Spiders

Diagrams in this book extensively use element-wise multiplication, written as a black circle.
But what does that mean and why this operation allowed in tensor networks?

Actually, this little dot is a tensor as well, specifically, a generalised identity tensor.
This operation is often referred to as a spider, because it has many legs.

A spider (eyes added for dramatic effect).

Spiders are identity (Kronecker) tensors; zero everywhere except ones on the diagonal.
For example, a second-order spider is exactly an identity matrix.

Connecting two wires with a spider does nothing.

In higher orders, this diagonal structure means wires can only interact through equal dimensions. For instance, the spider in the bilinear layer only allows interactions between the same dimension.

Matrices before and after perform all the 'shuffling' of information. Hence, spiders can be seen as summing over all indices of the connected tensor.

Comparison with full tensors

Merging spiders

Two adjacent spiders can be combined into a larger spider.

While this book often keeps spiders separated for visual clarity, this may be easier to reason about.

Spiders are undeniably the lifeblood of tensor networks, they allow matrices to be combined in interesting and expressive ways.