  This paper is the second in the series Commutative Scaling of Width and Depth
(WD) about commutativity of infinite width and depth limits in deep neural
networks. Our aim is to understand the behaviour of neural functions (functions
that depend on a neural network model) as width and depth go to infinity (in
some sense), and eventually identify settings under which commutativity holds,
i.e. the neural function tends to the same limit no matter how width and depth
limits are taken. In this paper, we formally introduce and define the
commutativity framework, and discuss its implications on neural network design
and scaling. We study commutativity for the neural covariance kernel which
reflects how network layers separate data. Our findings extend previous results
established in [55] by showing that taking the width and depth to infinity in a
deep neural network with skip connections, when branches are suitably scaled to
avoid exploding behaviour, result in the same covariance structure no matter
how that limit is taken. This has a number of theoretical and practical
implications that we discuss in the paper. The proof techniques in this paper
are novel and rely on tools that are more accessible to readers who are not
familiar with stochastic calculus (used in the proofs of WD(I))).
