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79
Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations
, 2005
"... How do real graphs evolve over time? What are “normal” growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include hea ..."
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Cited by 301 (39 self)
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How do real graphs evolve over time? What are “normal” growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include heavy tails for in and outdegree distributions, communities, smallworld phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time. Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time, with the number of edges growing superlinearly in the number of nodes. Second, the average distance between nodes often shrinks over time, in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) orO(log(log n)). Existing graph generation models do not exhibit these types of behavior, even at a qualitative level. We provide a new graph generator, based on a “forest fire” spreading process, that has a simple, intuitive justification, requires very few parameters (like the “flammability” of nodes), and produces graphs exhibiting the full range of properties observed both in prior work and in the present study.
A Brief History of Generative Models for Power Law and Lognormal Distributions
 INTERNET MATHEMATICS
"... Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a a lognormal distribution. In trying ..."
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Cited by 252 (7 self)
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Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a a lognormal distribution. In trying
Stochastic Models for the Web Graph
, 2000
"... The web may be viewed as a directed graph each of whose vertices is a static HTML web page, and each of whose edges corresponds to a hyperlink from one web page to another. In this paper we propose and analyze random graph models inspired by a series of empirical observations on the web. Our graph m ..."
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Cited by 217 (10 self)
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The web may be viewed as a directed graph each of whose vertices is a static HTML web page, and each of whose edges corresponds to a hyperlink from one web page to another. In this paper we propose and analyze random graph models inspired by a series of empirical observations on the web. Our graph models differ from the traditional Gn;p models in two ways: 1. Independently chosen edges do not result in the statistics (degree distributions, clique multitudes) observed on the web. Thus, edges in our model are statistically dependent on each other. 2. Our model introduces new vertices in the graph as time evolves. This captures the fact that the web is changing with time. Our results are two fold: we show that graphs generated using our model exhibit the statistics observed on the web graph, and additionally, that natural graph models proposed earlier do not exhibit them. This remains true even when these earlier models are generalized to account for the arrival of vertices over time. In particular, the sparse random graphs in our models exhibit properties that do not arise in far denser random graphs generated by ErdosR'enyi models.
The Web as a graph
, 2000
"... The pages and hyperlinks of the WorldWide Web maybe viewed as nodes and edges in a directed graph. This graph has about a billion nodes today,several billion links, and appears to grow exponentially with time. There are many reasonsmathematical, sociological, and commercialfor studying the e ..."
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Cited by 175 (2 self)
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The pages and hyperlinks of the WorldWide Web maybe viewed as nodes and edges in a directed graph. This graph has about a billion nodes today,several billion links, and appears to grow exponentially with time. There are many reasonsmathematical, sociological, and commercialfor studying the evolution of this graph. We first review a set of algorithms that operate on the Web graph, addressing problems from Web search, automatic community discovery, and classification. We then recall a number of measurements and properties of the Web graph. Noting that traditional random graph models do not explain these observations, we propose a new family of random graph models.
RMAT: A recursive model for graph mining
 In Fourth SIAM International Conference on Data Mining (SDM’ 04
, 2004
"... How does a ‘normal ’ computer (or social) network look like? How can we spot ‘abnormal ’ subnetworks in the Internet, or web graph? The answer to such questions is vital for outlier detection (terrorist networks, or illegal moneylaundering rings), forecasting, and simulations (“how will a computer ..."
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Cited by 138 (16 self)
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How does a ‘normal ’ computer (or social) network look like? How can we spot ‘abnormal ’ subnetworks in the Internet, or web graph? The answer to such questions is vital for outlier detection (terrorist networks, or illegal moneylaundering rings), forecasting, and simulations (“how will a computer virus spread?”). The heart of the problem is finding the properties of real graphs that seem to persist over multiple disciplines. We list such “laws ” and, more importantly, we propose a simple, parsimonious model, the “recursive matrix ” (RMAT) model, which can quickly generate realistic graphs, capturing the essence of each graph in only a few parameters. Contrary to existing generators, our model can trivially generate weighted, directed and bipartite graphs; it subsumes the celebrated ErdősRényi model as a special case; it can match the power law behaviors, as well as the deviations from them (like the “winner does not take it all ” model of Pennock et al. [21]). We present results on multiple, large real graphs, where we show that our parameter fitting algorithm (AutoMATfast) fits them very well. 1
Searching the Web
 ACM TRANSACTIONS ON INTERNET TECHNOLOGY
, 2001
"... We offer an overview of current Web search engine design. After introducing a generic search engine architecture, we examine each engine component in turn. We cover crawling, local Web page storage, indexing, and the use of link analysis for boosting search performance. The most common design and im ..."
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Cited by 124 (1 self)
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We offer an overview of current Web search engine design. After introducing a generic search engine architecture, we examine each engine component in turn. We cover crawling, local Web page storage, indexing, and the use of link analysis for boosting search performance. The most common design and implementation techniques for each of these components are presented. For this presentation we draw from the literature and from our own experimental search engine testbed. Emphasis is on introducing the fundamental concepts and the results of several performance analyses we conducted to compare different designs.
Graph evolution: Densification and shrinking diameters
 ACM TKDD
, 2007
"... How do real graphs evolve over time? What are “normal” growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include hea ..."
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Cited by 120 (13 self)
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How do real graphs evolve over time? What are “normal” growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include heavy tails for in and outdegree distributions, communities, smallworld phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time. Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time, with the number of edges growing superlinearly in the number of nodes. Second, the average distance between nodes often shrinks over time, in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) or O(log(log n)). Existing graph generation models do not exhibit these types of behavior, even at a qualitative level. We provide a new graph generator, based on a “forest fire” spreading process, that has a simple, intuitive justification, requires very few parameters (like the “flammability ” of nodes), and produces graphs exhibiting the full range of properties observed both in prior work and in the present study. We also notice that the “forest fire” model exhibits a sharp transition between sparse graphs and graphs that are densifying. Graphs with decreasing distance between the nodes are generated around this transition point. Last, we analyze the connection between the temporal evolution of the degree distribution and densification of a graph. We find that the two are fundamentally related. We also observe that real networks exhibit this type of r
Towards Compressing Web Graphs
 In Proc. of the IEEE Data Compression Conference (DCC
, 2000
"... In this paper, we consider the problem of compressing graphs of the link structure of the World Wide Web. We provide efficient algorithms for such compression that are motivated by recently proposed random graph models for describing the Web. ..."
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Cited by 80 (1 self)
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In this paper, we consider the problem of compressing graphs of the link structure of the World Wide Web. We provide efficient algorithms for such compression that are motivated by recently proposed random graph models for describing the Web.
Realistic, mathematically tractable graph generation and evolution, using kronecker multiplication
 In PKDD
, 2005
"... Abstract. How can we generate realistic graphs? In addition, how can we do so with a mathematically tractable model that makes it feasible to analyze their properties rigorously? Real graphs obey a long list of surprising properties: Heavy tails for the in and outdegree distribution; heavy tails f ..."
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Cited by 76 (21 self)
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Abstract. How can we generate realistic graphs? In addition, how can we do so with a mathematically tractable model that makes it feasible to analyze their properties rigorously? Real graphs obey a long list of surprising properties: Heavy tails for the in and outdegree distribution; heavy tails for the eigenvalues and eigenvectors; small diameters; and the recently discovered “Densification Power Law ” (DPL). All published graph generators either fail to match several of the above properties, are very complicated to analyze mathematically, or both. Here we propose a graph generator that is mathematically tractable and matches this collection of properties. The main idea is to use a nonstandard matrix operation, the Kronecker product, to generate graphs that we refer to as “Kronecker graphs”. We show that Kronecker graphs naturally obey all the above properties; in fact, we can rigorously prove that they do so. We also provide empirical evidence showing that they can mimic very well several real graphs. 1
Graph mining: Laws, generators, and algorithms
 ACM COMPUTING SURVEYS
, 2006
"... How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M : N relation i ..."
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Cited by 70 (6 self)
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How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M : N relation in database terminology can be represented as a graph. A lot of these questions boil down to the following: "How can we generate synthetic but realistic graphs?" To answer this, we must first understand what patterns are common in realworld graphs and can thus be considered a mark of normality/realism. This survey give an overview of the incredible variety of work that has been done on these problems. One of our main contributions is the integration of points of view from physics, mathematics, sociology, and computer science. Further, we briefly describe recent advances on some related and interesting graph problems.