Fig 1.
Random networks for different degree sequences constructed using the configuration model. Figure on the left shows configured random network from a degree sequence containing only degree 3. Figure in the middle shows a configured random network from a degree sequence containing degrees 3 and 4. Figure on the right shows a configured random network from degree sequence containing degrees 3, 4 and 5.
Fig 2.
Illustration of a spin field in the 2D Ising model. An upward arrow marks a node with an upward spin, i.e. s = + 1, and a downward arrow marks a node with a downward spin, i.e. s = −1. Nodes in the spin field seek spin alignment with their immediate neighborhood. The illustration depicts a regular 2D grid network structure. In our simulations, however, we use random degree mixed networks such as shown in Fig 1 as the underlying network structure.
Fig 3.
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].
Fig 4.
Critical transitions in degree mixed networks.
Shows tipping points in networks of 1000 nodes for all possible degree sequence substitution between nodes of degree 3 and 5. The reference line shows that the tipping points of degree homogeneous networks do not follow a perfect linear relationship.
Fig 5.
Shows tipping boundaries created by fitted curves for degree mixed networks between two degrees (k1, k2). The gray area marks forbidden tipping regions and the reference line shows that the tipping points of degree homogeneous networks do not follow a perfect linear relationship.
Fig 6.
Tipping points in mixed networks containing multiple degrees.
Figures show the statistical spread of tipping points HT generated for degree mixed networks with degree sequences containing three distinct degrees (k1, k2, k3).
Fig 7.
Transition comparison for different bifurcation types in the networked Ising model.
Figure on the left shows a smooth state transition along a pitchfork bifurcation where the system is initialized in the top left and the critical parameter T is iteratively increased from 0.01 to 10 to induce the transition. Figure on the right shows an abrupt state transition (critical transition) along a bifurcation with hysteresis where the system is initialized on the bottom left and the critical parameter H is iteratively increased from -5 to 5 to induce the transition. Tipping point candidates in the pitchfork system are indicated by TT and by HT in the hysteresis system. The red line corresponds to a transition on a homogeneous degree network with average degree 5, and the cyan line corresponds to a transition on a degree mixed network consisting of degrees 3 and 7, also with average degree 5.