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General Considerations

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Dynamical Systems on Networks

Part of the book series: Frontiers in Applied Dynamical Systems: Reviews and Tutorials ((FIADS,volume 4))

Abstract

Now that we have discussed several families of models as motivation (and because they are interesting in their own right), we present some general considerations for studying dynamical systems on networks. We alluded to several of these ideas in our prior discussions.

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Notes

  1. 1.

    Note that we previously used the notation \(\lambda\) in Sec. 3.2.1 to represent the transmission rate in the SI model. In this section, we use \(\lambda\) with appropriate subscripts to represent the eigenvalues of A.

  2. 2.

    Although we were able to separate the dependence on structure and dynamics in our example, note that the analysis is more complicated when the equilibrium points are different for different nodes and when considering other types of behavior (e.g., periodic or chaotic dynamics) [244].

  3. 3.

    See [175] for a discussion of tensors.

  4. 4.

    When calculating a sample mean using numerical simulations of a dynamical system on a network (or a family of networks), there are several possible sources of stochasticity: (1) choice of initial condition, (2) choice of which nodes to update (when considering asynchronous updating), (3) the update rule itself, (4) parameter values that are used in an update rule, and (5) selection of particular networks from a random-graph ensemble. Some or all of these sources of randomness can be present when studying dynamical systems on networks, and (when possible) it is also desirable to compare the sample means to ensemble averages (i.e., expectations over a suitable probability distribution).

  5. 5.

    It may also be useful to develop analogous class-based approximations that use structural characteristics other than degree.

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Authors and Affiliations

  1. Math Institute, Radcliffe Observatory Quarter, Andrew Wiles Building, Woodstock Road, Oxford, United Kingdom

    Mason A. Porter

  2. MACSI Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland

    James P. Gleeson

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  1. Mason A. Porter
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  2. James P. Gleeson
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Porter, M.A., Gleeson, J.P. (2016). General Considerations. In: Dynamical Systems on Networks. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-26641-1_4

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