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5- ConclusionsIn both population models, the homogeneous mixing (fully-connected graph) topology fosters more rapid growth and supports higher levels of population than the spatial and random-graph topologies. This is so because, unlike any other topology, it adds no propagative restrictions to the intrinsic rules of the model. Nearest-neighbor spatial topologies and other local topologies such as the hierarchically-clustered random graph do introduce additional constraints on propagation. The maximum propagation rate in any direction can not exceed the largest birth rate of any creature in the system (the so-called ``speed of light'' [18]). By contrast, in a fully-connected graph, each cell is just one step away from every other cell. Another way to view the constraints that local topologies place on propagation is that they force replicators to waste much of their effort on futile attempts to replicate in areas that are already densely occupied. We can expect a similar slowing of population growth in any topology which possesses a reasonable degree of locality. The slowing effects of local constraints can engender qualitatively different phenomena such as the population oscillations depicted in Figs. 4d and 5. Essential to the cyclic behavior was the formation of an unoccupied core in the center of Figs. 5a and 5b, which would not have existed if cells far from the core had been permitted to replicate into it.
The constraints placed by sparsely-connected random graphs upon propagation are of a different
nature. Consider a node x with two neighbors y and z, both of which are unoccupied. As
soon as x replicates into y, the rate at which it can successfully replicate in the future
is immediately
halved. In contrast, in a fully-connected graph of N nodes, replication of the
initially-occupied cell into
one of its N-1 neighbors only reduces the field of eligible targets by the
negligible fraction
Many interesting phenomena remain to be discovered through further exploration of other topologies and models. I hope that this paper will help to make potential creators of new artificial worlds more aware of the effects that their choice of topology might have on the behavior that they observe, and that it will encourage researchers in artificial life to adapt some of the very interesting worlds that they have developed to new topologies. Finally, I hope to convince biologists, sociologists, and computer scientists of the importance of understanding the topological structure of their respective worlds.
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Bottlenecks in sparsely-connected graphs not only slow population growth;
they diminish the equilibrium populations and
increase the likelihood that the population will become extinct. Similar effects can be expected
in any topologies
in which each cell has only a few neighbors.
In fact, I have observed this phenomenon in the
hierarchical topology for the one-species model [