A Random Graph Model for Power Law Graphs

Aiello, William; Chung, Fan; Lü, Linyuan

doi:10.1080/10586458.2001.10504428

Cited by 337 publications

(441 citation statements)

References 14 publications

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“…We use G(w) to denote a random graph generated in this manner. (Note that this model is different than the so-called "configuration model" in which the degree distribution is exactly specified and which was studied by Molloy and Reed [124,125] and also Aiello, Chung, and Lu [7,8]. This model is also different than generative models such as preferential attachment models [9,127,25] or models based on optimization [57,58,61], although common to all of these generative models is that they attempt to reproduce empirically-observed power-law behavior [11,62,27,129,50].…”

Section: Very Sparse Random Graphs Have Very Unbalanced Deep Cutsmentioning

confidence: 96%

“…(These particular networks were chosen to be representative of the wide range of networks we have examined, and for ease of comparison we will compute other properties for them in future sections. See Figures 7,8, and 9 in Section 3.4 for the NCP plots of other networks listed in Tables 1, 2 and 3, and for a discussion of them.) The most striking feature of these plots is that the NCP plot is steadily increasing for nearly its entire range.…”

Section: Community Profile Plots For Large Social and Information Netmentioning

confidence: 99%

“…Notice that in all cases the "best" communities are quite small (typically between 10 and 100 nodes) and that the network community profile plot steadily increases for nearly its entire range. See Figures 7,8, and 9 for the NCP plots of other networks.…”

Section: Community Profile Plots For Large Social and Information Netmentioning

confidence: 99%

“…3.4 More community profile plots for large social and information networks Figures 7,8, and 9 show additional examples of NCP plots for networks from Tables 1, 2 and 3. In the first two rows of Figure 7, we have several examples of purely Social networks and two email networks, in the third row we have patent and blog Information/citation networks, and in the final row we have three examples of actor and author Collaboration networks.…”

Section: Community Profile Plots For Large Social and Information Netmentioning

confidence: 99%

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Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,512

1,082

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A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions.Rather than defining a procedure to extract sets of nodes from a graph and then attempt to interpret these sets as a "real" communities, we employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the "best" possible community-according to the conductance measure-over a wide range of size scales. We study over 100 large real-world networks, ranging from traditional and on-line social networks, to technological and information networks and web graphs, and ranging in size from thousands up to tens of millions of nodes.Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. Our observations agree with previous work on small networks, but we show that large networks have a very different structure. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales (up to ≈ 100 nodes); and communities of size scale beyond ≈ 100 nodes gradually "blend into" the expanderlike core of the network and thus become less "community-like," with a roughly inverse relationship between community size and optimal community quality. This observation agrees well with the so-called Dunbar number which gives a limit to the size of a well-functioning community.However, this behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as testbeds of community detection algorithms. The relatively gradual increase of the network community profile plot as a function of increasing community size depends in a subtle manner on the way in which local clustering information is propagated from smaller to larger size scales in the network. We have found that a generative graph model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe i...

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Section: Very Sparse Random Graphs Have Very Unbalanced Deep Cutsmentioning

confidence: 96%

Section: Community Profile Plots For Large Social and Information Netmentioning

confidence: 99%

Section: Community Profile Plots For Large Social and Information Netmentioning

confidence: 99%

Section: Community Profile Plots For Large Social and Information Netmentioning

confidence: 99%

See 2 more Smart Citations

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,512

1,082

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show abstract

“…In that context, the web graph is a power-law graph, which means roughly that the probability that a degree is larger than d is at least d −β for some β > 0. Models for power-law graphs are developed in [42], [7], [78].…”

Section: Evaluating the Results Of Data Miningmentioning

confidence: 99%

On Approximation Algorithms for Data Mining Applications

Afrati

2006

Lecture Notes in Computer Science

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Mathematical results on scale‐free random graphs

Bollobás

Riordan

2002

Handbook of Graphs and Networks

336

315

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IntroductionRecently there has been much interest in studying large-scale real-world networks and attempting to model their properties using random graphs. Although the study of real-world networks as graphs goes back some time, recent activity perhaps started with the paper of Watts and Strogatz [55] about the 'smallworld phenomenon'. Since then the main focus of attention has shifted to the 'scale-free' nature of the networks concerned, evidenced by, for example, powerlaw degree distributions. It was quickly observed that the classical models of random graphs introduced by Erdős and Rényi [28] and Gilbert [33] are not appropriate for studying these networks, so many new models have been introduced. The work in this field falls very roughly into the following categories.1. Direct studies of the real-world networks themselves, measuring various properties such as degree-distribution, diameter, clustering, etc.

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A Random Graph Model for Power Law Graphs

Cited by 337 publications

References 14 publications

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

On Approximation Algorithms for Data Mining Applications

Mathematical results on scale‐free random graphs

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