Cyclic Structural Hawkes Processes for Learning Causal Structure from Discrete-Time Event Sequences

Learning causal structure from discrete-time event sequences is particularly challenging because the exact temporal order of events within the same time interval is unobserved. As a result, causal relations that occur within one interval cannot be captured by methods that rely only on strictly lagged dependence. To tackle this issue, existing approaches such as Structural Hawkes Processes (SHPs) explicitly model causal interactions within the same interval, but require the corresponding causal structure to be acyclic to ensure a well-defined recursive generative process. Such an assumption can be restrictive, since events within one interval may repeatedly excite each other and naturally form directed cycles. In this work, we propose Cyclic Structural Hawkes Processes (C-SHPs) for learning cyclic causal structures from discrete-time event sequences. By connecting the observed event counts to an underlying continuous-time Hawkes process through the probability generating functional (PGFL), we develop theoretical tools for characterizing self-loops, adjacency, and edge directions from temporally aggregated observations. We further establish the identifiability of the directed causal structure in the presence of self-loops, reciprocal edges, and arbitrary directed cycles. Based on these results, we develop a practical causal discovery algorithm that operates directly on discrete-time event counts. Experiments on cyclic graphs demonstrate robust performance across varying time resolutions, model parameters, sample sizes, graph dimensions, and graph densities.