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16
Geographic routing in social networks
, 2005
"... We live in a “small world,” where two arbitrary people are likely connected by a short chain of intermediate friends. With scant information about a target individual, people can successively forward a message along such a chain. Experimental studies have verified this property in real social networ ..."
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Cited by 232 (10 self)
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We live in a “small world,” where two arbitrary people are likely connected by a short chain of intermediate friends. With scant information about a target individual, people can successively forward a message along such a chain. Experimental studies have verified this property in real social networks, and theoretical models have been advanced to explain it. However, existing theoretical models have not been shown to capture behavior in realworld social networks. Here we introduce a richer model relating geography and socialnetwork friendship, in which the probability of befriending a particular person is inversely proportional to the number of closer people. In a large social network, we show that one third of the friendships are independent of geography, and the remainder exhibit the proposed relationship. Further, we prove analytically that short chains can be discovered in every network exhibiting the relationship.
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 111 (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.
The Internet ASLevel Topology: Three Data Sources and One Definitive Metric
"... We calculate an extensive set of characteristics for Internet AS topologies extracted from the three data sources most frequently used by the research community: traceroutes, BGP, and WHOIS. We discover that traceroute and BGP topologies are similar to one another but differ substantially from the W ..."
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Cited by 108 (15 self)
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We calculate an extensive set of characteristics for Internet AS topologies extracted from the three data sources most frequently used by the research community: traceroutes, BGP, and WHOIS. We discover that traceroute and BGP topologies are similar to one another but differ substantially from the WHOIS topology. Among the widely considered metrics, we find that the joint degree distribution appears to fundamentally characterize Internet AS topologies as well as narrowly define values for other important metrics. We discuss the interplay between the specifics of the three data collection mechanisms and the resulting topology views. In particular, we show how the data collection peculiarities explain differences in the resulting joint degree distributions of the respective topologies. Finally, we release to the community the input topology datasets, along with the scripts and output of our calculations. This supplement should enable researchers to validate their models against real data and to make more informed selection of topology data sources for their specific needs.
Systematic topology analysis and generation using degree correlations
 In SIGCOMM
"... Researchers have proposed a variety of metrics to measure important graph properties, for instance, in social, biological, and computer networks. Values for a particular graph metric may capture a graph’s resilience to failure or its routing efficiency. Knowledge of appropriate metric values may inf ..."
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Cited by 94 (7 self)
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Researchers have proposed a variety of metrics to measure important graph properties, for instance, in social, biological, and computer networks. Values for a particular graph metric may capture a graph’s resilience to failure or its routing efficiency. Knowledge of appropriate metric values may influence the engineering of future topologies, repair strategies in the face of failure, and understanding of fundamental properties of existing networks. Unfortunately, there are typically no algorithms to generate graphs matching one or more proposed metrics and there is little understanding of the relationships among individual metrics or their applicability to different settings. We present a new, systematic approach for analyzing network topologies. We first introduce the dKseries of probability distributions specifying all degree correlations within dsized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dKseries implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d =0, 1, 2, 3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d = 2 case is sufficient for most practical purposes, while d = 3 essentially reconstructs the Internet AS and routerlevel topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.
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 77 (7 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 41 (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.
Toward compact interdomain routing
, 2005
"... Despite prevailing concerns that the current Internet interdomain routing system will not scale to meet the needs of the 21st century global Internet, networking research has not yet led to the construction of a new routing architecture with satisfactory and mathematically provable scalability chara ..."
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Cited by 10 (0 self)
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Despite prevailing concerns that the current Internet interdomain routing system will not scale to meet the needs of the 21st century global Internet, networking research has not yet led to the construction of a new routing architecture with satisfactory and mathematically provable scalability characteristics. Worse, continuing empirical trends of the existing routing and topology structure of the Internet are alarming: the foundational principles of the current routing and addressing architecture are an inherently bad match for the naturally evolving structure of Internet interdomain topology. We are fortunate that a sister discipline, theory of distributed computation, has developed routing algorithms that offer promising potential for genuinely scalable routing on realistic Internetlike topologies. Indeed, there are many recent breakthroughs in the area of compact routing, which has been shown to drastically outperform, in terms of efficiency and scalability, even the boldest proposals developed in networking research. Many open questions remain, but we believe the applicability of compact routing techniques to Internet interdomain routing is a research area whose potential payoff for the future of networking is too high to ignore.
Characterizing and Modelling Clustering Features in ASLevel Internet Topology
"... Abstract—The ASlevel Internet topology has shown significant clustering features. In this paper, we propose a new set of clustering metrics and conduct extensive measurement on the ASlevel Internet topology. We give a thorough characterization on the clustering features and their evolution. We also ..."
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Cited by 4 (1 self)
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Abstract—The ASlevel Internet topology has shown significant clustering features. In this paper, we propose a new set of clustering metrics and conduct extensive measurement on the ASlevel Internet topology. We give a thorough characterization on the clustering features and their evolution. We also study the clustering features of different topological structures by comparing the Internet with various topology models. Due to the limitation of existing topology models on capturing clustering features, we design a new topology model based on clustering. Through extensive evaluations, we claim that our model can closely capture the clustering features as well as other common topological properties. I.
Embedding, Distance Estimation and Object Location in Networks
, 2006
"... Concurrent with numerous theoretical results on metric embeddings, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into l ..."
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Cited by 2 (1 self)
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Concurrent with numerous theoretical results on metric embeddings, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into lowdimensional Euclidean space. A fundamental distinction between the theoretical approaches to embeddings and this recent Internetrelated work is that the latter operates under the additional constraint that it is only feasible to measure a linear number of node pairs, and typically in a highly structured way. Indeed, the most common framework here is a beaconbased approach: one randomly chooses a small number of nodes (’beacons’) in the network, and each node measures its distance to these beacons only. Moreover, beaconbased algorithms are also designed for the more basic problem of triangulation, in which one uses the triangle inequality to infer the distances that have not been measured. We give algorithms with provable performance guarantees for triangulation and embedding. We show that in addition to multiplicative error in the distances, performance guarantees for beaconbased algorithms typically must include a notion of ”slack ” – a certain fraction of all distances may be arbitrarily distorted. For arbitrary metrics, we give a beaconbased embedding algorithm that achieves constant distortion on a (1 − ɛ)fraction of distances; this provides some theoretical justification for the success of the recent
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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Cited by 1 (0 self)
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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