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Best to start at the beginning and read through:

I am guessing these diagrams are taken from Electrical Engineering and not Quantum Physics.

The blog looks like an expansion of their paper http://arxiv.org/abs/1403.7048v3

Yes! That's where I have seen these diagrams before. They are Hopf algebras, which appear both in Electrical Engineering and Theoretical Physics.

Baez discussed the paper on the n-Category Café, and it may be easier for the inexperienced (like me!) to get started there: https://golem.ph.utexas.edu/category/2015/05/props_for_linea... .

Very impressive if the author developed this by himself. But this is a well known idea from category theory:

This reminds me of something I never understood fully: Penrose's tensor notation.

I am still trying to wrap my head around this, but the parallels with functional programming are striking.

String diagrams are a way to describe morphisms (think functions) in monoidal categories.

A construction using monoidal categories (or a slight generalisation) can be used to give a categorical model for arrows.

This gives rise to these diagrams:

Matrices, diagrammatically