Learning on Dynamic Hypergraphs from Incidence Streams

Hypergraphs model higher-order interactions, and in many systems their hyperedges persist: nodes join over time, carry attributes such as roles and may leave. Dynamic hypergraph learning addresses separate tasks on transient hyperedges, while persistent hyperedges are reduced to their star expansion for temporal graph networks. Neither line characterizes what an evaluation protocol reveals beyond the history or what a model must retain. A unified formulation of all tasks would make both questions precise. We represent a dynamic hypergraph as a stream of incidence records and factorize its likelihood, so that every task is a conditional of one factorization. Protocol leakage is then the conditional mutual information an exposure adds to the history, equal to the Bayes log-risk it removes. For persistent processes with exponential kernels and Markov roles (Dirichlet or known role kernels), a finite state of node, hyperedge, pair and incidence statistics suffices. First-hop readouts of the star expansion cannot represent its pair terms. The incidence state machine (ISM) maintains this state in one pass and reads it by attention pooling. On persistent streams from public datasets, the ISM performs best overall across join, role and departure prediction and outperforms memory-based and first-hop temporal graph networks on joining.