Full 1RSB analysis and possible discontinuous phase transition in SI epidemic inference

Solve and analyze the full one-step replica symmetry breaking (1RSB) cavity equations for the Bayesian inference of infection times in the Susceptible–Infectious (SI) model on random regular graphs under Nishimori (Bayes-optimal) conditions, and determine whether these equations admit a discontinuous 1RSB phase transition (i.e., a non-trivial 1RSB fixed point appearing discontinuously outside the replica-symmetric instability thresholds).

Background

The paper demonstrates an instability of the replica-symmetric (RS) cavity solution for epidemic inference on random regular graphs with deterministic spreading (λ=1) and noiseless observations, under Nishimori conditions. This establishes the presence of replica symmetry breaking (RSB) in a parameter range of the seed probability γ, challenging the common belief that Nishimori conditions preclude RSB.

However, the stability analysis performed is local and continuous: it reveals when the RS fixed point becomes unstable but cannot exclude the existence of a discontinuous transition to a non-trivial 1RSB solution occurring even when RS remains locally stable. To clarify whether a discontinuous 1RSB transition exists, the authors point to the need for solving the full 1RSB equations for this SI inference model and explicitly leave this task for future work.