Measurements of criticality in the Olami-Feder-Christensen model

The Olami-Feder-Christensen model is a simple lattice based cellular automaton model introduced as a prototype to study self-organization in systems with a continuous state variable. Despite its simplicity there remains controversy over whether the system is truly critical in the nonconservative regime. Here we address this issue by introducing the layer branching rate, which measures how contributions to the system branching rate vary across the lattice. By considering this quantity for layers far from the edges of the finite-sized lattices, we find that the model is only critical in the conservative limit, but that previous studies have underestimated the system branching rate in the nonconservative case. We further derive expressions for the branching rate in systems where the state variable across the lattice is described by a uniform distribution, in order to determine the effect of self-organization upon the level of criticality. We find that organization raises the branching rate in the nearest-neighbor case, but in contrast lowers the level of criticality in a random-neighbor model.