Tensor network notation is really useful for differentiating with respect to tensors. For example, where F is a real function of a matrix variable, think of how you'd differentiate F(A X) with respect to X. Conceptually this is easy, but I used to have to slow down to write it in Einstein notation. Thinking in terms of tensor diagrams, I just see F', X and A strung together in a triangle. Differentiating with respect to X means removing it from the triangle. The dangling edges are the indices of the derivative, and what's left is a matrix product of A and F' along the index that doesn't involve X.<p>This blog post made me realize that tensor diagrams are the same as the factor graphs we talk about in random field theory. Indices of a tensor network become variables of a factor graph and tensors become factors. The contraction of a tensor network with positive tensors is the partition function of a corresponding field and so on.