Deviation inequality for monotonic Boolean functions with application to the number of k-cycles in a random graph

Authors:
Dmitry Panchenko
Affiliations:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts
Venue:
Random Structures & Algorithms
Year:
2004

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Abstract

Using Talagrand's concentration inequality on the discrete cube {0, 1}m we show that given a real-valued function Z(x) on {0, 1}m that satisfies certain monotonicity conditions one can control the deviations of Z(x) above its median by a local Lipschitz norm of Z at the point x. As one application, we obtain a deviation inequality for the number of k-cycles in a random graph.