Randomized algorithms
Trawling the Web for emerging cyber-communities
WWW '99 Proceedings of the eighth international conference on World Wide Web
On power-law relationships of the Internet topology
Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication
The degree sequence of a scale-free random graph process
Random Structures & Algorithms
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Extracting Large-Scale Knowledge Bases from the Web
VLDB '99 Proceedings of the 25th International Conference on Very Large Data Bases
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Random evolution in massive graphs
Handbook of massive data sets
Stochastic models for the Web graph
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
On Certain Connectivity Properties of the Internet Topology
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
The Diameter of a Scale-Free Random Graph
Combinatorica
The web as a graph: measurements, models, and methods
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
The Strange Logic of Random Graphs
The Strange Logic of Random Graphs
Limits and power laws of models for the web graph and other networked information spaces
CAAN'04 Proceedings of the First international conference on Combinatorial and Algorithmic Aspects of Networking
Note: An explicit construction of (3,t )-existentially closed graphs
Discrete Applied Mathematics
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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 "preferential attachment" 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 Erdö;s and Rényi 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.