Question
Take a power-law network with an exponential cutoff as an example:
$$P(z)\sim z^{-\alpha} e^{-z/z_c}$$
where $P(z)$ is the degrees of nodes, $z$ is the order of nodes, $\alpha$ is the power law coefficient, and $z_c$ is the cutoff (Arbesman et al. 2010). Does the nodes with fewer degrees tend to connect to other low-degree nodes?
Detail
I saw a claim in Castro & Siew (2020): "... a truncated exponential degree distribution reflecting an upper limit to the maximum degree of words in memory, and exhibited assortative mixing by degree (i.e. high-degree nodes tended to connect to other high-degree nodes; low-degree nodes tended to connect to other low-degree nodes). These features make phonological networks robust to both random and degree-targeted node removal."
However the authors didn't give a justification for it. I kind of having got an idea to verify it by finding some real complex networks and computing the edge likelihood of high- and low-degree nodes. However I am still wondering whether the claim has already been verified empirically, or can be proved mathematically?
Refs
- Arbesman, S., Strogatz, S. H., & Vitevitch, M. S. (2010). The structure of phonological networks across multiple languages. International Journal of Bifurcation and Chaos, 20(03), 679–685.
- Castro, N., & Siew, C. S. (2020). Contributions of modern network science to the cognitive sciences: revisiting research spirals of representation and process. Proceedings of the Royal Society A, 476(2238), 20190825.