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International Symposium on Leveraging Applications of Formal Methods

ISoLA 2020: Leveraging Applications of Formal Methods, Verification and Validation: Engineering Principles pp 397–415Cite as

Centrality-Preserving Exact Reductions of Multi-Layer Networks

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12477))

Abstract

Multi-Layer Networks (MLN) generalise the traditional, single layered networks, by allowing to simultaneously express multiple aspects of relationships in collective systems, while keeping the description intuitive and compact. As such, they are increasingly gaining popularity for modelling Collective Adaptive Systems (CAS), e.g. engineered cyber-physical systems or animal collectives. One of the most important notions in network analysis are centrality measures, which inform us about the relative importance of nodes. Computing centrality measures is often challenging for large and dense single-layer networks. This challenge is even more prominent in the multi-layer setup, and thus motivates the design of efficient, centrality-preserving MLN reduction techniques. Network centrality does not naturally translate to its multi-layer counterpart, since the interpretation of the relative importance of nodes and layers may differ across application domains. In this paper, we take a notion of eigenvector-based centrality for a special type of MLNs (multiplex MLNs), with undirected, weighted edges, which was recently proposed in the literature. Then, we define and implement a framework for exact reductions for this class of MLNs and accompanying eigenvector centrality. Our method is inspired by the existing bisimulation-based exact model reductions for single-layered networks: the idea behind the reduction is to identify and aggregate nodes (resp. layers) with the same centrality score. We do so via efficient, static, syntactic transformations. We empirically demonstrate the speed up in the computation over a range of real-world MLNs from different domains including biology and social science.

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Notes

  1. 1.

    We refer the interested reader to the original reference, for a discussion on the error and rate of convergence.

  2. 2.

    In case of symmetric graphs, ingoing and outgoing edges will be indistinguishable and overall neighbours are accounted for.

References

  1. Banerjee, A., Chandrasekhar, A., Duflo, E., Jackson, M.: The diffusion of microfinance. Science 341, 1236498 (2013)

    Article  Google Scholar 

  2. Barrett, L., Henzi, P., Lusseau, D.: Taking sociality seriously: the structure of multi-dimensional social networks as a source of information for individuals. Phil. Trans. Roy. Soc. Lond. Ser. B Biol. Sci. 367, 2108–2118 (2012)

    Article  Google Scholar 

  3. Battiston, F., Nicosia, V., Latora, V.: Structural measures for multiplex networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 89, 032804 (2014)

    Article  Google Scholar 

  4. Bloch, F., Jackson, M., Tebaldi, P.: Centrality measures in networks. SSRN Electron. J. (2016)

    Google Scholar 

  5. Bonacich, P.: Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol. 2(1), 113–120 (1972)

    Article  Google Scholar 

  6. Bonacich, P.: Power and centrality: a family of measures. Am. J. Sociol. 92(5), 1170–1182 (1987)

    Article  Google Scholar 

  7. Brandes, U., et al.: On modularity clustering. IEEE Trans. Knowl. Data Eng. 20, 172–188 (2008)

    Article  Google Scholar 

  8. Brandes, U., Lerner, J.: Structural similarity: spectral methods for relaxed blockmodeling. J. Classif. 27(3), 279–306 (2010)

    Article  MathSciNet  Google Scholar 

  9. Cardelli, L., Tribastone, M., Tschaikowski, M., Vandin, A.: Efficient syntax-driven lumping of differential equations. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 93–111. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49674-9_6

    Chapter  MATH  Google Scholar 

  10. Cardelli, L., Tribastone, M.,Tschaikowski, M., Vandin, A.: Symbolic computation of differential equivalences. In: 43st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL) (2016)

    Google Scholar 

  11. Cardelli, L., Tribastone, M., Tschaikowski, M., Vandin, A.: ERODE: a tool for the evaluation and reduction of ordinary differential equations. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10206, pp. 310–328. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54580-5_19

    Chapter  MATH  Google Scholar 

  12. Cardelli, L., Tribastone, M., Tschaikowski, M., Vandin, A.: Maximal aggregation of polynomial dynamical systems. PNAS 114(38), 10029–10034 (2017)

    Article  Google Scholar 

  13. Cardillo, A., et al.: Emergence of network features from multiplexity. Sci. Rep. 3, 1344 (2013)

    Article  Google Scholar 

  14. Christakis, N., Fowler, J.: Social network sensors for early detection of contagious outbreaks. PloS One 5, e12948 (2010)

    Article  Google Scholar 

  15. De Domenico, M., Nicosia, V., Arenas, A., Latora, V.: Structural reducibility of multilayer networks. Nat. Commun. 6, 6864 (2015)

    Article  Google Scholar 

  16. De Domenico, M., Solé-Ribalta, A., Gómez, S., Arenas, A.: Navigability of interconnected networks under random failures. Proc. Natl. Acad. Sci. 111, 8351–8356 (2014)

    Article  MathSciNet  Google Scholar 

  17. De Domenico, M., et al.: Mathematical formulation of multi-layer networks. Phys. Rev. X 3, 07 (2013)

    Google Scholar 

  18. De Domenico, M., Solé-Ribalta, A., Omodei, E., Gomez, S., Arenas, A.: Centrality in interconnected multilayer networks. Phys. D Nonlinear Phenom. 323, 11 (2013)

    MathSciNet  MATH  Google Scholar 

  19. De Domenico, M., Solé-Ribalta, A., Omodei, E., Gomez, S., Arenas, A.: Ranking in interconnected multilayer networks reveals versatile nodes. Nat. Commun. 6, 6868 (2015)

    Article  Google Scholar 

  20. De Domenico, M., Lancichinetti, A., Arenas, A., Rosvall, M.: Identifying modular flows on multilayer networks reveals highly overlapping organization in social systems. ArXiv, abs/1408.2925 (2015)

    Google Scholar 

  21. Farine, D., Aplin, L., Sheldon, B., Hoppitt, W.: Interspecific social networks promote information transmission in wild songbirds. Proc. Biol. Sci. Roy. Soc. 282, 20142804 (2015)

    Google Scholar 

  22. Feret, J., Henzinger, T., Koeppl, H., Petrov, T.: Lumpability abstractions of rule-based systems. Theor. Comput. Sci. 431, 137–164 (2012)

    Article  MathSciNet  Google Scholar 

  23. Franz, M., Altmann, J., Alberts, S.: Knockouts of high-ranking males have limited impact on baboon social networks. Curr. Zool. 61, 107–113 (2015)

    Article  Google Scholar 

  24. Ganguly, A., Petrov, T., Koeppl, H.: Markov chain aggregation and its applications to combinatorial reaction networks. J. Math. Biol. 69(3), 767–797 (2014)

    Article  MathSciNet  Google Scholar 

  25. Gazda, S., Iyer, S., Killingback, T., Connor, R., Brault, S.: The importance of delineating networks by activity type in bottlenose dolphins (Tursiops truncatus) in cedar key, Florida. Roy. Soc. Open Sci. 2, 140263 (2015)

    Article  Google Scholar 

  26. Granell, C., Gomez, S., Arenas, A.: Dynamical interplay between awareness and epidemic spreading in multiplex networks. Phys. Rev. Lett. 111, 128701 (2013)

    Article  Google Scholar 

  27. Iacobelli, G., Tribastone, M., Vandin, A.: Differential bisimulation for a markovian process algebra. In: MFCS, pp. 293–306 (2015)

    Google Scholar 

  28. Jackson, M.: Social and Economic Networks (2008)

    Google Scholar 

  29. Johnson, J., Borgatti, S., Everett, M.: Analyzing Social Networks (2013)

    Google Scholar 

  30. Kapferer, B.: Strategy and transaction in an African factory: African workers and Indian management in a Zambian town. Manchester Univ. Press 43(4), 362–363 (1972)

    Google Scholar 

  31. Katz, L.: A new status index derived from sociometric analysis. Psychometrika 18(1), 39–43 (1953)

    Article  Google Scholar 

  32. KhudaBukhsh, W.R., Auddy, A., Disser, Y., Koeppl, H.: Approximate lumpability for markovian agent-based models using local symmetries. J. Appl. Probab. 56, 04 (2018)

    MathSciNet  MATH  Google Scholar 

  33. Krackhardt, D.: Cognitive social structures. Social Netw. 9(2), 109–134 (1987)

    Article  MathSciNet  Google Scholar 

  34. Lazega, E.: The Collegial Phenomenon: The Social Mechanisms of Cooperation among Peers in a Corporate Law Partnership. Oxford University Press, Oxford (2001)

    Book  Google Scholar 

  35. Lerner, J.: Role assignments. In: Network Analysis: Methodological Foundations [Outcome of a Dagstuhl Seminar, 13–16 April 2004], pp. 216–252 (2004)

    Google Scholar 

  36. Magnani, M., Micenková, B., Rossi, L.: Combinatorial analysis of multiple networks. CoRR, abs/1303.4986 (2013)

    Google Scholar 

  37. McKay, B.: Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)

    MathSciNet  MATH  Google Scholar 

  38. McKay, B., Piperno, A.: Practical graph isomorphism, II. J. Symb. Comput. 6009, 94–112 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Newman, M.: Networks: An Introduction. Oxford University Press Inc., New York (2010)

    Book  Google Scholar 

  40. Omodei, E., De Domenico, M., Arenas, A.: Characterizing interactions in online social networks during exceptional events. Front. Phys. 3, 06 (2015)

    Article  Google Scholar 

  41. Padgett, J.F., Ansell, C.K.: Robust action and the rise of the medici, 1400–1434. Am. J. Sociol. 98(6), 1259–1319 (1993)

    Article  Google Scholar 

  42. Page, L., Brin, S., Motwani, R., Winograd, T.: The pagerank citation ranking: bringing order to the web. In: WWW 1999 (1999)

    Google Scholar 

  43. Robert Paige and Robert Endre Tarjan: Three partition refinement algorithms. SIAM J. Comput. 16(6), 973–989 (1987)

    Article  MathSciNet  Google Scholar 

  44. Pappas, G.J.: Bisimilar linear systems. Automatica 39(12), 2035–2047 (2003)

    Article  MathSciNet  Google Scholar 

  45. Snijders, T.A.B., Pattison, P.E., Robins, G.L., Handcock, M.S.: New specifications for exponential random graph models. Sociol. Methodol. 36(1), 99–153 (2006)

    Article  Google Scholar 

  46. Conde, L.S., Romance, M., Herrero, R., Flores, J., del Amo, A.G., Boccaletti, S.: Eigenvector centrality of nodes in multiplex networks. Chaos (Woodbury, N.Y.) 23, 033131 (2013)

    Google Scholar 

  47. Stark, C., Breitkreutz, B.-J., Reguly, T., Boucher, L., Breitkreutz, A., Tyers, M.: Biogrid: a general repository for interaction datasets. Nucleic Acids Res. 34, D535–D539 (2006)

    Article  Google Scholar 

  48. Tognazzi, S., Tribastone, M., Tschaikowski, M., Vandin, A.: Differential equivalence yields network centrality. In: Margaria, T., Steffen, B. (eds.) ISoLA 2018, Part III. LNCS, vol. 11246, pp. 186–201. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03424-5_13

    Chapter  Google Scholar 

  49. Tschaikowski, M., Tribastone, M.: Exact fluid lumpability for Markovian process algebra. In: CONCUR, pp. 380–394 (2012)

    Google Scholar 

  50. Tudisco, F., Arrigo, F., Gautier, A.: Node and layer eigenvector centralities for multiplex networks. SIAM J. Appl. Math. 78(2), 853–876 (2018)

    Article  MathSciNet  Google Scholar 

  51. van der Schaft, A.J.: Equivalence of dynamical systems by bisimulation. IEEE Trans. Autom. Control 49, 2160–2172 (2004)

    Article  MathSciNet  Google Scholar 

  52. Vickers, M., Chan, M.: Representing classroom social structure. Victoria Institute of Secondary Education, Melbourne (1981)

    Google Scholar 

  53. Stanley, W., Katherine, F.: Social Network Analysis: Methods and Applications, vol. 8. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  54. Wei, X., Valler, N.B., Prakash, A., Neamtiu, I., Faloutsos, M., Faloutsos, C.: Competing memes propagation on networks: a case study of composite networks. ACM SIGCOMM Comput. Commun. Rev. 42, 5–11 (2012)

    Article  Google Scholar 

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Acknowledgements

The authors’ research is supported by the Ministry of Science, Research and the Arts of the state of Baden-Württemberg, and the DFG Centre of Excellence 2117 ‘Centre for the Advanced Study of Collective Behaviour’ (ID: 422037984). The authors would like to thank Ulrik Brandes and Giacomo Rapisardi for the inspiring discussions on the topic, Andrea Vandin for the support and the insights on the use of the tool ERODE and the anonymous reviewers for their suggestions and comments.

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  1. University of Konstanz, Konstanz, Germany

    Tatjana Petrov & Stefano Tognazzi

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  1. Tatjana Petrov
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  2. Stefano Tognazzi
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Correspondence to Tatjana Petrov or Stefano Tognazzi .

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  1. University of Limerick and Lero, Limerick, Ireland

    Tiziana Margaria

  2. TU Dortmund, Dortmund, Germany

    Bernhard Steffen

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Petrov, T., Tognazzi, S. (2020). Centrality-Preserving Exact Reductions of Multi-Layer Networks. In: Margaria, T., Steffen, B. (eds) Leveraging Applications of Formal Methods, Verification and Validation: Engineering Principles. ISoLA 2020. Lecture Notes in Computer Science(), vol 12477. Springer, Cham. https://doi.org/10.1007/978-3-030-61470-6_24

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  • DOI: https://doi.org/10.1007/978-3-030-61470-6_24

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