Recurrent dynamic message passing with loops for epidemics on networks

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