7 research outputs found
On the method of typical bounded differences
Concentration inequalities are fundamental tools in probabilistic
combinatorics and theoretical computer science for proving that random
functions are near their means. Of particular importance is the case where f(X)
is a function of independent random variables X=(X_1, ..., X_n). Here the well
known bounded differences inequality (also called McDiarmid's or
Hoeffding-Azuma inequality) establishes sharp concentration if the function f
does not depend too much on any of the variables. One attractive feature is
that it relies on a very simple Lipschitz condition (L): it suffices to show
that |f(X)-f(X')| \leq c_k whenever X,X' differ only in X_k. While this is easy
to check, the main disadvantage is that it considers worst-case changes c_k,
which often makes the resulting bounds too weak to be useful.
In this paper we prove a variant of the bounded differences inequality which
can be used to establish concentration of functions f(X) where (i) the typical
changes are small although (ii) the worst case changes might be very large. One
key aspect of this inequality is that it relies on a simple condition that (a)
is easy to check and (b) coincides with heuristic considerations why
concentration should hold. Indeed, given an event \Gamma that holds with very
high probability, we essentially relax the Lipschitz condition (L) to
situations where \Gamma occurs. The point is that the resulting typical changes
c_k are often much smaller than the worst case ones.
To illustrate its application we consider the reverse H-free process, where H
is 2-balanced. We prove that the final number of edges in this process is
concentrated, and also determine its likely value up to constant factors. This
answers a question of Bollob\'as and Erd\H{o}s.Comment: 25 page
-connectivity of the random graph with restricted budget
In this short note, we consider a graph process recently introduced by
Frieze, Krivelevich and Michaeli. In their model, the edges of the complete
graph are ordered uniformly at random and are then revealed consecutively
to a player called Builder. At every round, Builder must decide if they accept
the edge proposed at this round or not. We prove that, for every ,
Builder can construct a spanning -connected graph after
rounds by accepting edges with probability converging to 1 as
. This settles a conjecture of Frieze, Krivelevich and Michaeli.Comment: 15 pages, 2 figures, revised versio
A scaling limit for the length of the longest cycle in a sparse random graph
We discuss the length L-c,L-n of the longest cycle in a sparse random graph G(n,p), p = c/n, c constant. We show that for large c there exists a function f (c) such that L-c,L-n/n -> f (c) a.s. The function f (c) = 1 - Sigma(infinity)(k=1) p(k)(c)e(-kc) where pk(c) is a polynomial in c. We are only able to explicitly give the values p(1), p(2), although we could in principle compute any p(k). We see immediately that the length of the longest path is also asymptotic to f (c)n w.h.p