Fig 1.
A typical CIB matrix standardized with ScenarioWizard [22], with three descriptors and three states per descriptor.
Descriptors are for a social system conceptualized at a global or large continental-region level, i.e. an aggregation of countries. The top middle submatrix gives the (estimated) directed influences from descriptor “1” (Population) upon descriptor “2” (Income per capita) with the top right entry of this submatrix (“4”), denoting the extent to which Low population promotes high income. Rows highlighted in grey represent the possible scenario “Low population, High income per capita, Low educational attainment”. Below the impact matrix we sum these rows and determine which “target” scenario is most supported by the current scenario. Figure adapted with permission from Lloyd and Schweizer [23].
Table 1.
Comparison of notable scenarios for the Population-Income-Education system according to five succession rules.
Fig 2.
An example of a Markov chain with all transition probabilities equal to 1.
Here we have taken the example scenario described in Fig 1 under the local succession (with a deterministic tiebreaking rule). All initial states filter down to one of three self-consistent states in a purely deterministic manner.
Fig 3.
Plots of the three functions introduced in the text.
From left to right: Boltzmann, arctan, and logistic, each for three values of β: β = 1(continuous), β = ½ (dashed), β = 2 (dotted).
Fig 4.
Markov chain of Population-Income-Education example with the global succession rule.
L, M, H stands for Low, Medium, and High states respectively. The order of the states reflects the outcomes for the descriptors population, income per capita, and educational attainment respectively. Nodes are color-coded according to population (green-High; magenta-Medium; blue-Low). Transitions with probability 1 are black.
Fig 5.
Markov chain of the Population-Income-Education example with the adiabatic succession rule.
Color coding as in Fig 4.
Fig 6.
Markov chain of the Population-Income-Education example with the random descriptor succession rule.
Color coding as in Fig 4. In cases where multiple states have the same impact score, succession is chosen at random amongst these states (grey edges). Node size is proportional to long term probability (assuming initial node selected uniformly at random). This is equivalent to eigenvector centrality.
Fig 7.
Markov chain of the Population-Income-Education example with the local Boltzmann succession rule (β = 1).
Color coding as in Fig 6. Edges with weight less than 0.02 are not shown. Node size is proportional to the steady state probability of the node (long-term forecast). In this case, the initial state does not matter.
Fig 8.
Markov chain of the Population-Income-Education example with the local logistic succession rule (beta = 1; shift = 1).
Color coding as in Fig 6. To enhance clarity, edges with weights less than 0.1 are not shown in this figure. Node size is proportional to steady-state probability.
Fig 9.
Markov chain of the Population-Income-Education example with the local arctan succession rule (beta = 1).
Color coding as in Fig 6. For the sake of clarity, edges representing transitions with less than 10% probability are not shown.