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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 125 (10 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.
Microscopic Evolution of Social Networks
, 2008
"... We present a detailed study of network evolution by analyzing four large online social networks with full temporal information about node and edge arrivals. For the first time at such a large scale, we study individual node arrival and edge creation processes that collectively lead to macroscopic pr ..."
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Cited by 111 (5 self)
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We present a detailed study of network evolution by analyzing four large online social networks with full temporal information about node and edge arrivals. For the first time at such a large scale, we study individual node arrival and edge creation processes that collectively lead to macroscopic properties of networks. Using a methodology based on the maximumlikelihood principle, we investigate a wide variety of network formation strategies, and show that edge locality plays a critical role in evolution of networks. Our findings supplement earlier network models based on the inherently nonlocal preferential attachment. Based on our observations, we develop a complete model of network evolution, where nodes arrive at a prespecified rate and select their lifetimes. Each node then independently initiates edges according to a “gap” process, selecting a destination for each edge according to a simple triangleclosing model free of any parameters. We show analytically that the combination of the gap distribution with the node lifetime leads to a power law outdegree distribution that accurately reflects the true network in all four cases. Finally, we give model parameter settings that allow automatic evolution and generation of realistic synthetic networks of arbitrary scale.
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 82 (6 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
Kronecker Graphs: An Approach to Modeling Networks
 JOURNAL OF MACHINE LEARNING RESEARCH 11 (2010) 9851042
, 2010
"... How can we generate realistic networks? In addition, how can we do so with a mathematically tractable model that allows for rigorous analysis of network properties? Real networks exhibit a long list of surprising properties: Heavy tails for the in and outdegree distribution, heavy tails for the ei ..."
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Cited by 54 (3 self)
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How can we generate realistic networks? In addition, how can we do so with a mathematically tractable model that allows for rigorous analysis of network properties? Real networks exhibit a long list of surprising properties: Heavy tails for the in and outdegree distribution, heavy tails for the eigenvalues and eigenvectors, small diameters, and densification and shrinking diameters over time. Current network models and generators either fail to match several of the above properties, are complicated to analyze mathematically, or both. Here we propose a generative model for networks that is both mathematically tractable and can generate networks that have all the above mentioned structural properties. Our main idea here is to use a nonstandard matrix operation, the Kronecker product, to generate graphs which we refer to as “Kronecker graphs”. First, we show that Kronecker graphs naturally obey common network properties. In fact, we rigorously prove that they do so. We also provide empirical evidence showing that Kronecker graphs can effectively model the structure of real networks. We then present KRONFIT, a fast and scalable algorithm for fitting the Kronecker graph generation model to large real networks. A naive approach to fitting would take superexponential
Ten years in the evolution of the Internet ecosystem
 In ACM SIGCOMM IMC
, 2008
"... Our goal is to understand the evolution of the Autonomous System (AS) ecosystem over the last decade. Instead of focusing on abstract topological properties, we classify ASes into a number of “species ” depending on their function and business type. Further, we consider the semantics of interAS lin ..."
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Cited by 45 (11 self)
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Our goal is to understand the evolution of the Autonomous System (AS) ecosystem over the last decade. Instead of focusing on abstract topological properties, we classify ASes into a number of “species ” depending on their function and business type. Further, we consider the semantics of interAS links, in terms of customerprovider versus peering relations. We find that the available historic datasets from RouteViews and RIPE are not sufficient to infer the evolution of peering links, and so we restrict our focus to customerprovider links. Our findings highlight some important trends in the evolution of the Internet over the last decade, and hint at what the Internet is heading towards. After an exponential increase phase until 2001, the Internet now grows linearly in terms of both ASes and interAS links. The growth is mostly due to enterprise networks and content/access providers at the periphery of the Internet. The average path length remains almost constant mostly due to the increasing multihoming degree of transit and content/access providers. In recent years, enterprise networks prefer to connect to small transit providers, while content/access providers connect equally to both large and small transit providers. The AS species differ significantly from each other with respect to their rewiring activity; content/access providers are the most active. A few large transit providers act as “attractors ” or “repellers ” of customers. For many providers, strong attractiveness precedes strong repulsiveness by 39 months. Finally, in terms of regional growth, we find that the AS ecosystem is now larger and more dynamic in Europe than
The little engine(s) that could: scaling online social networks
 in ACM SIGCOMM Conference, 2010
"... The difficulty of scaling Online Social Networks (OSNs) has introduced new system design challenges that has often caused costly rearchitecting for services like Twitter and Facebook. The complexity of interconnection of users in social networks has introduced new scalability challenges. Convention ..."
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Cited by 33 (3 self)
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The difficulty of scaling Online Social Networks (OSNs) has introduced new system design challenges that has often caused costly rearchitecting for services like Twitter and Facebook. The complexity of interconnection of users in social networks has introduced new scalability challenges. Conventional vertical scaling by resorting to full replication can be a costly proposition. Horizontal scaling by partitioning and distributing data among multiples servers – e.g. using DHTs – can lead to costly interserver communication. We design, implement, and evaluate SPAR, a social partitioning and replication middleware that transparently leverages the social graph structure to achieve data locality while minimizing replication. SPAR guarantees that for all users in an OSN, their direct neighbor’s data is colocated in the same server. The gains from this approach are multifold: application developers can assume local semantics, i.e., develop as they would for a single server; scalability is achieved by adding commodity servers with low memory and network I/O requirements; and redundancy is achieved at a fraction of the cost. We detail our system design and an evaluation based on datasets from Twitter, Orkut, and Facebook, with a working implementation. We show that SPAR incurs minimum overhead, and can help a wellknown opensource Twitter clone reach Twitter’s scale without changing a line of its application logic and achieves higher throughput than Cassandra, Facebook’s DHT based keyvalue store database.
Inferring Networks of Diffusion and Influence
, 2012
"... Information diffusion and virus propagation are fundamental processes taking place in networks. While it is often possible to directly observe when nodes become infected with a virus or publish the information, observing individual transmissions (who infects whom, or who influences whom) is typicall ..."
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Cited by 31 (1 self)
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Information diffusion and virus propagation are fundamental processes taking place in networks. While it is often possible to directly observe when nodes become infected with a virus or publish the information, observing individual transmissions (who infects whom, or who influences whom) is typically very difficult. Furthermore, in many applications, the underlying network over which the diffusions and propagations spread is actually unobserved. We tackle these challenges by developing a method for tracing paths of diffusion and influence through networks and inferring the networks over which contagions propagate. Given the times when nodes adopt pieces of information or become infected, we identify the optimal network that best explains the observed infection times. Since the optimization problem is NPhard to solve exactly, we develop an efficient approximation algorithm that scales to large datasets and finds provably nearoptimal networks. We demonstrate the effectiveness of our approach by tracing information diffusion in a set of 170 million blogs and news articles over a one year period to infer how information flows through the online media space. We find that the diffusion network of news for the top 1,000 media sites and blogs tends to have a coreperiphery structure with a small set of core media sites that diffuse information to the rest of the Web. These sites tend to have stable circles of influence with more general news media sites acting as connectors between them.
No Free Lunch in Data Privacy
"... Differential privacy is a powerful tool for providing privacypreserving noisy query answers over statistical databases. It guarantees that the distribution of noisy query answers changes very little with the addition or deletion of any tuple. It is frequently accompanied by popularized claims that i ..."
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Cited by 30 (3 self)
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Differential privacy is a powerful tool for providing privacypreserving noisy query answers over statistical databases. It guarantees that the distribution of noisy query answers changes very little with the addition or deletion of any tuple. It is frequently accompanied by popularized claims that it provides privacy without any assumptions about the data and that it protects against attackers who know all but one record. In this paper we critically analyze the privacy protections offered by differential privacy. First, we use a nofreelunch theorem, which defines nonprivacy as a game, to argue that it is not possible to provide privacy and utility without making assumptions about how the data are generated. Then we explain where assumptions are needed. We argue that privacy of an individual is preserved when it is possible to limit the inference of an attacker about the participation of the individual in the data generating process. This is different from limiting the inference about the presence of a tuple (for example, Bob’s participation in a social network may cause edges to form between pairs of his friends, so that it affects more than just the tuple labeled as “Bob”). The definition of evidence of participation, in turn, depends on how the data are generated – this is how assumptions enter the picture. We explain these ideas using examples from social network research as well as tabular data for which deterministic statistics have been previously released. In both cases the notion of participation varies, the use of differential privacy can lead to privacy breaches, and differential privacy does not always adequately limit inference about participation.
Effective label acquisition for collective classification
 in International Conference on Knowledge Discovery and Data mining
, 2008
"... Information diffusion, viral marketing, and collective classification all attempt to model and exploit the relationships in a network to make inferences about the labels of nodes. A variety of techniques have been introduced and methods that combine attribute information and neighboring label inform ..."
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Cited by 25 (6 self)
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Information diffusion, viral marketing, and collective classification all attempt to model and exploit the relationships in a network to make inferences about the labels of nodes. A variety of techniques have been introduced and methods that combine attribute information and neighboring label information have been shown to be effective for collective labeling of the nodes in a network. However, in part because of the correlation between node labels that the techniques exploit, it is easy to find cases in which, once a misclassification is made, incorrect information propagates throughout the network. This problem can be mitigated if the system is allowed to judiciously acquire the labels for a small number of nodes. Unfortunately, under relatively general assumptions, determining the optimal set of labels to acquire is intractable. Here we propose an acquisition method that learns the cases when a given collective classification algorithm makes mistakes, and suggests acquisitions to correct those mistakes. We empirically show on both real and synthetic datasets that this method significantly outperforms a greedy approximate inference approach, a viral marketing approach, and approaches based on network structural measures such as node degree and network clustering. In addition to significantly improving accuracy with just a small amount of labeled data, our method is tractable on large networks.
The Internet is Flat: Modeling the Transition from a Transit Hierarchy to a Peering Mesh
 in Proceedings of ACM CoNEXT
, 2010
"... Recent measurements and anecdotal evidence indicate that the Internet ecosystem is rapidly evolving from a multitier hierarchybuiltmostlywithtransit(customerprovider)links toadensemeshformedwithmostlypeeringlinks. ThistransitioncanhavemajorimpactontheglobalInterneteconomy aswell asonthe trafficflo ..."
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Cited by 22 (5 self)
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Recent measurements and anecdotal evidence indicate that the Internet ecosystem is rapidly evolving from a multitier hierarchybuiltmostlywithtransit(customerprovider)links toadensemeshformedwithmostlypeeringlinks. ThistransitioncanhavemajorimpactontheglobalInterneteconomy aswell asonthe trafficflowandtopologicalstructureofthe Internet. In this paper, we study this evolutionarytransition with an agentbased network formation model that captures key aspectsof the interdomainecosystem, viz., interdomain traffic flow and routing, provider and peer selection strategies, geographical constraints, and the economics of transit and peering interconnections. The model predicts several substantial differences between the Hierarchical Internet and the Flat Internet in terms of topological structure, pathlengths,interdomaintrafficflow,andtheprofitabilityof transitproviders. Wealsoquantifytheeffectofthethreefactors driving this evolutionary transition. Finally, we examineahypotheticalscenarioinwhichalargecontentprovider producesmorethanhalfofthetotalInternettraffic. 1.