Simulation model : Group Dynamics, Social Networks and Cultural Capital
(c) Douglas R. White 1998 (Nov13)
site under development
1. Start with a large number n (e.g., n>10000) ancestral nodes.
2.Assign some number of nodes in the next generation greater than n (using a growth function).
3. For each node in the generation, assign two parental nodes from the preceding generation by a power law for shortest distance. Iterate back to step 2.
The power law for distance is simulated for each node by choosing one parental node randomly, and then choosing the other parent node according to the probability distribution of the power law, which selects the distance.
By definition, in this simplified model where all parents are one generation back and where ego always marries in the same generation, all distances to potential parents (or parents-in-law in a p-graph) are odd numbers. Hence, once the next node is chosen and its first parental node is assigned we do not need to recompute all the distances to other potential parent nodes but we simply draw the parental distance from the power law distribution and proceed as follows. Let the number drawn be 2n+1 (an odd number). Procede randomly from the node to any of its ancestors n generations above, then proceed randomly downward to any node n generations below this ancestor. Assign that node (always at the proper generation) as the other parental node.
All we have to do in addition is keep track of (1) the total generational depth g, and (2) the connected components of the graph and then (3) if n>g, pick a parent from an unconnected component of the graph.
The beauty of this simulation is that it runs in linear time and can perform simulations well into the 100s of millions of nodes (limited only by memory) on an ordinary PC.
The theory is that as the population grows under the Bose-Einstein process to large numbers, the distribution of distances becomes a stable Pareto-Levy distribution with infinite variance.
The interesting things about this model as a form of graphical evolution are that the use of the power law (where most distances are short) will generate local pockets of structural endogamy, with an occasional random link between distant clusters that will lead to exponential evolution of a giant component and of giant small worlds with larger diameters than the local population pockets.
We can also play with the form of the power law, so that as the "world population" in any generation begins to densify, or transport technology changes, then there is greater population mixing at a distance, and the power law coefficient shifts slightly to higher average distances. Then we can show simulation results at various populations, generational depth, and parameters of the power law.
As a later refinement of this model, we introduce a class-differentiation model. Using again a power-law model of resource accumulation by individual families, we then introduce biases towards wealth-status endogamy and rules of inheritance (divisible versus indivisible or single-heir estates) and observe over time the interaction between the power law distribution parameter that affects structural endogamy generally with that which affects resource distribution under different inheritance rules.
Similar models will be considered for succession to office within structurally endogamous and socially stratified groups.
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