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33
The structure and function of complex networks
 SIAM REVIEW
, 2003
"... Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, ..."
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Cited by 1467 (9 self)
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Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the smallworld effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
What is Twitter, a Social Network or a News Media?
"... Twitter, a microblogging service less than three years old, commands more than 41 million users as of July 2009 and is growing fast. Twitter users tweet about any topic within the 140character limit and follow others to receive their tweets. The goal of this paper is to study the topological charac ..."
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Cited by 391 (9 self)
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Twitter, a microblogging service less than three years old, commands more than 41 million users as of July 2009 and is growing fast. Twitter users tweet about any topic within the 140character limit and follow others to receive their tweets. The goal of this paper is to study the topological characteristics of Twitter and its power as a new medium of information sharing. We have crawled the entire Twitter site and obtained 41.7 million user profiles, 1.47 billion social relations, 4, 262 trending topics, and 106 million tweets. In its followerfollowing topology analysis we have found a nonpowerlaw follower distribution, a short effective diameter, and low reciprocity, which all mark a deviation from known characteristics of human social networks [28]. In order to identify influentials on Twitter, we have ranked users by the number of followers and by PageRank and found two rankings to be similar.
Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint
 In SRDS
, 2003
"... Abstract How will a virus propagate in a real network?Does an epidemic threshold exist for a finite powerlaw graph, or any finite graph? How long does ittake to disinfect a network given particular values of infection rate and virus death rate? We answer the first question by providing equations th ..."
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Cited by 81 (18 self)
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Abstract How will a virus propagate in a real network?Does an epidemic threshold exist for a finite powerlaw graph, or any finite graph? How long does ittake to disinfect a network given particular values of infection rate and virus death rate? We answer the first question by providing equations that accurately model virus propagation in any network including real and synthesized networkgraphs. We propose a general epidemic threshold condition that applies to arbitrary graphs: weprove that, under reasonable approximations, the epidemic threshold for a network is closely relatedto the largest eigenvalue of its adjacency matrix. Finally, for the last question, we show that infections tend to zero exponentially below the epidemic threshold. We show that our epidemic threshold modelsubsumes many known thresholds for specialcase graphs (e.g., Erd&quot;osR'enyi, BA powerlaw, homogeneous); we show that the threshold tends to zero for infinite powerlaw graphs. Finally, we illustrate thepredictive power of our model with extensive experiments on real and synthesized graphs. We show thatour threshold condition holds for arbitrary graphs.
A Measurementdriven Analysis of Information Propagation in the Flickr Social Network
"... Online social networking sites like MySpace, Facebook, and Flickr have become a popular way to share and disseminate content. Their massive popularity has led to viral marketing techniques that attempt to spread content, products, and ideas on these sites. However, there is little data publicly avai ..."
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Cited by 69 (3 self)
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Online social networking sites like MySpace, Facebook, and Flickr have become a popular way to share and disseminate content. Their massive popularity has led to viral marketing techniques that attempt to spread content, products, and ideas on these sites. However, there is little data publicly available on viral propagation in the real world and few studies have characterized how information spreads over current online social networks. In this paper, we collect and analyze largescale traces of information dissemination in the Flickr social network. Our analysis, based on crawls of the favorite markings of 2.5 million users on 11 million photos, aims at answering three key questions: (a) how widely does information propagate in the social network? (b) how quickly does information propagate? and (c) what is the role of wordofmouth exchanges between friends in the overall propagation of information in the network? Contrary to viral marketing “intuition, ” we find that (a) even popular photos do not spread widely throughout the network, (b) even popular photos spread slowly through the network, and (c) information exchanged between friends is likely to account for over 50 % of all favoritemarkings, but with a significant delay at each hop.
Inoculation Strategies for Victims of Viruses and the SumofSquares Partition Problem
 PROCEEDINGS OF THE 16TH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2005
"... We propose a simple game for modeling containment of the spread of viruses in a graph of n nodes. Each node must choose to either install antivirus software at some known cost C, or risk infection and a loss L if a virus that starts at a random initial point in the graph can reach it without being ..."
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Cited by 43 (2 self)
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We propose a simple game for modeling containment of the spread of viruses in a graph of n nodes. Each node must choose to either install antivirus software at some known cost C, or risk infection and a loss L if a virus that starts at a random initial point in the graph can reach it without being stopped by some intermediate node. The goal of individual nodes is to minimize their individual expected cost. We prove many game theoretic properties of the model, including an easily applied characterization of Nash equilibria, culminating in our showing that allowing selfish users to choose Nash equilibrium strategies is highly undesirable, because the price of anarchy is an unacceptable Θ(n) in the worst case. This shows in particular that a centralized solution can give a much better total cost than an equilibrium solution. Though it is NPhard to compute such a social optimum, we show that the problem can be reduced to a previously unconsidered combinatorial problem that we call the sumofsquares partition problem. Using a greedy algorithm based on sparse cuts, we show that this problem can be approximated to within a factor of O(log² n), giving the same approximation ratio for the inoculation game.
Epidemic Thresholds in Real Networks
"... How will a virus propagate in a real network? How long does it take to disinfect a network given particular values of infection rate and virus death rate? What is the single best node to immunize? Answering these questions is essential for devising networkwide strategies to counter viruses. In addi ..."
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Cited by 43 (9 self)
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How will a virus propagate in a real network? How long does it take to disinfect a network given particular values of infection rate and virus death rate? What is the single best node to immunize? Answering these questions is essential for devising networkwide strategies to counter viruses. In addition, viral propagation is very similar in principle to the spread of rumors, information, and “fads, ” implying that the solutions for viral propagation would also offer insights into these other problem settings. We answer these questions by developing a nonlinear dynamical system (NLDS) that accurately models viral propagation in any arbitrary network, including real and synthesized network graphs. We propose a general epidemic threshold condition for the NLDS system: we prove that the epidemic threshold for a network is exactly the inverse of the largest eigenvalue of its adjacency matrix. Finally, we show that below the epidemic threshold, infections die out at an exponential rate. Our epidemic threshold model subsumes many known thresholds for specialcase graphs (e.g., Erdös–Rényi, BA powerlaw, homogeneous). We demonstrate the predictive power of our model with extensive experiments on real and synthesized graphs, and show that our threshold condition holds for arbitrary graphs. Finally, we show how to utilize our threshold condition for practical uses: It can dictate which nodes to immunize; it can assess the effects of a throttling
On the spread of viruses on the internet
 In SODA
, 2005
"... We analyze the contact process on random graphs generated according to the preferential attachment scheme as a model for the spread of viruses in the Internet. We show that any virus with a positive rate of spread from a node to its neighbors has a nonvanishing chance of becoming epidemic. Quantita ..."
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Cited by 39 (5 self)
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We analyze the contact process on random graphs generated according to the preferential attachment scheme as a model for the spread of viruses in the Internet. We show that any virus with a positive rate of spread from a node to its neighbors has a nonvanishing chance of becoming epidemic. Quantitatively, we discover an interesting dichotomy: for a virus with effective spread rate λ, if the infection starts at a typical vertex, then it develops log(1/λ) into an epidemic with probability λ Θ ( log log(1/λ)), but on average the epidemic probability is λΘ(1). 1
On achieving software diversity for improved network security using distributed coloring algorithms
 In Proceedings of the 11 th ACM Conference on Computer and Communications Security (CCS
, 2004
"... It is widely believed that diversity in operating systems, software packages, and hardware platforms will decrease the virulence of worms and the effectiveness of repeated applications of single attacks. Research efforts in the field have focused on introducing diversity using a variety of technique ..."
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Cited by 28 (2 self)
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It is widely believed that diversity in operating systems, software packages, and hardware platforms will decrease the virulence of worms and the effectiveness of repeated applications of single attacks. Research efforts in the field have focused on introducing diversity using a variety of techniques on a systembysystem basis. This paper, on the other hand, assumes the availability of diverse software packages for each system and then seeks to increase the intrinsic value of available diversity by considering the entire computer network. We present several distributed algorithms for the assignment of distinct software packages to individual systems and analyze their performance. Our goal is to limit the ability of a malicious node to use a single attack to compromise its neighboring nodes, and by extension, the rest of the nodes in the network. The algorithms themselves are analyzed for attack tolerance, and strategies for improving the security of the individual software assignment schemes are presented. We present a comparative analysis of our algorithms using simulation results on a topology obtained from email traffic logs between users at our institution. We find that hybrid versions of our algorithms incorporating multiple assignment strategies achieve better attack tolerance than any given assignment strategy. Our work thus shows that diversity must be introduced at all levels of system design, including any scheme that is used to introduce diversity itself.
2003a. Epidemic spreading in complex networks with degree correlations
 Physics and Astronomy
"... ..."
Characterizing Social Cascades in Flickr
, 2008
"... Online social networking sites like MySpace and Flickr have become a popular way to share and disseminate content. Their massive popularity has led to the viral marketing of content, products, and political campaigns on the sites themselves. Despite the excitement, the precise mechanisms by which in ..."
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Cited by 17 (0 self)
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Online social networking sites like MySpace and Flickr have become a popular way to share and disseminate content. Their massive popularity has led to the viral marketing of content, products, and political campaigns on the sites themselves. Despite the excitement, the precise mechanisms by which information is exchanged over these networks are not well understood. In this paper, we investigate social cascades, or how information disseminates through social links in online social networks. Using real traces of 1,000 popular photos and a social network collected from Flickr, and a theoretical framework borrowed from epidemiology, we show that social cascades are an important factor in the dissemination of content. Our work provides an important first step in understanding how information disseminates in social networks.