## Laws of Graph Evolution: Densification and Shrinking Diameters (2006)

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Venue: | ACM Transactions on knowledge discovery from data |

Citations: | 134 - 13 self |

### BibTeX

@ARTICLE{Leskovec06lawsof,

author = {Jure Leskovec and Jon Kleinberg and Christos Faloutsos},

title = {Laws of Graph Evolution: Densification and Shrinking Diameters},

journal = {ACM Transactions on knowledge discovery from data},

year = {2006},

volume = {1},

pages = {1--40}

}

### Years of Citing Articles

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### Abstract

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 out-degree distributions, communities, small-world 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.

### Citations

2314 | Emergence of scaling in random networks
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- 1999
(Show Context)
Citation Context ...wing diameter assumption: The diameter is a slowly growing function of the network size, as in “small world” graphs [6, 11, 38, 50]. For example, the intensively-studied preferential attachment model =-=[5, 40]-=- posits a network in which each new node, when it arrives, attaches to the existing network by a constant number of out-links, according to a “rich-get-richer” rule. Recent work has given tight asympt... |

2077 |
Collective dynamics of ’small-world’ networks
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- 1998
(Show Context)
Citation Context ...valently, the number of edges grows linearly in the number of nodes.) (B) Slowly growing diameter assumption: The diameter is a slowly growing function of the network size, as in “small world” graphs =-=[6, 11, 38, 50]-=-. For example, the intensively-studied preferential attachment model [5, 40] posits a network in which each new node, when it arrives, attaches to the existing network by a constant number of out-link... |

1556 | The Structure and Function of Complex Networks
- Newman
- 2003
(Show Context)
Citation Context ...ted together); networks of users exchanging e-mail or instant messages; citation networks and hyperlink networks; social networks (who-trustswhom, who-talks-to-whom, and so forth); and countless more =-=[40]-=-. The study of such networks has proceeded along two related tracks: the measurement of large network datasets, and the development of random graph models that approximate the observed properties. Man... |

1306 | On power-law relationships of the Internet topology
- Faloutsos, Faloutsos, et al.
- 1999
(Show Context)
Citation Context ...eas of focus has been on degree power laws, showing that the set of node degrees has a heavy-tailed distribution. Such degree distributions have been identified in phone call graphs [2], the Internet =-=[20]-=-, the Web [5, 23, 32], click-stream data [8] and for a who-trusts-whom social network [13]. Other properties include the “small-world phenomenon,” popularly known as “six degrees of separation”, which... |

879 |
The small world problem
- Milgram
- 1967
(Show Context)
Citation Context ...valently, the number of edges grows linearly in the number of nodes.) (B) Slowly growing diameter assumption: The diameter is a slowly growing function of the network size, as in “small world” graphs =-=[6, 11, 38, 50]-=-. For example, the intensively-studied preferential attachment model [5, 40] posits a network in which each new node, when it arrives, attaches to the existing network by a constant number of out-link... |

387 |
Evolution of Networks: From Biological Nets to the Internet and WWW
- Dorogovtsev, Mendes
- 2003
(Show Context)
Citation Context ...twork grew; our findings and associated model challenge this assumption, by showing that networks across a number of domains are becoming denser over time. Dorogovtsev and Mendes in a series of works =-=[16, 17, 19]-=- analyzed possible scenarios of nonlinearly growing networks while maintaining scale-free structure. Among considered hypothetical scenarios were also those where the number of links grows polynomiall... |

322 | Faloutsos. Graphs over time: Densification laws, shrinking diameters, and possible explanations
- Leskovec, C
- 2005
(Show Context)
Citation Context ...n of this work: we present two families of probabilistic generative models for graphs that capture aspects of these properties. The first model, which we refer to as Community Guided Attachment (CGA) =-=[36]-=-, argues that graph densification can have a simple underlying basis; it is based on a decomposition of the nodes into a nested set of communities, such that the difficulty of forming links between co... |

313 | The Web as a graph: Measurements, models and methods
- Kleinberg, Kumar, et al.
- 1999
(Show Context)
Citation Context ... of degree power laws led to the development of random graph models that exhibited such degree distributions, including the family of models based on preferential attachment [5, 3, 15], copying model =-=[27, 31]-=-, and the related growing network with redirection model [29], which produces graphs with constant diameter and logarithmically increasing average degree [30]. Similar to our Forest Fire Model is the ... |

306 | Trawling the web for emerging cyber-communities
- Kumar, Rajagopalan, et al.
- 1999
(Show Context)
Citation Context ...as been on degree power laws, showing that the set of node degrees has a heavy-tailed distribution. Such degree distributions have been identified in phone call graphs [2], the Internet [20], the Web =-=[5, 23, 32]-=-, click-stream data [8] and for a who-trusts-whom social network [13]. Other properties include the “small-world phenomenon,” popularly known as “six degrees of separation”, which states that real gra... |

274 | The dynamics of viral marketing
- Leskovec, Adamic, et al.
- 2005
(Show Context)
Citation Context ...utonomous Systems and lower (but still significant) densification exponent (≈ 1.1) for affiliation and collaboration type networks. 3.1.7 Product recommendation network We also report the analysis of =-=[33]-=-, where they measured the densification of a large person-to-person recommendation network from a large on-line retailer. Nodes represent people and edges represent recommendations. The network genera... |

272 | A brief history of generative models for power law and lognormal distributions. Internet Mathematics
- Mitzenmacher
(Show Context)
Citation Context ...tion experiments also indicated that the diameter of networks generated by the recursive search does not decrease over time, but it either slowly increases or remains constant. We point the reader to =-=[39, 40, 7]-=- for overviews of this area. Recent work of Chakrabarti and Faloutsos [12] gives a survey of the properties of real world graphs and the underlying generative models for graphs. It is important to not... |

236 | Stochastic models for the Web graph
- Kumar, Raghavan, et al.
(Show Context)
Citation Context ... of degree power laws led to the development of random graph models that exhibited such degree distributions, including the family of models based on preferential attachment [5, 3, 15], copying model =-=[27, 31]-=-, and the related growing network with redirection model [29], which produces graphs with constant diameter and logarithmically increasing average degree [30]. Similar to our Forest Fire Model is the ... |

234 | Graph structure in the web
- Broder, Kumar, et al.
- 2000
(Show Context)
Citation Context ...ification is forced by introducing an additional “node attractiveness” factor that is not only degree-dependent but also time-dependent. The motivation for their work comes from the fact that authors =-=[10, 20]-=- reported the increase of the average degree over time on the Web and the Internet. Our work here differs in that it presents measurements on many time evolving networks to support our findings, and p... |

197 | The average distance in a random graph with given expected degree sequences
- Chung, Lu
- 2002
(Show Context)
Citation Context ...hes to the existing network by a constant number of out-links, according to a “rich-get-richer” rule. Recent work has given tight asymptotic bounds on the diameter of preferential attachment networks =-=[9, 14]-=-; depending on the precise model, these bounds grow logarithmically [30] or even slower than logarithmically in the number of nodes. How are assumptions (A) and (B) reflected in data on network growth... |

195 | Power laws, Pareto distributions and Zipf’s law
- Newman
- 2005
(Show Context)
Citation Context ...sification Power Law with exponent a = 2/γ. arises. In this case the Densification Power Law is the consequence of the fact that a power-law distribution with exponent γ < 2 has no finite expectation =-=[41]-=-, and thus the average degree grows as degree exponent is constant. Our second result is for the case when temporally evolving graph densifies with densification exponent a, and follows a power-law de... |

178 |
Empirical analysis of an evolving social network
- Kossinets, Watts
- 2006
(Show Context)
Citation Context ...ng, mistyped or spam. Given a set of email messages we need to create a graph. Since there can be multiple emails sent between same two addresses (nodes) we follow the practice of Kossinets and Watts =-=[28]-=-. Given a set of email messages, each node corresponds to an email address. We create an edge between nodes i and j, if they exchanged messages both ways, i.e. i sent at least one message to j, and j ... |

176 | What’s new on the web? The evolution of the web from a search engine perspective
- Ntoulas, Cho, et al.
- 2004
(Show Context)
Citation Context ...ifying properties of a single snapshot or a very small number of snapshots of a large network. For example, despite the intense interest in the Web’s link structure, the recent work of Ntoulas et al. =-=[42]-=- noted the lack of prior empirical research on the evolution of the Web. Thus, while one can assert based on these studies that, qualitatively, real networks have relatively small average node degrees... |

149 | R-mat: A recursive model for graph mining
- CHAKRABARTI, ZHAN, et al.
- 2004
(Show Context)
Citation Context ...vy-tailed distribution. Such degree distributions have been identified in phone call graphs [2], the Internet [20], the Web [5, 23, 32], click-stream data [8] and for a who-trusts-whom social network =-=[13]-=-. Other properties include the “small-world phenomenon,” popularly known as “six degrees of separation”, which states that real graphs have surprisingly small (average or effective) diameter (see [6, ... |

142 | Small-World Phenomena and the Dynamics of Information
- Kleinberg
(Show Context)
Citation Context ...[13]. Other properties include the “small-world phenomenon,” popularly known as “six degrees of separation”, which states that real graphs have surprisingly small (average or effective) diameter (see =-=[6, 9, 11, 14, 26, 38, 50, 51]-=-). In parallel with empirical studies of large networks, there has been considerable work on probabilistic models for graph generation. The discovery of degree power laws led to the development of ran... |

137 | Identity and search in social networks
- Watts, Dodds, et al.
- 2002
(Show Context)
Citation Context ...[13]. Other properties include the “small-world phenomenon,” popularly known as “six degrees of separation”, which states that real graphs have surprisingly small (average or effective) diameter (see =-=[6, 9, 11, 14, 26, 38, 50, 51]-=-). In parallel with empirical studies of large networks, there has been considerable work on probabilistic models for graph generation. The discovery of degree power laws led to the development of ran... |

126 |
Connectivity of growing random networks
- Krapivsky, Redner, et al.
- 2000
(Show Context)
Citation Context ...ls that exhibited such degree distributions, including the family of models based on preferential attachment [5, 3, 15], copying model [27, 31], and the related growing network with redirection model =-=[29]-=-, which produces graphs with constant diameter and logarithmically increasing average degree [30]. Similar to our Forest Fire Model is the work of Vazquez [48, 49] where ideas based on random walks an... |

116 |
Sampling from large graphs
- Leskovec, Faloutsos
- 2006
(Show Context)
Citation Context ... is to maintain structural properties of the network. Densification laws can help discard bad sampling methods, by providing means to reject sampled subgraphs. A recent work of Leskovec and Faloutsos =-=[35]-=- proposed two views on sampling from large graphs. For Back-in-time sampling the goal is to find a sequence of sampled subgraphs that matches the evolution of the original graph and thus obey the temp... |

114 |
The diameter of a scale-free random graph
- Bollobás, Riordan
- 2004
(Show Context)
Citation Context ...hes to the existing network by a constant number of out-links, according to a “rich-get-richer” rule. Recent work has given tight asymptotic bounds on the diameter of preferential attachment networks =-=[9, 14]-=-; depending on the precise model, these bounds grow logarithmically [30] or even slower than logarithmically in the number of nodes. How are assumptions (A) and (B) reflected in data on network growth... |

105 |
Graph structure in the web: Experiments and models
- Broder, Kumar, et al.
- 2000
(Show Context)
Citation Context ...valently, the number of edges grows linearly in the number of nodes.) (B) Slowly growing diameter assumption: The diameter is a slowly growing function of the network size, as in “small world” graphs =-=[6, 11, 38, 50]-=-. For example, the intensively-studied preferential attachment model [5, 40] posits a network in which each new node, when it arrives, attaches to the existing network by a constant number of out-link... |

104 | The NBER patent citations data file: Lessons, insights and methodological tools. NBER working paper series
- Hall, Jaffe, et al.
- 2001
(Show Context)
Citation Context ...we show the summary of results on all 11 datasets we considered in table 1. 3.1.2 Patents citation graph Next, we consider a U.S. patent dataset maintained by the National Bureau of Economic Research =-=[22]-=-. The data set spans 37 years (January 1, 1963 to DecemberGraphs Over Time 9 10 6 10 8 Apr 2003 Number of edges 10 2 10 2 10 3 10 4 10 5 Jan 1993 10 3 10 4 Number of nodes (a) arXiv Edges = 0.0113 x ... |

100 | ANF: a fast and scalable tool for data mining in massive graphs
- Palmer, Gibbons, et al.
- 2002
(Show Context)
Citation Context ...graphs of our scale. We used several different approximate methods, obtaining almost identical results from all of them. In particular, we applied the Approximate Neighborhood Function (ANF) approach =-=[44]-=- (in two different implementations), which can estimate effective diameters for very large graphs, as well as a basic sampling approach in which we ran exhaustive breadth-first search from a subset of... |

96 |
Internet: Diameter of the world-wide web
- Albert, Jeong, et al.
- 1999
(Show Context)
Citation Context |

90 | A functional approach to external graph algorithms
- Abello, Buchsbaum, et al.
(Show Context)
Citation Context ...One of the main areas of focus has been on degree power laws, showing that the set of node degrees has a heavy-tailed distribution. Such degree distributions have been identified in phone call graphs =-=[2]-=-, the Internet [20], the Web [5, 23, 32], click-stream data [8] and for a who-trusts-whom social network [13]. Other properties include the “small-world phenomenon,” popularly known as “six degrees of... |

89 | Towards a theory of scale-free graphs: definition, properties, and implications
- Li, Alderson, et al.
(Show Context)
Citation Context ...tion experiments also indicated that the diameter of networks generated by the recursive search does not decrease over time, but it either slowly increases or remains constant. We point the reader to =-=[39, 40, 7]-=- for overviews of this area. Recent work of Chakrabarti and Faloutsos [12] gives a survey of the properties of real world graphs and the underlying generative models for graphs. It is important to not... |

87 | A general model of web graph
- Cooper, Frieze
- 2003
(Show Context)
Citation Context ... generation. The discovery of degree power laws led to the development of random graph models that exhibited such degree distributions, including the family of models based on preferential attachment =-=[5, 3, 15]-=-, copying model [27, 31], and the related growing network with redirection model [29], which produces graphs with constant diameter and logarithmically increasing average degree [30]. Similar to our F... |

78 | Realistic, mathematically tractable graph generation and evolution, using kronecker multiplication
- Leskovec, Chakrabarti, et al.
- 2005
(Show Context)
Citation Context ... produces densifying graphs, is an example of a hierarchical graph generation model, in which the linkage probability between nodes decreases as a function of their relative distance in the hierarchy =-=[13, 26, 51, 36, 34, 1]-=-. Again, there is a distinction between the aims of this past work and our model here; where these earlier network models were seeking to capture properties of individual snapshots of a graph, we seek... |

77 | Graph mining: Laws, generators, and algorithms
- CHAKRABARTI, FALOUTSOS
- 2006
(Show Context)
Citation Context ...cursive search does not decrease over time, but it either slowly increases or remains constant. We point the reader to [39, 40, 7] for overviews of this area. Recent work of Chakrabarti and Faloutsos =-=[12]-=- gives a survey of the properties of real world graphs and the underlying generative models for graphs. It is important to note the fundamental contrast between one of our main findings here — that th... |

69 | A simple conceptual model for the Internet topology
- Tauro, Palmer, et al.
- 2001
(Show Context)
Citation Context ...irs whose undirected shortest connecting path in a graph G has length at most d. And let D be an integer for which g(D − 1) < 0.9 and g(D) ≥ 0.9. Then the graph G has the integer effective diameter D =-=[47]-=-. In other words, the integer effective diameter is the smallest number of hops D at which at least 90% of all connected pairs of nodes can be reached. Last we give the definition of the effective dia... |

57 |
Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations, Phys
- Vázquez
(Show Context)
Citation Context ...d growing network with redirection model [29], which produces graphs with constant diameter and logarithmically increasing average degree [30]. Similar to our Forest Fire Model is the work of Vazquez =-=[48, 49]-=- where ideas based on random walks and recursive search for generating networks were introduced. In a random walk model the walk starts at a random node, follows links, and for each visited node with ... |

50 |
Growth dynamics of the world-wide web
- Adamic, Huberman
- 1999
(Show Context)
Citation Context ...as been on degree power laws, showing that the set of node degrees has a heavy-tailed distribution. Such degree distributions have been identified in phone call graphs [2], the Internet [20], the Web =-=[5, 23, 32]-=-, click-stream data [8] and for a who-trusts-whom social network [13]. Other properties include the “small-world phenomenon,” popularly known as “six degrees of separation”, which states that real gra... |

49 | The DGX distribution for mining massive, skewed data
- BI, FALOUTSOS, et al.
(Show Context)
Citation Context ...howing that the set of node degrees has a heavy-tailed distribution. Such degree distributions have been identified in phone call graphs [2], the Internet [20], the Web [5, 23, 32], click-stream data =-=[8]-=- and for a who-trusts-whom social network [13]. Other properties include the “small-world phenomenon,” popularly known as “six degrees of separation”, which states that real graphs have surprisingly s... |

42 |
Laws: Minutes from an Infinite Paradise
- Fractals, Power
- 1991
(Show Context)
Citation Context ...densification behavior. We take the following approach. Power laws often appear in combination with selfsimilar structures. Intuitively, a self-similar object consists of miniature replicas of itself =-=[46]-=-. Our approach involves two steps, both of which are based on self-similarity. We begin by searching for self-similar, recursive structures. In fact, we can easily find several such recursive sets: Fo... |

39 |
Growing and navigating the small world web by local content
- Menczer
(Show Context)
Citation Context ...re only related by membership in a larger community. In a different domain, Menczer studied the frequency of links among Web pages that are organized into a topic hierarchy such as the Open Directory =-=[37]-=-. He showed that link density among pages decreases with the height of their least common ancestor in the hierarchy. That is, two pages on closely related topics are more likely to be hyperlinked than... |

31 |
Language as an evolving word web
- Dorogovtsev, Mendes
- 2001
(Show Context)
Citation Context ...twork grew; our findings and associated model challenge this assumption, by showing that networks across a number of domains are becoming denser over time. Dorogovtsev and Mendes in a series of works =-=[16, 17, 19]-=- analyzed possible scenarios of nonlinearly growing networks while maintaining scale-free structure. Among considered hypothetical scenarios were also those where the number of links grows polynomiall... |

20 | Citation statistics from more than a century of physical review. APS Meeting Abstracts
- Redner
- 2004
(Show Context)
Citation Context ...ussion of this point by Ntoulas et al. [42]). Two exceptions are the very recent work of Katz [25], who independently discovered densification power laws for citation networks, and the work of Redner =-=[45]-=-, who studied the evolution of the citation graph of Physical Review over the past century. Katz’s work builds on his earlier research on power-law relationships between the size and the recognition o... |

19 |
power-laws, and pareto - a ranking tutorial. http://www.hpl.hp.com/research/ idl/papers/ranking/ranking.html
- Zipf
- 2002
(Show Context)
Citation Context ...ith exponents γ = 1.75 and γ = 2.24, respectively. The actual slope of the plotted line is 1/(γ − 1), which is the relation between the power-law exponent γ and the slope of the rank degree plot (see =-=[4]-=- for more details on these relationships). In both plots of figure 13 we observe linearity which suggests a power-law relationship for a part of the degree distribution. For the email network we obser... |

17 |
Effect of the accelerating growth of communications networks on their structure. arXiv: cond-mat/0009065
- Dorogovtsev, Mendes
- 2000
(Show Context)
Citation Context ...twork grew; our findings and associated model challenge this assumption, by showing that networks across a number of domains are becoming denser over time. Dorogovtsev and Mendes in a series of works =-=[16, 17, 19]-=- analyzed possible scenarios of nonlinearly growing networks while maintaining scale-free structure. Among considered hypothetical scenarios were also those where the number of links grows polynomiall... |

16 | Overview of the 2003 kdd cup
- Gehrke, Ginsparg, et al.
- 2003
(Show Context)
Citation Context ...fer to the log-log plot of number of edges e(t)8 J. Leskovec et al. versus number of nodes n(t). 3.1.1 ArXiv citation graph We first investigate a citation graph provided as part of the 2003 KDD Cup =-=[21]-=-. The HEP–TH (high energy physics theory) citation graph from the e-print arXiv covers all the citations within a dataset of n=29,555 papers with e= 352,807 edges. If a paper i cites paper j, the grap... |

16 | F.: 'Accelerated growth of networks', in
- Dorogovtsev, Mendes
- 2002
(Show Context)
Citation Context ...ation and the power-law degree distribution over time, and find evidence that some of the real world graphs obey the relations we find. A similar analysis was also performed by Dorogovtsev and Mendes =-=[18]-=- although without specific measurements or comparison to empirical data. We analyze the following two cases: If the degree distribution of a time evolving graph is power-law, and it maintains constant... |

14 |
editors. Handbook of massive data sets
- Abello, Pardalos, et al.
- 2002
(Show Context)
Citation Context ... generation. The discovery of degree power laws led to the development of random graph models that exhibited such degree distributions, including the family of models based on preferential attachment =-=[5, 3, 15]-=-, copying model [27, 31], and the related growing network with redirection model [29], which produces graphs with constant diameter and logarithmically increasing average degree [30]. Similar to our F... |

13 |
The self-similar science system
- Katz
- 1999
(Show Context)
Citation Context ... of the citation graph of Physical Review over the past century. Katz’s work builds on his earlier research on power-law relationships between the size and the recognition of professional communities =-=[24]-=-; his work on densification is focused specifically on citations, and he does not propose a generative network model to account for the densification phenomenon, as we do here. Redner’s work focuses o... |

13 | Scale Independent Bibliometric Indicators
- Katz
- 2005
(Show Context)
Citation Context ...aphs, identifying patterns in a single snapshot, or a small number of network snapshots (see also the discussion of this point by Ntoulas et al. [42]). Two exceptions are the very recent work of Katz =-=[25]-=-, who independently discovered densification power laws for citation networks, and the work of Redner [45], who studied the evolution of the citation graph of Physical Review over the past century. Ka... |

11 |
Network growth by copying
- Krapivsky, Redner
- 2005
(Show Context)
Citation Context ... a “rich-get-richer” rule. Recent work has given tight asymptotic bounds on the diameter of preferential attachment networks [9, 14]; depending on the precise model, these bounds grow logarithmically =-=[30]-=- or even slower than logarithmically in the number of nodes. How are assumptions (A) and (B) reflected in data on network growth? Empirical studies of large networks to date have mainly focused on sta... |

9 |
Hierarchical Graph Maps
- Abello
(Show Context)
Citation Context ... produces densifying graphs, is an example of a hierarchical graph generation model, in which the linkage probability between nodes decreases as a function of their relative distance in the hierarchy =-=[13, 26, 51, 36, 34, 1]-=-. Again, there is a distinction between the aims of this past work and our model here; where these earlier network models were seeking to capture properties of individual snapshots of a graph, we seek... |

6 |
Disordered networks generated by recursive searches, Europhys
- Vázquez
(Show Context)
Citation Context ...d growing network with redirection model [29], which produces graphs with constant diameter and logarithmically increasing average degree [30]. Similar to our Forest Fire Model is the work of Vazquez =-=[48, 49]-=- where ideas based on random walks and recursive search for generating networks were introduced. In a random walk model the walk starts at a random node, follows links, and for each visited node with ... |