## R-MAT: A recursive model for graph mining (2004)

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Venue: | In Fourth SIAM International Conference on Data Mining (SDM’ 04 |

Citations: | 138 - 17 self |

### BibTeX

@INPROCEEDINGS{Chakrabarti04r-mat:a,

author = {Deepayan Chakrabarti and Yiping Zhan and Christos Faloutsos},

title = {R-MAT: A recursive model for graph mining},

booktitle = {In Fourth SIAM International Conference on Data Mining (SDM’ 04},

year = {2004}

}

### Years of Citing Articles

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### Abstract

How does a ‘normal ’ computer (or social) network look like? How can we spot ‘abnormal ’ sub-networks in the Internet, or web graph? The answer to such questions is vital for outlier detection (terrorist networks, or illegal money-laundering 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 ” (R-MAT) 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 Erdős-Rényi 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 (AutoMAT-fast) fits them very well. 1

### Citations

1401 | The structure and function of complex networks
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(Show Context)
Citation Context ...nt graph generators can be grouped in two classes: degree based and procedural. Given a degree distribution (typically following a power-law), the degreebased ones try to find a graph that matches it =-=[2, 17]-=-, but without giving any insights about the graph or trying to match other criteria (like small diameter, eigenvalues etc.). On the other hand, procedural generators try to find simple mechanisms to g... |

1241 | On power-law relationships of the internet topology
- Faloutsos, Faloutsos, et al.
- 1999
(Show Context)
Citation Context ...s, networks and their surprising regularities/laws have been attracting significant interest recently. The World Wide Web, the Internet topology and Peer-toPeer networks follows surprising power-laws =-=[5, 10, 3]-=-, exhibit strange “bow-tie” or “jellyfish” structures [5, 23], while still having a small diameter [2]. Finding patterns, laws and regularities in large real networks has numerous applications, from c... |

337 | Inferring Web communities from link topology
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Citation Context ...ved deviations from power-laws for the Web graph, which are well-modeled by the truncated, discretized lognormal (“DGX”) distribution of Bi et al. [4]. Graphs also exhibit a strong “community” effect =-=[11, 14]-=-. Most real graphs have surprisingly small diameters: the “six degrees of separation” for the social network, 19 for the Web, and small values for the Internet AS graph [2, 23]. Apart from these, ther... |

304 | The web as a graph: Measurements, models and methods
- Kleinberg, Kumar, et al.
- 1999
(Show Context)
Citation Context ...e been observed for the degree distributions of the Internet, the WWW and the citation graph, the distribution of “bipartite cores” (≈ communities), the eigenvalues of the adjacency matrix and others =-=[10, 13, 2]-=-. Recently, Pennock et al. [20] observed deviations from power-laws for the Web graph, which are well-modeled by the truncated, discretized lognormal (“DGX”) distribution of Bi et al. [4]. Graphs also... |

277 | Epidemic spreading in scale-free networks
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- 2001
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Citation Context ...having a small diameter [2]. Finding patterns, laws and regularities in large real networks has numerous applications, from criminology and law enforcement [8] to analyzing virus propagation patterns =-=[19]-=-and understanding networks of regulatory genes and interacting proteins [3] and so on. Discovering and listing such laws is only the first step. Ideally, we would like a generative model with the foll... |

250 |
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Citation Context ...e Web, and small values for the Internet AS graph [2, 23]. Apart from these, there are many other measures such as clustering coefficient, expansion, resilience, prestige, influence, stress and so on =-=[7, 12, 22, 18]-=-. Broder et al. [5] show that the WWW has a “bow-tie” structure with 4 roughly equal parts. Tauro et al. [23] find that the Internet topology is organized as a set of concentric circles around a small... |

209 | Mining knowledge-sharing sites for viral marketing - Richardson, Domingos |

207 | On distinguishing between internet power law topology generators
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Citation Context ...wever, this only gives power laws of exponent 3. Many modifications and alternatives to the basic idea have been proposed; some generators also include the geometrical layout of nodes in their models =-=[1, 2, 17, 20, 6]-=-. The BRITE graph generator [15] uses components from several of the above models. In general, all of the above generators fail to meet one or more of the following goals: (a) the generator should be ... |

189 | On the Origin of Power Laws in Internet Topologies
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Citation Context ... 3. Many modifications and alternatives to the basic idea have been proposed; some generators also include the geometrical layout of nodes in their models [1, 2, 17, 20, 6]. The BRITE graph generator =-=[15]-=- uses components from several of the above models. In general, all of the above generators fail to meet one or more of the following goals: (a) the generator should be procedural (b) it should be able... |

138 |
and Albert-Laszlo Barabasi. Statistical mechanics of complex networks
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Citation Context ...rld Wide Web, the Internet topology and Peer-toPeer networks follows surprising power-laws [5, 10, 3], exhibit strange “bow-tie” or “jellyfish” structures [5, 23], while still having a small diameter =-=[2]-=-. Finding patterns, laws and regularities in large real networks has numerous applications, from criminology and law enforcement [8] to analyzing virus propagation patterns [19]and understanding netwo... |

113 | Winners don’t take all: Characterizing the competition for links on the web
- Pennock, Flake, et al.
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Citation Context ... subsumes the celebrated Erdős-Rényi 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. =-=[20]-=-). We present results on multiple, large real graphs, where we show that our parameter fitting algorithm (AutoMAT-fast) fits them very well. 1 Introduction Graphs, networks and their surprising regula... |

103 | Extracting large-scale knowledge bases from the web
- Kumar, Raghavan, et al.
- 1999
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Citation Context ...ved deviations from power-laws for the Web graph, which are well-modeled by the truncated, discretized lognormal (“DGX”) distribution of Bi et al. [4]. Graphs also exhibit a strong “community” effect =-=[11, 14]-=-. Most real graphs have surprisingly small diameters: the “six degrees of separation” for the social network, 19 for the Web, and small values for the Internet AS graph [2, 23]. Apart from these, ther... |

102 |
Graph structure in the web: experiments and models
- Broder, Kumar, et al.
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Citation Context ...s, networks and their surprising regularities/laws have been attracting significant interest recently. The World Wide Web, the Internet topology and Peer-toPeer networks follows surprising power-laws =-=[5, 10, 3]-=-, exhibit strange “bow-tie” or “jellyfish” structures [5, 23], while still having a small diameter [2]. Finding patterns, laws and regularities in large real networks has numerous applications, from c... |

93 | Anf: A fast and scalable tool for data mining in massive graphs
- Palmer, Gibbons, et al.
- 2002
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Citation Context ...e Web, and small values for the Internet AS graph [2, 23]. Apart from these, there are many other measures such as clustering coefficient, expansion, resilience, prestige, influence, stress and so on =-=[7, 12, 22, 18]-=-. Broder et al. [5] show that the WWW has a “bow-tie” structure with 4 roughly equal parts. Tauro et al. [23] find that the Internet topology is organized as a set of concentric circles around a small... |

85 | Generating network topologies that obey power laws
- Palmer, Steffan
- 2000
(Show Context)
Citation Context ...er duplicate elimination (a, b, c, d) Probabilities of an edge falling into partitions in the R-MAT model. a + b + c + d = 1. Table 1: Table of Symbols generators have been proposed in passing before =-=[19]-=-, but the emphasis was on network issues. 3.1 Fast Algorithm to generate Directed Graphs: The adjacency matrix A of a graph of N nodes is an N ∗ N matrix, with entry a(i, j) = 1 if the edge (i, j) exi... |

74 | Spectral analysis of Internet topologies
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- 2003
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Citation Context ...e Web, and small values for the Internet AS graph [2, 23]. Apart from these, there are many other measures such as clustering coefficient, expansion, resilience, prestige, influence, stress and so on =-=[7, 12, 22, 18]-=-. Broder et al. [5] show that the WWW has a “bow-tie” structure with 4 roughly equal parts. Tauro et al. [23] find that the Internet topology is organized as a set of concentric circles around a small... |

69 | A simple conceptual model for the Internet topology
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Citation Context ...racting significant interest recently. The World Wide Web, the Internet topology and Peer-toPeer networks follows surprising power-laws [5, 10, 3], exhibit strange “bow-tie” or “jellyfish” structures =-=[5, 23]-=-, while still having a small diameter [2]. Finding patterns, laws and regularities in large real networks has numerous applications, from criminology and law enforcement [8] to analyzing virus propaga... |

49 | The DGX distribution for mining massive, skewed data
- Bi, Faloutsos, et al.
- 2001
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Citation Context ...hool of Computer Science, CMU ‡School of Computer Science, CMU This is exactly the main part of this work. We propose the Recursive Matrix (R-MAT) model, which naturally generates power-law (or “DGX” =-=[4]-=- ) degree distributions. We show that it naturally leads to smallworld graphs; it is recursive (=self-similar), and it has only a small number of parameters. The rest of this paper is organized as fol... |

40 |
Identifying web browsing trends and patterns
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Citation Context ...m epinions.com [21]: N = 75, 879; E = 508, 960. Epinions-U: An undirected version of the Epinions graph: N = 75, 879; E = 811, 602. Clickstream: A bipartite graph of Internet users’ browsing behavior =-=[16]-=-. An edge (u, p) denotes that user u accessed page p. It has 23, 396 users, 199, 308 pages and 952, 580 edges. Apart from degree distributions, we compare the models on their singular value vs. rank p... |

31 | Network topologies, power laws and hierarchy
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Citation Context |

22 | COPLINK Connect: information and knowledge management for law enforcement", Decision Support Systems
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Citation Context ...“jellyfish” structures [5, 23], while still having a small diameter [2]. Finding patterns, laws and regularities in large real networks has numerous applications, from criminology and law enforcement =-=[8]-=- to analyzing virus propagation patterns [19]and understanding networks of regulatory genes and interacting proteins [3] and so on. Discovering and listing such laws is only the first step. Ideally, w... |

15 |
Linked: The New Science of Networks, Perseus Publishing
- Barabási
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Citation Context ...s, networks and their surprising regularities/laws have been attracting significant interest recently. The World Wide Web, the Internet topology and Peer-toPeer networks follows surprising power-laws =-=[5, 10, 3]-=-, exhibit strange “bow-tie” or “jellyfish” structures [5, 24], while still having a small diameter [2]. Finding patterns, laws and regularities in large real networks has numerous applications, from c... |

9 |
Topology of complex networks: Local events and universality
- Albert, Barabasi
- 2000
(Show Context)
Citation Context ...wever, this only gives power laws of exponent 3. Many modifications and alternatives to the basic idea have been proposed; some generators also include the geometrical layout of nodes in their models =-=[1, 2, 17, 20, 6]-=-. The BRITE graph generator [15] uses components from several of the above models. In general, all of the above generators fail to meet one or more of the following goals: (a) the generator should be ... |