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## Kronecker Graphs: An Approach to Modeling Networks (2010)

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Venue: | JOURNAL OF MACHINE LEARNING RESEARCH 11 (2010) 985-1042 |

Citations: | 123 - 3 self |

### Citations

3932 | Emergence of scaling in random networks
- Barabási, Albert
- 1999
(Show Context)
Citation Context ...ch real networks in a number of respects (for example, it does not produce heavy-tailed degree distributions). The vast majority of recent network models involve some form of preferential attachment (=-=Barabási and Albert, 1999-=-; Albert and Barabási, 2002; Winick and Jamin, 2002; Kleinberg et al., 1999; Kumar et al., 1999; Flaxman et al., 2007) that employs a simple rule: new node joins the graph at each time step, and then ... |

3323 |
Collective dynamics of ’small-world’ networks
- Watts, Strogatz
- 1998
(Show Context)
Citation Context ...andom surfer model” (Blum et al., 2006), etc. A different family of network methods strives for small diameter and local clustering in networks. Examples of such models include the small-world model (=-=Watts and Strogatz, 1998-=-) and the Waxman generator (Waxman, 1988). Another family of models shows that heavy tails emerge if nodes try to optimize their connectivity under resource constraints (Carlson and Doyle, 1999; Fabri... |

3064 | On the evolution of random graphs
- Erdős, Rényi
- 1960
(Show Context)
Citation Context ...t hold as the diameter shrinks and then seems to stabilize as the network grows. 2.2 Generative models of network structure The earliest probabilistic generative model for graphs was the Erdős-Rényi (=-=Erdős and Rényi, 1960-=-) random graph model, where each pair of nodes has an identical, independent probability of being joined by an edge. The study of this model has led to a rich mathematical theory. However, as the mode... |

2214 |
On information and sufficiency
- Kullback, Leibler
- 1951
(Show Context)
Citation Context ... problem: One could first identify the set of network properties (statistics) to match, then define a quality of fit 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... |

2143 | Statistical mechanics of complex networks - Albert, Barabási - 2002 |

1564 |
Modelling by Shortest Data Description
- Rissanen
- 1978
(Show Context)
Citation Context ...iple global maxima of the likelihood space but as N1 is small the additional term would make no real difference. As an alternative 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. 6. Experiments on real and synthetic data Next we described ... |

1416 | Self-similarity in world wide web traffic: Evidence and possible causes
- Crovella, Bestavros
- 1996
(Show Context)
Citation Context ... to extract the network hierarchical structure have also been developed (Sales-Pardo et al., 2007; Clauset et al., 2008). Moreover, especially web graphs (Dill et al., 2002; Dorogovtsev et al., 2002; =-=Crovella and Bestavros, 1997-=-) and biological networks (Ravasz and Barabási, 2003) were found to be self-similar and “fractal”. The second intuition comes from viewing every node of Pk as being described with an ordered sequence ... |

1286 |
The small world problem
- Milgram
- 1967
(Show Context)
Citation Context ...ne social networks (Chakrabarti et al., 2004) and many others. Small diameter: Most real-world graphs exhibit relatively small diameter (the “small- world” phenomenon, or “six degrees of separation” (=-=Milgram, 1967-=-)): A graph has diameter D if every pair of nodes can be connected by a path of length at most D edges. The diameter D is susceptible to outliers. Thus, a more robust measure of the pair wise distance... |

823 | The small-world phenomenon: an algorithm perspective. In:
- Kleinberg
- 2000
(Show Context)
Citation Context ...component and another phase transition for connectivity, and proved that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm (=-=Kleinberg, 1999-=-). General overview of Kronecker product is given in (Imrich and Klavˇzar, 2000) and properties of Kronecker graphs related to graph minors, planarity, cut vertex and cut edge have been explored in (B... |

777 |
Routing of multipoint connections
- Waxman
- 2002
(Show Context)
Citation Context ...erent family of network methods strives for small diameter and local clustering in networks. Examples of such models include the small-world model (Watts and Strogatz, 1998) and the Waxman generator (=-=Waxman, 1988-=-). Another family of models shows that heavy tails emerge if nodes try to optimize their connectivity under resource constraints (Carlson and Doyle, 1999; Fabrikant et al., 2002). In summary, most cur... |

734 |
How Nature Works: The Science of Self-organized Criticality,
- Bak
- 1996
(Show Context)
Citation Context ...work. 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” or phase transition (=-=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... |

607 | Power-law distributions in empirical data
- Clauset, Shalizi, et al.
(Show Context)
Citation Context ...portant, because a careful choice of the initial graph K1 makes the resulting multinomial distribution to behave like a power law or Discrete Gaussian Exponential (DGX) distribution (Bi et al., 2001; =-=Clauset et al., 2007-=-). Theorem 5 (Multinomial degree distribution) Kronecker graphs have multinomial degree distributions, for both in- and out-degrees. Proof Let the initiator K1 have the degree sequence d1,d2,... ,dN1 ... |

537 | Graphs over time: densification laws, shrinking diameters and possible explanations - Leskovec, Kleinberg, et al. - 2005 |

496 | Group formation in large social networks: Membership, growth, and evolution. - Backstrom, Huttenlocher, et al. - 2006 |

420 | Hierarchical organization of modularity in metabolic networks - Ravasz, Somera, et al. |

400 | Structure and evolution of online social networks. - Kumar - 2006 |

373 | The Web as a graph: Measurements, Models, and Methods
- Kleinberg, Kumar, et al.
- 1999
(Show Context)
Citation Context ... power law if the number of nodes Nd with degree d is given by Nd ∝ d−γ (γ > 0) where γ is called the power law exponent. Power laws have been found in the Internet (Faloutsos et al., 1999), the Web (=-=Kleinberg et al., 1999-=-; Broder et al., 2000), citation graphs (Redner, 1998), online social networks (Chakrabarti et al., 2004) and many others. Small diameter: Most real-world graphs exhibit relatively small diameter (the... |

360 | Mapping the gnutella network: Properties of large-scale peer-to-peer systems and implications for system design - Ripeanu, Foster, et al. |

337 |
Markov chain Monte Carlo: Stochastic Simulation for Bayesian Inference
- Gamerman
- 1997
(Show Context)
Citation Context ...σu,σv] P[σ ′ u ,σ′ v ] (1 − P[σu,σv]) (1 − P[σ ′ u,σ ′ v]) ∏ (u,v)/∈G (σu,σv)̸=(σ ′ u,σ ′ v) (1 − P[σu,σv]) (1 − P[σ ′ u ,σ′ v ]) This immediately suggests the use of a Metropolis sampling algorithm (=-=Gamerman, 1997-=-) to simulate draws from the permutation distribution since Metropolis is solely based on such ratios (where normalizing constants cancel out). In particular, suppose that in the Metropolis algorithm ... |

321 | Logit models and logistic regression for social networks: I. An introduction to Markov graphs and p
- Wasserman, Pattison
- 1996
(Show Context)
Citation Context ...ed. Most work in estimating network models comes from the area of social sciences, statistics and social network analysis where the exponential random graphs, also known as p∗ model, were introduced (=-=Wasserman and Pattison, 1996-=-). The model essentially defines a log linear model over all possible graphs G, p(G|θ) ∝ exp(θ T s(G)), where G is a graph, and s is a set of functions, that can be viewed as summary statistics for th... |

289 | Stochastic models for the web graph.
- Kumar, Raghavan, et al.
- 2000
(Show Context)
Citation Context ... “shrinking diameter” property mentioned above. There are also many variations of preferential attachment model, all somehow employing the “rich get richer” type mechanism, e.g., the “copying model” (=-=Kumar et al., 2000-=-), the “winner does not take all” model (Pennock et al., 2002), the “forest fire” model (Leskovec et al., 2005b), the “random surfer model” (Blum et al., 2006), etc. A different family of network meth... |

286 |
Limit Theorems of Probability Theory
- Petrov
- 1995
(Show Context)
Citation Context ...e 6(c) we first have to choose b following by c. The probability of each individual edge of Pk follows a Bernoulli distribution, as the edge occurrences are independent. By the Central Limit Theorem (=-=Petrov, 1995-=-) the number of edges Instead of flipping O(N2 ) = O(N2k 1 in Pk tends to a normal distribution with mean ( ∑ N1 i,j=1 θij) k = E k 1 , where θij ∈ P1. So, given a stochastic initiator matrix P1 we fi... |

271 | Trust management for the semantic web
- Richardson, Agrawal, et al.
- 2003
(Show Context)
Citation Context ...We consider several different networks, like a graph of connectivity among Internet Autonomous systems (AS-ROUTEVIEWS) with N =6,474 and E =26,467 a who-trusts-whom type social network from Epinions (=-=Richardson et al., 2003-=-) (EPINIONS) with N =75,879 and E =508,960 and 40KRONECKER GRAPHS: AN APPROACH TO MODELING NETWORKS many others. The largest network we consider for fitting is FLICKR photo-sharing online social netw... |

266 | Graph evolution: Densification and shrinking diameters - LESKOVEC, KLEINBERG, et al. - 2007 |

265 | Scale-Free Networks," - Barabási, Bonabeau - 2003 |

260 |
How popular is your paper? an empirical study of the citation distribution.
- Redner
- 1998
(Show Context)
Citation Context ...Nd ∝ d−γ (γ > 0) where γ is called the power law exponent. Power laws have been found in the Internet (Faloutsos et al., 1999), the Web (Kleinberg et al., 1999; Broder et al., 2000), citation graphs (=-=Redner, 1998-=-), online social networks (Chakrabarti et al., 2004) and many others. Small diameter: Most real-world graphs exhibit relatively small diameter (the “small- world” phenomenon, or “six degrees of separa... |

256 | On Distinguishing between Internet power law topology generators,"
- Bu, Towsley
- 2002
(Show Context)
Citation Context ...Our model also obeys recently discovered temporal evolution patterns (Leskovec et al., 2005b, 2007a). And, contrary to other models that match this combination of network properties (as for example, (=-=Bu and Towsley, 2002-=-; Klemm and Eguíluz, 2002; Vázquez, 2003; Leskovec et al., 2005b; Zheleva et al., 2009)), Kronecker graphs also lead to tractable analysis and rigorous proofs. Furthermore, the Kronecker graphs genera... |

248 | Hierarchical structure and the prediction of missing links in networks - Clauset, Moore, et al. |

245 | R-MAT: A recursive model for graph mining.
- Chakrabarti, Zhan, et al.
- 2004
(Show Context)
Citation Context ...his paper we introduce Kronecker graphs, a generative network model which obeys all the main static network patterns that have appeared in the literature (Faloutsos et al., 1999; Albert et al., 1999; =-=Chakrabarti et al., 2004-=-; Farkas et al., 2001; Mihail and Papadimitriou, 2002; Watts and Strogatz, 1998). Our model also obeys recently discovered temporal evolution patterns (Leskovec et al., 2005b, 2007a). And, contrary to... |

244 | Statistical properties of community structure in large social and information networks - Leskovec, Lang, et al. - 2008 |

230 | Product Graphs: Structure and Recognition - Imrich, Klavžar - 2000 |

213 | Hierarchical organization in complex networks, - Ravasz, Barabasi - 2003 |

208 | Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters.
- Leskovec, Lang, et al.
- 2009
(Show Context)
Citation Context ...ts recursively — the core again has the dense core and the periphery, and so on. And similarly the periphery itself has the core and the periphery. This suggest an “onion” like nested core-periphery (=-=Leskovec et al., 2008-=-a,b) network structure as illustrated in Figure 24(c), where the network is composed of denser and denser layers as one moves towards the center of the network. We also observe similar structure of th... |

178 | Heuristically Optimized Tradeoffs: A New Paradigm for Power Laws in the Internet
- Fabrikant, Koutsoupias, et al.
- 2002
(Show Context)
Citation Context ... 1998) and the Waxman generator (Waxman, 1988). Another family of models shows that heavy tails emerge if nodes try to optimize their connectivity under resource constraints (Carlson and Doyle, 1999; =-=Fabrikant et al., 2002-=-). In summary, most current models focus on modeling only one (static) network property, and neglect the others. In addition, it is usually hard to analytically analyze properties of the network model... |

176 | Highly optimized tolerance: a mechanism for power laws in designed systems
- Carlson, Doyle
- 1999
(Show Context)
Citation Context ...odel (Watts and Strogatz, 1998) and the Waxman generator (Waxman, 1988). Another family of models shows that heavy tails emerge if nodes try to optimize their connectivity under resource constraints (=-=Carlson and Doyle, 1999-=-; Fabrikant et al., 2002). In summary, most current models focus on modeling only one (static) network property, and neglect the others. In addition, it is usually hard to analytically analyze propert... |

168 | Inet-3.0: Internet Topology Generator
- Winick, Jamin
- 2002
(Show Context)
Citation Context ... it does not produce heavy-tailed degree distributions). The vast majority of recent network models involve some form of preferential attachment (Barabási and Albert, 1999; Albert and Barabási, 2002; =-=Winick and Jamin, 2002-=-; Kleinberg et al., 1999; Kumar et al., 1999) that employs a simple rule: new node joins the graph at each time step, and then creates a connection to an existing node u with the probability proportio... |

154 | Models of core/periphery structures
- Borgatti, Everett
- 1999
(Show Context)
Citation Context ...s the top left element is the largest and then the values on the diagonal decay faster than off the diagonal (Leskovec, 2009). This suggests a network structure which is also known as core-periphery (=-=Borgatti and Everett, 2000-=-; Holme, 2005), the jellyfish (Tauro et al., 2001; Siganos et al., 2006), or the octopus (Chung and Lu, 2006) structure of the network as illustrated in Figure 24(c). All of the above basically say th... |

154 | Winners Don't Take All: Characterizing the Competition for Links on the Web, - Pennock, Flake, et al. - 2002 |

139 |
Diameter of the world-wide web.
- Albert, Jeong, et al.
- 1999
(Show Context)
Citation Context ...to be realistic. In this paper we introduce Kronecker graphs, a generative network model which obeys all the main static network patterns that have appeared in the literature (Faloutsos et al., 1999; =-=Albert et al., 1999-=-; Chakrabarti et al., 2004; Farkas et al., 2001; Mihail and Papadimitriou, 2002; Watts and Strogatz, 1998). Our model also obeys recently discovered temporal evolution patterns (Leskovec et al., 2005b... |

132 | Cascading behavior in large blog graphs. - Leskovec, McGlohon, et al. - 2007 |

121 | Extracting large-scale knowledge bases from the web.
- Kumar, Raghavan, et al.
- 1999
(Show Context)
Citation Context ...utions). The vast majority of recent network models involve some form of preferential attachment (Barabási and Albert, 1999; Albert and Barabási, 2002; Winick and Jamin, 2002; Kleinberg et al., 1999; =-=Kumar et al., 1999-=-; Flaxman et al., 2007) that employs a simple rule: new node joins the graph at each time step, and then creates a connection to an existing node u with the probability proportional to the degree of t... |

120 | Anf: a fast and scalable tool for data mining in massive graphs
- PALMER, GIBBONS, et al.
(Show Context)
Citation Context ...arabási, 2002; Milgram, 1967; Leskovec et al., 2005b). Hop-plot: It extends the notion of diameter by plotting the number of reachable pairs g(h) within h hops, as a function of the number of hops h (=-=Palmer et al., 2002-=-). It gives us a sense of how quickly nodes’ neighborhoods expand with the number of hops. Scree plot: This is a plot of the eigenvalues (or singular values) of the graph adjacency matrix, versus thei... |

115 |
Graph structure in the Web: experiments and models
- Broder, Kumar, et al.
(Show Context)
Citation Context ... of nodes Nd with degree d is given by Nd ∝ d−γ (γ > 0) where γ is called the power law exponent. Power laws have been found in the Internet (Faloutsos et al., 1999), the Web (Kleinberg et al., 1999; =-=Broder et al., 2000-=-), citation graphs (Redner, 1998), online social networks (Chakrabarti et al., 2004) and many others. Small diameter: Most real-world graphs exhibit relatively small diameter (the “small- world” pheno... |

106 | Realistic, mathematically tractable graph generation and evolution, using kronecker multiplication - LESKOVEC, CHAKRABARTI, et al. - 2005 |

106 | Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations. - Vazquez - 2003 |

101 | The ubiquitous Kronecker product - Loan - 2000 |

91 | A simple conceptual model for the internet topology
- Tauro, Palmer, et al.
- 2001
(Show Context)
Citation Context ...be connected by a path of length at most D edges. The diameter D is susceptible to outliers. Thus, a more robust measure of the pair wise distances between nodes in a graph is the effective diameter (=-=Tauro et al., 2001-=-), which is the minimum number of links (steps/hops) in which some fraction (or quantile q, say q = 0.9) of all connected pairs of nodes can reach each other. The effective diameter has been found to ... |

85 | A Tomkins. Self-similarity in the web. - Dill, Kumar, et al. - 2001 |

72 | Scalable modeling of real graphs using kronecker multiplication
- Leskovec, Faloutsos
(Show Context)
Citation Context ...system. Yet, we would like to share our network. Our work gives ways to such a realistic, ’similar’ network. The current paper builds on our previous work on Kronecker graphs (Leskovec et al., 2005a; =-=Leskovec and Faloutsos, 2007-=-) and is organized as follows: Section 2 briefly surveys the related literature. In section 3 we introduce the Kronecker graph model, and give formal statements about the properties of networks it gen... |

70 |
Estimating the Dimension of a Model. The Annals of Statistics, 6(2): 461–464
- Schwarz
- 1978
(Show Context)
Citation Context ...e appropriate value of N1, the size of matrix Θ, and choose the right tradeoff between the complexity of the model and the quality of the fit, we propose to use the Bayes Information Criterion (BIC) (=-=Schwarz, 1978-=-). Stochastic Kronecker graph model the presence of edges with independent Bernoulli random variables, where the canonical number of , which is a function of a lower-dimensional parameter Θ. This is t... |

66 | Spectra of “Real-world” graphs: beyond the semi-circle law, cond-mat/100235 - Farkas, Derenyi, et al. - 2001 |

66 |
Extracting the hierarchical organization of complex systems.
- Sales-Pardo, Guimera, et al.
- 2007
(Show Context)
Citation Context ...ers have argued that real networks are hierarchically organized (Ravasz et al., 2002; Ravasz and Barabási, 2003) and algorithms to extract the network hierarchical structure have also been developed (=-=Sales-Pardo et al., 2007-=-; Clauset et al., 2008). Moreover, especially web graphs (Dill et al., 2002; Dorogovtsev et al., 2002; Crovella and Bestavros, 1997) and biological networks (Ravasz and Barabási, 2003) were found to b... |

64 | Eigenvalues of random power law graphs - Chung, Lu, et al. - 2003 |

62 | On the Eigenvalue Power Law
- Mihail, Papadimitriou
- 2002
(Show Context)
Citation Context ...rative network model which obeys all the main static network patterns that have appeared in the literature (Faloutsos et al., 1999; Albert et al., 1999; Chakrabarti et al., 2004; Farkas et al., 2001; =-=Mihail and Papadimitriou, 2002-=-; Watts and Strogatz, 1998). Our model also obeys recently discovered temporal evolution patterns (Leskovec et al., 2005b, 2007a). And, contrary to other models that match this combination of network ... |

58 | The Kronecker Product of Graphs - Weichsel - 1962 |

56 | The DGX distribution for mining massive, skewed data,”
- Bi, Faloutsos, et al.
- 2001
(Show Context)
Citation Context ...model. This is important, because a careful choice of the initial graph K1 makes the resulting multinomial distribution to behave like a power law or Discrete Gaussian Exponential (DGX) distribution (=-=Bi et al., 2001-=-; Clauset et al., 2007). Theorem 5 (Multinomial degree distribution) Kronecker graphs have multinomial degree distributions, for both in- and out-degrees. Proof Let the initiator K1 have the degree se... |

49 | Fast counting of triangles in large real networks without counting: Algorithms and laws
- TSOURAKAKIS
- 2008
(Show Context)
Citation Context ...ighbors of a node. 4KRONECKER GRAPHS: AN APPROACH TO MODELING NETWORKS The plot of the number of triangles ∆ versus the number of nodes participating in ∆ triangles has also been found to be skewed (=-=Tsourakakis, 2008-=-). Densification Power Law: The relation between the number of edges E(t) and the number of nodes N(t) in evolving network at time t obeys the densification power law (DPL), which states that E(t) ∝ N... |

43 |
Highly clustered scale-free networks”,
- Klemm, Eguiluz
- 2002
(Show Context)
Citation Context ...ecently discovered temporal evolution patterns (Leskovec et al., 2005b, 2007a). And, contrary to other models that match this combination of network properties (as for example, (Bu and Towsley, 2002; =-=Klemm and Eguíluz, 2002-=-; Vázquez, 2003; Leskovec et al., 2005b; Zheleva et al., 2009)), Kronecker graphs also lead to tractable analysis and rigorous proofs. Furthermore, the Kronecker graphs generative process also has a n... |

42 |
Signs of life: How complexity pervades biology.
- Sole, Goodwin
- 2000
(Show Context)
Citation Context ...ion 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. Implications to the structu... |

33 |
BPseudofractal scale-free web
- Dorogovtsev, Goltsev, et al.
- 2002
(Show Context)
Citation Context ...bási, 2003) and algorithms to extract the network hierarchical structure have also been developed (Sales-Pardo et al., 2007; Clauset et al., 2008). Moreover, especially web graphs (Dill et al., 2002; =-=Dorogovtsev et al., 2002-=-; Crovella and Bestavros, 1997) and biological networks (Ravasz and Barabási, 2003) were found to be self-similar and “fractal”. The second intuition comes from viewing every node of Pk as being descr... |

30 |
Complex graphs and networks, volume 107
- Chung, Lu
- 2006
(Show Context)
Citation Context ...ec, 2009). This suggests a network structure which is also known as core-periphery (Borgatti and Everett, 2000; Holme, 2005), the jellyfish (Tauro et al., 2001; Siganos et al., 2006), or the octopus (=-=Chung and Lu, 2006-=-) structure of the network as illustrated in Figure 24(c). All of the above basically say that the network is composed of a densely linked network core and the periphery. In our case this would imply ... |

29 | A likelihood approach to analysis of network data,” - Wiuf, Brameier, et al. - 2006 |

27 | Stochastic Kronecker graphs.
- Mahdian, Xu
- 2007
(Show Context)
Citation Context ... 6 for the illustration of the recursive structure of P. 3.4 Additional properties of Kronecker graphs Stochastic Kronecker Graphs with initiator matrix of size N1 = 2 were studied by Mahdian and Xu (=-=Mahdian and Xu, 2007-=-). The authors showed a phase transition for the emergence of the giant component and another phase transition for connectivity, and proved that such graphs have constant diameters beyond the connecti... |

26 | Core-periphery organization of complex networks.
- Holme
- 2005
(Show Context)
Citation Context ...eft element is the largest and then the values on the diagonal decay faster than off the diagonal. This suggests a network structure which is also known as core-periphery (Borgatti and Everett, 2000; =-=Holme, 2005-=-), the jellyfish (Tauro et al., 2001; Siganos et al., 2006), or the octopus (Chung and Lu, 2006) structure of the network as illustrated in Figure 24(c). All of the above basically say that the networ... |

26 | Random dot product graph models for social networks. Workshop on Algorithms and Models for the Web-graph, - Young, Scheinerman - 2007 |

22 |
Factoring cardinal product graphs in polynomial time,
- Imrich
- 1998
(Show Context)
Citation Context ...behavior of several real networks (social networks, citations, web, internet, and others). While Kronecker products have been studied by the algebraic combinatorics community (see, e.g., (Chow, 1997; =-=Imrich, 1998-=-; Imrich and Klavˇzar, 2000; Hammack, 2009)), the present work is the first to employ this operation in the design of network models to match real data. Then we also make a step further and tackle the... |

19 | A random-surfer web-graph model”,
- Blum, Rwebangira
- 2006
(Show Context)
Citation Context ...pe mechanism, e.g., the “copying model” (Kumar et al., 2000), the “winner does not take all” model (Pennock et al., 2002), the “forest fire” model (Leskovec et al., 2005b), the “random surfer model” (=-=Blum et al., 2006-=-), etc. A different family of network methods strives for small diameter and local clustering in networks. Examples of such models include the small-world model (Watts and Strogatz, 1998) and the Waxm... |

19 | Characterization and modeling of proteinprotein interaction networks - Colizza, Flammini, et al. |

19 | Overview of the 2003 kdd cup - Gehrke, Ginsparg, et al. |

19 |
Random dot product graphs: A model for social networks
- Nickel
- 2006
(Show Context)
Citation Context ...set of features and the initiator matrix encodes the probability of linking given the attribute values of two nodes somewhat resembles the Random dot product graph model (Young and Scheinerman, 2007; =-=Nickel, 2008-=-). The important difference here is that we multiply individual linking probabilities, while in Random dot product graphs one takes the sum of individual probabilities which seems somewhat less natura... |

13 | Graph model selection using maximum likelihood
- Bezáková, Kalai, et al.
- 2006
(Show Context)
Citation Context ... of nodes in one network to nodes in the other network. For example, Butts (Butts, 2005) used permutation sampling to determine similarity between two graph adjacency matrices, while Bezáková et al. (=-=Bezáková et al., 2006-=-) used permu6KRONECKER GRAPHS: AN APPROACH TO MODELING NETWORKS SYMBOL DESCRIPTION G Real network N Number of nodes in G E Number of edges in G K Kronecker graph (synthetic estimate of G) K1 Initiato... |

5 |
Some remarks on the Kronecker product of graphs
- Bottreou, Métivier
- 1998
(Show Context)
Citation Context ...9). General overview of Kronecker product is given in (Imrich and Klavˇzar, 2000) and properties of Kronecker graphs related to graph minors, planarity, cut vertex and cut edge have been explored in (=-=Bottreau and Metivier, 1998-=-). Moreover, recently (Tsourakakis, 2008) gave a closed form expression for the number of triangles in a Kronecker graph that depends on the eigenvalues of the initiator graph K1. (2) 16KRONECKER GRA... |

5 | Proof of a conjecture concerning the direct product of bipartite graphs - Hammack |

4 | of Oregon Route Views Project. Online data and reports. http://www.routeviews.org - University - 1997 |

3 | http://www-personal.umich.edu/ ∼ mejn/netdata - data - 2007 |

2 |
Permutation models for relational data. (tech. rep. mbs 05-02
- Butts
- 2005
(Show Context)
Citation Context ... so we aim to find an accurate mapping between the nodes of the two graphs. An ordering or a permutation defines the mapping of nodes in one network to nodes in the other network. For example, Butts (=-=Butts, 2005-=-) used permutation sampling to determine similarity between two graph adjacency matrices, while Bezáková et al. (Bezáková et al., 2006) used permu6KRONECKER GRAPHS: AN APPROACH TO MODELING NETWORKS S... |

1 |
Networks, communities and kronecker products
- Leskovec
- 2009
(Show Context)
Citation Context ...hown in Figure 24, large a,d and small b,c would imply that the network is composed of hierarchically nested communities, where there are many edges inside each community and few edges crossing them (=-=Leskovec, 2009-=-). One could think of this structure as some kind of organizational or university hierarchy, where one expects the most friendships between people within same lab, a bit less between people in the sam... |

1 | Inet-3.0: Internet Topology Generator - CHAKRABARTI, FALOUTSOS, et al. - 2002 |

1 | Pseudofractal scale-free web - CHAKRABARTI, FALOUTSOS, et al. - 2002 |

1 | Stochastic kronecker graphs - CHAKRABARTI, FALOUTSOS, et al. - 2007 |

1 | Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters - CHAKRABARTI, FALOUTSOS, et al. - 2008 |