Isomorphism and embedding problems for infinite limits of scale-free graphs

The study of random graphs has traditionally been dominated by the closely-related models <i>G</i>(<i>n, m</i>), in which a graph is sampled from the uniform distribution on graphs with <i>n</i> vertices and <i>m</i> edges, and <i>G</i>(<i>n, p</i>), in which each of the (<i>n</i>/2) edges is sampled independently with probability <i>p.</i> 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 "preferential attachment" and "copying" models, in which vertices arrive one-by-one in sequence and attach at random in "rich-get-richer" fashion to <i>d</i> 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, <i>PA graphs</i>), as well as the infinite graphs obtained by continuing the process indefinitely. We are guided in part by a striking result of Erdö;s and Rényi on countable graphs produced by the infinite analogue of the <i>G</i>(<i>n, p</i>) 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 <i>d</i> of the model. For <i>d</i> = 1 and <i>d</i> = 2, there exist infinite graphs <i>R</i>^∞<inf><i>d</i></inf> such that a random graph generated according to the infinite preferential attachment process is isomorphic to <i>R</i>^∞<inf><i>d</i></inf> with probability 1. For <i>d</i> ≥ 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 <i>H</i> for which the expected number of subgraph embeddings of <i>H</i> in an <i>n</i>-node PA graph remains bounded as <i>n</i> goes to infinity.