  Deep Neural Networks (DNNs) are often examined at the level of their response
to input, such as analyzing the mutual information between nodes and data sets.
Yet DNNs can also be examined at the level of causation, exploring "what does
what" within the layers of the network itself. Historically, analyzing the
causal structure of DNNs has received less attention than understanding their
responses to input. Yet definitionally, generalizability must be a function of
a DNN's causal structure since it reflects how the DNN responds to unseen or
even not-yet-defined future inputs. Here, we introduce a suite of metrics based
on information theory to quantify and track changes in the causal structure of
DNNs during training. Specifically, we introduce the effective information (EI)
of a feedforward DNN, which is the mutual information between layer input and
output following a maximum-entropy perturbation. The EI can be used to assess
the degree of causal influence nodes and edges have over their downstream
targets in each layer. We show that the EI can be further decomposed in order
to examine the sensitivity of a layer (measured by how well edges transmit
perturbations) and the degeneracy of a layer (measured by how edge overlap
interferes with transmission), along with estimates of the amount of integrated
information of a layer. Together, these properties define where each layer lies
in the "causal plane" which can be used to visualize how layer connectivity
becomes more sensitive or degenerate over time, and how integration changes
during training, revealing how the layer-by-layer causal structure
differentiates. These results may help in understanding the generalization
capabilities of DNNs and provide foundational tools for making DNNs both more
generalizable and more explainable.
