  The causal connectivity of a network is often inferred to understand the
network function. It is arguably acknowledged that the inferred causal
connectivity relies on causality measure one applies, and it may differ from
the network's underlying structural connectivity. However, the interpretation
of causal connectivity remains to be fully clarified, in particular, how causal
connectivity depends on causality measures and how causal connectivity relates
to structural connectivity. Here, we focus on nonlinear networks with pulse
signals as measured output, $e.g.$, neural networks with spike output, and
address the above issues based on four intensively utilized causality measures,
$i.e.$, time-delayed correlation, time-delayed mutual information, Granger
causality, and transfer entropy. We theoretically show how these causality
measures are related to one another when applied to pulse signals. Taking the
simulated Hodgkin-Huxley neural network and the real mouse brain network as two
illustrative examples, we further verify the quantitative relations among the
four causality measures and demonstrate that the causal connectivity inferred
by any of the four well coincides with the underlying network structural
connectivity, therefore establishing a direct link between the causal and
structural connectivity. We stress that the structural connectivity of networks
can be reconstructed pairwisely without conditioning on the global information
of all other nodes in a network, thus circumventing the curse of
dimensionality. Our framework provides a practical and effective approach for
pulse-output network reconstruction.
