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Lumping the Approximate Master Equation for Multistate Processes on Complex Networks

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11024)

Abstract

Complex networks play an important role in human society and in nature. Stochastic multistate processes provide a powerful framework to model a variety of emerging phenomena such as the dynamics of an epidemic or the spreading of information on complex networks. In recent years, mean-field type approximations gained widespread attention as a tool to analyze and understand complex network dynamics. They reduce the model’s complexity by assuming that all nodes with a similar local structure behave identically. Among these methods the approximate master equation (AME) provides the most accurate description of complex networks’ dynamics by considering the whole neighborhood of a node. The size of a typical network though renders the numerical solution of multistate AME infeasible. Here, we propose an efficient approach for the numerical solution of the AME that exploits similarities between the differential equations of structurally similar groups of nodes. We cluster a large number of similar equations together and solve only a single lumped equation per cluster. Our method allows the application of the AME to real-world networks, while preserving its accuracy in computing estimates of global network properties, such as the fraction of nodes in a state at a given time.

Keywords

  • Complex networks
  • Multistate processes
  • AME
  • Model reduction
  • Lumping

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Notes

  1. 1.

    https://github.com/gerritgr/LumPyQest.

  2. 2.

    We omit t for the ease of notation.

  3. 3.

    We omit once more t for easiness.

  4. 4.

    https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.integrate.ode.html.

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Acknowledgments

This research was been partially funded by the German Research Council (DFG) as part of the Collaborative Research Center “Methods and Tools for Understanding and Controlling Privacy”. We thank James P. Gleeson for his comments regarding the performance of AME on specific models and Michael Backenköhler for his comments on the manuscript.

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Correspondence to Gerrit Großmann .

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Großmann, G., Kyriakopoulos, C., Bortolussi, L., Wolf, V. (2018). Lumping the Approximate Master Equation for Multistate Processes on Complex Networks. In: McIver, A., Horvath, A. (eds) Quantitative Evaluation of Systems. QEST 2018. Lecture Notes in Computer Science(), vol 11024. Springer, Cham. https://doi.org/10.1007/978-3-319-99154-2_10

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  • DOI: https://doi.org/10.1007/978-3-319-99154-2_10

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