The diameter of a cycle plus a random matching
SIAM Journal on Discrete Mathematics
The Structure and Dynamics of Networks: (Princeton Studies in Complexity)
The Structure and Dynamics of Networks: (Princeton Studies in Complexity)
Mathematical aspects of mixing times in Markov chains
Foundations and Trends® in Theoretical Computer Science
Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics)
Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics)
The diameter of sparse random graphs
Combinatorics, Probability and Computing
Maximum hitting time for random walks on graphs
Random Structures & Algorithms
Estimating clustering coefficients and size of social networks via random walk
Proceedings of the 22nd international conference on World Wide Web
Binary Opinion Dynamics with Stubborn Agents
ACM Transactions on Economics and Computation
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"Small worlds" are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman--Watts small world, the mixing time is of order log2 n. This confirms a prediction of Richard Durrett, who proved a lower bound of order log2 n and an upper bound of order log3 n.