through the analysis of the infinitesimal vicinity of the origin. We further derive an estimator which allows to apply the intrinsic degree model to graphs. In particular, we can locally estimate the growth rate of the neighborhood size by observing the close neighborhood of some query points in a graph. We evaluate the performance of the estimator through experiments on both artificial and real networks.


Intrinsic dimensionality Graph Degree Estimation 



This research was supported in part by the Technical University of Munich - Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement no 291763, co-funded by the European Union. M. E. Houle was supported by JSPS Kakenhi Kiban (B) Research Grant 18H03296.


  1. 1.
    Albert, R., Barabási, A.L.: Topology of evolving networks: local events and universality. Phys. Rev. Lett. 85(24), 5234–5237 (2000)CrossRefGoogle Scholar
  2. 2.
    Amsaleg, L., Bailey, J., Barbe, D., Erfani, S.M., Houle, M.E., Nguyen, V., Radovanović, M.: The vulnerability of learning to adversarial perturbation increases with intrinsic dimensionality. In: WIFS 2017, pp. 1–6 (2017)Google Scholar
  3. 3.
    Amsaleg, L., Chelly, O., Furon, T., Girard, S., Houle, M.E., Kawarabayashi, K., Nett, M.: Estimating local intrinsic dimensionality. In: SIGKDD, pp. 29–38 (2015)Google Scholar
  4. 4.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bu, D.: Topological structure analysis of the protein-protein interaction network in budding yeast. Nucl. Acids Res. 31(9), 2443–2450 (2003)CrossRefGoogle Scholar
  6. 6.
    Caron, F., Fox, E.B.: Sparse graphs using exchangeable random measures. J. Roy. Stat. Soc.: Ser. B (Stat. Methodol.) 79(5), 1295–1366 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Casanova, G., Englmeier, E., Houle, M., Kroeger, P., Nett, M., Schubert, E., Zimek, A.: Dimensional testing for reverse \(k\)-nearest neighbor search. PVLDB 10(7), 769–780 (2017)Google Scholar
  8. 8.
    Erdős, P., Rényi, A.: On random graphs I. Publicationes Mathematicae Debrecen 6, 290 (1959)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gilbert, E.N.: Random graphs. Ann. Math. Stat. 30(4), 1141–1144 (1959)CrossRefGoogle Scholar
  10. 10.
    Houle, M.E.: Dimensionality, discriminability, density and distance distributions. In: ICDMW, pp. 468–473 (2013)Google Scholar
  11. 11.
    Houle, M.E., Kashima, H., Nett, M.: Generalized expansion dimension. In: ICDMW, pp. 587–594 (2012)Google Scholar
  12. 12.
    Houle, M.E., Ma, X., Nett, M., Oria, V.: Dimensional testing for multi-step similarity search. In: ICDM, pp. 299–308 (2012)Google Scholar
  13. 13.
    Houle, M.E., Ma, X., Oria, V.: Effective and efficient algorithms for flexible aggregate similarity search in high dimensional spaces. TKDE 27(12), 3258–3273 (2015)Google Scholar
  14. 14.
    Houle, M.E., Ma, X., Oria, V., Sun, J.: Efficient algorithms for similarity search in axis-aligned subspaces. In: Traina, A.J.M., Traina, C., Cordeiro, R.L.F. (eds.) SISAP 2014. LNCS, vol. 8821, pp. 1–12. Springer, Cham (2014). Scholar
  15. 15.
    Houle, M.E., Ma, X., Oria, V., Sun, J.: Query expansion for content-based similarity search using local and global features. TOMM 13(3), 25:1–25:23 (2017)CrossRefGoogle Scholar
  16. 16.
    Houle, M.E., Oria, V., Wali, A.M.: Improving \(k\)-nn graph accuracy using local intrinsic dimensionality. In: Beecks, C., Borutta, F., Kröger, P., Seidl, T. (eds.) SISAP 2017. LNCS, vol. 10609, pp. 110–124. Springer, Cham (2017). Scholar
  17. 17.
    Houle, M.E.: Local intrinsic dimensionality I: an extreme-value-theoretic foundation for similarity applications. In: Beecks, C., Borutta, F., Kröger, P., Seidl, T. (eds.) SISAP 2017. LNCS, vol. 10609, pp. 64–79. Springer, Cham (2017). Scholar
  18. 18.
    Houle, M.E.: Local intrinsic dimensionality II: multivariate analysis and distributional support. In: Beecks, C., Borutta, F., Kröger, P. (eds.) SISAP 2017. LNCS, vol. 10609, pp. 80–95. Springer, Cham (2017). Scholar
  19. 19.
    Klimt, B., Yang, Y.: The Enron corpus: a new dataset for email classification research. In: Boulicaut, J.-F., Esposito, F., Giannotti, F., Pedreschi, D. (eds.) ECML 2004. LNCS (LNAI), vol. 3201, pp. 217–226. Springer, Heidelberg (2004). Scholar
  20. 20.
    Lovász, L.: Large Networks and Graph Limits, Colloquium Publications, vol. 60. American Mathematical Society (2012)Google Scholar
  21. 21.
    Ma, X., Li, B., Wang, Y., Erfani, S.M., Wijewickrema, S.N.R., Schoenebeck, G., Song, D., Houle, M.E., Bailey, J.: Characterizing adversarial subspaces using local intrinsic dimensionality. In: ICLR, pp. 1–15 (2018)Google Scholar
  22. 22.
    Ma, X., Wang, Y., Houle, M.E., Zhou, S., Erfani, S.M., Xia, S., Wijewickrema, S.N.R., Bailey, J.: Dimensionality-driven learning with noisy labels. In: ICML, pp. 1–10 (2018)Google Scholar
  23. 23.
    Price, D.D.S.: A general theory of bibliometric and other cumulative advantage processes. J. Am. Soc. Inf. Sci. 27(5), 292–306 (1976)CrossRefGoogle Scholar
  24. 24.
    Romano, S., Chelly, O., Nguyen, V., Bailey, J., Houle, M.E.: Measuring dependency via intrinsic dimensionality. In: ICPR, pp. 1207–1212 (2016)Google Scholar
  25. 25.
    Travers, J., Milgram, S.: An experimental study of the small world problem. Sociometry 32(4), 425 (1969)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Lorenzo von Ritter
    • 1
    • 2
    Email author
  • Michael E. Houle
    • 1
  • Stephan Günnemann
    • 2
  1. 1.National Institute of InformaticsChiyoda-kuJapan
  2. 2.Technical University of MunichMunichGermany

Personalised recommendations