IDEAS home Printed from
   My bibliography  Save this paper

Modelling fraud detection by attack trees and Choquet integral


  • Silvia Bortot
  • Mario Fedrizzi
  • Silvio Giove


Modelling an attack tree is basically a matter of associating a logical "and" and a logical "or" but in most of real world applications related to fraud management the "and/or"logic is not adequate to effectively represent the relationship between a parent node and its children, most of all when information about attributes is associated to the nodes and the main problem to solve is how to promulgate attribute values up the tree through recursive aggregation operations occurring at the "and/or"nodes. OWA-based aggregations have been introduced to generalize "and" and "or" operators starting from the observation that in between the extremes "or all"(and) and "or any"(or), terms (quantifiers) like "several" "most" "few" "some" etc. can be introduced to represent the different weights associated to the nodes in the aggregation. The aggregation process taking place at an OWA node depends on the ordered position of the child nodes but it doesn't take care of the possible interactions between the nodes. In this paper, we propose to overcome this drawback introducing the Choquet integral whose distinguished feature is to be able to take into account the interaction between nodes. At first, the attack tree is valuated recursively through a bottom-up algorithm whose complexity is linear versus the number of nodes and exponential for every node. Then, the algorithm is extended assuming that the attribute values in the leaves are unimodal LR fuzzy numbers and the calculation of Choquet integral is carried out using the alpha-cuts.

Suggested Citation

  • Silvia Bortot & Mario Fedrizzi & Silvio Giove, 2011. "Modelling fraud detection by attack trees and Choquet integral," DISA Working Papers 2011/09, Department of Computer and Management Sciences, University of Trento, Italy, revised 31 Aug 2011.
  • Handle: RePEc:trt:disawp:2011/09

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. Grabisch, Michel & Labreuche, Christophe & Vansnick, Jean-Claude, 2003. "On the extension of pseudo-Boolean functions for the aggregation of interacting criteria," European Journal of Operational Research, Elsevier, vol. 148(1), pages 28-47, July.
    2. Grabisch, Michel & Kojadinovic, Ivan & Meyer, Patrick, 2008. "A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the Kappalab R package," European Journal of Operational Research, Elsevier, vol. 186(2), pages 766-785, April.
    3. Ghirardato, Paolo & Le Breton, Michel, 2000. "Choquet Rationality," Journal of Economic Theory, Elsevier, vol. 90(2), pages 277-285, February.
    4. Michel Grabisch & Christophe Labreuche, 2010. "A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid," Annals of Operations Research, Springer, vol. 175(1), pages 247-286, March.
    5. Marc Roubens & Michel Grabisch, 1999. "An axiomatic approach to the concept of interaction among players in cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(4), pages 547-565.
    6. Cardin, Marta & Giove, Silvio, 2008. "Aggregation Functions With Non-Monotonic Measures," Fuzzy Economic Review, International Association for Fuzzy-set Management and Economy (SIGEF), vol. 0(2), pages 3-15, November.
    7. De Waegenaere, Anja & Wakker, Peter P., 2001. "Nonmonotonic Choquet integrals," Journal of Mathematical Economics, Elsevier, vol. 36(1), pages 45-60, September.
    8. Chateauneuf, Alain & Jaffray, Jean-Yves, 1989. "Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 263-283, June.
    9. Grabisch, Michel, 1996. "The application of fuzzy integrals in multicriteria decision making," European Journal of Operational Research, Elsevier, vol. 89(3), pages 445-456, March.
    Full references (including those not matched with items on IDEAS)


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:trt:disawp:2011/09. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Roberto Gabriele). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.