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Quantum networks (QNs) exhibit stronger connectivity than predicted by classical percolation, yet the origin of this phenomenon remains unexplored. We apply a statistical physics model—concurrence percolation—to uncover the origin of stronger connectivity on hierarchical scale-free networks, the (U,V) flowers. These networks allow full analytical control over path connectivity through two adjustable path-length parameters, ≤V. This precise control enables us to determine critical exponents well beyond current simulation limits, revealing that classical and concurrence percolations, while both satisfying the hyperscaling relation, fall into distinct universality classes. This distinction arises from how they “superpose” parallel, nonshortest path contributions into overall connectivity. Concurrence percolation, unlike its classical counterpart, is sensitive to nonshortest paths and shows higher resilience to detours as these paths lengthen. This enhanced resilience is also observed in real-world hierarchical, scale-free internet networks. Our findings highlight a crucial principle for QN design: When nonshortest paths are abundant, they notably enhance QN connectivity beyond what is achievable with classical percolation. 

Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network’s indirect connectivity. This realization leads to the emergence of an alternative theory called “concurrence percolation”, which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design. 

We are now exposed daily to more information than we can process and this has substantial costs. We argue that the information space should be recognized as part of our environment and call for research into the effects and management of information overload. 

Abstract The global spread of the COVID-19 pandemic has followed complex pathways, largely attributed to the high virus infectivity, human travel patterns, and the implementation of multiple mitigation measures. The resulting geographic patterns describe the evolution of the epidemic and can indicate areas that are at risk of an outbreak. Here, we analyze the spatial correlations of new active cases in the USA at the county level and characterize the extent of these correlations at different times. We show that the epidemic did not progress uniformly and we identify various stages which are distinguished by significant differences in the correlation length. Our results indicate that the correlation length may be large even during periods when the number of cases declines. We find that correlations between urban centers were much more significant than between rural areas and this finding indicates that long-range spreading was mainly facilitated by travel between cities, especially at the first months of the epidemic. We also show the existence of a percolation transition in November 2020, when the largest part of the country was connected to a spanning cluster, and a smaller-scale transition in January 2021, with both times corresponding to the peak of the epidemic in the country. 

Abstract An excellent method for predicting links in multiplex networks is reflected in its ability to reconstruct them accurately. Although link prediction methods perform well on estimating the existence probability of each potential link in monoplex networks by the set of partially observed links, we lack a mathematical tool to reconstruct the multiplex network from the observed aggregate topology and partially observed links in multiplex networks. Here, we fill this gap by developing a theoretical and computational framework that builds a probability space containing possible structures with a maximum likelihood estimation. Then, we discovered that the discrimination, an indicator quantifying differences between layers from an entropy perspective, determines the reconstructability, i.e., the accuracy of such reconstruction. This finding enables us to design the optimal strategy to allocate the set of observed links in different layers for promoting the optimal reconstruction of multiplex networks. Finally, the theoretical analyses are corroborated by empirical results from biological, social, engineered systems, and a large volume of synthetic networks. 

Abstract Quantum networks describe communication networks that are based on quantum entanglement. A concurrence percolation theory has been recently developed to determine the required entanglement to enable communication between two distant stations in an arbitrary quantum network. Unfortunately, concurrence percolation has been calculated only for very small networks or large networks without loops. Here, we develop a set of mathematical tools for approximating the concurrence percolation threshold for unprecedented large-scale quantum networks by estimating the path-length distribution, under the assumption that all paths between a given pair of nodes have no overlap. We show that our approximate method agrees closely with analytical results from concurrence percolation theory. The numerical results we present include 2D square lattices of 200²nodes and complex networks of up to 10⁴nodes. The entanglement percolation threshold of a quantum network is a crucial parameter for constructing a real-world communication network based on entanglement, and our method offers a significant speed-up for the intensive computations involved. 

The code and data for the paper 'Estimating comparable distances to tipping points across mutualistic systems by scaled recovery rates' that published on Nat. Ecol. Evol, whose DOI is 10.1038/s41559-022-01850-8