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Symmetries, due to their fundamental importance to dynamical processes on networks, have attracted a great deal of current research. Finding all symmetric nodes in large complex networks typically relies on automorphism groups from algebraic-group theory, which are solvable in quasipolynomial time. We articulate a conceptually appealing and computationally extremely efficient approach to finding and characterizing all symmetric nodes by introducing a structural position vector (SPV) for each node in networks. We establish the mathematical result that symmetric nodes must have the same SPV value and demonstrate, using six representative complex networks from the real world, that all symmetric nodes in these networks can be found in linear time. Furthermore, the SPVs not only characterize the similarity of nodes but also quantify the nodal influences in propagation dynamics. A caveat is that the proved mathematical result relating the SPV values to nodal symmetries is not sufficient; i.e., nodes having the same SPV values may not be symmetric, which arises in regular networks or networks with a dominant regular component. We point out with an analysis that this caveat is, in fact, shared by the known existing approaches to finding symmetric nodes in the literature. We further argue, with the aid of a mathematical analysis, that our SPV method is generally effective for finding the symmetric nodes in real-world networks that typically do not have a dominant regular component. Our SPV-based framework, therefore, provides a physically intuitive and computationally efficient way to uncover, understand, and exploit symmetric structures in complex networks arising from real-world applications.

According to the official report, the first case of COVID-19 and the first death in the United States occurred on January 20 and February 29, 2020, respectively On April 21, California reported that the first death in the state occurred on February 6, implying that community spreading of COVID-19 might have started earlier than previously thought Exactly what is time zero, i e , when did COVID-19 emerge and begin to spread in the U S and other countries? We develop a comprehensive predictive modeling framework to address this question Using available data of confirmed infections to obtain the optimal values of the key parameters, we validate the model and demonstrate its predictive power We then carry out an inverse inference analysis to determine time zero for 10 representative states in the U S , plus New York City, United Kingdom, Italy, and Spain The main finding is that, in both the U S and Europe, COVID-19 started around the new year day

We uncover a phenomenon in coupled nonlinear networks with a symmetry: as a bifurcation parameter changes through a critical value, synchronization among a subset of nodes can deteriorate abruptly, and, simultaneously, perfect synchronization emerges suddenly among a different subset of nodes that are not directly connected. This is a synchronization metamorphosis leading to an explosive transition to remote synchronization. The finding demonstrates that an explosive onset of synchrony and remote synchronization, two phenomena that have been studied separately, can arise in the same system due to symmetry, providing another proof that the interplay between nonlinear dynamics and symmetry can lead to a surprising phenomenon in physical systems.

According to the official report, the first case of COVID-19 and the first death in the United States occurred on January 20 and February 29, 2020, respectively. On April 21, California reported that the first death in the state occurred on February 6, implying that community spreading of COVID-19 might have started earlier than previously thought. Exactly what is time ZERO, i.e., when did COVID-19 emerge and begin to spread in the US and other countries? We develop a comprehensive predictive modeling framework to address this question. Using available data of confirmed infections to obtain the optimal values of the key parameters, we validate the model and demonstrate its predictive power. We then carry out an inverse inference analysis to determine time ZERO for ten representative States in the US, plus New York city, UK, Italy, and Spain. The main finding is that, in both the US and Europe, COVID-19 started around the new year day.

Due to the heterogeneity among the States in the US, predicting COVID-19 trends and quantitatively assessing the effects of government testing capability and control measures need to be done via a State-by-State approach. We develop a comprehensive model for COVID-19 incorporating time delays and population movements. With key parameter values determined by empirical data, the model enables the most likely epidemic scenarios to be predicted for each State, which are indicative of whether testing services and control measures are vigorous enough to contain the disease. We find that government control measures play a more important role than testing in suppressing the epidemic. The vast disparities in the epidemic trends among the States imply the need for long-term placement of control measures to fully contain COVID-19.

Coarse graining technology is one of the important methods to study large-scale complex networks currently. Here, we propose a generalized-degree-based coarse graining (GDCG) approach to extract respectively the undirected or directed coarse-grained networks by merging the nodes with same or similar generalized degree. The new approach provides an adjustable generalized degree by parameter p for preserving some significant properties of the initial networks during the coarse-graining processes. Compared with the existing coarse-graining methods, the GDCG method is only based on the generalized degree, which is not only simple and operable, but also keeps some statistical properties and the synchronizability of the original networks. Moreover, the size of the coarse-grained networks can be chosen freely in the proposed method. Finally, extensive numerical simulations demonstrate the effectiveness of our approach.