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STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
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SIAM Journal on Computing
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STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
The diameter of long-range percolation clusters on finite cycles
Random Structures & Algorithms
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Random Structures & Algorithms
How many random edges make a dense graph Hamiltonian?
Random Structures & Algorithms
Compact representations of separable graphs
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
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Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
On Certain Connectivity Properties of the Internet Topology
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Adding random edges to dense graphs
Random Structures & Algorithms
Spectral techniques applied to sparse random graphs
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On smoothed analysis in dense graphs and formulas
Random Structures & Algorithms
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This paper studies the expansion properties of randomly perturbed graphs. These graphs are formed by, for example, adding a random $1{\text{-out}}$ or very sparse Erdős-Rényi graph to an arbitrary connected graph.The central results show that there exists a constant 驴such that when any connected n-vertex base graph $\bar{G}$ 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驴 Vwith $|S| \leq \frac34 n$. When $\bar{G}$ is perturbed by adding a random Erdős-Rényi graph, $\mathbb{G}_{n,\epsilon/n}$, 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 [Nature 292(1998), 440---442] and by Kleinberg in [Proc. of 32nd ACM Symposium on Theory of Computing(2000), 163---170]. 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 way of summing over subsets of vertices which allows an argument based on the First Moment Method to succeed.