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60
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
Object Location Using Path Separators
, 2006
"... We study a novel separator property called kpath separable. Roughly speaking, a kpath separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is kpath separable. We then show that kpat ..."
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Cited by 35 (11 self)
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We study a novel separator property called kpath separable. Roughly speaking, a kpath separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is kpath separable. We then show that kpath separable graphs can be used to solve several object location problems: (1) a smallworldization with an average polylogarithmic number of hops; (2) an (1 + ε)approximate distance labeling scheme with O(log n) space labels; (3) a stretch(1 + ε) compact routing scheme with tables of polylogarithmic space; (4) an (1+ε)approximate distance oracle with O(n log n) space and O(log n) query time. Our results generalizes to much wider classes of weighted graphs, namely to boundeddimension isometric sparable graphs.
Stochastic kronecker graphs
 Proceedings of the 5th Workshop on Algorithms and Models for the WebGraph
, 2007
"... A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for largescale realworld networks such as the web. This model simultaneously captures several wellknown properties of realworld networks; in particular, it gives rise to a he ..."
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Cited by 20 (2 self)
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A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for largescale realworld networks such as the web. This model simultaneously captures several wellknown properties of realworld networks; in particular, it gives rise to a heavytailed degree distribution, has a low diameter, and obeys the densification power law. Most properties of Kronecker products of graphs (such as connectivity and diameter) are only rigorously analyzed in the deterministic case. In this paper, we study the basic properties of stochastic Kronecker products based on an initiator matrix of size two (which is the case that is shown to provide the best fit to many realworld networks). We will show a phase transition for the emergence of the giant component and another phase transition for connectivity, and prove that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm. 1
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.
Universal augmentation schemes for network navigability: overcoming the √ nbarrier, in "Proceedings of the nineteenth annual ACM symposium on parallelism and architectures
 ÉtatsUnis, ACM
"... Augmented graphs were introduced for the purpose of analyzing the ”six degrees of separation between individuals” observed experimentally by the sociologist Standley Milgram in the 60’s. Formally, an augmented graph is a pair (G, ϕ) where G is a graph, and ϕ is a collection of probability distributi ..."
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Cited by 13 (6 self)
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Augmented graphs were introduced for the purpose of analyzing the ”six degrees of separation between individuals” observed experimentally by the sociologist Standley Milgram in the 60’s. Formally, an augmented graph is a pair (G, ϕ) where G is a graph, and ϕ is a collection of probability distributions {ϕu, u ∈ V (G)}. Every node u ∈ V (G) is given an extra link, called a long range link, pointing to some node v, called the long range contact of u. The head v of this link is chosen at random by Pr{u → v} = ϕu(v). In augmented graphs, greedy routing is the oblivious routing process in which every intermediate node chooses among all its neighbors (including its long range contact) the one that is closest to the target according to the distance measured in the underlying graph G, and forwards to it. Roughly, augmented graphs aim at modeling the structure of social networks, while greedy routing aims at modeling the searching procedure applied in Milgram’s experiment. Our objective is to design efficient universal augmentation schemes, i.e., augmentation schemes that give to any graph G a collection of probability distributions ϕ such that greedy routing in (G, ϕ) is fast. It is known that the uniform scheme is a universal scheme ensuring that, for any nnode ϕunif This work was partially done while Zvi Lotker was visiting LRI
On the Windfall of Friendship: Inoculation Strategies on Social Networks
 EC'08
, 2008
"... This paper studies a virus inoculation game on social networks. A framework is presented which allows the measuring of the windfall of friendship, i.e., how much players benefit if they care about the welfare of their direct neighbors in the social network graph compared to purely selfish environmen ..."
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Cited by 12 (2 self)
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This paper studies a virus inoculation game on social networks. A framework is presented which allows the measuring of the windfall of friendship, i.e., how much players benefit if they care about the welfare of their direct neighbors in the social network graph compared to purely selfish environments. We analyze the corresponding equilibria and show that the computation of the worst and best Nash equilibrium is N Phard. Intriguingly, even though the windfall of friendship can never be negative, the social welfare does not increase monotonically with the extent to which players care for each other. While these phenomena are known on an anecdotal level, our framework allows us to quantify these effects analytically.
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
Pragmatic Evaluation of Folksonomies
"... Recently, a number of algorithms have been proposed to obtain hierarchical structures — socalled folksonomies — from social tagging data. Work on these algorithms is in part driven by a belief that folksonomies are useful for tasks such as: (a) Navigating social tagging systems and (b) Acquiring se ..."
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Cited by 9 (6 self)
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Recently, a number of algorithms have been proposed to obtain hierarchical structures — socalled folksonomies — from social tagging data. Work on these algorithms is in part driven by a belief that folksonomies are useful for tasks such as: (a) Navigating social tagging systems and (b) Acquiring semantic relationships between tags. While the promises and pitfalls of the latter have been studied to some extent, we know very little about the extent to which folksonomies are pragmatically useful for navigating social tagging systems. This paper sets out to address this gap by presenting and applying a pragmatic framework for evaluating folksonomies. We model exploratory navigation of a tagging system as decentralized search on a network of tags. Evaluation is based on the fact that the performance of a decentralized search algorithm depends on the quality of the background knowledge used. The key idea of
Center of attention: How facebook users allocate attention across friends
 In International Conference on Weblogs and Social Media (ICWSM
, 2011
"... An individual’s personal network — their set of social contacts — is a basic object of study in sociology. Studies of personal networks have focused on their size (the number of contacts) and their composition (in terms of categories such as kin and coworkers). Here we propose a new measure for the ..."
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Cited by 9 (1 self)
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An individual’s personal network — their set of social contacts — is a basic object of study in sociology. Studies of personal networks have focused on their size (the number of contacts) and their composition (in terms of categories such as kin and coworkers). Here we propose a new measure for the analysis of personal networks, based on the way in which an individual divides his or her attention across contacts. This allows us to contrast people who focus a large fraction of their interactions on a small set of close friends with people who disperse their attention more widely. Using data from Facebook, we find that this balance of attention is a relatively stable property of an individual over time, and that it displays interesting variation across both different groups of people and different modes of interaction. In particular, activities based on communication involve a much higher focus of attention than activities based simply on observation, and these two modalities also exhibit different forms of variation in interaction patterns both within and across groups. Finally, we contrast the amount of attention paid by individuals to their most frequent contacts with the rate of change in the actual identities of these contacts, providing a measure of churn for this set. 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