Dynamic Hierarchies in Temporal Directed Networks

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11052)


The outcome of interactions in many real-world systems can be often explained by a hierarchy between the participants. Discovering hierarchy from a given directed network can be formulated as follows: partition vertices into levels such that, ideally, there are only forward edges, that is, edges from upper levels to lower levels. In practice, the ideal case is impossible, so instead we minimize some penalty function on the backward edges. One practical option for such a penalty is agony, where the penalty depends on the severity of the violation. In this paper we extend the definition of agony to temporal networks. In this setup we are given a directed network with time stamped edges, and we allow the rank assignment to vary over time. We propose 2 strategies for controlling the variation of individual ranks. In our first variant, we penalize the fluctuation of the rankings over time by adding a penalty directly to the optimization function. In our second variant we allow the rank change at most once. We show that the first variant can be solved exactly in polynomial time while the second variant is NP-hard, and in fact inapproximable. However, we develop an iterative method, where we first fix the change point and optimize the ranks, and then fix the ranks and optimize the change points, and reiterate until convergence. We show empirically that the algorithms are reasonably fast in practice, and that the obtained rankings are sensible. Code related to this paper is available at:

Supplementary material

478890_1_En_4_MOESM1_ESM.pdf (389 kb)
Supplementary material 1 (pdf 388 KB)


  1. 1.
    Bellman, R.: On the approximation of curves by line segments using dynamic programming. Commun. ACM 4(6), 284 (1961)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162(1), 439–485 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Elo, A.E.: The Rating of Chessplayers, Past and Present. Arco Publisher, La Palma (1978)Google Scholar
  4. 4.
    Even, G., (Seffi) Naor, J., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20(2), 151–174 (1998)Google Scholar
  5. 5.
    Gama, J., Zliobaite, I., Bifet, A., Pechenizkiy, M., Bouchachia, A.: A survey on concept drift adaptation. ACM Comput. Surv. 46(4), 44:1–44:37 (2014)CrossRefGoogle Scholar
  6. 6.
    Gupte, M., Shankar, P., Li, J., Muthukrishnan, S., Iftode, L.: Finding hierarchy in directed online social networks. In: WWW, pp. 557–566 (2011)Google Scholar
  7. 7.
    Henderson, K., et al.: RolX: structural role extraction & mining in large graphs. In: KDD, pp. 1231–1239 (2012)Google Scholar
  8. 8.
    Jameson, K.A., Appleby, M.C., Freeman, L.C.: Finding an appropriate order for a hierarchy based on probabilistic dominance. Anim. Behav. 57, 991–998 (1999)CrossRefGoogle Scholar
  9. 9.
    Leskovec, J., Krevl, A.: SNAP Datasets: stanford large network dataset collection, January 2005.
  10. 10.
    Macchia, L., Bonchi, F., Gullo, F., Chiarandini, L.: Mining summaries of propagations. In: ICDM, pp. 498–507 (2013)Google Scholar
  11. 11.
    Maiya, A.S., Berger-Wolf, T.Y.: Inferring the maximum likelihood hierarchy in social networks. In: ICSE, pp. 245–250 (2009)Google Scholar
  12. 12.
    McCallum, A., Wang, X., Corrada-Emmanuel, A.: Topic and role discovery in social networks with experiments on enron and academic email. J. Artif. Int. Res. 30(1), 249–272 (2007)Google Scholar
  13. 13.
    Orlin, J.B.: A faster strongly polynomial minimum cost flow algorithm. Oper. Res. 41(2), 338–350 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Roopnarine, P.D., Hertog, R.: Detailed food web networks of three Greater Antillean coral reef systems: the Cayman Islands, Cuba, and Jamaica. Dataset Pap. Ecol. 2013, 9 (2013)Google Scholar
  15. 15.
    Tatti, N.: Faster way to agony–discovering hierarchies in directed graphs. In: ECML PKDD, pp. 163–178 (2014)Google Scholar
  16. 16.
    Tatti, N.: Hierarchies in directed networks. In: ICDM, pp. 991–996 (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.F-SecureHelsinkiFinland
  2. 2.Aalto UniversityEspooFinland

Personalised recommendations