In the human immune system, there is a coevolutionary arms race occurring between pathogens and the host. Pathogens, especially viruses like HIV and SARS-CoV-2, evolve to escape the immune challenge presented by the human immune system. In return, the human immune system can re-organize and through processes that resemble Darwinian evolution, produce novel antibodies that target and neutralize the evolved pathogens. However, it is unclear who leads and who follows in this coevolutionary arms race. I introduce a bipartite, Markovian model to study a coevolving network of species and investigate causality in stochastically evolving systems. My results include novel analytical expressions for observables that discern causality, e.g., the rate of change of partial mutual information, and characterize how causal relationships change with the dimension and topology of the network. I compare the theory to simulations and describe statistical features of these processes. The tools I develop will be useful for distinguishing drivers of evolution in fitness seascapes and, more broadly, detecting causality in any type of dynamic network that undergoes nonequilibrium stochastic dynamics.