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Competition-Induced Preferential Attachment

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By N. Berger, C. Borgs, J. T. Chayes, R. M. D.Souza, and R. D. Kleinberg

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Presented by: Jenny Lam

Abstract:
Models based on preferential attachment have had much success
in reproducing the power law degree distributions which seem ubiquitous
in both natural and engineered systems. Here, rather than assuming
preferential attachment, we give an explanation of how it can arise from
a more basic underlying mechanism of competition between opposing
forces.

We introduce a family of one-dimensional geometric growth models,
constructed iteratively by locally optimizing the tradeoffs between two
competing metrics. This family admits an equivalent description as a
graph process with no reference to the underlying geometry. Moreover,
the resulting graph process is shown to be preferential attachment with
an upper cutoff. 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 introduce and rigorously analyze a generalized version of our
graph process, with two natural parameters, one corresponding to the
cutoff and the other a .fertility. parameter. Limiting cases of this
process include the standard preferential attachment model (introduced
by Price and by Barabasi-Albert) and the uniform attachment model. In
the general case, we prove that the process has a power law degree
distribution up to a cutoff, and establish monotonicity of the power as
a function of the two parameters.