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Statistical properties of community structure in large social and information networks
"... A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structur ..."
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Cited by 246 (14 self)
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A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structural properties of such sets of nodes. We define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales, and we study over 70 large sparse realworld networks taken from a wide range of application domains. Our results suggest a significantly more refined picture of community structure in large realworld networks than has been appreciated previously. Our most striking finding is that in nearly every network dataset we examined, we observe tight but almost trivial communities at very small scales, and at larger size scales, the best possible communities gradually “blend in ” with the rest of the network and thus become less “communitylike.” This behavior is not explained, even at a qualitative level, by any of the commonlyused network generation models. Moreover, this behavior is exactly the opposite of what one would expect based on experience with and intuition from expander graphs, from graphs that are wellembeddable in a lowdimensional structure, and from small social networks that have served as testbeds of community detection algorithms. We have found, however, that a generative model, in which new edges are added via an iterative “forest fire” burning process, is able to produce graphs exhibiting a network community structure similar to our observations.
Community structure in large networks: Natural cluster sizes and the absence of large welldefined clusters
, 2008
"... A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins wit ..."
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Cited by 208 (17 self)
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A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempt to interpret these sets as a “real ” communities, we employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales. We study over 100 large realworld networks, ranging from traditional and online social networks, to technological and information networks and
Towards a theory of scalefree graphs: Definition, properties, and implications
 Internet Mathematics
, 2005
"... Abstract. There is a large, popular, and growing literature on “scalefree ” networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortunately no consistent, precise definition of scale ..."
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Cited by 137 (12 self)
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Abstract. There is a large, popular, and growing literature on “scalefree ” networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortunately no consistent, precise definition of scalefree graphs and few rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and that the most celebrated claims regarding the Internet and biology are verifiably false. In this paper, we introduce a structural metric that allows us to differentiate between all simple, connected graphs having an identical degree sequence, which is of particular interest when that sequence satisfies a power law relationship. We demonstrate that the proposed structural metric yields considerable insight into the claimed properties of SF graphs and provides one possible measure of the extent to which a graph is scalefree. This structural view can be related to previously studied graph properties such as the various notions of selfsimilarity, likelihood, betweenness and assortativity. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the current literature, and offers a rigorous and quantitative alternative, while suggesting the potential for a rich and interesting theory. This paper is aimed at readers familiar with the basics of Internet technology and comfortable with a theoremproof style of exposition, but who may be unfamiliar with the existing literature on scalefree networks. 1.
Modeling Model Uncertainty
 Journal of the European Economic Assocation
, 2003
"... Recently there has been a great deal of interest in studying monetary policy under model uncertainty. We point out that different assumptions about the uncertainty may result in drastically different “robust ” policy recommendations. Therefore, we develop new methods to analyze uncertainty about the ..."
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Cited by 82 (9 self)
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Recently there has been a great deal of interest in studying monetary policy under model uncertainty. We point out that different assumptions about the uncertainty may result in drastically different “robust ” policy recommendations. Therefore, we develop new methods to analyze uncertainty about the parameters of a model, the lag specification, the serial correlation of shocks, and the effects of real time data in one coherent structure. We consider both parametric and nonparametric specifications of this structure and use them to estimate the uncertainty in a small model of the US economy. We then use our estimates to compute robust Bayesian and minimax monetary policy rules, which are designed to perform well in the face of uncertainty. Our results suggest that the aggressiveness recently found in robust policy rules is likely to be caused by overemphasizing uncertainty about economic dynamics at low frequencies.
Measuring the Robustness of a Resource Allocation
"... Parallel and distributed systems may operate in an environment that undergoes unpredictable changes causing certain system performance features to degrade. Such systems need robustness to guarantee limited degradation despite fluctuations in the behavior of its component parts or environment. This r ..."
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Cited by 81 (48 self)
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Parallel and distributed systems may operate in an environment that undergoes unpredictable changes causing certain system performance features to degrade. Such systems need robustness to guarantee limited degradation despite fluctuations in the behavior of its component parts or environment. This research investigates the robustness of an allocation of resources to tasks in parallel and distributed systems. The main contributions of this paper are (1) a mathematical description of a metric for the robustness of a resource allocation with respect to desired system performance features against multiple perturbations in multiple system and environmental conditions, and (2) a procedure for deriving a robustness metric for an arbitrary system. For illustration, this procedure is employed to derive robustness metrics for three example distributed systems. Such a metric can help researchers evaluate a given resource allocation for robustness against uncertainties in specified perturbation parameters.
Robustness and fragility of boolean models for genetic regulatory networks
 J. Theoretical Biology
, 2005
"... Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. We propose a new modeling framework for gene ..."
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Cited by 71 (10 self)
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Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. We propose a new modeling framework for gene regulatory networks, that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights into the development of gene expression patterns. For example, we show that very short synthesis and decay times can perturb the wild type pattern. On the other hand, separation of timescales between pre and posttranslational processes and a minimal prepattern ensure convergence to the wild type expression pattern regardless of fluctuations.
Understanding internet topology: principles, models, and validation
 IEEE/ACM TRANSACTIONS ON NETWORKING
, 2005
"... Building on a recent effort that combines a firstprinciples approach to modeling routerlevel connectivity with a more pragmatic use of statistics and graph theory, we show in this paper that for the Internet, an improved understanding of its physical infrastructure is possible by viewing the phys ..."
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Cited by 51 (8 self)
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Building on a recent effort that combines a firstprinciples approach to modeling routerlevel connectivity with a more pragmatic use of statistics and graph theory, we show in this paper that for the Internet, an improved understanding of its physical infrastructure is possible by viewing the physical connectivity as an annotated graph that delivers raw connectivity and bandwidth to the upper layers in the TCP/IP protocol stack, subject to practical constraints (e.g., router technology) and economic considerations (e.g., link costs). More importantly, by relying on data from Abilene, a Tier1 ISP, and the Rocketfuel project, we provide empirical evidence in support of the proposed approach and its consistency with networking reality. To illustrate its utility, we: 1) show that our approach provides insight into the origin of high variability in measured or inferred routerlevel maps; 2) demonstrate that it easily accommodates the incorporation of additional objectives of network design (e.g., robustness to router failure); and 3) discuss how it complements ongoing community efforts to reverseengineer the Internet.
Mathematics and the Internet: A Source of Enormous Confusion and Great Potential
"... For many mathematicians and physicists, the Internet has become a popular realworld domain for the application and/or development of new theories related to the organization and behavior of largescale, complex, and dynamic systems. In some cases, the Internet has served both as inspiration and just ..."
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Cited by 47 (6 self)
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For many mathematicians and physicists, the Internet has become a popular realworld domain for the application and/or development of new theories related to the organization and behavior of largescale, complex, and dynamic systems. In some cases, the Internet has served both as inspiration and justification for the popularization of new models and mathematics within the scientific enterprise. For example, scalefree network models of the preferential attachment type [8] have been claimed to describe the Internet’s connectivity structure, resulting in surprisingly general and strong claims about the network’s resilience to random failures of its components and its vulnerability to targeted attacks against its infrastructure [2]. These models have, as their trademark, powerlaw type node degree distributions that drastically distinguish them from the classical ErdősRényi type random graph models [13]. These “scalefree ” network models have attracted significant attention within the scientific community and have been partly responsible for launching and fueling the new field of network science [42, 4]. To date, the main role that mathematics has played in network science has been to put the physicists’ largely empirical findings on solid grounds Walter Willinger is at AT&T LabsResearch in Florham Park, NJ. His email address is walter@research.att. com.
Dynamics of Large Networks
, 2008
"... A basic premise behind the study of large networks is that interaction leads to complex collective behavior. In our work we found very interesting and counterintuitive patterns for time evolving networks, which change some of the basic assumptions that were made in the past. We then develop models ..."
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Cited by 33 (0 self)
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A basic premise behind the study of large networks is that interaction leads to complex collective behavior. In our work we found very interesting and counterintuitive patterns for time evolving networks, which change some of the basic assumptions that were made in the past. We then develop models that explain processes which govern the network evolution, fit such models to real networks, and use them to generate realistic graphs or give formal explanations about their properties. In addition, our work has a wide range of applications: it can help us spot anomalous graphs and outliers, forecast future graph structure and run simulations of network evolution. Another important aspect of our research is the study of “local ” patterns and structures of propagation in networks. We aim to identify building blocks of the networks and find the patterns of influence that these blocks have on information or virus propagation over the network. Our recent work included the study of the spread of influence in a large persontoperson