This paper studies the expansion properties of randomly perturbed graphs. These graphs are formed by adding, for example, a random $1$-out or very sparse Erdős-Rényi graph to an arbitrary connected graph.

The central results show that there exists a constant $\delta$ such that when any connected $n$-vertex base graph $G_0$ is perturbed by adding a random $1$-out then, with high probability, the resulting graph has $e(S,\bar S) \geq \delta |S|$ for all $S\subseteq V$ with $|S| \leq \frac34 n$. When $G_0$ is perturbed by adding a random Erdős-Rényi graph, $\mathbb{G}_{n,e/n}$, the expansion of the perturbed graph depends on the structure of the base graph. A necessary condition for the base graph is given under which the resulting graph is an expander with high probability.

The proof techniques are also applied to study rapid mixing in the small-worlds graphs described by Watts and Strogatz in Watts and Strogatz, 1998 and by Kleinberg in Kleinberg, 2000. Analysis of Kleinberg's model shows that the graph stops being an expander exactly at the point where a decentralized algorithm is effective in constructing a short path.

The proofs of expansion rely on a method of summing over subsets of vertices that allows an argument based on the first-moment method to succeed.

We address the question of locality in random graphs. In particular, we study a geometric random tree model $\tau_{\alpha,n}$ which is a variant of the FKP model proposed in [Fabrikant et al., 2002]. We choose vertices $v_1, \ldots, v_n$ in some convex body uniformly and fix a point $\mathfrak{o}$. We then build our tree inductively, where at time $t$ we add an edge from $v_t$ to the vertex in $v_1, \ldots, v_{t-1}$ that minimizes $\alpha \| v_t - v_i \| + \| v_i - \mathfrak{o}\| $ for $i \lt t$, where $\alpha \gt 0$. We categorize an edge $v_i \to v_j$ in this graph as local or global depending on the edge length relative to the distance from $v_i$ to $\mathfrak{o}$. We study the extent to which the tree is composed of either global or local edges and, in particular, show that it undergoes a transition at $\alpha=1$.

PageRank is a popularity measure designed by Google to rank Web pages. Experiments confirm that PageRank values obey a power law with the same exponent as In-Degree values. This paper presents a novel mathematical model that explains this phenomenon. The relation between PageRank and In-Degree is modeled through a stochastic equation, which is inspired by the original definition of PageRank, and is analogous to the well-known distributional identity for the busy period in the $M/G/1$ queue. Further, we employ the theory of regular variation and Tauberian theorems to prove analytically that the tail distributions of PageRank and In-Degree differ only by a multiplicative constant, for which we derive a closed-form expression. Our analytical results are in good agreement with experimental data.

We present a new model for self-organizing networks such as the World Wide Web graph and analyze its limit behavior. In the model, new vertices are introduced over time that copy the neighborhood 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 $\rho $ 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 $\rho $. We describe certain infinite deterministic graphs that arise naturally from our model and that embed in all graphs generated by the model.

We examine the relationship of a graph $G$ and its random subgraphs, which are defined by independently choosing each edge with probability $p$. Suppose that $G$ has a spectral gap $\lambda$ (in terms of its normalized Laplacian) and minimum degree $d_{\min}$. Then we can show that a random subgraph of $G$ on $n$ vertices with edge-selection probability $p$ almost surely has as its spectral gap $\lambda - O\big(\sqrt{\frac{\log n }{pd_{\min}}}+\frac{(\log n)^{3/2}}{pd_{\min}(\log \log n)^{3/2}}\big)$.

PageRank is a key element in the success of search engines, allowing the display of the most relevant hits in the first screen of results. One key aspect that distinguishes PageRank from other prestige measures such as in-degree is its global nature. From the information provider perspective, this makes it difficult or even impossible to predict how their pages will be ranked. Consequently, a market has emerged for the optimization of search engine results. Here we study the accuracy with which PageRank can be approximated by in-degree, a local measure made freely available by search engines. Theoretical and empirical analyses lead us to conclude that, given the weak degree of correlations in the Web link graph, the approximation can be relatively accurate, giving service and information providers an effective new marketing tool.

We survey machine learning techniques to learn ranking functions for entities represented as feature vectors as well as nodes in a social network. In the feature-vector scenario, an entity, e.g., a document $x$, is mapped to a feature vector $\psi(x)\in\mathbb{R}^d$ in a $d$-dimensional space, and we have to search for a weight vector $\beta\in\mathbb{R}^d$. The ranking is then based on the values of $\beta\cdot\psi(x)$. This case corresponds to information retrieval in the ``vector space'' model. Training data consists of a partial order of preference among entities. We study probabilistic Bayesian and maximum-margin approaches to solving this problem, including recent efficient near-linear-time approximate algorithms. In the graph node-ranking scenario, we briefly review PageRank, generalize it to arbitrary Markov conductance matrices, and consider the problem of learning conductance parameters from partial orders between nodes. In another class of formulation, the graph does not establish PageRank or prestige-flow relationships between nodes, but encourages a certain smoothness between the scores (ranks) of neighboring nodes. Some of these techniques have been used by Web search companies with very large query logs. We review some of the issues that arise when applying the theory to practical systems. Finally, we review connections between the stability of a score/rank-learning algorithm and its power to generalize to unforeseen test data.