Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2005-02-08
Physics
Condensed Matter
Disordered Systems and Neural Networks
24 pages, one figure. To appear in the journal: Combinatorics, Probability and Computing. Note, this is a long version, with c
Scientific paper
We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs 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 the 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 a finite 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.
Berger Noam
Borgs Christian
Chayes Jennifer T.
D'Souza Raissa M.
Kleinberg Robert D.
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