Infinite Limits and Adjacency Properties of a Generalized Copying Model

We present a new model for self-organizing networks such as the web graph, and analyze its limit behaviour. In the model, new vertices are introduced over time that copy the neighbourhood structure of existing vertices, and a certain number of extra edges may be added to the new vertex that randomly join to any of the existing vertices. A function ρ parameterizes the number of extra edges. We study the model by considering the infinite limit graphs it generates. The limit graphs satisfy with high probability certain adjacency properties similar to but not as strong as the ones satisfied by the infinite random graph. We prove that the strength of the adjacency properties satisfied by the limit are governed by the choice of ρ. We describe certain infinite deterministic graphs which arise naturally from our model, and that embed in all graphs generated by the model.