Networked Ising model.

Shows two types of bifurcations observed in the networked Ising model. Simulations were performed on random networks of 1000 nodes with homogeneous degree sequence (4, …, 4). The solid and dashed lines indicate the mean transition curve of 1000 runs for which a critical parameter was iteratively increased or decreased to induce the transition. Solid lines mark stable states, dotted lines mark unstable states, and dashed lines mark the critical transition on which we define the tipping point HT. The figure on the left depicts a pitchfork bifurcation with respect to changes in the critical parameter T and the figure on the right depicts a bifurcation showing the phenomenon of hysteresis with respect to changes in the critical parameter H where the temperature T is fixed at 2. Note that this temperature value was selected for illustrative purposes only. In all subsequent simulations, the temperature is set to 1 to ensure a bistable stability landscape for a larger range of degrees. Additional information on how the two bifurcation diagrams relate to each other, for example, in the form of a cusp diagram, can be obtained from [27, 28].