Recently many new "scale-free'' random graph models have been introduced, motivated by the power-law degree sequences observed in many large-scale, real-world networks. Perhaps the best known, the Barabási-Albert model, has been extensively studied from heuristic and experimental points of view. Here we consider mathematically two basic characteristics of a precise version of this model, the LCD model, namely robustness to random damage, and vulnerability to malicious attack. We show that the LCD graph is much more robust than classical random graphs with the same number of edges, but also more vulnerable to attack. In particular, if vertices of the n-vertex LCD graph are deleted at random, then as long as any positive proportion remains, the graph induced on the remaining vertices has a component of order n. In contrast, if the deleted vertices are chosen maliciously, a constant fraction less then 1 can be deleted to destroy all large components. For the Barabási-Albert model, these questions have been studied experimentally and heuristically by several groups.

Measuring the properties of a large, unstructured network can be difficult: One may not have full knowledge of the network topology, and detailed global measurements may be infeasible. A valuable approach to such problems is to take measurements from selected locations within the network and then aggregate them to infer large-scale properties. One sees this notion applied in settings that range from Internet topology discovery tools to remote software agents that estimate the download times of popular web pages. Some of the most basic questions about this type of approach, however, are largely unresolved at an analytical level. How reliable are the results? How much does the choice of measurement locations affect the aggregate information one infers about the network?

We describe algorithms that yield provable guarantees for a particular problem of this type: detecting a network failure. Suppose we want to detect events of the following form in an n-node network: An adversary destroys up to k nodes or edges, after which two subsets of the nodes, each of size at least $\epsilon n$, are disconnected from one another. We call such an event an $(\epsilon,k)$-partition. One method for detecting such events would be to place "agents'' at a set D of nodes, and record a fault whenever two of them become separated from each other. To be a good detection set, D should become disconnected whenever there is an $(\epsilon,k)$-partition; in this way, it "witnesses'' all such events.

We show that every graph has a detection set of size polynomial in k and $\epsilon^{-1}$, and independent of the size of the graph itself. Moreover, random sampling provides an effective way to construct such a set. Our analysis establishes a connection between graph separators and the notion of VC-dimension, using techniques based on matchings and disjoint paths.

We consider the problem of searching a randomly growing graph by a random walk. In particular we consider two simple models of "web-graphs." Thus at each time step a new vertex is added and it is connected to the current graph by randomly chosen edges. At the same time a "spider" S makes a number of steps of a random walk on the current graph. The parameter we consider is the expected proportion of vertices that have been visited by S up to time t.

Random graph theory is used to examine the "small-world phenomenon"---any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order $\log n / \log \tilde d$ where $\tilde d$ is the weighted average of the sum of squares of the expected degrees. Of particular interest are power law random graphs in which the number of vertices of degree k is proportional to $1/k^{\beta}$ for some fixed exponent $\beta $. For the case of $\beta > 3$, we prove that the average distance of the power law graphs is almost surely of order $\log n / \log \tilde d$. However, many Internet, social, and citation networks are power law graphs with exponents in the range $2 < \beta < 3$ for which the power law random graphs have average distance almost surely of order $\log \log n$, but have diameter of order $\log n$ (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph, that we call the core, having $n^{c/\log \log n} $ vertices. Almost all vertices are within distance $\log \log n$ of the core although there are vertices at distance $\log n$ from the core.

In this paper, we describe six algorithmic problems that arise in web search engines and that are not or only partially solved: (1) Uniformly sampling of web pages; (2) modeling the web graph; (3) finding duplicate hosts; (4) finding top gainers and losers in data streams; (5) finding large dense bipartite graphs; and (6) understanding how eigenvectors partition the web.