Although the hypothesis that nestedness determines mutualistic ecosystem dynamics is accepted in general, results of some recent data analyses and theoretical studies have begun to cast doubt on the impact of nestedness on ecosystem stability. However, definite conclusions have not yet been reached because previous studies are mainly based on numerical simulations. Therefore, we reveal a mathematical architecture in the relationship between ecological mutualistic networks and local stability based on spectral graph analysis. In particular, we propose a theoretical method for estimating the dominant eigenvalue (i.e., spectral radius) of quantitative (or weighted) bipartite networks by extending spectral graph theory, and provide a theoretical prediction that the heterogeneity of node degrees and link weights primarily determines the local stability; on the other hand, nestedness additionally affects it. Numerical simulations demonstrate the validity of our theory and prediction. This study emphasizes the importance of ecological network heterogeneity in ecosystem dynamics, and it enhances our understanding of structure–stability relationships.

Ecological networks exhibit non-random structural patterns, such as modularity and nestedness, which determine ecosystem stability with species diversity and connectance. Such structure-stability relationships are well known. However, another important perspective is less well understood: the relationship between the environment and structure. Inspired by theoretical studies that suggest that network structure can change due to environmental variability, we collected data on a number of empirical food webs and mutualistic networks and evaluated the effect of climatic seasonality on ecological network structure. As expected, we found that climatic seasonality affects ecological network structure. In particular, an increase in modularity due to climatic seasonality was observed in food webs; however, it is debatable whether this occurs in mutualistic networks. Interestingly, the type of climatic seasonality that affects network structure differs with ecosystem type. Rainfall and temperature seasonality influence freshwater food webs and mutualistic networks, respectively; food webs are smaller, and more modular, with increasing rainfall seasonality. Mutualistic networks exhibit a higher diversity (particularly of animals) with increasing temperature seasonality. These results confirm the theoretical prediction that the stability increases with greater perturbation. Although these results are still debatable because of several limitations in the data analysis, they may enhance our understanding of environment-structure relationships.

Conserving ecosystem function and associated services requires deep understanding of the underlying basis of system stability. While the study of ecological dynamics is a mature and diverse field, the lack of a general model that predicts a broad range of theoretical and empirical observations has allowed unresolved contradictions to persist. Here we provide a general model of mutualistic ecological interactions between two groups and show for the first time how the conditions for bi-stability, the nature of critical transitions, and identifiable leading indicators in time-series can be derived from the basic parameters describing the underlying ecological interactions. Strong mutualism and nonlinearity in handling-time are found to be necessary conditions for the occurrence of critical transitions. We use the model to resolve open questions concerning the effects of heterogeneity in inter-species interactions on both resilience and abundance, and discuss these in terms of potential trade-offs in real systems. This framework provides a basis for rich investigations of ecological system dynamics, and may be generalizable across many ecological contexts.

Although neural-network architectures are critical for their performance, how the structural characteristics of a neural network affect its performance has still not been fully explored. Here, we map architectures of neural networks to directed acyclic graphs (DAGs), and find that incoherence, a structural characteristic to measure the order of DAGs, is a good indicator for the performance of corresponding neural networks. Therefore, we propose a deep isotropic neural-network architecture by folding a chain of the same blocks and then connecting the blocks with skip connections at different distances. Our model, named FoldNet, has two distinguishing features compared with traditional residual neural networks. First, the distances between block pairs connected by skip connections increase from always equal to one to specially selected different values, which lead to more incoherent graphs and let the neural network explore larger receptive fields and, thus, enhance its multi-scale representation ability. Second, the number of direct paths increases from one to multiple, which leads to a larger proportion of shorter paths and, thus, improves the direct propagation of information throughout the entire network. Image-classification results on CIFAR-10 and Tiny ImageNet benchmarks suggested that our new network architecture performs better than traditional residual neural networks. FoldNet with 25.4M parameters can achieve 72.67% top-1 accuracy on the Tiny ImageNet after 100 epochs, which is competitive compared with the-state-of-art results on the Tiny ImageNet.