Degree Distribution of Competition-Induced Preferential Attachment Graphs

We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeofs between two competing metrics, and show that this family is equivalent to a family of preferential attachment random graph models with upper cutoffs. This is first explanation of how preferential attachment can arise from a more basic underlying mechanism of local competition. We rigorously determine the degree distribution for the family of random graph models, showing that it obeys a power law up to afinite threshold and decays exponentially above this threshold. We also rigorously analyze a generalized version of our graph process, with two natural parameters, one corresponding to the cutoff and the other a \fertility” parameter. We prove that the general model has a power-law degree distribution up to a cutoff, and establish monotonicity of the power as a function of the two parameters. Limiting cases of the general model include the standard preferential attachment model without cutoff and the uniform attachment model.