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The Complexity of Finding Effectors


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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  1. 1.

    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.

  2. 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.


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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.

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Corresponding author

Correspondence to Nimrod Talmon.

Additional information

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).

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  • Probabilistic information propagation
  • Influence maximization
  • Network activation
  • Social networks
  • Exact algorithms
  • Parameterized complexity