Abstract
The study of random graphs has traditionally been dominated by the closely-related models G(n, m), in which a graph is sampled from the uniform distribution on graphs with n vertices and m edges, and G(n, p), in which each of the (n/2) edges is sampled independently with probability p. Recently, however, there has been considerable interest in alternate random graph models designed to more closely approximate the properties of complex real-world networks such as the Web graph, the Internet, and large social networks. Two of the most well-studied of these are the closely related and copying models, in which vertices arrive one-by-one in sequence and attach at random in rich-get-richer fashion to d earlier vertices.Here we study the infinite limits of the preferential attachment process --- namely, the asymptotic behavior of finite graphs produced by preferential attachment (brie y, PA graphs), as well as the infinite graphs obtained by continuing the process indefinitely. We are guided in part by a striking result of Erdo;s and Renyi on countable graphs produced by the infinite analogue of the G(n, p) model, showing that any two graphs produced by this model are isomorphic with probability 1; it is natural to ask whether a comparable result holds for the preferential attachment process.We find, somewhat surprisingly, that the answer depends critically on the out-degree d of the model. For d = 1 and d = 2, there exist infinite graphs R∞d such that a random graph generated according to the infinite preferential attachment process is isomorphic to R∞d with probability 1. For d ≥ 3, on the other hand, two different samples generated from the infinite preferential attachment process are non-isomorphic with positive probability. The main technical ingredients underlying this result have fundamental implications for the structure of finite PA graphs; in particular, we give a characterization of the graphs H for which the expected number of subgraph embeddings of H in an n-node PA graph remains bounded as n goes to infinity.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.