On Asymptotic Expansion in the Random Allocation of Particles by Sets

We consider a scheme of equiprobable allocation of particles into cells by sets. The Edgeworth type asymptotic expansion in the local central limit theorem for a number of empty cells left after allocation of all sets of particles is derived.Comment: 15 page

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

Silence is Golden with High Probability: Maintaining a Connected Backbone in Wireless Sensor Networks

Reducing node energy consumption to extend network lifetime is a vital requirement in wireless sensor networks. In this paper, we present and analyze the energy consumption of a class of cell-based energy conservation protocols. The goal of our protocols is to alternately turn o/on the transceivers of the nodes, while maintaining a connected backbone of active nodes. The protocols presented in this paper are shown to be optimal, in the sense that they extend the network lifetime by a factor which is proportional to the node density

Random subgraphs of the 2D Hamming graph: the supercritical phase

We study random subgraphs of the 2-dimensional Hamming graph H(2, n), which is the Cartesian product of two complete graphs on n vertices. Let p be the edge probability, and write p = (1 + ε)/(2(n − 1)) for some ε ∈ R. In Borgs et al. (Random Struct Alg 27:137–184, 2005; Ann Probab 33:1886–1944, 2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2, n) for ε ≤ �V−1/3, where � > 0 is a constant and V = n2 denotes the number of vertices in H(2, n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when ε � (log V)1/3V−1/3, then the largest connected component has size close to 2εV with high probability.We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of p are supercritical. Barring the factor (log V)1/3, this identifies the size of the largest connected component all the way down to the critical p window

Extreme nash equilibria

Abstract. We study the combinatorial structure and computational complexity of extreme Nash equilibria, ones that maximize or minimize a certain objective function, in the context of a selfish routing game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel links, to control the routing of its own assigned traffic. InaNash equilibrium, each user routes its traffic on links that minimize its expected latency cost. Our structural results provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria and that under a certain condition, the social cost of any Nash equilibrium is within a factor of 6 + ε, of that of the fully mixed Nash equilibrium, assuming that link capacities are identical