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From my understanding, dynamic networks are similar to traditional models except that they function in continuous time and have edges and nodes that evolve over time? Is this a correct understanding? Also why are they probabilistic?

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I'd suggest you to read this link. It explains very well what a Dynamic Network is: http://www.ccs.neu.edu/home/rraj/Talks/DynamicNetworks/DYNAMO/IntroDynamicNetworks.pdf

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A dynamic network is any time-dependent network $G_{t} := \left(V_{t},E_{t}\right)$, whose vertex and/or edge sets undergo some variation as time evolves.

A myriad of deterministic and stochastic real-world processes have been successfully modelled using dynamics networks, and the possibilities for defining complex dynamic behaviours in terms of varying $V_{t}$ and $E_{t}$ sets is truly endless.

You are indeed correct in thinking that dynamics networks "have edges and nodes that evolve over time", but you need not restrict it to probabilistic or continuous time functions. Discrete time evolution is common within neural network models, and deterministic networks play a role in information theory research.

However, if a dynamic network does display stochastic variation, probabilistic network metrics and/or structures are considered, in order to gain insights into the average/predicted behaviours of the modelled real-world process.

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