The NP-hard Effectors problem on directed graphs is motivated by applications in network mining, particularly concerning the analysis of probabilistic information-propagation processes in social networks. In the corresponding model the arcs carry probabilities and there is a probabilistic diffusion process activating nodes by neighboring activated nodes with probabilities as specified by the arcs. The point is to explain a given network activation state as well as possible by using a minimum number of “effector nodes”; these are selected before the activation process starts. We correct, complement, and extend previous work from the data mining community by a more thorough computational complexity analysis of Effectors, identifying both tractable and intractable cases. To this end, we also exploit a parameterization measuring the “degree of randomness” (the number of ‘really’ probabilistic arcs) which might prove useful for analyzing other probabilistic network diffusion problems as well.
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We conjecture that both models coincide if we are allowed to choose an unlimited number of effectors, that is, if the number of chosen effectors does not matter. On the contrary, they do not coincide if the number of effectors is bounded, see Section 2.
Notably, in our model it actually remains active. The point is that before the whole computation starts (and after it ends) nodes may (have) become inactive again. Still, “temporary activeness” may make a node an effector that helps explaining the currently observed network activation state.
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall (1993)
Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press (2009)
Askalidis, G., Berry, R.A., Subramanian, V.G.: Explaining snapshots of network diffusions Structural and hardness results. In: proceedings of the 20th International Conference on Computing and Combinatorics, volume 8591 of LNCS, pp 616–625. Springer (2014)
Bazgan, C., Chopin, M., Nichterlein, A., Sikora, F.: Parameterized inapproximability of target set selection and generalizations. Computability 3(2), 135–145 (2014)
Bazgan, C., Chopin, M., Nichterlein, A., Sikora, F.: Parameterized approximability of maximizing the spread of influence in networks. J. Discret. Algorithm. 27, 54–65 (2014)
Ben-Zwi, O., Hermelin, D., Lokshtanov, D., Newman, I.: Treewidth governs the complexity of target set selection. Discret. Optim. 8(1), 87–96 (2011)
Bharathi, S., Kempe, D., Salek, M.: Competitive influence maximization in social networks. In: proceedings of the Third International Workshop on Internet and Network Economics, volume 4858 of LNCS, pp 306–311. Springer (2007)
Chopin, M., Nichterlein, A., Niedermeier, R., Weller, M.: Constant thresholds can make target set selection tractable. Theory Comput. Syst. 55(1), 61–83 (2014)
Cygan, M., Fomin, F. V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized algorithms. springer (2015)
Domingos, P., Richardson, M.: Mining the network value of customers. In: proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp 57–66. ACM (2001)
Downey, R.G., Fellows, M.R.: Fundamentals of parameterized complexity. springer (2013)
Fellows, M.R., Hermelin, D., Rosamond, F., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009)
Flum, J., Grohe, M.: Parameterized complexity theory, springer (2006)
Kempe, D., Kleinberg, J.M., Tardos, É.: Maximizing the spread of influence through a social network. Theory Comput. 11, 105–147 (2015)
Lappas, T., Terzi, E., Gunopulos, D., Mannila, H.: Finding effectors in social networks. In: proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp 1059–1068. ACM (2010)
Nichterlein, A., Niedermeier, R., Uhlmann, J., Weller, M.: On tractable cases of target set selection. Soc. Netw. Anal. Min. 3(2), 233–256 (2013)
Niedermeier, R.: Invitation to Fixed-Parameter algorithms. oxford university press (2006)
Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)
Wang, C., Chen, W., Wang, Y.: Scalable influence maximization for independent cascade model in large-scale social networks. Data Min. Knowl. Discov. 25(3), 545–576 (2012)
We are grateful to two anonymous reviewers of Theory of Computing Systems whose careful and constructive feedback helped to significantly improve the presentation of the paper.
An extended abstract appeared in Proceedings of the 12th Annual Conference on Theory and Applications of Models of Computation (TAMC ’15), Volume 9076 of LNCS, pages 224–235, Springer, 2015. This article provides all proofs in full detail.
Laurent Bulteau was supported by the Alexander von Humboldt Foundation, Bonn, Germany. Main work done while affiliated with TU Berlin.
Stefan Fafianie was supported by the DFG Emmy Noether-program (KR 4286/1). Main work done while affiliated with TU Berlin.
Vincent Froese was supported by the DFG, project DAMM (NI 369/13).
Nimrod Talmon was supported by DFG Research Training Group “Methods for Discrete Structures” (GRK 1408). Main work done while affiliated with TU Berlin.
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Bulteau, L., Fafianie, S., Froese, V. et al. The Complexity of Finding Effectors. Theory Comput Syst 60, 253–279 (2017). https://doi.org/10.1007/s00224-016-9670-8
- Probabilistic information propagation
- Influence maximization
- Network activation
- Social networks
- Exact algorithms
- Parameterized complexity