Picture yourself strolling down this Polynesian beach
Tensors are like multi-legged creatures. A leg (index) of one creature (tensor) can be connected (contracted, summed over) with the leg of another creature. Tensor networks consisting of many interconnected creatures are very common in Physics. For example:
‘t Hooft and Veltman used tensor networks to explain some aspects of quantum field theory (Ward-Takahashi identities, renormalization, etc) in their famous 1973 paper entitled “Diagrammar”.
Predrag Cvitanovic used tensor networks (he calls them birdtracks) to explain both, quantum field theory in this “webBook”, and group representation theory in this webBook.
Sir Roger Penrose used tensor networks to do general relativity calculations. See here
Quantum circuit diagrams are tensor networks too. In my 2004 paper, “QC Paulinesia”, I tried to compile all the tensor network (circuit diagram) identities I had encountered while learning about quantum computing. Sort of like a table of integrals for quantum computerists. These identities almost always entail Pauli matrices, and I was reading a travel book about the Polynesian islands at the time, hence the title. (My original plan was to publish periodic updates of QC Paulinesia, with corrections and additions. I haven’t done a single update yet though, because, because…okay, because I’m damn lazy.)
