Abstract
For simplicity, we frame our discussions in terms of unweighted, undirected networks. When such a network is time-independent, it can be represented using a symmetric adjacency matrix A = A T with elements \(A_{ij} = A_{ji}\) that are equal to 1 if nodes i and j are connected (or, more properly, “adjacent”) and 0 if they are not.
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Notes
- 1.
This is a standard assumption, but it is not always desirable. For example, one may wish to investigate narcissism in people tagging themselves in pictures on Facebook, a set of coupled oscillators can include self-interactions, and so on.
- 2.
By analogy with statistical physics, the N → ∞ limit is often called a “thermodynamic limit.”
- 3.
Reference [212] gives one illustration of how considering a very unrealistic random-graph ensemble can be crucial for developing understanding of the behavior of a dynamical process on networks.
- 4.
Strictly speaking, one also needs to ensure appropriate conditions on the moments of P k as N → ∞. For example, one could demand that the second moment remains finite as N → ∞.
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Porter, M.A., Gleeson, J.P. (2016). A Few Basic Concepts. 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_2
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