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# 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if the network topology is highly heterogeneous, like in scale-free networks, so that the random network assumption is no longer applicable?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).