Gated Graph Neural Networks for Learning Hidden Independent Cascade Dynamics

Information and infectious diseases spread through social networks, but the spreading probabilities driving them are hard to estimate without per-node activation times. Applications seldom supply these, and inference must instead proceed through indirect and noisy proxies for the terminal infection states. We study this inverse problem on a fixed, known graph under the Hidden Independent Cascade (HIC) model, with one spreading probability per node rather than a single global rate, so the number of unknowns scales with the number of nodes. Seed sets and observation parameters are known, while activation times and terminal infection states are latent, and the observed-data likelihood requires marginalizing over every spreading outcome. We propose a simulation-based amortized estimator that recovers the full node-level parameter vector without reconstructing individual latent cascades. Repeated seed-conditioned symptom observations are summarized as Symptom-Aware Cascade Features (SACF), which combine empirical symptom statistics with neighborhood and structural information. SACF are mapped to parameters by SAGE-HC, a permutation-equivariant gated graph neural network whose learned gates attenuate neighborhood messages corrupted by false positives and false negatives, and training on simulated HIC realizations yields a reusable inverse map. As a benchmark under the same hidden observations, we extend the Dynamic Message Passing learning framework to the HIC emission model. On synthetic and empirical graphs the two methods separate by topology. DMP is highly accurate on trees, whereas SAGE-HC is substantially better on heterogeneous, loopy graphs under noisy terminal symptoms.