Evolution unfolds in populations of reproducing individuals, with the dynamics shaped by numerous factors, including population structure, update rules, mutation rates, traits (‘strategies’) and the reproductive success derived from traits (‘payoffs’). Developing a unifying principle to predict which traits will evolve and when, remains a major challenge. While rigorous mathematical results exist for static population structures, the general case of dynamic structures, where connections between individuals change over time, is more difficult and less understood. Here we present a structure-coefficient theorem for evolutionary games on dynamic networks with exogenous stochastic transitions. The framework enables arbitrary Markovian network dynamics independent of strategic interactions, together with arbitrary update mechanisms, mutation rates, strategy spaces and payoff matrices. We demonstrate that the combined effects of structure, update rules and mutation on evolutionary outcomes can be fully captured by three universal structure coefficients. Further analysis reveals that, although most sampled dynamic networks do not promote prosocial behavior, a class of dynamic core–periphery networks consistently promotes multiple forms of prosocial behavior, including cooperation, altruism, fairness and trust. The authors establish a structure-coefficient theorem, showing that three universal coefficients capture evolutionary outcomes in changing networks, and identify dynamic core–periphery networks that promote cooperation, altruism, fairness and trust.
Structure coefficients provide a unified way to identify dynamic network structures that promote cooperation, altruism, fairness and trust.
Abstract Predicting properties from coordinate–category data—sets of vectors paired with categorical information—is fundamental to computational science, manifesting in materials science as predicting formation energies or elastic moduli from crystal structures. While large language models (LLMs) have increasingly been applied to such tasks, optimal strategies for achieving reliable predictions remain elusive. Here we report fundamental limitations in LLM’s ability to learn from coordinate information in coordinate–category data. Through systematic experiments using synthetic datasets with tunable coordinate and category contributions, and a comprehensive benchmarking framework (MatText), we find that LLMs consistently fail to capture coordinate information while excelling at category patterns. This geometric blindness persists regardless of model size (up to 70 billion parameters), dataset scale (up to 2 × 10 6 structures) or text representation strategy. For material property prediction tasks dominated by structural effects, specialized geometric architectures consistently outperform LLMs, as evidenced by a clear ‘Graph Neural Network Language Model (GNN-LM) wall’ in performance benchmarks. Our analysis provides concrete guidelines for architecture selection in scientific machine learning.
The success of generative molecular design depends on the synthetic accessibility of the molecules it creates. While computational models can help predict chemical reactions, their current effectiveness is limited by a lack of reference data and the difficulty of accurately predicting reaction regioselectivity. Here, to address this, we present an integrated closed-loop workflow that combines efficient reaction data acquisition with high-accuracy deep learning. We developed an active learning strategy, driven by a tree-based ensemble, to guide the acquisition of the most informative laboratory experiments. This experimental campaign yielded a diverse benchmark for C–H borylation. The bespoke dataset allowed us to train and evaluate geometric graph neural networks. We demonstrate that augmenting these symmetry-aware models with self-supervised auxiliary tasks consistently improves performance for both reaction outcome prediction and regioselectivity prediction. Prospective tests on unseen substrates, featuring challenging N-heteroaryl motifs, pinpointed the correct borylation positions in all cases, illustrating the practical utility of our workflow for efficient drug optimization. Researchers pair active learning with geometric deep learning to predict the outcome and site selectivity of chemical reactions from scarce data, guiding experiments that accelerate drug discovery and expand accessible chemical space.
Artificial intelligence is enabling epidemic models to learn across outbreaks. The next challenge is ensuring that these models learn and transfer not mere statistical patterns, but epidemiological mechanisms.
