The coronavirus known as COVID-19 has spread worldwide since December 2019. Without any vaccination or medicine, the means of controlling it are limited to quarantine and social distancing. Here we study the spatio-temporal propagation of the first wave of the COVID-19 virus in China and compare it to other global locations. We provide a comprehensive picture of the spatial propagation from Hubei to other provinces in China in terms of distance, population size, and human mobility and their scaling relations. Since strict quarantine has been usually applied between cities, more insight into the temporal evolution of the disease can be obtained by analyzing the epidemic within cities, especially the time evolution of the infection, death, and recovery rates which affected by policies. We compare the infection rate in different cities in China and provinces in Italy and find that the disease spread is characterized by a two-stages process. In early times, of the order of few days, the infection rate is close to a constant probably due to the lack of means to detect infected individuals before infection symptoms are observed. Then at later times it decays approximately exponentially due to quarantines. This exponential decay allows us to define a characteristic time of controlling the disease which we found to be approximately 20 days for most cities in China in marked contrast to different provinces in Italy which are characterized with much longer controlling time indicating less efficient controlling policies. Moreover, we study the time evolution of the death and recovery rates which we found to show similar behavior as the infection rate and reflect the health system situation which could be overloaded.

The new coronavirus known as COVID-19 is spread world-wide since December 2019. Without any vaccination or medicine, the means of controlling it are limited to quarantine and social distancing. Here we study the spatio-temporal propagation of the first wave of the COVID-19 virus in China and compare it to other global locations. We provide a comprehensive picture of the spatial propagation from Hubei to other provinces in China in terms of distance, population size, and human mobility and their scaling relations. Since strict quarantine has been usually applied between cities, more insight about the temporal evolution of the disease can be obtained by analyzing the epidemic within cities, especially the time evolution of the infection, death, and recovery rates which affected by policies. We study and compare the infection rate in different cities in China and provinces in Italy and find that the disease spread is characterized by a two-stages process. At early times, at order of few days, the infection rate is close to a constant probably due to the lack of means to detect infected individuals before infection symptoms are observed. Then at later times it decays approximately exponentially due to quarantines. The time evolution of the death and recovery rates also distinguish between these two stages and reflect the health system situation which could be overloaded.

We study a spatial network model with exponentially distributed link-lengths on an underlying grid of points, undergoing a structural crossover from a random, Erdős-Rényi graph to a 2D lattice at the characteristic interaction range ζ. We find that, whilst far from the percolation threshold the random part of the incipient cluster scales linearly with ζ, close to criticality it extends in space until the universal length scale ζ 3/2 before crossing over to the spatial one. We demonstrate this critical stretching phenomenon in percolation and in dynamical processes, and we discuss its implications to real-world phenomena, such as neural activation, traffic flows or epidemic spreading.

We study a realistic spatial network model constructed by randomly linking lattice sites with linklengths following an exponential distribution with a characteristic scale ζ. We find that this simple spatial network topology does not fulfill any single universality class, but exhibits a new multiuniversality with two sets of critical exponents. This bi-universality is characterized by random-like scaling laws for measurements on a scale smaller than ζ but spatial scaling for measurements on a larger scale. We further explore this topology by studying the resilience of a two-layer multiplex under localized attack. We find that for a broad range of the control parameters, our system is metastable. In this metastable region, a localized attack larger than a critical size -that does not depends on the size of the system -induces a propagating cascade of failures leading to the system collapse.

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Many interdependent, real-world infrastructures involve interconnections between different communities or cities. Here we study if and how the effects of such interconnections can be described as an external field for interdependent networks experiencing first-order percolation transitions. We find that the critical exponents γ and δ, related to the external field can also be defined for firstorder transitions but that they have different values than those found for second-order transitions. Surprisingly, we find that both sets of different exponents can be found even within a single model of interdependent networks, depending on the dependency coupling strength. Specifically, the exponent γ in the first-order regime (high coupling) does not obey the fluctuation dissipation theorem, whereas in the continuous regime (for low coupling) it does. Nevertheless, in both cases they satisfy Widom's identity, δ − 1 = γ/β which further supports the validity of their definitions. Our results provide physical intuition into the nature of the phase transition in interdependent networks and explain the underlying reasons for two distinct sets of exponents.

Understanding the resilience of infrastructures, such as a transportation network, has significant importance for our daily life. Recently, a homogeneous spatial network model was developed for studying spatial embedded networks with characteristic link length such as power-grids and the brain. However, although many real-world networks are spatially embedded and their links have characteristics length such as pipelines, power lines or ground transportation lines they are not homogeneous but rather heterogeneous. For example, density of links within cities are significantly higher than between cities. Here we develop and study numerically and analytically a similar realistic heterogeneous spatial modular model using percolation process to better understand the effect of heterogeneity on such networks. The model assumes that inside a city there are many lines connecting different locations, while long lines between the cities are sparse and usually directly connecting only a few nearest neighbours cities in a two dimensional plane. We find that our heterogeneous model experiences two distinct continuous transitions, one when the cities disconnect from each other and the second when each city breaks apart. This is in contrast to the homogeneous model where a single transition is found. Although the critical threshold for site percolation in 2D grid remains an open question we analytically find the critical threshold for site percolation in this model. In addition, it has been found that the homogeneous model experience a single transition having a unique phenomenon called critical stretching where a geometric crossover from random to spatial structure in different scales found to stretch non-linearly with the characteristic length at criticality. In marked contrast, we show here that the heterogeneous model does not experience such a phenomenon indicating that critical stretching strongly depends on the network structure.

We study the effect of localized attacks on a multiplex network, where each layer is a network of communities embedded in space. We assume that nodes are densely connected within a community and sparsely connected to the nodes in the neighboring communities. To investigate percolation processes in this realistic system we develop an analytical scheme, applying the finite-element method. We find, both by simulation and theory, that in many cases there is a critical size of localized damage above which it will spread and the entire system will collapse. In addition, we show that for a constant number of links, networks with less connectivity between communities are surprisingly more robust.