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 ourheterogeneousmodel 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 thehomogeneousmodel 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 calledcritical stretchingwhere 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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- May 01, 2020
- Source ID
- 10.1088/1367-2630/ab8263
Entities
People
- Bnaya Gross
- Dana Vaknin
- Sergey V. Buldyrev
- Shlomo Havlin
Organizations
- Defense Threat Reduction Agency
- Israel Science Foundation