The lower tail: Poisson approximation revisited

The well-known "Janson's inequality" gives Poisson-like upper bounds for the lower tail probability \Pr(X \le (1-\eps)\E X) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case \eps=1) of Janson, Luczak and Rucinski.Comment: 21 page

Block Sensitivity of Minterm-Transitive Functions

Boolean functions with symmetry properties are interesting from a complexity theory perspective; extensive research has shown that these functions, if nonconstant, must have high `complexity' according to various measures. In recent work of this type, Sun gave bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f) = Omega(N^{1/3}), and that there exists such a function for which bs(f) = O(N^{3/7}ln N). His example function belongs to a subclass of transitively invariant functions called the minterm-transitive functions (defined in earlier work by Chakraborty). We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy bs(f) = Omega(N^{3/7}). Thus Sun's example function has nearly minimal block sensitivity for this subclass. Second, we give an improved example: a minterm-transitive function for which bs(f) = O(N^{3/7}ln^{1/7}N).Comment: 10 page

Moment inequalities for functions of independent random variables

A general method for obtaining moment inequalities for functions of independent random variables is presented. It is a generalization of the entropy method which has been used to derive concentration inequalities for such functions [Boucheron, Lugosi and Massart Ann. Probab. 31 (2003) 1583-1614], and is based on a generalized tensorization inequality due to Latala and Oleszkiewicz [Lecture Notes in Math. 1745 (2000) 147-168]. The new inequalities prove to be a versatile tool in a wide range of applications. We illustrate the power of the method by showing how it can be used to effortlessly re-derive classical inequalities including Rosenthal and Kahane-Khinchine-type inequalities for sums of independent random variables, moment inequalities for suprema of empirical processes and moment inequalities for Rademacher chaos and U-statistics. Some of these corollaries are apparently new. In particular, we generalize Talagrand's exponential inequality for Rademacher chaos of order 2 to any order. We also discuss applications for other complex functions of independent random variables, such as suprema of Boolean polynomials which include, as special cases, subgraph counting problems in random graphs.Comment: Published at http://dx.doi.org/10.1214/009117904000000856 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the limiting distribution of the metric dimension for random forests

The metric dimension of a graph G is the minimum size of a subset S of vertices of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension for two different models of random forests, in each case obtaining normal limit distributions for this parameter.Comment: 22 pages, 5 figure

Identifying and locating-dominating codes in (random) geometric networks

International audienceWe model a problem about networks built from wireless devices using identifying and locating-dominating codes in unit disk graphs. It is known that minimising the size of an identifying code is NP-complete even for bipartite graphs. First, we improve this result by showing that the problem remains NP-complete for bipartite planar unit disk graphs. Then, we address the question of the existence of an identifying code for random unit disk graphs. We derive the probability that there exists an identifying code as a function of the radius of the disks and we find that for all interesting ranges of r this probability is bounded away from one. The results obtained are in sharp contrast with those concerning random graphs in the Erdos-Renyi model. Another well-studied class of codes are locating-dominating codes, which are less demanding than identifying codes. A locating-dominating code always exists, but minimising its size is still NP-complete in general. We extend this result to our setting by showing that this question remains NP-complete for arbitrary planar unit disk graphs. Finally, we study the minimum size of such a code in random unit disk graphs, and we prove that with probability tending to one, it is of size (n/r)^{2/3+o(1)} if r ≤ √2/2 − ε is chosen such that nr^2 → ∞ and of size n^{1+o(1)} if nr^2 ≪ ln n