1. This result extends your network theory and GNN methods by proving that the root of a growing tree can be recovered with high confidence even when obscured by random noise, directly improving your ability to trace the origin of propagation processes in noisy biological or cryptographic networks.

For a tree generated by a discrete-time stochastic growth process (e.g., uniform random recursive tree, preferential attachment tree, or Pólya urn tree) with n nodes, given only the unlabeled tree topology (or a noise-corrupted observation of it, where noise consists of edge deletions/insertions or mislabeling with probability p per edge), there exists an estimator (e.g., "rumor centrality" / Jordan-centrality-based or GNN-based) that identifies a root candidate set of size O(1) or O(log n) such that the true root lies in this set with probability ≥ 1 − δ, for fixed δ ∈ (0,1), as n → ∞, and this probability degrades gracefully (quantifiably, not catastrophically) as noise level p increases from 0 to some critical threshold p*. The claim is falsifiable: it predicts a specific functional form (e.g., root-finding accuracy ~ C/log n for uniform recursive trees, or a specific decay curve in p) that can be numerically estimated and compared against theory.

Simulate synthetic trees under 3 growth models × 5 sizes (n = 500, 1000, 2000, 5000, 10000) × 6 noise levels (p = 0, 0.05, 0.1, 0.2, 0.3, 0.4) × 200 independent trials each = 54,000 simulated trees minimum. For each tree, compute root-recovery accuracy using (a) a closed-form centrality estimator (Jordan center / rumor centrality) as the theory baseline, and (b) a trained GNN (GraphSAGE or GIN backbone) predicting root probability per node. Compare empirical inclusion probability in top-1, top-5, top-log(n) sets against theoretical predictions; fit degradation curves in p.

for model in [URT, PA, PolyaUrn]:
    for n in [500,1000,2000,5000,10000]:
        for p in [0,0.05,0.1,0.2,0.3,0.4]:
            for trial in range(200):
                T = generate_tree(model, n)
                T_noisy = inject_noise(T, p)
                root_true = T.root
                scores_baseline = rumor_centrality(T_noisy)
                scores_gnn = gnn_model.predict(T_noisy)  # pretrained/fine-tuned
                rank_baseline = get_rank(scores_baseline, root_true)
                rank_gnn = get_rank(scores_gnn, root_true)
                log_result(model,n,p,trial,rank_baseline,rank_gnn)
fit_curves(results)  # nonlinear least squares to C/log(n), decay-in-p models
run_statistical_tests(results)  # paired t-test, bootstrap CI

GNN architecture: 3-layer GraphSAGE, hidden_dim=64, node features = [degree, normalized BFS-eccentricity, local clustering proxy], output = per-node root-probability via softmax over graph, trained with cross-entropy loss against true root label across 10,000 training trees per growth model, 80/20 train/test split, Adam optimizer, lr=1e-3, 50 epochs, early stopping on validation top-K accuracy.