Inferring the Hidden Cascade Infection over Erdös-Rényi (ER) Random Graph

Finding hidden infected nodes is extremely important when information or diseases spread rapidly in a network because hints regarding the global properties of the diffusion dynamics can be provided, and effective control strategies for mitigating such spread can be derived. In this study, to understand the impact of the structure of the underlying network, a cascade infection-recovery problem is considered over an Erdos-Renyi (ER) random graph when a subset of infected nodes is partially observed. The goal is to reconstruct the underlying cascade that is likely to generate these observations. To address this, two algorithms are proposed: (i) a Neighbor-based recovery algorithm (NBRA(α)), where 0≤α≤1 is a control parameter, and (ii) a BFS tree-source-based recovery algorithm (BSRA). The first one simply counts the number of infected neighbors for candidate hidden cascade nodes and computes the possibility of infection from the neighbors by controlling the parameter α. The latter estimates the cascade sources first and computes the infection probability from the sources. A BFS tree approximation is used for the underlying ER random graph with respect to the sources for computing the infection probability because of the computational complexity in general loopy graphs. We then conducted various simulations to obtain the recovery performance of the two proposed algorithms. As a result, although the NBRA(α) uses only local information of the neighboring infection status, it recovers the hidden cascade infection well and is not significantly affected by the average degree of the ER random graph, whereas the BSRA works well on a local tree-like structure.