Showing papers by "Abraham D. Flaxman published in 2003"

A spectral technique for random satisfiable 3CNF formulas

Abstract: Let I be a random 3CNF formula generated by choosing a truth assignment φ for variables x1, ..., xn uniformly at random and including every clause with i literals set true by φ with probability pi, independently. We show that for any 0 ≤ η2, η3 ≤ 1 there is a constant dmin so that for all d ≥ dmin, a spectral algorithm similar to the graph coloring algorithm of [1] will find a satisfying assignment with high probability for p1 = d/n2, p2 = η2d/n2, and p3 = η3d/n2. Appropriately setting η2 and η3 yields natural distributions on satisfiable 3CNFs, not-all-equal-sat 3CNFs, and exactly-one-sat 3CNFs.

High degree vertices and eigenvalues in the preferential attachment graph

Abstract: The preferential attachment graph is a random graph formed by adding a new vertex at each time step, with a single edge which points to a vertex selected at random with probability proportional to its degree. Every m steps the most recently added m vertices are contracted into a single vertex, so at time t there are roughly t/m vertices and exactly t edges. This process yields a graph which has been proposed as a simple model of the world wide web [BA99]. For any constant k, let Δ1 ≥ Δ2 ≥ ⋯ ≥ Δ k be the degrees of the k highest degree vertices. We show that at time t, for any function f with f(t)→ ∞ as t→ ∞, \(\frac{t^{1/2}}{f(t)} \leq \Delta_1 \leq t^{1/2}f(t),\) and for i = 2,..., k, \(\frac{t^{1/2}}{f(t)} \leq \Delta_i \leq \Delta_{i-1} -- \frac{t^{1/2}}{f(t)},\) with high probability (whp). We use this to show that at time t the largest k eigenvalues of the adjacency matrix of this graph have λ k = (1± o(1))Δ k 1/2 whp.

High degree vertices and eigenvalues in the preferential attachment graph

Abstract: The preferential attachment graph is a random graph formed by adding a new vertex at each time step, with a single edge which points to a vertex selected at random with probability proportional to its degree. Every m steps the most recently added m vertices are contracted into a single vertex, so at time t there are roughly t/m vertices and exactly t edges. This process yields a graph which has been proposed as a simple model of the world wide web [BA99]. For any constant k, let Δ 1 ≥ Δ 2 ≥ ... ≥ Δ k be the degrees of the k highest degree vertices. We show that at time t, for any function f with f(t) → ∞ as t → ∞, t /f(t) ≤ Δ 1 ≤ t 1/2 f(t), and for i = 2,..., k, t /f(t) ≤ Δ i ≤ Δ i-1 - t /f(t), with high probability (whp). We use this to show that at time t the largest k eigenvalues of the adjacency matrix of this graph have λ k = (1 ± o(1))Δ 1/2 k whp.