Motivated by the problem of graph structure compression under realistic source models, we study the symmetry behavior of preferential and uniform attachment graphs. These are two dynamic models of network growth in which new nodes attach to a constant number m of existing ones according to some attachment scheme. We prove symmetry results for m = 1 and 2, and we conjecture that for m > 3, both ...

The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are multiple players in each interested party. While all players in a party share the same set of resources, each player has her own preferences. Under additive valuations drawn random...

There exist an elliptical confidence region for the w, as described in the following theorem Theorem 1. ([2], Theorem 2) Assuming ‖w‖ ≤ √ d and ‖xt‖ ≤ √ d, with probably 1− δ, we have w ∈ Ct, where Ct = { z : ‖z − ŵt‖Mt ≤ 2 √ d log Td δ } For any x ∈ A, we define UCBx,t = maxz∈Ct z′x if w ∈ Ct (which holds with high probability). At each time, the UCB algorithm then simply picks the bandit with...

We show that provided log n/n ≤ p ≤ 1 − n−1/4 log n we can with high probability find a collection of bδ(G)/2c edge-disjoint Hamilton cycles in G ∼ Gn,p, plus an additional edge-disjoint matching of size bn/2c if δ(G) is odd. This is clearly optimal and confirms, for the above range of p, a conjecture of Frieze and Krivelevich.

In this paper we introduce a new sequence of quantities for random polytopes. Let KN = conv{X1, . . . ,XN} be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body K of R. We prove that the so-called k-th mean outer radius R̃k(KN ) has order max{ √ k, √ logN}LK with high probability if n ≤ N ≤ e √ . We also show that this is also the right or...

Let P(n, m) be the class of simple labelled planar graphs with n nodes and m edges, and let Rn,q be a graph drawn uniformly at random from P(n, bqnc). We show properties that hold with high probability (w.h.p.) for Rn,q when 1 < q < 3. For example, we show that Rn,q contains w.h.p. linearly many nodes of each given degree and linearly many node disjoint copies of each given fixed connected plan...

A common assumption in sensor networks is that the sensors are located according to a uniform random distribution. In this paper we show that uniform random points on the two dimensional unit square are almost a “grid”. In particular, for a synchronous geographic sensor network we show how to emulate any grid protocol on random sensor networks, with high probability. This suggests the following...

The retention of communication data has recently attracted much public interest, mostly because of the possibility of its misuse. In this paper, we present protocols that address the privacy concerns of the communication partners. Our data retention protocols store streams of encrypted data items, some of which may be flagged as critical (representing misbehavior). The frequent occurrence of cr...

We show that the absolute value of the determinant of a matrix with random independent (but not necessarily i.i.d.) entries is strongly concentrated around its mean. As an application, we show that Godsil–Gutman and Barvinok estimators for the permanent of a strictly positive matrix give subexponential approximation ratios with high probability. A positive answer to the main conjecture of the p...

A k-uniform, d-regular instance of Exact Cover is a family of m sets Fn,d,k = {Sj ⊆ {1, . . . , n}}, where each subset has size k and each 1 ≤ i ≤ n is contained in d of the Sj . It is satisfiable if there is a subset T ⊆ {1, . . . , n} such that |T ∩ Sj | = 1 for all j. Alternately, we can consider it a d-regular instance of Positive 1-in-k SAT, i.e., a Boolean formula with m clauses and n var...