Existence and conditions for discontinuous transitions in the hypergraph rumor model

Determine whether the threshold-based rumor model on hypergraphs admits discontinuous phase transitions (hybrid or first-order), and derive sufficient and necessary conditions—expressed in terms of model parameters and hypergraph structural properties—for the emergence of such discontinuities.

Background

In contrast to several higher-order contagion models that exhibit discontinuous or hybrid transitions, the authors consistently observe continuous phase transitions in their threshold rumor model. They hypothesize that the non-spontaneous nature of annihilation may be responsible but do not provide a formal proof.

Establishing whether discontinuous transitions can occur within this framework, and if so, under which precise structural and parametric conditions, is highlighted as an unresolved problem requiring formal verification.