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13
Consistency of spectral clustering
, 2004
"... Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spe ..."
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Cited by 289 (15 self)
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Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spectral clustering algorithms, which cluster the data with the help of eigenvectors of graph Laplacian matrices. We show that one of the two of major classes of spectral clustering (normalized clustering) converges under some very general conditions, while the other (unnormalized), is only consistent under strong additional assumptions, which, as we demonstrate, are not always satisfied in real data. We conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering in practical applications. We believe that methods used in our analysis will provide a basis for future exploration of Laplacianbased methods in a statistical setting.
Distributed graph layout for sensor networks
 In 12th Symposium on Graph Drawing (GD
, 2004
"... Sensor network applications frequently require that the sensors know their physical locations in some global coordinate system. This is usually achieved by equipping each sensor with a location measurement device, such as GPS. However, lowend systems or indoor systems, which cannot use GPS, must lo ..."
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Cited by 29 (2 self)
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Sensor network applications frequently require that the sensors know their physical locations in some global coordinate system. This is usually achieved by equipping each sensor with a location measurement device, such as GPS. However, lowend systems or indoor systems, which cannot use GPS, must locate themselves based only on crude information available locally, such as intersensor distances. We show how a collection of sensors, capable only of measuring distances to close neighbors, can compute their locations in a purely distributed manner, i.e. where each sensor communicates only with its neighbors. This can be viewed as a distributed graph drawing algorithm. We experimentally show that our algorithm consistently produces good results under a variety of simulated realworld conditions, and is relatively robust to the presence of noise in the distance measurements.
Energy Models for Graph Clustering
"... The cluster structure of many realworld graphs is of great interest, as the clusters may correspond e.g. to communities in social networks or to cohesive modules in software systems. Layouts can naturally represent the cluster structure of graphs by grouping densely connected nodes and separating s ..."
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Cited by 15 (1 self)
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The cluster structure of many realworld graphs is of great interest, as the clusters may correspond e.g. to communities in social networks or to cohesive modules in software systems. Layouts can naturally represent the cluster structure of graphs by grouping densely connected nodes and separating sparsely connected nodes. This article introduces two energy models whose minimum energy layouts represent the cluster structure, one based on repulsion between nodes (like most existing energy models) and one based on repulsion between edges. The latter model is not biased towards grouping nodes with high degrees, and is thus more appropriate for the many realworld graphs with rightskewed degree distributions. The two energy models are shown to be closely related to widely used quality criteria for graph clusterings – namely the density of the cut, Shi and Malik’s normalized cut, and Newman’s modularity – and to objective functions optimized by eigenvectorbased graph drawing methods.
Structural Properties of the Caenorhabditis elegans Neuronal Network, PLoS
 Comput Biol
, 2011
"... Neuronal wiring diagrams and analysis of their structural properties can provide insights into the function of nervous systems. Using materials from White et al. and new electron micrographs, we assemble a whole neuronal wiring diagram of hermaphrodite Caenorhabditis elegans. We catalog various stat ..."
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Cited by 9 (3 self)
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Neuronal wiring diagrams and analysis of their structural properties can provide insights into the function of nervous systems. Using materials from White et al. and new electron micrographs, we assemble a whole neuronal wiring diagram of hermaphrodite Caenorhabditis elegans. We catalog various statistical and topological properties of the neuronal network and also propose a convenient method for visualization. The C. elegans neuronal network is far from random yet is statistically similar in many respects to other natural networks. We apply spectral analysis to network dynamics and provide a theoretical framework for predicting the propagation of signals in the network in response to sensory or artificial stimulation. These results should help plan experimental investigations of the network and facilitate discovery of principles governing network structure and function.
Unreliable and ResourceConstrained Decoding
, 2010
"... Traditional information theory and communication theory assume that decoders are noiseless and operate without transient or permanent faults. Decoders are also traditionally assumed to be unconstrained in physical resources like materiel, memory, and energy. This thesis studies how constraining reli ..."
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Cited by 3 (3 self)
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Traditional information theory and communication theory assume that decoders are noiseless and operate without transient or permanent faults. Decoders are also traditionally assumed to be unconstrained in physical resources like materiel, memory, and energy. This thesis studies how constraining reliability and resources in the decoder limits the performance of communication systems. Five communication problems are investigated. Broadly speaking these are communication using decoders that are wiring costlimited, that are memorylimited, that are noisy, that fail catastrophically,
Dynamic spectral layout of small worlds
 IN PROC. 13TH INT. SYMP. GRAPH DRAWING, GD
, 2005
"... Spectral methods are naturally suited for dynamic graph layout, because moderate changes of a graph yield moderate changes of the layout under weak assumptions. We discuss some general principles for dynamic graph layout and derive a dynamic spectral layout approach for the animation of smallworld ..."
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Cited by 2 (0 self)
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Spectral methods are naturally suited for dynamic graph layout, because moderate changes of a graph yield moderate changes of the layout under weak assumptions. We discuss some general principles for dynamic graph layout and derive a dynamic spectral layout approach for the animation of smallworld models.
Dynamic Spectral Layout with an Application to Small Worlds
, 2007
"... Spectral methods are naturally suited for dynamic graph layout because, usually, moderate changes of a graph yield moderate changes of the layout. We discuss some general principles for dynamic graph layout and derive a dynamic spectral layout approach for the animation of smallworld models. ..."
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Cited by 1 (1 self)
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Spectral methods are naturally suited for dynamic graph layout because, usually, moderate changes of a graph yield moderate changes of the layout. We discuss some general principles for dynamic graph layout and derive a dynamic spectral layout approach for the animation of smallworld models.
Graph Embedding and Nonlinear Dimensionality Reduction
"... Traditionally, spectral methods such as principal component analysis (PCA) have been applied to many graph embedding and dimensionality reduction tasks. These methods aim to find lowdimensional representations of data that preserve its inherent structure. However, these methods often perform poorly ..."
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Traditionally, spectral methods such as principal component analysis (PCA) have been applied to many graph embedding and dimensionality reduction tasks. These methods aim to find lowdimensional representations of data that preserve its inherent structure. However, these methods often perform poorly when applied to data which does not lie exactly near a linear manifold. In this thesis, I present a set of novel graph embedding algorithms which extend spectral methods, allowing graph representations of highdimensional data or networks to be accurately embedded in a lowdimensional space. I first propose minimum volume embedding (MVE) which, like other leading dimensionality reduction algorithms, first encodes the highdimensional data as a nearestneighbor graph, where the edge weights between neighbors correspond to kernel values between points, and then embeds this graph in a lowdimensional space. Next I present structure preserving embedding (SPE), an algorithm for embedding unweighted graphs where similarity between nodes is not known. SPE finds lowdimensional embeddings which explicitly preserve graph topology, meaning a connectivity algorithm, such as knearest neighbors, will recover the edges of the input graph from only the coordinates of the nodes after embedding. I further explore preserving graph