In a number of multi-agent artificial life studies where agents interact over limited distances, the emergence and/or evolution of a specific behavior may depend critically upon interagent distances. Little theoretical analysis has been done previously concerning how to predict such distances. In this paper, we derive a probabilistic method that, for an agent at an arbitrary location in a two-dimensional cellular world, predicts the expected distance to a nearest other agent. Our method works for many world topologies, and we apply it to determine the expected distance for six commonly used ones. Further, the method is readily adapted to handle special restrictions. Over a wide variety of agent densities we show that the theoretically predicted distances are largely in agreement with the distances measured in computational experiments with randomly placed agents. We then utilize our prediction method to interpret recent observations that an imprecise threshold in the density of agents exists for the evolution of communication. We thus illustrate that, despite its conceptual simplicity, our method can aid the analysis and even the design of complex artificial environments populated by agents that have the potential to interact with one another.

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