## Dynamics of Large Networks (2008)

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Citations: | 23 - 0 self |

### BibTeX

@TECHREPORT{Leskovec08dynamicsof,

author = {Jurij Leskovec},

title = {Dynamics of Large Networks},

institution = {},

year = {2008}

}

### OpenURL

### Abstract

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 person-to-person

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Citation Context ... = (R1(A), . . .,Rm(A)). Here, the situation can arise that two placements A1 and A2 are incomparable, e.g., R1(A1) > R1(A2), but R2(A1) < R2(A2). So all we can hope for are Pareto-optimal solutions [=-=Boyd and Vandenberghe, 2004-=-]. A placement A is called Pareto-optimal, if there does not exist another placement A ′ such that Ri(A ′ ) ≥ Ri(A) for all i, and Rj(A ′ ) > Rj(A) for some j (i.e., there 2 The objective R is similar... |

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2831 |
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Citation Context ...o find out the network of referrals. They found that strong ties, those between family or friends, were more likely to be activated for information flow and were also more influential than weak ties [=-=Granovetter, 1973-=-] between acquaintances. In the context of the internet, word-of-mouth advertising is not restricted to pairwise or small-group interactions between individuals. Rather, customers can share their expe... |

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Citation Context ...egree exponent. Such degree distributions have been identified in phone call graphs [Abello et al., 1998], the Internet [Faloutsos et al., 1999], the Web [Kleinberg et al., 1999, Broder et al., 2000, =-=Barabási and Albert, 1999-=-, Huberman and Adamic, 1999, Kumar et al., 1999b], citation graphs [Redner, 1998], online social networks [Chakrabarti et al., 2004], click-stream data [Bi et al., 2001], and many others. Typically fo... |

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Citation Context ...ey insight coming to the fore with the pursuit of effective web search is that inferences about relevance can be enhanced by considering the hyperlink relationships among documents [Kleinberg, 1999a, =-=Page et al., 1998-=-]. We present a methodology we refer to as web projections that centers on the creation and the use of graphical properties of subgraphs of the web. With the approach, we project a set of web pages of... |

2230 |
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Citation Context ...d Barabási, 2002, Newman, 2003, Li et al., 2005, Boccaletti et al., 2006, Chakrabarti and Faloutsos, 2006] for overviews of the structural properties of networks. The book on social network analysis [=-=Wasserman and Faust, 1994-=-] is also useful reading. 2.3 Models of network structure and evolution In parallel with empirical studies of large networks, there has been considerable work on models for graph generation. Both dete... |

2198 |
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Citation Context ...rge real-world graphs, 16like Internet, Web, and social networks [Albert and Barabási, 2002, Milgram, 1967, Albert et al., 1999, Bollobas and Riordan, 2004, Broder et al., 2000, Chung and Lu, 2002a, =-=Watts and Strogatz, 1998-=-]). Scree plot: This is a plot of the eigenvalues (or singular values) of the graph adjacency matrix of the graph, versus their rank, using a log-log scale. The scree plot for real networks is often f... |

2073 | On the evolution of random graphs
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Citation Context ...istical property or distribution to arise in the network. Erdős–Rényi random graph model The earliest probabilistic generative model for graphs was a random graph model introduced by Erdős and Rényi [=-=Erdős and Rényi, 1960-=-]. The model states that given a number of nodes each pair of nodes has an identical, independent probability of being joined by an edge. There are two variants of the model: Gn,p is defined to have n... |

1957 | Random Graphs
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Citation Context ...f interest to us here. (The other interesting special case, in which all the expected degrees wi are equal to np, for some p ∈ [0, 1], corresponds to the classical Erdös-Renyi Gnp random graph model [=-=Bollobás, 1985-=-].) Given the number of nodes n, the power-law exponent β, and the parameters w and wmax, Chung and Lu [Chung and Lu, 2006a] give the degree sequence for a power-law graph: wi = ci −1/(β−1) for i s.t.... |

1957 |
The Diffusion of Innovations
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Citation Context ...nfluence by others [Bikhchandani et al., 1992]. Cascades are also known as “fads” or “resonance.” Cascades have been studied for many years by sociologists concerned with the diffusion of innovation [=-=Rogers, 1995-=-]; more recently, researchers in several fields have investigated cascades for the purpose of selecting trendsetters for viral marketing [Domingos and Richardson, 2001], finding inoculation targets in... |

1808 | A global geometric framework for nonlinear dimensionality reduction - Tenenbaum, Silva, et al. - 2000 |

1711 | The structure and function of complex networks - Newman - 2003 |

1528 |
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Citation Context ...s a set of connection nodes, i.e., minimal set of nodes that makes graph Gp connected. Note that finding the minimal set of connection nodes C is NP-hard, since the problem of finding a Steiner tree [=-=Karp, 1972-=-] reduces to this problem. In our experiments, we used a heuristic policy to find the set C, i.e., to connect the components of Gp. We found that the heuristic policy was reliable and performed well o... |

1449 |
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Citation Context ... very direct approach to this problem: One could first identify the set of statistics to match, then define an error metric and somehow optimize over it. For example, one could use the KL divergence [=-=Kullback and Leibler, 1951-=-], or the sum of squared differences between the degree distribution of the real network G and its synthetic counterpart K. Moreover, as we are interested in matching several such statistics between t... |

1441 | Statistical mechanics of complex networks - Albert, Barabási |

1435 | Data clustering: A review
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Citation Context ...densely linked set of nodes, i.e., a community. For example, see the reviews on community identification [Newman, 2004, Danon et al., 2005, Palla et al., 2005, Clauset et al., 2008], data clustering [=-=Jain et al., 1999-=-], and graph and spectral clustering [Gaertler, 2005, Schaeffer, 2007, von Luxburg, 2006]. It has been observed that community-like sets of nodes tend to correspond to organizational units in social n... |

1356 | On Power-Law Relationships of the Internet Topology
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Citation Context ...s Nd of degree d is given by Nd ∝ d−γ (γ > 1) where γ is called the power law degree exponent. Such degree distributions have been identified in phone call graphs [Abello et al., 1998], the Internet [=-=Faloutsos et al., 1999-=-], the Web [Kleinberg et al., 1999, Broder et al., 2000, Barabási and Albert, 1999, Huberman and Adamic, 1999, Kumar et al., 1999b], citation graphs [Redner, 1998], online social networks [Chakrabarti... |

1261 |
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Citation Context ...here ˆ Θ are maximum likelihood parameters under the model with ˆ Θ of size N1 × N1, and N is the number of nodes in G. Similarly, to BIC one could also consider the Minimum Description Length (MDL) [=-=Rissanen, 1978-=-] principle where the model is scored by the quality of the fit plus the size of the description that encodes the model and the parameters. 5.6 Experiments on real and synthetic data We divide the exp... |

1129 |
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Citation Context ...urther subdivide the new groups until the desired number of clusters groups is achieved. This may be combined with local improvement methods like the Kernighan-Lin and Fiduccia-Mattheyses procedures [=-=Kernighan and Lin, 1970-=-, Fiduccia and Mattheyses, 1982], which are fast and can climb out of some local minima. The latter was combined with a multi-resolution framework to create Metis [Karypis and Kumar, 1998b,a], a very ... |

1110 |
Human Behavior and the Principle of Least Effort
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Citation Context ... if the limit < ∞ for some ɛ. Examples of heavy-tailed distributions include power law distributions, Pareto and others which we examine next. 14Relation to Zipf and Pareto distributions Zipf’s law [=-=Zipf, 1949-=-] usually refers to the rank-frequency plots, i.e., “size” or magnitude y of an occurrence of an event relative to its rank r. Zipf’s law is named after George Kingsley Zipf, a Harvard linguistics pro... |

935 |
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Citation Context ... al., 2001] as we defined it in definition 2.1.3. The effective diameter has been found to be small for large real-world graphs, 16like Internet, Web, and social networks [Albert and Barabási, 2002, =-=Milgram, 1967-=-, Albert et al., 1999, Bollobas and Riordan, 2004, Broder et al., 2000, Chung and Lu, 2002a, Watts and Strogatz, 1998]). Scree plot: This is a plot of the eigenvalues (or singular values) of the graph... |

860 |
Community structure in social and biological networks
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Citation Context ...sters are most often thought as sets of nodes that has more and/or better-connected edges between its members than between members of that set and the remainder of the network [Radicchi et al., 2004, =-=Girvan and Newman, 2002-=-]. Many times it is also naturally assumed that the communities observe a recursive structure, where bigger communities can further be split into smaller and smaller communities [Clauset et al., 2006,... |

857 | Finding and evaluating community structure in networks - Newman, Girvan |

700 | Routing of multipoint connections - Waxman - 1988 |

687 | The small-world phenomenon: An algorithmic perspective - KLEINBERG - 2000 |

631 | Error and attack tolerance of complex networks
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Citation Context ...ess than 20 are on the fringe of the network, and that the core starts to rapidly decrease as nodes of degree 20 or more are deleted. 9.7.3 Strength of the ties It has been observed by Albert et al. [=-=Albert et al., 2000-=-] that many real-world networks are robust to node-level changes or attacks. Researchers have showed that networks like the World Wide Web, Internet, and several social networks display a high degree ... |

630 |
A theory of fads, fashion, custom, and cultural change as informational cascades
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Citation Context ...erlying generative models for graphs. 2.4 Diffusion and cascading behavior in networks Information cascades are phenomena in which an action or idea becomes widely adopted due to influence by others [=-=Bikhchandani et al., 1992-=-]. Cascades are also known as “fads” or “resonance.” Cascades have been studied for many years by sociologists concerned with the diffusion of innovation [Rogers, 1995]; more recently, researchers in ... |

629 |
Applied Probability and Queues
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Citation Context ... distribution. This means that its tails are not exponentially bounded; that is, they have heavier tails than the exponential distribution. More precisely, we define heavy-tails in the following way [=-=Asmussen, 2003-=-]: Definition 2.1.5. The distribution of a random variable X is heavy-tailed if for all ɛ > 0. Pr(X > x) lim x→∞ e−ɛx = ∞ In contrast, we say a distribution is light-tailed if the limit < ∞ for some ɛ... |

572 |
A new product growth model for consumer durables
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Citation Context ...ptible to recommendations of a particular product. There are numerous other models of influence spread in social networks. One of the first and most influential diffusion models was proposed by Bass [=-=Bass, 1969-=-]. The model of product diffusion predicts the number of people who will adopt an innovation over time. It does not explicitly account for the structure of the social network but it rather assumes tha... |

560 | A new approach to the maximum flow problem
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Citation Context ...hat scales to the sizes we need. Ordinary max flow is a very thoroughly studied problem. 271Currently, the best theoretical time bounds are [Goldberg and Rao, 1998], the most practical algorithm is [=-=Goldberg and Tarjan, 1988-=-], while the best implementation is hi pr by [Cherkassky and Goldberg, 1995]. Since Metis+MQI using the hi pr code is very fast and scalable, while the method empirically seems to usually find the low... |

540 | Detecting community structure in networks - Newman |

539 | B.: Focused Crawling: a New Approach to Topic-Specific Web Resource Discovery. Computer Networks 31(11-16 - Chakrabarti, Berg, et al. - 1623 |

538 | The link prediction problem for social networks - Liben-Nowell, Kleinberg - 2003 |

525 | Network motifs: Simple building blocks of complex networks - Milo, Shen-Orr, et al. - 2002 |

515 | Multilevel k-way partitioning scheme for irregular graphs
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Citation Context ...graph. Moreover, since there exist a rich suite of both theoretical and practical algorithms [Hendrickson and Leland, 1995, Spielman and Teng, 1996, Leighton and Rao, 1988, 1999, Arora et al., 2004b, =-=Karypis and Kumar, 1998-=-b,a, Zhao and Karypis, 2004, Dhillon et al., 2007], we can for point (4) compare and contrast several methods to approximately optimize it. To illustrate conductance, note that of the three 5-node set... |

500 |
How Nature Works: The Science of Self-Organized Criticality
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(Show Context)
Citation Context ... densification power law. Phase transition phenomena: The Erdős-Rényi graphs exhibit phase transitions [Erdős and Rényi, 1960]. Several researchers argue that real systems are “at the edge of chaos” [=-=Bak, 1996-=-, Sole and Goodwin, 2000]. Stochastic Kronecker Graphs also exhibit phase transitions [Mahdian and Xu, 2007] for the emergence of the giant component and another phase transition for connectivity. Imp... |

466 |
R.W.: A multi-level algorithm for partitioning graphs
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(Show Context)
Citation Context ...binatorial quantity; and it has a very natural interpretation in terms of random walks on the interaction graph. Moreover, since there exist a rich suite of both theoretical and practical algorithms [=-=Hendrickson and Leland, 1995-=-, Spielman and Teng, 1996, Leighton and Rao, 1988, 1999, Arora et al., 2004b, Karypis and Kumar, 1998b,a, Zhao and Karypis, 2004, Dhillon et al., 2007], we can for point (4) compare and contrast sever... |

458 |
An analysis of approximations for maximizing submodular set functions
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(Show Context)
Citation Context ...cements A ⊆ B ⊆ V and sensors s ∈ V \ B, it holds that R(A ∪ {s}) − R(A) ≥ R(B ∪ {s}) − R(B). A set function R with this property is called submodular. Proof. Our proof is similar to the analysis of [=-=Nemhauser et al., 1978-=-]. Fix scenario i. We first show that the function Ri(A) = πi(∞) − πi(T(A, i)) is submodular. Consider A ⊆ B ⊆ V. Let s ∈ V \ B. We have three cases. (i) T(s, i) ≥ T(A, i). Then T(A ∪ {s}) = T(A) and ... |

457 | Topic-sensitive pagerank
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(Show Context)
Citation Context ...PageRank that work with subgraphs of the web have also been explored in prior research. This work includes explorations of domain-specific or person-specific PageRank [Richardson and Domingos, 2002a, =-=Haveliwala, 2002-=-], and on the use of non-random jump vectors for personalization [Jeh and Widom, 2003]. Related work on identifying web spam has examined methods for propagating from trustedpages [Wu et al., 2006]. R... |

451 | A Linear-Time Heuristic for Improving Network Partitions - Fiduccia, Mattheyses - 1982 |

438 |
Algebraic connectivity of graphs
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(Show Context)
Citation Context ...eigenvector of the graph’s Laplacian matrix to find a cut whose conductance is no bigger than φ if the graph actually contains a cut with conductance O(φ 2 ) [Cheeger, 1969, Donath and Hoffman, 1972, =-=Fiedler, 1973-=-, Mohar, 1991, Chung, 1997]. The spectral method also produces lower bounds which can show that the solution for a given graph is closer to optimal than promised by the worst-case guarantee. Second, t... |

429 |
Infectious Diseases of Humans: Dynamics and Control
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(Show Context)
Citation Context ...f the previous research on the flow of information or influence through the networks has been done in the context of epidemiology and the spread of diseases or viruses over the network [Bailey, 1975, =-=Anderson and May, 2002-=-]. Classical disease propagation models are based on the stages of a disease in a host: a person is first susceptible to a disease, then if she is exposed to an infectious contact she can become infec... |

429 | Network motifs in the transcriptional regulation network of Escherichia coli. Nat Genet 31(1 - Shen-Orr, Milo, et al. - 2002 |

422 |
Exploring complex networks
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- 2001
(Show Context)
Citation Context ...s of static or evolving networks such as degree distributions, diameter, clustering coefficient, communities, densification and shrinking diameters [Faloutsos et al., 1999, Albert and Barabási, 2002, =-=Strogatz, 2001-=-, Newman, 2003, Dorogovtsev and Mendes, 2003, Broder et al., 2000, Fetterly et al., 2004, Leskovec et al., 2007b, Ntoulas et al., 2004, Kumar et al., 2006]. In contrast the following chapter focuses o... |

420 |
Threshold models of collective behavior
- Granovetter
- 1978
(Show Context)
Citation Context ...nder which the disease will either dominate or die out. Diffusion models that try to model the process of adoption of an idea or a product can generally be divided into two groups: • Threshold model [=-=Granovetter, 1978-=-] where each node in the network has a threshold t ∈ [0, 1], typically drawn from some probability distribution. We also assign connection weights wu,v on the edges of the network. A node adopts the b... |

416 |
Trust among strangers in internet transactions: Empirical analysis of ebay’s reputation system
- Resnick, Zeckhauser
(Show Context)
Citation Context ...techniques have been proposed [Montgomery, 2001] to describe product information flow online, and the rating of products and merchants has been shown to effect the likelihood of an item being bought [=-=Resnick and Zeckhauser, 2002-=-, Chevalier and Mayzlin, 2006]. More sophisticated online recommendation systems allow users to rate others’ reviews, or directly rate other reviewers to implicitly form a trusted reviewer network tha... |

414 |
Evolution of networks: From Biological Nets to the Internet and WWW
- Dorogovtsev, Mendes
- 2003
(Show Context)
Citation Context ...ks such as degree distributions, diameter, clustering coefficient, communities, densification and shrinking diameters [Faloutsos et al., 1999, Albert and Barabási, 2002, Strogatz, 2001, Newman, 2003, =-=Dorogovtsev and Mendes, 2003-=-, Broder et al., 2000, Fetterly et al., 2004, Leskovec et al., 2007b, Ntoulas et al., 2004, Kumar et al., 2006]. In contrast the following chapter focuses on local microscopic processes that give rais... |

388 | Learning to rank using gradient descent
- Burges, Shaked, et al.
- 2005
(Show Context)
Citation Context ...r of textual and global graph features to rank search results. For the classification tasks, we learn a threshold on the score to predict the class. The baseline ranking algorithm we used is RankNet [=-=Burges et al., 2005-=-], a supervised machine-learning technique developed to learn a ranking function. The learning methodology is a neural net algorithm that optimizes feature weights to best match explicitly provided pa... |