Omega-N: Interpretable Structural Node Descriptors and Their Applicability Domain

A composite structural index summarises a network in one number; for a triangle-based index it is spectrally redundant: Tr(A^3) is the third moment of the adjacency spectrum. The non-redundant content sits one level down, in diag(A^3), which depends on eigenvectors and is not spectrally determined. A corollary in the theory paper for this index family stated that, and predicted: the global scalar should tie sharpened spectral baselines rather than beat them, while the node-wise attribution should do better where the number of structural epicentres is unknown. This paper tests it. We construct Omega-N by localizing each of the four factors. The direct localization is badly conditioned; two corrections from published practice fix it, a configuration-null excess for every local factor and a personalized-PageRank neighbourhood at several scales, giving ten interpretable features per node from the graph alone, with no attributes, training or embeddings. Against a recursive feature engine at five levels of recursion, Omega-N wins on one and ties on four of six in-domain node-classification evaluations, with ten features against up to 252. Two statistics computed from the graph and labels, not from performance, partition the eight benchmarks without error, and the two they exclude are the two it loses. The strongest application is drug-target prioritisation on protein interaction networks: +0.073 to +0.144 AUPRC over a centrality battery across four constructions, replicated on an independent AP-MS network and label source, surviving three bias controls (degree-matched, ten repetitions: +0.1047 and +0.1030, both 10/10, p=0.00195). The clearest negative sits in the same application: adding Omega-N to centralities plus Node2Vec changes nothing (+0.0014, p=0.31). The claim is narrow: ten interpretable features

Evaluating GNNs for Success Prediction in Artist Collaboration Networks

As the music industry becomes an increasingly collaborative effort, understanding the underlying structures of the artist network has become a focal point in cultural data analytics. This study expands on the previous analyses of the Italian and Danish networks by introducing a novel dataset of the Polish music scene. By utilizing methodologies used in the prior studies, this work enables a direct comparison between three distinct European music landscapes and allows to merged the created networks into one. Furthermore, this research introduces a framework to test the efficacy of Graph Neural Networks (GNNs) for artist popularity predictions based on the metadata and the position in the network. The statistical analysis revealed that the Polish and tri-national network exhibit similar properties and clustering behaviours, consistent with prior models. An evaluation of the predictive architectures reveals that while GNN models achieve a comparable F1-macro scores to the Multilayer Perceptron (MLP) in specific cases however the MLP remains a superior model regarding the success metric. The results suggest that internal node features - such as genre and label affiliation might carry more predictive capabilities than the topology of the network. The higher performance of the GNN models in the tri-national network might also suggests that the relational features become more informative when the network spans multiple linguistic and geographic boundaries, with the GNNs potentially capturing complex 'bridge' structures between the merged networks.

Trajectory-Aware Node Contributions and the Limits of Static Controllability

A recurring data mining task in complex networks is to determine how individual nodes contribute to system behavior. Existing approaches rely on either static-graph centralities or control-theoretic quantities such as controllability Gramians, which assume linear, time-invariant dynamics. Estimated systems, however, are typically nonlinear and time-varying. We define "emergent contribution (EC)," a finite-horizon measure of a node's dynamical leverage: the metric-weighted energy of its impulse response accumulated along the system trajectory. Computed from the Jacobians of any differentiable model, EC is estimator-agnostic and reduces exactly to average controllability in the linear, time-invariant limit. Our contribution is a characterization of when the two measures agree and diverge. Using a controlled synthetic family with known ground-truth contribution, we construct a phase diagram spanning nonlinearity, regime structure, persistence, and perturbation amplitude. EC and average controllability agree under static or smoothly drifting dynamics and both track ground truth. Divergence emerges under persistent regime switching, is strongest under persistent sign reversal, and disappears when the sign reversal is removed. At extreme perturbation amplitudes, both measures degrade, identifying the limits of local linearization. We place five estimated real systems from several domains within this phase space. Their placement serves as a diagnostic of when EC provides information beyond static controllability and therefore justifies its additional computational cost. On one panel examined in depth, a twenty-seed retraining ensemble reveals a robust variance--leverage dissociation: nodes whose perturbations propagate widely despite low within-system variance, which is not recovered by static centralities nor variance-based summaries.

Network Learning with Semi-relaxed Gromov-Wasserstein

Estimating the generative mechanism of large-scale networks is a fundamental challenge in statistical machine learning. It requires the identification of the latent connectivity structure, which is in general an NP-hard combinatorial problem due to the absence of canonical node labels. We address this challenge by allowing for probabilistic couplings, thereby relaxing the assignment problem. Our estimation framework can be formulated as a semi-relaxed Gromov-Wasserstein objective and provides a low-dimensional representation of the generative structure. We solve this via a block-coordinate conditional gradient algorithm. Despite the relaxation, the resulting solution is typically deterministic: in fact, we show that the optimality gap between the relaxed solution and the deterministic assignment vanishes at rate $O(1/n)$, where $n$ is the number of nodes. This allows for tractable recovery of the underlying model and enables rigorous statistical analysis: we establish consistency and minimax-optimal convergence rates for both stochastic block models and Holder-smooth graphons. Our implementation scales efficiently with $n$, as demonstrated on both synthetic and real-world datasets.