On the spectrum of the normalized graph Laplacian

Banerjee, Anirban; Jost, Jürgen

doi:10.1016/j.laa.2008.01.029

Cited by 112 publications

(145 citation statements)

References 8 publications

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“…Depending on whether one is considering the adjacency matrix or the Laplacian matrix, localized eigenvectors can correspond to structural inhomogeneities such as very high degree nodes or very small cluster-like sets of nodes. In addition, localization is often preserved or modified in characteristic ways when a graph is generated by modifying an existing graph in a structured manner; and thus it has been used as a diagnostic in certain network applications [193,194]. The implications of the algorithms described in this review remain to be explored for these and other applications where eigenvector localization is a significant phenomenon.…”

Section: Statistical Leverage In Large-scale Data Analysismentioning

confidence: 99%

Randomized Algorithms for Matrices and Data

Mahoney¹

2012

Chapman &Amp; Hall/CRC Data Mining and Knowledge Discovery Series

335

373

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Randomized algorithms for very large matrix problems have received a great deal of attention in recent years. Much of this work was motivated by problems in large-scale data analysis, largely since matrices are popular structures with which to model data drawn from a wide range of application domains, and this work was performed by individuals from many different research communities. While the most obvious benefit of randomization is that it can lead to faster algorithms, either in worst-case asymptotic theory and/or numerical implementation, there are numerous other benefits that are at least as important. For example, the use of randomization can lead to simpler algorithms that are easier to analyze or reason about when applied in counterintuitive settings; it can lead to algorithms with more interpretable output, which is of interest in applications where analyst time rather than just computational time is of interest; it can lead implicitly to regularization and more robust output; and randomized algorithms can often be organized to exploit modern computational architectures better than classical numerical methods.This monograph will provide a detailed overview of recent work on the theory of randomized matrix algorithms as well as the application of those ideas to the solution of practical problems in large-scale data analysis. Throughout this review, an emphasis will be placed on a few simple core ideas that underlie not only recent theoretical advances but also the usefulness of these tools in large-scale data applications. Crucial in this context is the connection with the concept of statistical leverage. This concept has long been used in statistical regression diagnostics to identify outliers; and it has recently proved crucial in the development of improved worst-case matrix algorithms that are also amenable to high-quality numerical implementation and that are useful to domain scientists. This connection arises naturally when one explicitly decouples the effect of randomization in these matrix algorithms from the underlying linear algebraic structure. This decoupling also permits much finer control in the application of randomization, as well as the easier exploitation of domain knowledge.Most of the review will focus on random sampling algorithms and random projection algorithms for versions of the linear least-squares problem and the low-rank matrix approximation problem. These two problems are fundamental in theory and ubiquitous in practice. Randomized methods solve these problems by constructing and operating on a randomized sketch of the input matrix Afor random sampling methods, the sketch consists of a small number of carefully-sampled and rescaled columns/rows of A, while for random projection methods, the sketch consists of a small number of linear combinations of the columns/rows of A. Depending on the specifics of the situation, when compared with the best previously-existing deterministic algorithms, the resulting randomized algorithms have worst-case running time that is asympt...

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Section: Statistical Leverage In Large-scale Data Analysismentioning

confidence: 99%

Randomized Algorithms for Matrices and Data

Mahoney¹

2012

Chapman &Amp; Hall/CRC Data Mining and Knowledge Discovery Series

335

373

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“…The spreading of physical or chemical substances, rather than information content, requires imposing mass conservation, which results in a different formulation of the Laplacian operator 30 , where W ij is formally replaced by W ji in the definition of D ij , as it can be readily obtained from a simple microscopic derivation. Clearly, the two operators coincide, when defined on a symmetric network.…”

Section: Introductionmentioning

confidence: 99%

The theory of pattern formation on directed networks

et al. 2014

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Dynamical processes on networks have generated widespread interest in recent years. The theory of pattern formation in reaction-diffusion systems defined on symmetric networks has often been investigated, due to its applications in a wide range of disciplines. Here we extend the theory to the case of directed networks, which are found in a number of different fields, such as neuroscience, computer networks and traffic systems. Owing to the structure of the network Laplacian, the dispersion relation has both real and imaginary parts, at variance with the case for a symmetric, undirected network. The homogeneous fixed point can become unstable due to the topology of the network, resulting in a new class of instabilities, which cannot be induced on undirected graphs. Results from a linear stability analysis allow the instability region to be analytically traced. Numerical simulations show travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph.

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“…Examples include the cases where X is the adjacency matrix [9], the (in-degree or out-degree) Laplacian [13], normalized Laplacian [1], etc.…”

Section: Introductionmentioning

confidence: 99%

Strong targeted controllability of dynamical networks

Monshizadeh

Camlibel

Trentelman

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-Network controllability is the ability to control the entire network, meaning that we can drive the network from any initial state to any desired final state in finite time by using appropriate inputs which are applied to a subset of nodes of the network. Despite obvious advantages, network controllability is not always feasible as it may ask for a considerable portion of the nodes to be controlled. Moreover, there are cases where controllability of the entire network is not of interest, but rather we are interested in controllability properties of certain parts of the network. This motivates us to investigate the so-called "targeted controllability" of the network, where controllability is only required for a subset of nodes in the network. Noting that targeted controllability can be treated as an output controllability problem, we investigate the (strong) structural output controllability properties of the network from a topological viewpoint. In addition, we examine the structural properties of the reachable subspace of the network. To this end, we use the notion of zero forcing sets, which has been recently exploited in the context of structural controllability.

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On the spectrum of the normalized graph Laplacian

Cited by 112 publications

References 8 publications

Randomized Algorithms for Matrices and Data

Randomized Algorithms for Matrices and Data

The theory of pattern formation on directed networks

Strong targeted controllability of dynamical networks

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