Time evolution of dense multigraph limits under edge-conservative preferential attachment dynamics

TL;DR

This paper studies the evolution of dense multigraph limits under a preferential attachment dynamic, providing a complete characterization of their time evolution and revealing subaging phenomena.

Contribution

It introduces a new model for evolving multigraphs with preferential attachment and characterizes the limit object dynamics over time.

Findings

Complete description of the limit object evolution

Identification of subaging behavior

Analysis using exchangeable arrays, queuing, and diffusion processes

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.