Spatial Diffusion Processes 1: A Model and an Approximation Method

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Abstract
This paper describes a model of the diffusion of a message among the cities of a system. The diffusion process depends upon the flow of information about the message from prior adopters. Two cases are identified: (1) in which adopters are individuals within cities, so that information relevant to adoption may flow from adopters in one city to other individuals in that city, and (2) in which the city is the adopting unit, so that information relevant to adoption may only arrive from other adopter cities. In both cases an approximation must be made in order that the differential equations can be solved. The simpler case is case (1), which is analysed fully; calculations reveal a close negative relationship between the total interaction into a city and the mean and variance of the time of arrival of the message in that city. Case (2) can only be solved in series form, and so the second paper in this set examines this case numerically.