Almost Exact Recovery in Label Spreading

  • Konstantin Avrachenkov
  • Maximilien DrevetonEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11631)


In semi-supervised graph clustering setting, an expert provides cluster membership of few nodes. This little amount of information allows one to achieve high accuracy clustering using efficient computational procedures. Our main goal is to provide a theoretical justification why the graph-based semi-supervised learning works very well. Specifically, for the Stochastic Block Model in the moderately sparse regime, we prove that popular semi-supervised clustering methods like Label Spreading achieve asymptotically almost exact recovery as long as the fraction of labeled nodes does not go to zero and the average degree goes to infinity.


Semi-supervised clustering Community detection Label spreading Random graphs Stochastic Block Model 



This work has been done within the project of Inria – Nokia Bell Labs “Distributed Learning and Control for Network Analysis”.


  1. 1.
    Abbe, E.: Community detection and stochastic block models. Found. Trends® Commun. Inf. Theory 14(1–2), 1–162 (2018)Google Scholar
  2. 2.
    Avrachenkov, K., Gonçalves, P., Mishenin, A., Sokol, M.: Generalized optimization framework for graph-based semi-supervised learning. In: SIAM International Conference on Data Mining (SDM 2012) (2012)Google Scholar
  3. 3.
    Avrachenkov, K., Kadavankandy, A., Litvak, N.: Mean field analysis of personalized pagerank with implications for local graph clustering. J. Stat. Phys. 173(3–4), 895–916 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Avrachenkov, K.E., Filar, J.A., Howlett, P.G.: Analytic Perturbation Theory and Its Applications, vol. 135. SIAM, Philadelphia (2013)CrossRefGoogle Scholar
  5. 5.
    Chapelle, O., Schölkopf, B., Zien, A.: Semi-supervised Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge (2006)CrossRefGoogle Scholar
  6. 6.
    Condon, A., Karp, R.M.: Algorithms for graph partitioning on the planted partition model. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds.) APPROX/RANDOM-1999. LNCS, vol. 1671, pp. 221–232. Springer, Heidelberg (1999). Scholar
  7. 7.
    Erdős, P., Rényi, A.: On random graphs. Publ. Math. (Debr.) 6, 290–297 (1959)zbMATHGoogle Scholar
  8. 8.
    Gilbert, E.N.: Random graphs. Ann. Math. Statist. 30(4), 1141–1144 (1959)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Holland, P.W., Laskey, K.B., Leinhardt, S.: Stochastic blockmodels: first steps. Soc. Netw. 5(2), 109–137 (1983)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  11. 11.
    Johnson, R., Zhang, T.: On the effectiveness of Laplacian normalization for graph semi-supervised learning. J. Mach. Learn. Res. 8(Jul), 1489–1517 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Le, C.M., Levina, E., Vershynin, R.: Concentration and regularization of random graphs. Random Struct. Algorithms 51(3), 538–561 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mai, X., Couillet, R.: A random matrix analysis and improvement of semi-supervised learning for large dimensional data. J. Mach. Learn. Res. 19(1), 3074–3100 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Zhou, D., Bousquet, O., Lal, T.N., Weston, J., Schölkopf, B.: Learning with local and global consistency. In: Advances in Neural Information Processing Systems, pp. 321–328 (2004)Google Scholar
  15. 15.
    Zhu, X.: Semi-supervised learning literature survey. Technical report, Computer Science Department, University of Wisconsin-Madison (2006)Google Scholar
  16. 16.
    Zhu, X., Ghahramani, Z., Lafferty, J.D.: Semi-supervised learning using Gaussian fields and harmonic functions. In: ICML (2003)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Inria Sophia AntipolisValbonneFrance

Personalised recommendations