Degree Distribution of Competition-Induced Preferential Attachment Graphs

Authors:
N. Berger;C. Borgs;J. T. Chayes;R. M. D'souza;R. D. Kleinberg
Affiliations:
Microsoft Research, One Microsoft Way, Redmond WA 98052, USA (e-mail: berger@its.caltech.edu, borgs@microsoft.com, jchayes@microsoft.com, raissa@alum.mit.edu);Microsoft Research, One Microsoft Way, Redmond WA 98052, USA (e-mail: berger@its.caltech.edu, borgs@microsoft.com, jchayes@microsoft.com, raissa@alum.mit.edu);Microsoft Research, One Microsoft Way, Redmond WA 98052, USA (e-mail: berger@its.caltech.edu, borgs@microsoft.com, jchayes@microsoft.com, raissa@alum.mit.edu);Microsoft Research, One Microsoft Way, Redmond WA 98052, USA (e-mail: berger@its.caltech.edu, borgs@microsoft.com, jchayes@microsoft.com, raissa@alum.mit.edu);MIT CSAIL, 77 Massachusetts Ave, Cambridge MA 02139, USA (e-mail: rdk@math.mit.edu)
Venue:
Combinatorics, Probability and Computing
Year:
2005

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Abstract

We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the trade-offs between two competing metrics, and show that this family is equivalent to a family of preferential attachment random graph models with upper cut-offs. 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 analyse a generalized version of our graph process, with two natural parameters, one corresponding to the cut-off and the other a ‘fertility’ parameter. We prove that the general model has a power-law degree distribution up to a cut-off, 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 cut-off and the uniform attachment model.