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Regarding the importance of the degree distribution, we have pretty solid theoretical results on how it's actually the spectrum of the adjacency matrix which acts as a global metric for how well epidemics spread. I always like to recommend Ganesh et al.'s article [1] for how we came to understand this phenomenon but also Prakash et al.'s paper [2] for their theorem which holds for a very large class of epidemic models.

What's pretty interesting is that the largest eigenvalue of the adjacency matrix of an undirected graph lies between the average degree and the maximum degree so the gut feeling you get when playing with the degree of a graph is legitimate.

I will jump on the opportunity do shamelessly self-plug our most recent work [3] on how to modify the topology of a network to have the epidemic go subcritical and quickly disappear.

The basic idea in our paper is to keep the maximum number of edges from the original graph under the constraint that the adjacency spectrum is bounded. Since that's a NP-hard problem we go for an approximation algorithm.

In any case, Melting Asphalt's essays are really an example to follow! A gold standard for expository material!

[1] Ganesh et al. https://ieeexplore.ieee.org/abstract/document/1498374

[2] Prakash et al. https://link.springer.com/article/10.1007/s10115-012-0520-y

[3] Bazgan et al. https://link.springer.com/chapter/10.1007/978-3-030-04651-4_...

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