  Several theoretical methods have been developed to approximate prevalence and
threshold of epidemics on networks. Among them, the recurrent dynamic
message-passing (rDMP) theory offers a state-of-the-art performance by
preventing the echo chamber effect in network edges. However, the rDMP theory
was derived in an intuitive ad-hoc way, lacking a solid theoretical foundation
and resulting in a probabilistic inconsistency flaw. Furthermore, real-world
networks are clustered and full of local loops like triangles, whereas rDMP is
based on the assumption of a locally tree-like network structure, which makes
rDMP potentially inefficient on real applications. In this work, for the
recurrent-state epidemics, we first demonstrate that the echo chamber effect
exits not only in edges but also in local loops, which rDMP-like method can not
avoid. We then correct the deficiency of rDMP in a principled manner, leading
to the natural introduction of new higher-order dynamic messages, extending
rDMP to handle local loops. By linearizing the extended message-passing
equations, a new epidemic threshold estimation is given by the inverse of the
leading eigenvalue of a matrix named triangular non-backtracking matrix.
Numerical experiments conducted on synthetic and real-world networks to
evaluate our method, the efficacy of which is validated in epidemic prevalence
and threshold prediction tasks. In addition, our method has the potential to
speed up the solution of the immunization, influence maximization, and
robustness optimization problems in the networks.
