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Centrality-Preserving Exact Reductions of Multi-Layer Networks

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Leveraging Applications of Formal Methods, Verification and Validation: Engineering Principles (ISoLA 2020)

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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.

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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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