Gated Graph Neural Networks for Learning Hidden Independent Cascade Dynamics

Anubha Goel; Illia Oleksiienko; Mateusz Wilinski; Alexandros Iosifidis; Juho Kanniainen

cs.SIOct 2, 2026

Gated Graph Neural Networks for Learning Hidden Independent Cascade Dynamics

Authors: Anubha Goel, Illia Oleksiienko, Mateusz Wilinski, Alexandros Iosifidis, Juho Kanniainen

Abstract

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.

Figures & tables

(A) Noise-free observations
Graph	DMP	SAGE-HC
Balanced tree	$0.01154$ / $0.93012$	$0.00872\pm 0.00001$ / $0.95874\pm 0.00016$
BA(2)	$0.18616$ / $0.33141$	$0.00708\pm 0.00004$ / $0.97814\pm 0.00033$
BA(3)	$0.25926$ / $0.24653$	$0.01086\pm 0.00008$ / $0.95082\pm 0.00081$
BA(4)	$0.28200$ / $0.20722$	$0.01672\pm 0.00008$ / $0.90181\pm 0.00059$
Karate Club	$0.21747$ / $0.28993$	$0.01404\pm 0.00013$ / $0.92962\pm 0.00123$

Table 1: Performance for the benchmark setting under noise-free and noisy observations. Each SAGE-HC entry reports Smooth L1 / [email protected] . Values are mean

\pm

standard deviation across 5 independent training runs.

Graph	No noise		Noisy
Graph	Mean interval width	Mean pred. std	Mean interval width	Mean pred. std
Balanced tree	$0.04291\pm 0.00015$	$0.03462\pm 0.00012$	$0.11540\pm 0.00057$	$0.09309\pm 0.00046$
Karate Club	$0.04886\pm 0.00037$	$0.03941\pm 0.00030$	$0.07294\pm 0.00038$	$0.05884\pm 0.00030$
BA(2)	$0.03603\pm 0.00015$	$0.02907\pm 0.00012$	$0.05761\pm 0.00031$	$0.04648\pm 0.00025$
BA(3)	$0.04182\pm 0.00016$	$0.03374\pm 0.00013$	$0.05819\pm 0.00023$	$0.04694\pm 0.00018$
BA(4)	$0.04625\pm 0.00045$	$0.03731\pm 0.00037$	$0.06641\pm 0.00046$	$0.05357\pm 0.00037$

Table 2: Empirical prediction stability of SAGE-HC corresponding to 5000 spreading parameter assignments, 10 feature realizations per assignment, and 50 cascades per seed. Values are reported as mean

\pm

standard deviation across 5 independent training runs.

Figure 1: Sensitivity of SAGE-HC to the number of spreading parameter assignments on BA(2), with the number of feature realizations and cascades per seed held fixed. Increasing the number of assignments substantially improves estimation at small and moderate sample sizes, after which both Smooth L1 error and [email protected] approach a plateau. The noisy regime remains more challenging throughout. Error bars denote the standard deviation across repeated training runs.

Figure 2: Sensitivity of SAGE-HC to the number of cascades per seed on BA(2), with the number of spreading parameter assignments and feature realizations held fixed. Increasing the cascade budget sharply reduces Smooth L1 error and improves [email protected], with the largest gains occurring at low and moderate cascade counts. At sufficiently large cascade budgets, the noisy and noise-free regimes both approach saturation, showing that additional spreading evidence can substantially mitigate observation noise. Error bars denote the standard deviation across repeated training runs.

Figure 3: Sensitivity of SAGE-HC to graph size on BA(2), with the number of spreading parameter assignments, feature realizations per assignment, and cascades per seed held fixed. Performance improves with increasing graph size, particularly under noisy observations, and exhibits diminishing returns for larger graphs. Since all nodes are used as seeds, the experiment measures end-to-end scaling under a fixed per-seed cascade budget rather than graph size under a fixed total observation budget. Error bars denote the standard deviation across repeated training runs where multiple runs are available.

Figure 4: Effect of training-time feature replication on BA(2) with 1,000 spreading parameter assignments. The number of feature realizations used during training is varied, while the strict unseen-test budget is fixed at

F_{\mathrm{test}}=2

for every model. Fixing the test budget isolates the effect of training-time replication from changes in the amount of information available at inference. Performance improves primarily for small to moderate training FPA, particularly under hidden/noisy observations, and shows diminishing returns thereafter. Error bars denote variability across independent training runs.

Figure 5: Sensitivity to the number of cascades available at test time for a frozen SAGE-HC model on BA(2). The model is trained once with

C_{\mathrm{train}}=1000

cascades per seed. The same held-out test assignments are then evaluated using different test-time cascade budgets. The left panel reports prediction error and the right panel reports [email protected]. Performance improves rapidly as additional cascade evidence becomes available, particularly under hidden/noisy observations, and approaches a plateau at larger test-time budgets.

Figure 6: Comparison of accuracies at the $0.1$ threshold of the noise-free and noisy training settings for a tree graph with progressively added edges to increase its loopiness

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