Two transitions in spatial modular networks

Abstract

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.

Spatio-temporal propagation of COVID-19 pandemics

“…Since quarantine is usually applied on a city level and the interactions of people within cities is significantly different than the one between cities [20], more insight can be obtained in analyzing the epidemic in cities. Our results suggest that the temporal propagation of the disease in most of the cities in China is similar.…”

“…As a result of the heterogeneous structure of cities in a country [20], quarantine has been usually applied strictly between cities and the mobility within cities was less restricted. Thus, more insight can be obtained in analyzing the epidemic within cities while studying the disease decay on a country scale might lead to uncertain conclusion regarding the disease situation since the disease may propagate in one city and decay in another.…”

Epidemic spreading and control strategies in spatial modular network

“…The second (higher) transition at β 2D c when the epidemic spreads in the entire country. These two epidemic transitions are analogues to the two percolation transitions found in Gross et al (2020c) in the same model although the analytical solution is different. While the position of the first transition does not depend on ζ , the position of the second decreases with ζ , and at large ζ it almost coalesces with the first one.…”

“…The spatial community model (Vaknin et al 2019;Gross et al 2020c) illustrated in Fig. 1, represents the infection channels within and between communities on a 2-dimensional square lattice with N = L × L lattice sites, where L is the linear size of the lattice and the lattice sites are the nodes of the network.…”

“…In this paper, we applied the susceptible-infected-recovered (SIR) model to study the epidemic spreading in a 2D spatial community network model (Vaknin et al 2019;Gross et al 2020c), see Fig. 1, to better describe epidemic spreading in human social community organization.…”