Competition-Induced Preferential Attachment
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 Barabási-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.
KeywordsDegree Distribution Preferential Attachment Degree Sequence Random Graph Model Attachment Model
Unable to display preview. Download preview PDF.
- 1.Aiello, W., Chung, F., Lu, L.: Random evolution of massive graphs. In: Handbook of Massive Data Sets, pp. 97–122. Kluwer, Dordrecht (2002)Google Scholar
- 5.Berger, N., Bollobás, B., Borgs, C., Chayes, J.T., Riordan, O.: Degree distribution of the FKP network model. In: International Colloquium on Automata, Languages and Programming (2003)Google Scholar
- 6.Bollobás, B., Borgs, C., Chayes, J., Riordan, O.: Directed scale-free graphs. In: Proceedings of the 14th ACM-SIAM Symposium on Discrete Algorithms, pp. 132–139 (2003)Google Scholar
- 7.Bollobás, B., Riordan, O.: Mathematical results on scale-free random graphs. In: Handbook of Graphs and Networks, Berlin, 2002, Wiley-VCH, Chichester (2002)Google Scholar
- 10.Cooper, C., Frieze, A.M.: A general model of web graphs. In: Proceedings of 9th European Symposium on Algorithms, pp. 500–511 (2001)Google Scholar
- 13.Fabrikant, A., Koutsoupias, E., Papadimitriou, C.H.: Heuristically optimized trade-offs: a new paradigm for power laws in the internet. In: International Colloquium on Automata, Languages and Programming, pp. 110–122 (2002)Google Scholar
- 15.Govindan, R., Tangmunarunkit, H.: Heuristics for Internet map discovery. In: Proceedings of INFOCOM, pp. 1371–1380 (2000)Google Scholar
- 16.Kenyon, C., Schabanel, N.: Personal communicationGoogle Scholar
- 17.Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: Stochastic models for the web graph. In: Proc. 41st IEEE Symp. on Foundations of Computer Science, pp. 57–65 (2000)Google Scholar
- 22.Zipf, G.K.: Human Behavior and the Principle of Least Effort. Addison-Wesley, Cambridge (1949)Google Scholar