Mixture models and exploratory analysis in networks

Newman, M. E. J.; Leicht, Elizabeth

doi:10.1073/pnas.0610537104

Cited by 477 publications

(436 citation statements)

References 28 publications

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“…On the other hand, Guimerà, Sales-Pardo, and Amaral [85] and Fortunato and Barthélemy [73] showed that random graphs have high-modularity subsets and that there exists a size scale below which communities cannot be identified. In part as a response to this, some recent work has had a more statistical flavor [86,140,144,94,133]. In light of our results, this work seems promising, both due to potential "overfitting" issues arising from the extreme sparsity of the networks, and also due to the empirically-promising regularization properties exhibited by local spectral methods.…”

Section: Relationship With Community Identification Methodsmentioning

confidence: 77%

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

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,502

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: Relationship With Community Identification Methodsmentioning

confidence: 77%

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

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,502

1,082

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“…Runtime for the main loop in MATLAB on a 2 GHz laptop is ~6 min for N = 10 6 nodes with average degree 16 and K = 4.] Furthermore, we note that previous methods in which parameter inference is performed by optimizing a likelihood function via expectation maximization (EM) [11,18] are also special cases of the framework presented here. EM is a limiting case of VB in which one collapses the distributions over parameters to point estimates at the mode of each distribution; however EM is prone to over-fitting and cannot be used to determine the appropriate number of modules, as the likelihood of observed data increases with the number of modules in the model.…”

Section: Main Loopmentioning

confidence: 99%

“…Update the variational distribution over parameters from the expected counts and pseudocounts (11) (12) (15) where C = N(N − 1)/2, M =Σ i>j A ij , and u⃗ is a N-by-1 vector of 1's;…”

Section: Main Loopmentioning

confidence: 99%

Bayesian Approach to Network Modularity

2008

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We present an efficient, principled, and interpretable technique for inferring module assignments and for identifying the optimal number of modules in a given network. We show how several existing methods for finding modules can be described as variant, special, or limiting cases of our work, and how the method overcomes the resolution limit problem, accurately recovering the true number of modules. Our approach is based on Bayesian methods for model selection which have been used with success for almost a century, implemented using a variational technique developed only in the past decade. We apply the technique to synthetic and real networks and outline how the method naturally allows selection among competing models.Large-scale networks describing complex interactions among a multitude of objects have found application in a wide array of fields, from biology to social science to information technology [1,2]. In these applications one often wishes to model networks, suppressing the complexity of the full description while retaining relevant information about the structure of the interactions [3]. One such network model groups nodes into modules, or "communities," with different densities of intra-and interconnectivity for nodes in the same or different modules. We present here a computationally efficient Bayesian framework for inferring the number of modules, model parameters, and module assignments for such a model.The problem of finding modules in networks (or "community detection") has received much attention in the physics literature, wherein many approaches [4,5] focus on optimizing an energy-based cost function with fixed parameters over possible assignments of nodes into modules. The particular cost functions vary, but most compare a given node partitioning to an implicit null model, the two most popular being the configuration model and a limited version of the stochastic block model (SBM) [6,7]. While much effort has gone into how to optimize these cost functions, less attention has been paid to what is to be optimized. In recent studies which emphasize the importance of the latter question it was shown that there are inherent problems with existing approaches regardless of how optimization is performed, wherein parameter choice sets a lower limit on the size of detected modules, referred to as the "resolution limit" problem [8,9]. We extend recent probabilistic treatments of modular networks [10,11] to develop a solution to this problem that relies on inferring distributions over the model parameters, as opposed to asserting parameter values a priori, to determine the modular structure of a given network. The developed techniques are principled, interpretable, computationally efficient, and can be shown to generalize several previous studies on module detection.

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“…Along with the rapid development of network clustering techniques, the ability of revealing overlaps between communities has become very important as well [86,9,39,83,31,89,57,71,52]. Indeed, communities in realworld graphs are often inherently overlapping: each person in a social web belongs usually to several groups (family, colleagues, friends, etc.…”

Section: Applications: Community Finding and Clusteringmentioning

confidence: 99%

k-Clique Percolation and Clustering

Palla

Ábel

Farkas

et al. 2008

Bolyai Society Mathematical Studies

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We summarise recent results connected to the concept of k-clique percolation. This approach can be considered as a generalisation of edge percolation with a great potential as a community finding method in real-world graphs. We present a detailed study of the critical point for the appearance of a giant kclique percolation cluster in the Erdős-Rényi-graph. The observed transition is continuous and at the transition point the scaling of the giant component with the number of vertices is highly non-trivial. The concept is extended to weighted and directed graphs as well. Finally, we demonstrate the effectiveness of k-clique percolation as a community finding method via a series of real-world applications.

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Mixture models and exploratory analysis in networks

Cited by 477 publications

References 28 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

Bayesian Approach to Network Modularity

k-Clique Percolation and Clustering

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