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To better understand the temporal characteristics and the lifetime of fluctuations in stochastic processes in networks, we investigated diffusive persistence in various graphs. Global diffusive persistence is defined as the fraction of nodes for which the diffusive field at a site (or node) has not changed sign up to time t (or, in general, that the node remained active or inactive in discrete models). Here we investigate disordered and random networks and show that the behavior of the persistence depends on the topology of the network. In two-dimensional (2D) disordered networks, we find that above the percolation threshold diffusive persistence scales similarly as in the original 2D regular lattice, according to a power law P(t , L) ∼ t−θ with an exponent θ ~ 0.186, in the limit of large linear system size L. At the percolation threshold, however, the scaling exponent changes to θ ~ 0.141, as the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. Moreover, studying finite-size effects for 2D lattices at and above the percolation threshold, we find that at the percolation threshold, the long-time asymptotic value obeys a power law P(t , L) ∼ L−zθ with z ~ 2.86 instead of the value of z = 2 normally associated with finite-size effects on 2D regular lattices. In contrast, we observe that in random networks without a local regular structure, such as Erdos-Rényi networks, no simple power-law scaling behavior exists above the percolation threshold 

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.

 

Many systems may switch to an undesired state due to internal failures or external perturbations, of which critical transitions toward degraded ecosystem states are prominent examples. Resilience restoration focuses on the ability of spatially-extended systems and the required time to recover to their desired states under stochastic environmental conditions. The difficulty is rooted in the lack of mathematical tools to analyze systems with high dimensionality, nonlinearity, and stochastic effects. Here we show that nucleation theory can be employed to advance resilience restoration in spatially-embedded ecological systems. We find that systems may exhibit single-cluster or multi-cluster phases depending on their sizes and noise strengths. We also discover a scaling law governing the restoration time for arbitrary system sizes and noise strengths in two-dimensional systems. This approach is not limited to ecosystems and has applications in various dynamical systems, from biology to infrastructural systems.