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Mathematics LibreTexts

18: Dynamical Networks III - Analysis of Network Dynamics

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    • 18.1: Dynamics of Continuous-State Networks
      We will now switch gears to the analysis of dynamical properties of networks. We will first discuss how some of the analytical techniques we already covered in earlier chapters can be applied to dynamical network models, and then we will move onto some additional topics that are specific to networks.
    • 18.2: Diffusion on Networks
      Many important dynamical network models can be formulated as a linear dynamical system. The first example is the diffusion equation on a network that we discussed in Chapter 16:
    • 18.3: Synchronizability
      An interesting application of the spectral gap/algebraic connectivity is to determine the synchronizability of linearly coupled dynamical nodes, which can be formulated as follows:
    • 18.4: Mean-Field Approximation of Discrete-State Networks
      Analyzing the dynamics of discrete-state network models requires a different approach, because the assumption of smooth, continuous state space, on which the linear stability analysis is based on, no longer applies. This difference is similar to the difference between continuous field models and cellular automata (CA).
    • 18.5: Mean-Field Approximation on Random Networks
      If we can assume that the network topology is random with connection probability pe, then infection occurs with a joint probability of three events: that a node is connected to another neighbor node (pe), that the neighbor node is infected by the disease (q), and that the disease is actually transmitted to the node (pi).
    • 18.6: Mean-Field Approximation on Scale-Free Networks
      What if the network topology is highly heterogeneous, like in scale-free networks, so that the random network assumption is no longer applicable? A natural way to reconcile such heterogeneous topology and mean-field approximation is to adopt a specific degree distribution P(k). It is still a non-spatial summary of connectivities within the network, but you can capture some heterogeneous aspects of the topology in P(k).

    18: Dynamical Networks III - Analysis of Network Dynamics is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Hiroki Sayama.

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