Flow of dynamical causal structures with an application to correlations

Causal models capture cause-effect relations both qualitatively - via the graphical causal structure - and quantitatively - via the model parameters. They offer a powerful framework for analyzing and constructing processes. Here, we introduce a tool - the flow of causal structures - to visualize and explore the dynamical aspect of classical-deterministic processes, arguably like those present in general relativity. The flow describes all possible ways in which the causal structure of a process can evolve. We also present an algorithm to construct its supergraph - the superflow - from the causal structure only, without invoking the model parameters. As an application, we show that if all leaves of a flow are trivial, then the corresponding process produces causal correlations only, i.e., correlations where future data cannot influence past events. This strengthens the result that processes, where every directed cycle in their causal structure is chordless, establish causal correlations only. We also discuss the main difficulties for the quantum generalization of the present algorithms.