Aggregation Processes on Networks: Deterministic Equations, Stochastic Model and Numerical Simulation
- Technische Universitaet Dortmund, Facultaet fuer Mathematik, Lehrstuhl LSI, Vogelpothsweg 87, 44221 Dortmund (Germany)
We introduce an infinite system of equations modeling the time evolution of the growth process of a network. The nodes are characterized by their degree k(set-membership sign)N and a fitness parameter f(set-membership sign)[0,h]. Every new node which emerges becomes a fitness f' according to a given distribution P and attaches to an existing node with fitness f and degree k at rate fA{sub k}, where A{sub k} are positive coefficients, growing sub-linearly in k. If the parameter f takes only one value, the dynamics of this process can be described by a variant of the Becker-Doering equations, where the l growth of the size of clusters of size k occurs only with increment 1. In contrast l to the established Becker-Doering equations, the system considered here is nonconservative, since mass (i.e. links) is continuously added. Nevertheless, it has the property of linearity, which is a natural consequence of the process which is being modeled. The purpose of this paper is to construct a solution of the system based on a stochastic approximation algorithm, which allows also a numerical simulation in order to get insight into its qualitative behaviour. In particular we show analytically and numerically the property of Bose-Einstein condensation, which was observed in the literature on random graphs and which can be described as an emergence of a huge cluster which captures a macroscopic fraction of the total link density.
- OSTI ID:
- 21251362
- Journal Information:
- AIP Conference Proceedings, Vol. 1048, Issue 1; Conference: International conference on numerical analysis and applied mathematics 2008, Psalidi, Kos (Greece), 16-20 Sep 2008; Other Information: DOI: 10.1063/1.2991085; (c) 2008 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0094-243X
- Country of Publication:
- United States
- Language:
- English
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