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202
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.
BotGraph: Large Scale Spamming Botnet Detection
"... Network security applications often require analyzing huge volumes of data to identify abnormal patterns or activities. The emergence of cloudcomputing models opens up new opportunities to address this challenge by leveraging the power of parallel computing. In this paper, we design and implement a ..."
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Cited by 51 (4 self)
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Network security applications often require analyzing huge volumes of data to identify abnormal patterns or activities. The emergence of cloudcomputing models opens up new opportunities to address this challenge by leveraging the power of parallel computing. In this paper, we design and implement a novel system called BotGraph to detect a new type of botnet spamming attacks targeting major Web email providers. BotGraph uncovers the correlations among botnet activities by constructing large useruser graphs and looking for tightly connected subgraph components. This enables us to identify stealthy botnet users that are hard to detect when viewed in isolation. To deal with the huge data volume, we implement BotGraph as a distributed application on a computer cluster, and explore a number of performance optimization techniques. Applying it to two months of Hotmail log containing over 500 million users, BotGraph successfully identified over 26 million botnetcreated user accounts with a low false positive rate. The running time of constructing and analyzing a 220GB Hotmail log is around 1.5 hours with 240 machines. We believe both our graphbased approach and our implementations are generally applicable to a wide class of security applications for analyzing large datasets. 1
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.
Aggregate fluctuations and the network structure of intersectoral trade
, 2010
"... This paper analyzes the ow of intermediate inputs across sectors by adopting a network perspective on sectoral interactions. I apply these tools to show how fluctuations in aggregate economic activity can be obtained from independent shocks to individual sectors. First, I characterize the network st ..."
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Cited by 40 (4 self)
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This paper analyzes the ow of intermediate inputs across sectors by adopting a network perspective on sectoral interactions. I apply these tools to show how fluctuations in aggregate economic activity can be obtained from independent shocks to individual sectors. First, I characterize the network structure of input trade in the U.S.. On the demand side, a typical sector relies on a small number of key inputs and sectors are homogeneous in this respect. However, in their role as inputsuppliers sectors do di¤er: many specialized input suppliers coexist alongside general purpose sectors functioning as hubs to the economy. I then develop a model of intersectoral linkages that can reproduce these connectivity features. In a standard multisector setup, I use this model to provide analytical expressions linking aggregate volatility to the network structure of input trade. I show that the presence of sectoral hubs by coupling production decisions across sectors leads to fluctuations in aggregates.
Distances in random graphs with finite mean and infinite variance degrees.
 Electron. J. Probab.,
, 2007
"... Abstract In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent τ ∈ (2, 3). The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph wi ..."
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Cited by 38 (13 self)
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Abstract In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent τ ∈ (2, 3). The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph with N nodes is investigated when N → ∞. When τ ∈ (2, 3), this graph distance grows like 2 log log N  log(τ −2) . In different papers, the cases τ > 3 and τ ∈ (1, 2) have been studied. We also study the fluctuations around these asymptotic means, and describe their distributions. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.
The early evolution of the Hfree process
, 2009
"... The Hfree process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal Hfree graph obtained at t ..."
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Cited by 33 (4 self)
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The Hfree process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal Hfree graph obtained at the end of the process. When H is strictly 2balanced, we show that for some c> 0, with high probability as n → ∞, the minimum degree in G is at least cn 1−(vH −2)/(eH−1) (log n) 1/(eH −1). This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the complete bipartite graphs Kr,r with r ≥ 5. When H is a complete graph Ks with s ≥ 5 we show that for some C> 0, with high probability the independence number of G is at most Cn 2/(s+1) (log n) 1−1/(eH −1). This gives new lower bounds for Ramsey numbers R(s, t) for fixed s ≥ 5 and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.
Epidemics on random graphs with tunable clustering
, 2007
"... In this paper, a branching process approximation for the spread of a ReedFrost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quant ..."
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Cited by 30 (3 self)
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In this paper, a branching process approximation for the spread of a ReedFrost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if they share at least one group.
Dynamics of conversations
 In Proc. KDD
, 2010
"... How do online conversations build? Is there a common model that is followed in human communication? In this work we explore these questions in detail. By considering three different social datasets, namely, Usenet groups, Yahoo! Groups, and Twitter, we analyze the structure of conversations in each ..."
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Cited by 30 (1 self)
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How do online conversations build? Is there a common model that is followed in human communication? In this work we explore these questions in detail. By considering three different social datasets, namely, Usenet groups, Yahoo! Groups, and Twitter, we analyze the structure of conversations in each of these datasets. We propose simple mathematical models for the generation of basic conversation structures and then refine this model to take into account the identities of each member of the conversation. 1.
Diffusion and cascading behavior in random networks
, 2009
"... We consider a model of diffusion on graphs generalizing both the contact process and the bootstrap percolation. The initial seed of active nodes is chosen at random and remain active forever. Then each node (not in the seed) is activated if the number of active nodes in a random subset of its neighb ..."
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Cited by 28 (12 self)
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We consider a model of diffusion on graphs generalizing both the contact process and the bootstrap percolation. The initial seed of active nodes is chosen at random and remain active forever. Then each node (not in the seed) is activated if the number of active nodes in a random subset of its neighborhood exceeds a random threshold. We study the final set of active nodes for a random graph on n vertices with a given degree sequence. We let n tends to infinity. Under some regularity conditions on the degree sequence, we show that the number of final active nodes satisfy a law of large numbers. We also consider the case of a seed with a single active node and give conditions under which it can trigger a large cascade, i.e. the final set of active nodes contains a positive fraction of the size of the graph. Our results allow to study games with local interactions on a complex network. In particular, we compute the contagion threshold for random networks. Our method is based on the properties of empirical distributions of independent random variables and leads to simple proofs unifying and extending results in the random graphs literature and social science literature. 1