Criticality of the nonconservative earthquake model on random spatial networks

We study the nonconservative earthquake model on random spatial networks. The spatial networks are composed of sites on a two-dimensional (2D) plane which are connected locally. Differently from a regular lattice, the locations of sites are modeled in the way that sites are randomly placed on the plane. Using the same connectivity degree as a 2D lattice, however, the spatial network cannot exhibit critical earthquake behavior. Mimicking long range energy transfer, the connection radius is increased and the connectivity degree of the spatial network is increased. Then we show that the model exhibits self-organized criticality. The mechanism of the structural effect is presented. The spatial network includes many modules when connectivity degree is very small. The effect of modular structure on the avalanche dynamics is to limit the spreading of avalanches in the whole network. When the connectivity degree is larger, the long range energy transfer can overcome the effect of local modularity and criticality can be reached.