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36
Complex Networks and Decentralized Search Algorithms
 In Proceedings of the International Congress of Mathematicians (ICM
, 2006
"... The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that ..."
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Cited by 73 (1 self)
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The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that capture some of the qualitative properties observed in largescale network data; such models have the potential to help us reason, at a general level, about the ways in which realworld networks are organized. We survey one particular line of network research, concerned with smallworld phenomena and decentralized search algorithms, that illustrates this style of analysis. We begin by describing a wellknown experiment that provided the first empirical basis for the "six degrees of separation" phenomenon in social networks; we then discuss some probabilistic network models motivated by this work, illustrating how these models lead to novel algorithmic and graphtheoretic questions, and how they are supported by recent empirical studies of large social networks.
Distance Estimation and Object Location via Rings of Neighbors
 In 24 th Annual ACM Symposium on Principles of Distributed Computing (PODC
, 2005
"... We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: lowstretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulationbased distance estimation [33]. Fo ..."
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Cited by 64 (4 self)
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We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: lowstretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulationbased distance estimation [33]. Focusing on metrics of low doubling dimension, we approach these problems with a common technique called rings of neighbors, which refers to a sparse distributed data structure that underlies all our constructions. Apart from improving the previously known bounds for these problems, our contributions include extending Kleinberg’s small world model to doubling metrics, and a short proof of the main result in Chan et al. [14]. Doubling dimension is a notion of dimensionality for general metrics that has recently become a useful algorithmic concept in the theoretical computer science literature. 1
Distributed Routing in SmallWorld Networks
, 2007
"... So called smallworld networks – clustered networks with small diameters – are thought to be prevalent in nature, especially appearing in people’s social interactions. Many models exist for this phenomenon, with some of the most recent explaining how it is possible to find short routes between nodes ..."
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Cited by 18 (3 self)
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So called smallworld networks – clustered networks with small diameters – are thought to be prevalent in nature, especially appearing in people’s social interactions. Many models exist for this phenomenon, with some of the most recent explaining how it is possible to find short routes between nodes in such networks. Searching for such routes, however, always depends on nodes knowing what their and their neighbors positions are relative to the destination. In real applications where one may wish to search a smallworld network, such as peertopeer computer networks, this cannot always be assumed to be true. We propose and explore a method of routing that does not depend on such knowledge, and which can be implemented in a completely distributed way without any global elements. The Markov Chain MonteCarlo based algorithm takes only a graph as input, and requires no further information about the nodes themselves. The proposed method is tested against simulated and real world data.
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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Cited by 17 (7 self)
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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.
An Algorithmic Approach to Social Networks
 PhD thesis at MIT References 118 Science and Artificial Intelligence Laboratory
, 2005
"... ..."
Navigating LowDimensional and Hierarchical Population Networks
 In 14th European Symposium on Algorithm (ESA), LNCS 4168
, 2006
"... 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 p ..."
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Cited by 9 (4 self)
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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 rankbased 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 treebased metrics with varying population distributions. 1
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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Cited by 7 (0 self)
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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. Research supported in part by the ANR projects AL
Proximity Breeds Danger: Emerging Threats in Metroarea Wireless Networks
 In Proceedings of the 16 th USENIX Security Symposium
, 2007
"... The growing popularity of wireless networks and mobile devices is starting to attract unwanted attention especially as potential targets for malicious activities reach critical mass. In this study, we try to quantify the threat from largescale distributed attacks on wireless networks, and, more spe ..."
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Cited by 6 (0 self)
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The growing popularity of wireless networks and mobile devices is starting to attract unwanted attention especially as potential targets for malicious activities reach critical mass. In this study, we try to quantify the threat from largescale distributed attacks on wireless networks, and, more specifically, wifi networks in densely populated metropolitan areas. We focus on three likely attack scenarios: “wildfire ” worms that can spread contagiously over and across wireless LANs, coordinated citywide phishing campaigns based on wireless spoofing, and rogue systems for compromising location privacy in a coordinated fashion. The first attack illustrates how dense wifi deployment may provide opportunities for attackers who want to quickly compromise large numbers
Searching in a Small World
, 2005
"... The smallworld phenomenon, that the world’s social network is tightly connected, and that any two people can be linked by a short chain of friends, has long been a subject of interest. Famously, the psychologist Stanley Milgram performed an experiment where he asked people to deliver a letter to a ..."
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Cited by 5 (0 self)
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The smallworld phenomenon, that the world’s social network is tightly connected, and that any two people can be linked by a short chain of friends, has long been a subject of interest. Famously, the psychologist Stanley Milgram performed an experiment where he asked people to deliver a letter to a stranger by forwarding it to an acquaintance, who could forward it to one his acquaintances, and so on until the destination was reached. The results seemed to confirm that the smallworld phenomenon is real. Recently it has been shown by Jon Kleinberg that in order to search in a network, that is to actually find the short paths in the manner of the Milgram experiment, a very special type of a graph model is needed. In this thesis, we present two ideas about searching in the small world stemming from Kleinberg’s results. In the first we study the formation of networks of this type, attempting to see why the kind