Showing papers by "Abraham D. Flaxman published in 2006"

SybilGuard: defending against sybil attacks via social networks

Abstract: Peer-to-peer and other decentralized,distributed systems are known to be particularly vulnerable to sybil attacks. In a sybil attack,a malicious user obtains multiple fake identities and pretends to be multiple, distinct nodes in the system. By controlling a large fraction of the nodes in the system,the malicious user is able to "out vote" the honest users in collaborative tasks such as Byzantine failure defenses. This paper presents SybilGuard, a novel protocol for limiting the corruptive influences of sybil attacks.Our protocol is based on the "social network "among user identities, where an edge between two identities indicates a human-established trust relationship. Malicious users can create many identities but few trust relationships. Thus, there is a disproportionately-small "cut" in the graph between the sybil nodes and the honest nodes. SybilGuard exploits this property to bound the number of identities a malicious user can create.We show the effectiveness of SybilGuard both analytically and experimentally.

A Geometric Preferential Attachment Model of Networks II

Abstract: We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. The vertices of Gn are n sequentially generated points x 1, x 2, . . . , x n chosen uniformly at random from the unit sphere in ℝ3. After generating xt , we randomly connect that point to m points from those points in x 1, x 2, . . . , xt -1 that are within distance r of xt . Neighbors are chosen with probability proportional to their current degree, and a parameter a biases the choice towards self loops. We show that if m is sufficiently large, if r ≥ ln n/n 1/2-β for some constant β, and if α > 2, then with high probabilty (whp) at time n the number of vertices of degree k follows a power law with exponent α + 1. Unlike the preferential attachment graph, this geometric preferential attachment graph has small separators, similar to experimental observations of [Blandford et al. 03]. We further show that if m ≥ K ln n, for K sufficiently large, then Gn is connected and has diameter O(ln ...

Randomly coloring sparse random graphs with fewer colors than the maximum degree

Abstract: We analyze Markov chains for generating a random k-coloring of a random graph Gn,d/n. When the average degree d is constant, a random graph has maximum degree Θ(log n/log log n), with high probability. We show that, with high probability, an efficient procedure can generate an almost uniformly random k-coloring when k = Θ(log log n/log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but always require more colors than the maximum degree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

On the random 2-stage minimum spanning tree

Abstract: It is known [A. M. Frieze, Discrete Appl Math 10 (1985), 47–56] that if the edge costs of the complete graph Kn are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to $\zeta(3)=\sum_{i=1}^{\infty}i^{-3}$. Here we consider the following stochastic two-stage version of this optimization problem. There are two sets of edge costs cM: E → R and cT: E → R, called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs cM(e) and cT(e) are independent random variables, uniformly distributed in [0, 1]. The Monday costs are revealed first. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price cM(e), or to wait until its Tuesday price cT(e) appears. The set of edges XM bought on Monday is then completed by the set of edges XT bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost ζ(3)/2 + o(1). We show that, in the case of two-stage optimization, the expected value of the optimal cost exceeds ζ(3)/2 by an absolute constant e > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than α and completes them on Tuesday in an optimal way, and show that the optimal choice for α is α = 1/n with the expected cost ζ(3) - 1/2 + o(1). The threshold heuristic is shown to be sub-optimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning out-arborescence rooted at a fixed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 - 1/e + o(1). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

Average-case analysis for combinatorial problems

Abstract: This thesis considers the average case analysis of algorithms, focusing primarily on NP-hard combinatorial optimization problems. It includes a catalog of distributions frequently used in average-case analysis and a collection of mathematical tools that have been useful in studying these distributions. The bulk of the thesis consists of case-studies in average-case analysis of algorithms. Algorithms for 3-SAT, Subset Sum, Strong Connectivity, Stochastic Minimum Spanning Tree, and Uncapacitated Facility Location are analyzed on random instances.

First-passage percolation on a width-2 strip and the path cost in a VCG auction

Abstract: We study both the time constant for first-passage percolation, and the Vickery-Clarke-Groves (VCG) payment for the shortest path, on a width-2 strip with random edge costs. These statistics attempt to describe two seemingly unrelated phenomena, arising in physics and economics respectively: the first-passage percolation time predicts how long it takes for a fluid to spread through a random medium, while the VCG payment for the shortest path is the cost of maximizing social welfare among selfish agents. However, our analyses of the two are quite similar, and require solving (slightly different) recursive distributional equations. Using Harris chains, we can characterize distributions, not just expectations.

First-passage percolation on a width-2 strip and the path cost in a VCG auction

Abstract: We study both the time constant for first-passage percolation, and the Vickery-Clarke-Groves (VCG) payment for the shortest path, on a width-2 strip with random edge costs. These statistics attempt to describe two seemingly unrelated phenomena, arising in physics and economics respectively: the first-passage percolation time predicts how long it takes for a fluid to spread through a random medium, while the VCG payment for the shortest path is the cost of maximizing social welfare among selfish agents. However, our analyses of the two are quite similar, and require solving (slightly different) recursive distributional equations. Using Harris chains, we can characterize distributions, not just expectations.