Predicting future dynamics on networks is challenging, especially when the complete and accurate network topology is difficult to obtain in real-world scenarios. Moreover, the higher-order interactions among nodes, which have been found in a wide range of systems in recent years, such as the nets connecting multiple modules in circuits, further complicate accurate prediction of dynamics on hypergraphs. In this work, we proposed a two-step method called the topology-agnostic higher-order dynamics prediction (TaHiP) algorithm. The observations of nodal states of the target hypergraph are used to train a surrogate matrix, which is then employed in the dynamical equation to predict future nodal states in the same hypergraph, given the initial nodal states. TaHiP outperforms three latest Transformer-based prediction models in different real-world hypergraphs. Furthermore, experiments in synthetic and real-world hypergraphs show that the prediction error of the TaHiP algorithm increases with mean hyperedge size of the hypergraph, and could be reduced if the hyperedge size distribution of the hypergraph is known.

Multi-pass-dual-indenter (MPDI) scratch tests with various loading conditions were performed on two newly developed titanium alloys (Ti–10V–1Fe–3Al and Ti–10V–2Cr–3Al) to investigate their abrasion resistance under repetitive local sliding contact. A technically pure titanium sample was used as the reference. Various microstructures were established by different heat treatments, such as to turn the β-phase into a stable phase or a meta-stable phase showing Stress Induced Martensite (SIM) formation. The influence of phase evolution on the scratch resistance and corresponding failure mechanisms was unravelled. It was found that the phase morphology and fraction have a significant impact on the scratch resistance and that effect is applied load dependent. The scratch behaviour is closely related to the work hardening ability of the material surface especially at high loading conditions, while the original surface hardness is more relevant at low loading conditions. The observations definitely prove that, not withstanding the modest hardness level, a microstructure showing a combination of metastable β (trigger the stress-induced martensitic transformation) and flake α (enhancing the initial surface hardness) is the best route to improve the scratch resistance for these two titanium alloys, in particular for high load conditions.

The prevalence, which is the average fraction of infected nodes, has been studied to evaluate the robustness of a network subject to the spread of epidemics. We explore the vulnerability (infection probability) of each node in the metastable state with a given effective infection rate τ. Specifically, we investigate the ranking of the nodal vulnerability subject to a susceptible-infected-susceptible epidemic, motivated by the fact that the ranking can be crucial for a network operator to assess which nodes are more vulnerable. Via both theoretical and numerical approaches, we unveil that the ranking of nodal vulnerability tends to change more significantly as τ varies when τ is smaller or in Barabási-Albert than Erdos-Rényi random graphs.