Time evolution of dense multigraph limits under edge-conservative preferential attachment dynamics
Abstract
We define the edge reconnecting model, a random multigraph evolving in time. At each time step we change one endpoint of a uniformly chosen edge: the new endpoint is chosen by linear preferential attachment. We consider a sequence of edge reconnecting models where the sequence of initial multigraphs is convergent in a sense which is a natural generalization of the notion of convergence of dense graph sequences, defined by Lovasz and Szegedy in arXiv:math/0408173. We investigate how the limit object evolves under the edge reconnecting dynamics if we rescale time properly: we give the complete characterization of the time evolution of the limit object from its initial state up to the stationary state, which is described in the companion paper arXiv:1106.2058. In our proofs we use the theory of exchangeable arrays, queuing and diffusion processes. The number of parallel edges and the degrees evolve on different timescales and because of this the model exhibits subaging.
Cite
@article{arxiv.0912.3904,
title = {Time evolution of dense multigraph limits under edge-conservative preferential attachment dynamics},
author = {Balazs Rath},
journal= {arXiv preprint arXiv:0912.3904},
year = {2012}
}
Comments
The present paper is accepted for publication at Random Structures and Algorithms. This is a shorter version of the paper, the results about the stationary model are now in the separate paper: Balazs Rath, Laszlo Szakacs: Multigraph limit of the dense configuration model and the preferential attachment graph, arXiv:1106.2058