  This paper considers the growth in the length of one-dimensional trajectories
as they are passed through deep ReLU neural networks, which, among other
things, is one measure of the expressivity of deep networks. We generalise
existing results, providing an alternative, simpler method for lower bounding
expected trajectory growth through random networks, for a more general class of
weights distributions, including sparsely connected networks. We illustrate
this approach by deriving bounds for sparse-Gaussian, sparse-uniform, and
sparse-discrete-valued random nets. We prove that trajectory growth can remain
exponential in depth with these new distributions, including their sparse
variants, with the sparsity parameter appearing in the base of the exponent.
