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An Algorithmic Approach to Social Networks
 PHD THESIS AT MIT REFERENCES 118 SCIENCE AND ARTIFICIAL INTELLIGENCE LABORATORY
, 2005
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Nisan: Network information service for anonymization networks
 ACM CCS
, 2009
"... Network information distribution is a fundamental service for any anonymization network. Even though anonymization and information distribution about the network are two orthogonal issues, the design of the distribution service has a direct impact on the anonymization. Requiring each node to know ab ..."
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Network information distribution is a fundamental service for any anonymization network. Even though anonymization and information distribution about the network are two orthogonal issues, the design of the distribution service has a direct impact on the anonymization. Requiring each node to know about all other nodes in the network (as in Tor and AN.ON – the most popular anonymization networks) limits scalability and offers a playground for intersection attacks. The distributed designs existing so far fail to meet security requirements and have therefore not been accepted in real networks. In this paper, we combine probabilistic analysis and simulation to explore DHTbased approaches for distributing network information in anonymization networks. Based on our findings we introduce NISAN, a novel approach that tries to scalably overcome known security problems. It allows for selecting nodes uniformly at random from the full set of all available peers, while each of the nodes has only limited knowledge about the network. We show that our scheme has properties similar to a centralized directory in terms of preventing malicious nodes from biasing the path selection. This is done, however, without requiring to trust any third party. At the same time our approach provides high scalability and adequate performance. Additionally, we analyze different design choices and come up with diverse proposals depending on the attacker model. The proposed combination of security, scalability, and simplicity, to the best of our knowledge, is not available in any other existing network information distribution system.
The Rainbow Skip Graph: A FaultTolerant ConstantDegree Distributed Data Structure
"... We present a distributed data structure, which we call the rainbow skip graph. To our knowledge, this is the first peertopeer data structure that simultaneously achieves high faulttolerance, constantsized nodes, and fast update and query times for ordered data. It is a nontrivial adaptation of ..."
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We present a distributed data structure, which we call the rainbow skip graph. To our knowledge, this is the first peertopeer data structure that simultaneously achieves high faulttolerance, constantsized nodes, and fast update and query times for ordered data. It is a nontrivial adaptation of the SkipNet/skipgraph structures of Harvey et al. and Aspnes and Shah, so as to provide faulttolerance as these structures do, but to do so using constantsized nodes, as in the family tree structure of Zatloukal and Harvey. It supports successor queries on a set of n items using O(log n) messages with high probability, an improvement over the expected O(log n) messages of the family tree. Our structure achieves these results by using the following new constructs: • Rainbow connections: parallel sets of pointers between related components of nodes, so as to achieve good connectivity between “adjacent ” components, using constantsized nodes. • Hydra components: highlyconnected, highly faulttolerant components of constantsized nodes, which will contain relatively large connected subcomponents even under the failure of a constant fraction of the nodes in the component. We further augment the hydra components in the rainbow skip graph by using erasureresilient codes to ensure that any large subcomponent of nodes in a hydra component is sufficient to reconstruct all the data stored in that component. By carefully maintaining the size of related components and hydra components to be O(log n), we are able to achieve fast times for updates and queries in the rainbow skip graph. In addition, we show how to make the communication complexity for updates and queries be worst case, at the expense of more conceptual complexity and a slight degradation in the node congestion of the data structure.
A doubling dimension threshold Θ(log log n) for augmented graph navigability
 In 14th European Symposium on Algorithm (ESA), LNCS 4168
, 2006
"... Abstract. In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining wether such an augmentation i ..."
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Abstract. In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining wether such an augmentation is possible for all graphs. In this paper, we answer negatively to this question by exhibiting a threshold on the doubling dimension, above which an infinite family of graphs cannot be augmented to become navigable whatever the distribution of random edges is. Precisely, it was known that graphs of doubling dimension at most O(log log n) are navigable. We show that for doubling dimension ≫ log log n, an infinite family of graphs cannot be augmented to become navigable. Finally, we complete our result by studying the special case of square meshes, that we prove to always be augmentable to become navigable.
A New Perspective on the SmallWorld Phenomenon: Greedy Routing in TreeDecomposed Graphs
, 2005
"... Milgram’s experiment (1967) demonstrated that there are short chains of acquaintances between individuals, and that these chains can be discovered in a greedy manner. Kleinberg (2000) gave formal support to this socalled “small world phenomenon” by using meshes augmented with longrange links chose ..."
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Milgram’s experiment (1967) demonstrated that there are short chains of acquaintances between individuals, and that these chains can be discovered in a greedy manner. Kleinberg (2000) gave formal support to this socalled “small world phenomenon” by using meshes augmented with longrange links chosen randomly according to harmonic distributions. In this paper, we propose a new perspective on the small world phenomenon by considering arbitrary graphs augmented according to distributions guided by treedecompositions of the graphs. We show that, for any nnode graph G of treewidth ≤ k, there exists a treedecompositionbased distribution D such that greedy routing in the augmented graph (G, D) performs in O(k log 2 n) expected number of steps. We argue that augmenting a graph with longrange links chosen according to a treedecompositionbased distribution is plausible in the context of social networks. However, social networks can have unbounded treewidth. Nevertheless, we note that these networks have few long chordless cycles because of their high clustering coefficient. We prove that if G has chordality ≤ k, then the treedecompositionbased distribution D insures that greedy routing in (G, D) performs in O((k + log n)log n) expected number of steps. In particular, for any nnode graph G of chordality O(log n) (e.g., chordal graphs), greedy routing in the augmented graph (G, D) performs in O(log 2 n) expected number of steps. This latter result stresses the fact that our model may well explain why greedy routing is so efficient in social networks, such as observed in Milgram’s experiment.
D2B: a de Bruijn Based ContentAddressable Network
, 2005
"... We show that the de Bruijn graph is appropriate for maintaining dynamic connections, e.g., between the members of a P2P application who join and leave the system at their convenience. We describe the contentaddressable network D2B, based on an overlay network preserving de Bruijn connections dynami ..."
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Cited by 15 (1 self)
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We show that the de Bruijn graph is appropriate for maintaining dynamic connections, e.g., between the members of a P2P application who join and leave the system at their convenience. We describe the contentaddressable network D2B, based on an overlay network preserving de Bruijn connections dynamically, and on a distributed hash table (DHT) supporting efficient publish and search procedures. The overlay network has constant expected degree, and any publish or search operation in the DHT takes a logarithmic expected number of steps.
On Static and Dynamic Partitioning Behavior of LargeScale P2P Networks
, 2008
"... In this paper, we analyze the problem of network disconnection in the context of largescale P2P networks and understand how both static and dynamic patterns of node failure affect the resilience of such graphs. We start by applying classical results from random graph theory to show that a large va ..."
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In this paper, we analyze the problem of network disconnection in the context of largescale P2P networks and understand how both static and dynamic patterns of node failure affect the resilience of such graphs. We start by applying classical results from random graph theory to show that a large variety of deterministic and random P2P graphs almost surely (i.e., with probability 1 (1)) remain connected under random failure if and only if they have no isolated nodes. This simple, yet powerful, result subsequently allows us to derive in closedform the probability that a P2P network develops isolated nodes, and therefore partitions, under both types of node failure. We finish the paper by demonstrating that our models match simulations very well and that dynamic P2P systems are extremely resilient under node churn as long as the neighbor replacement delay is much smaller than the average user lifetime.
On the searchability of smallworld networks with arbitrary underlying structure
 IN PROCEEDINGS OF THE 42ND ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2010
"... Revisiting the“smallworld”experiments of the ’60s, Kleinberg observed that individuals are very effective at constructing short chains of acquaintances between any two people, and he proposed a mathematical model of this phenomenon. In this model, individuals are the nodes of a base graph, the squa ..."
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Revisiting the“smallworld”experiments of the ’60s, Kleinberg observed that individuals are very effective at constructing short chains of acquaintances between any two people, and he proposed a mathematical model of this phenomenon. In this model, individuals are the nodes of a base graph, the square grid, capturing the underlying structure of the social network; and this base graph is augmented with additional edges from each node to a few longrange contacts of this node, chosen according to some natural distancebased distribution. In this augmented graph, a greedy search algorithm takes only a polylogarithmic number of steps in the graph size. Following this work, several papers investigated the correlations between underlying structure and longrange connections that yield efficient decentralized search, generalizing Kleinberg’s results to broad classes of underlying structures, such as metrics of bounded doubling dimension, and minorexcluding graphs. We focus on the case of arbitrary base graphs. We show that for a simple longrange contact distribution consistent with empirical observations on social networks, a slight variation of greedy search, where the next hop is to a distant node only if it yields sufficient progress towards the target, requires n o(1) steps, where n is the number of nodes. Precisely, the expected number of steps for any source–target pair is at most 2 (log n)1 / 2 +o(1). This bound almost matches the best known lower bound of Ω(2 √ logn) steps, which applies to a general class of search algorithms. In the context of social networks, our result could be interpreted as: individuals may well be able to construct short chains between people regardless of the underlying structure of the social network.
On small world graphs in nonuniformly distributed key spaces
 In Proceedings of the 21st International Conference on Data Engineering Workshops (ICDEW
, 2005
"... In this paper we show that the topologies of most logarithmicstyle P2P systems like Pastry, Tapestry or PGrid resemble smallworld graphs. Inspired by Kleinberg’s smallworld model [6] we extend the model of building “routingefficient ” smallworld graphs and propose two new models. We show that ..."
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Cited by 14 (11 self)
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In this paper we show that the topologies of most logarithmicstyle P2P systems like Pastry, Tapestry or PGrid resemble smallworld graphs. Inspired by Kleinberg’s smallworld model [6] we extend the model of building “routingefficient ” smallworld graphs and propose two new models. We show that the graph, constructed according to our model for uniform key distribution and logarithmic outdegree, will have similar properties as the topologies of structured P2P systems with logarithmic outdegree. Moreover, we propose a novel model of building graphs which support uneven node distributions and preserves all desired properties of Kleinberg’s smallworld model. With such a model we are setting a reference base for nowadays emerging P2P systems that need to support uneven key distributions.