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The effect of powerlaw degrees on the navigability of small worlds
 Proceedings of the 28th ACM Symposium on Principles of Distributed Computing
, 2009
"... We analyze decentralized routing in smallworld networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variation of Kleinberg’s augmentedlattice model (STOC 2000), where the number of longrange contacts for each node is drawn from a p ..."
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We analyze decentralized routing in smallworld networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variation of Kleinberg’s augmentedlattice model (STOC 2000), where the number of longrange contacts for each node is drawn from a powerlaw distribution. This model is motivated by the experimental observation that many “realworld” networks have powerlaw degrees. In such networks, the exponent α of the power law is typically between 2 and 3. We prove that, in our model, for this range of values, 2 < α < 3, the expected number of steps of greedy routing from any source to any target is O(log α−1 n) steps. This bound is tight in a strong sense. Indeed, we prove that the expected number of steps of greedy routing for a uniformlyrandom pair of source–target nodes is Ω(log α−1 n) steps. We also show that for α < 2 or α ≥ 3, greedy routing performs in Θ(log 2 n) expected steps, and for α = 2, Θ(log 1+ε n) expected steps are required, where 1/3 ≤ ε ≤ 1/2. To the best of our knowledge, these results are the first to formally quantify the effect of the powerlaw degree distribution on the navigability of small worlds. Moreover, they show that this effect is significant. In particular, as α approaches 2 from above, the expected number of steps of greedy routing in the augmented lattice with powerlaw degrees approaches the squareroot of the expected number of steps of greedy routing in the augmented lattice with fixed degrees, although both networks have the same average degree.
Small Worlds as Navigable Augmented Networks — Model, Analysis, and Validation —
"... Abstract. The small world phenomenon, a.k.a. the six degree of separation between individuals, was identified by Stanley Milgram at the end of the 60s. Milgram experiment demonstrated that letters from arbitrary sources and bound to an arbitrary target can be transmitted along short chains of closel ..."
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Abstract. The small world phenomenon, a.k.a. the six degree of separation between individuals, was identified by Stanley Milgram at the end of the 60s. Milgram experiment demonstrated that letters from arbitrary sources and bound to an arbitrary target can be transmitted along short chains of closely related individuals, based solely on some characteristics of the target (professional occupation, state of leaving, etc.). In his paper on small world navigability, Jon Kleinberg modeled this phenomenon in the framework of augmented networks, and analyzed the performances of greedy routing in augmented multidimensional meshes. This paper objective is to survey the results that followed up Kleinberg seminal work, including results about: – extensions of the augmented network model, and variants of greedy routing, – designs of polylognavigable graph classes, – the quest for universal augmentation schemes, and
Networks Become Navigable as Nodes Move and Forget
, 2008
"... Abstract. We propose a dynamic process for network evolution, aiming at explaining the emergence of the small world phenomenon, i.e., the statistical observation that any pair of individuals are linked by a short chain of acquaintances computable by a simple decentralized routing algorithm, known as ..."
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Abstract. We propose a dynamic process for network evolution, aiming at explaining the emergence of the small world phenomenon, i.e., the statistical observation that any pair of individuals are linked by a short chain of acquaintances computable by a simple decentralized routing algorithm, known as greedy routing. Our model is based on the combination of two dynamics: a random walk (spatial) process, and an harmonic forgetting (temporal) process. Both processes reflect natural behaviors of the individuals, viewed as nodes in the network of interindividual acquaintances. We prove that, in kdimensional lattices, the combination of these two processes generates longrange links mutually independently distributed as a kharmonic distribution. We analyze the performances of greedy routing at the stationary regime of our process, and prove that the expected number of steps for routing from any source to any target in any multidimensional lattice is a polylogarithmic function of the distance between the two nodes in the lattice. Up to our knowledge, these results are the first formal proof that navigability in small worlds can emerge from a dynamic process for network evolution. Our dynamica process can find practical applications to the design of spatial gossip and resource location protocols.
Lowdistortion inference of latent similarities from a multiplex social network
, 2012
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Depth of field and cautiousgreedy routing in social networks
 In Proceedings of the 18th International Symposium on Algorithms and Computation
, 2007
"... Social networks support efficient decentralized search: people can collectively construct short paths to a specified target in the network. Rankbased friendship—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—is ..."
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Social networks support efficient decentralized search: people can collectively construct short paths to a specified target in the network. Rankbased friendship—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—is an empirically validated model of acquaintanceship that provably results in efficient decentralized search via greedy routing, even in networks with variable population densities. In this paper, we introduce cautiousgreedy routing, a variant of greedy that avoids taking large jumps unless they make substantial progress towards the target. Our main result is that cautiousgreedy routing finds a path of short expected length from an arbitrary source to a randomly chosen target, independent of the population densities. To quantify the expected length of the path, we define the depth of field of a metric space, a new quantity that intuitively measures the “width ” of directions that leave a point in the space. Our main result is that cautiousgreedy routing finds a path of expected length O(log 2 n) in nperson networks that have aspect ratio polynomial in n, bounded doubling dimension, and bounded depth of field. Specifically, in kdimensional grids under Manhattan distance with arbitrary population densities, the O(log 2 n) expected path length that we achieve with the cautiousgreedy algorithm improves the best previous bound of O(log 3 n) with greedy routing. 1
Recovering the long range links in Augmented graphs
 n o RR6197, Institut National de Recherche en Informatique et Automatique (INRIA), 2007, https:// hal.inria.fr/inria00147536. References in notes
"... Abstract. The augmented graph model, as introduced by Kleinberg (STOC 2000), is an appealing model for analyzing navigability in social networks. Informally, this model is defined by a pair (H, ϕ), where H is a graph in which internode distances are supposed to be easy to compute or at least easy t ..."
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Abstract. The augmented graph model, as introduced by Kleinberg (STOC 2000), is an appealing model for analyzing navigability in social networks. Informally, this model is defined by a pair (H, ϕ), where H is a graph in which internode distances are supposed to be easy to compute or at least easy to estimate. This graph is ”augmented ” by links, called longrange links, which are selected according to the probability distribution ϕ. The augmented graph model enables the analysis of greedy routing in augmented graphs G ∈ (H, ϕ). In greedy routing, each intermediate node handling a message for a target t selects among all its neighbors in G the one that is the closest to t in H and forwards the message to it. This paper addresses the problem of checking whether a given graph G is an augmented graph. It answers part of the questions raised by Kleinberg in his Problem 9 (Int. Congress of Math. 2006). More precisely, given G ∈ (H, ϕ), we aim at extracting the base graph H and
Deterministic Decentralized Search in Random Graphs
"... We study a general framework for decentralized search in random graphs. Our main focus is on deterministic memoryless search algorithms that use only local information to reach their destination in a bounded number of steps in expectation. This class includes (with small modifications) the search ..."
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We study a general framework for decentralized search in random graphs. Our main focus is on deterministic memoryless search algorithms that use only local information to reach their destination in a bounded number of steps in expectation. This class includes (with small modifications) the search algorithms used in Kleinberg’s pioneering work on longrange percolation graphs and hierarchical network models. We give a characterization of searchable graphs in this model, and use this characterization to prove a monotonicity property for searchability.
Wayfinding in Social Networks
"... With the recent explosion of popularity of commercial socialnetworking sites like Facebook and MySpace, the size of social networks that can be studied scientifically has passed from the scale traditionally studied by sociologists and anthropologists to the scale of networks more typically studied ..."
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With the recent explosion of popularity of commercial socialnetworking sites like Facebook and MySpace, the size of social networks that can be studied scientifically has passed from the scale traditionally studied by sociologists and anthropologists to the scale of networks more typically studied by computer scientists. In this chapter, I will highlight a recent line of computational research into the modeling and analysis of the smallworld phenomenon—the observation that typical pairs of people in a social network are connected by very short chains of intermediate friends—and the ability of members of a large social network to collectively find efficient routes to reach individuals in the network. I will survey several recent mathematical models of social networks that account for these phenomena, with an emphasis both on provable properties of these socialnetwork models and on the empirical validation of the models against real largescale socialnetwork data.
Augmented Graph Models for SmallWorld Analysis with Geographical Factors
"... Smallworld properties, such as smalldiameter and clustering, and the powerlaw property are widely recognized as common features of largescale realworld networks. Recent studies also notice two important geographical factors which play a significant role, particularly in Internet related setting ..."
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Smallworld properties, such as smalldiameter and clustering, and the powerlaw property are widely recognized as common features of largescale realworld networks. Recent studies also notice two important geographical factors which play a significant role, particularly in Internet related setting. These two are the distancebias tendency (links tend to connect to closer nodes) and the property of bounded growth in localities. However, existing formal models for realworld complex networks usually don’t fully consider these geographical factors. We describe a flexible approach using a standard augmented graph model (e.g. Watt and Strogatz’s [33], and Kleinberg’s [20] models) and present important initial results on a refined model where we focus on the smalldiameter characteristic and the above two geographical factors. We start with a general model where an arbitrary initial nodeweighted graph H is augmented with additional random links specified by a generic ‘distribution rule ’ τ and the weights of nodes in H. We consider a refined setting where the initial graph H is associated with a growthbounded metric, and τ has a distancebias characteristic, specified by parameters as follows. The base graph H has neighborhood growth bounded from both below and above, specified by parameters β1, β2> 0. (These parameters can be thought of as the dimension of the graph, e.g. β1 = 2 and β2 = 3 for a graph modeling a setting with nodes in both 2D and 3D settings.) That is 2β1 Nu(2r) ≤ Nu(r) ≤ 2β2 where Nu(r) is the number of nodes v within metric distance r from u: d(u, v) ≤ r. When we add random links using distribution τ, this distribution is specified by parameter α> 0 such that the probability that 1 a link from u goes to v � = u is ∝ dα (u,v). We show which parameters produce a smalldiameter graph and how the diameter changes depending on the relationship between the distancebias parameter α and the two bounded growth parameters β1, β2> 0. In particular, for most connected base graphs, the diameter of our aug
Derterministic decentralized search in random graphs
, 2007
"... We study a general framework for decentralized search in random graphs. Our main focus is on deterministic memoryless search algorithms that use only local information to reach their destination in a bounded number of steps in expectation. This class includes (with small modifications) the search al ..."
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We study a general framework for decentralized search in random graphs. Our main focus is on deterministic memoryless search algorithms that use only local information to reach their destination in a bounded number of steps in expectation. This class includes (with small modifications) the search algorithms used in Kleinberg’s pioneering work on longrange percolation graphs and hierarchical network models. We give a characterization of searchable graphs in this model, and use this characterization to prove a monotonicity property for searchability.