David Liben-Nowell

Chord: a scalable peer-to-peer lookup protocol for Internet applications

A fundamental problem that confronts peer-to-peer applications is the efficient location of the node that stores a desired data item. This paper presents Chord, a distributed lookup protocol that addresses this problem. Chord provides support for just one operation: given a key, it maps the key onto a node. Data location can be easily implemented on top of Chord by associating a key with each data item, and storing the key/data pair at the node to which the key maps. Chord adapts efficiently as nodes join and leave the system, and can answer queries even if the system is continuously changing. Results from theoretical analysis and simulations show that Chord is scalable: Communication cost and the state maintained by each node scale logarithmically with the number of Chord nodes.

Analysis of the evolution of peer-to-peer systems

In this paper, we give a theoretical analysis of peer-to-peer (P2P) networks operating in the face of concurrent joins and unexpected departures. We focus on Chord, a recently developed P2P system that implements a distributed hash table abstraction, and study the process by which Chord maintains its distributed state as nodes join and leave the system. We argue that traditional performance measures based on run-time are uninformative for a continually running P2P network, and that the rate at which nodes in the network need to participate to maintain system state is a more useful metric. We give a general lower bound on this rate for a network to remain connected, and prove that an appropriately modified version of Chords maintenance rate is within a logarithmic factor of the optimum rate.

Tracing information flow on a global scale using Internet chain-letter data

Although information, news, and opinions continuously circulate in the worldwide social network, the actual mechanics of how any single piece of information spreads on a global scale have been a mystery. Here, we trace such information-spreading processes at a person-by-person level using methods to reconstruct the propagation of massively circulated Internet chain letters. We find that rather than fanning out widely, reaching many people in very few steps according to “small-world” principles, the progress of these chain letters proceeds in a narrow but very deep tree-like pattern, continuing for several hundred steps. This suggests a new and more complex picture for the spread of information through a social network. We describe a probabilistic model based on network clustering and asynchronous response times that produces trees with this characteristic structure on social-network data.

Tetris is hard, even to approximate

In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p1-Ɛ, when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of 2-Ɛ, for any Ɛ > 0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piecesets.

TETRIS IS HARD, EVEN TO APPROXIMATE

In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all filled squares above it drop by one row. We prove that in the offline version of Tetris, it is -complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p1-e, when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of 2-e, for any e>0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piece sets.

Best Friends Alliances, Friend Ranking, and the MySpace Social Network

Like many topics of psychological research, the explanation for friendship is at once intuitive and difficult to address empirically. These difficulties worsen when one seeks, as we do, to go beyond “obvious” explanations (“humans are social creatures”) to ask deeper questions, such as “What is the evolved function of human friendship?” In recent years, however, a new window into human behavior has opened as a growing fraction of people’s social activity has moved online, leaving a wealth of digital traces behind. One example is a feature of the MySpace social network that allows millions of users to rank their “Top Friends.” In this study, we collected over 10 million people’s friendship decisions from MySpace to test predictions made by hypotheses about human friendship. We found particular support for the alliance hypothesis, which holds that human friendship is caused by cognitive systems that function to create alliances for potential disputes. Because an ally’s support can be undermined by a stronger outside relationship, the alliance model predicts that people will prefer partners who rank them above other friends. Consistent with the alliance model, we found that an individual’s choice of best friend in MySpace is strongly predicted by how partners rank that individual.

Computing Shapley Value in Supermodular Coalitional Games

Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i’s share is precisely equal to i’s expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), and give a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ±e) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ±e)-approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic.

Navigating low-dimensional and hierarchical population networks

Social networks are navigable small worlds, in which two arbitrary people are likely connected by a short path of intermediate friends that can be found by a decentralized routing algorithm using only local information. We develop a model of social networks based on an arbitrary metric space of points, with population density varying across the points. We consider rank-based friendships, where the probability that person u befriends person v is inversely proportional to the number of people who are closer to u than v is. Our main result is that greedy routing can find a short path (of expected polylogarithmic length) from an arbitrary source to a randomly chosen target, independent of the population densities, as long as the doubling dimension of the metric space of locations is low. We also show that greedy routing finds short paths with good probability in tree-based metrics with varying population distributions.

On the structure of syntenic distance.

This paper examines some of the rich structure of the syntenic distance model of evolutionary distance, introduced by Ferretti et al. (1996). The syntenic distance between two genomes is the minimum number of fissions, fusions, and translocations required to transform one into the other, ignoring gene order within chromosomes. We prove that the previously unanalyzed algorithm given by Ferretti et al. (1996) is a 2-approximation and no better, and that, further, it always outperforms the algorithm presented by DasGupta et al. (1998). We also prove the same results for an improved version of the Ferretti et al. algorithm. We then prove a number of properties which give insight into the structure of optimal move sequences. We give instances in which any move sequence working solely within connected components is nearly twice optimal and prove a general lower bound based on the spread of genes from each chromosome. We then prove a monotonicity property for the syntenic distance, and bound the difficulty of the hardest instance of any size. We discuss the results of implementing these algorithms and testing them on real and simulated synteny data.

On threshold behavior in query incentive networks

Motivated by the role of incentives in large-scale information systems, Kleinberg and Raghavan (FOCS 2005) studied strategic games in decentralized information networks. Given a branching process that specifies the network, the rarity of answers to a specific question, and a desired probability of success, how much reward does the root node need to offer so that it receives an answer with this probability, when all of the nodes are playing strategically? For a specific family of branching processes and a constant failure probability, they showed that the reward function exhibited a threshold behavior that depends on the branching parameter b. In this paper we study two factors that can contribute to this transition behavior, namely, the branching process itself and the failure probability. On one hand we show that the threshold behavior is robust with respect to the branching process: for all branching processes and any constant failure probability, if b > 2 then the required reward is linear in the expected depth of the search tree, and if b < 2 then the required reward is exponential in that depth. On the other hand we show that the threshold behavior is fragile with respect to the failure probability σ: if σ is inversely polynomial in the rarity of the answer, then all branching processes require rewards exponential in the depth of the search tree.