Analysis of the spatial cascading effect in networks

Reality networks such as power grids and social networks can be spatially embedded. In this paper, we focus on the spatial cascading effect in such networks. The spatial cascading effect is that the failure of one node may cause other nodes that are close to it in space to fail. The phenomenon is very common, such that a person is more likely to have an impact on his neighbors even if he is not connected with his neighbors via social networks. Based on this, we construct a spatial cascading model to simulate the spatial cascading effect. In addition, we apply the exponential distribution P(l)∼exp−lζ to fit the real link distances. The networks are generated by two-dimensional lattices. We define two kinds of connections, namely actual spatial connections. The actual connections are links generated by the exponential distribution. The spatial connections are links in the lattice. Simulations show that the spatial embeddedness makes networks more robust in our model, which is different from previous research results. We put forward an algorithm to alter the link distances in the networks without changing node degree values. Using the algorithm verifies our conclusion that if nodes tend to connect with local nodes, networks will be robust to the spatial cascading effect. We further extend our model to a more general form. The nodes embedded in lattice can be sparse, which means that the existing probability of nodes in the lattice is not always 1. The networks in the extension model are more vulnerable compared to those in the original model.