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Graph Weight Allocation to Meet Laplacian Spectral Constraints
2012
IEEE Transactions on Automatic Control
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We adjust the node and edge weightings of graphs using convex optimization to impose bounds on their Laplacian spectra. ...
Finally, we suggest efficient ways to accommodate larger graphs, and show that dual formulations lead to substantial improvement in the size of graphs that can be addressed. ...
DUAL FORMULATION OF THE QUASICONVEX LARGEST EIGENVALUE MINIMIZATION PROBLEM Popular interior point methods, such as SDPT3 [19] , when applied to the convex problems derived Section IV, are limited as ...
doi:10.1109/tac.2011.2181795
fatcat:w3taqoxwybezzaqgtj6n6gowx4
Designing Optimal Multiplex Networks for Certain Laplacian Spectral Properties
[article]
2020
arXiv
pre-print
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Minimizing the largest eigenvalue is driven by the spectral radius of the individual networks and its corresponding eigenvector. ...
Before a threshold, the total budget is distributed among interlayer edges corresponding to the nodal lines of this eigenvector, and the optimal largest eigenvalue of the Laplacian remains constant. ...
MINIMIZING λn In this section, we study the problem of minimizing the largest eigenvalue, λ n , of the Laplacian in multiplex networks. ...
arXiv:1903.01073v2
fatcat:2nnlo2dg4jdoji35mcmfxkjtru
Graph spectral techniques in computer sciences
2012
Applicable Analysis and Discrete Mathematics
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We have described the applications of some important graph eigenvalues (spectral radius, algebraic connectivity, the least eigenvalue etc.), eigenvectors (principal eigenvector, Fiedler eigenvector and ...
The survey consists of a description of particular topics from the theory of graph spectra independently of the areas of Computer science in which they are used. ...
The authors are grateful to the referees for the many useful comments and suggestions which have led to an improvement of this paper. Sup- ...
doi:10.2298/aadm111223025a
fatcat:msxavfozrzddlc5ipbcbw7ithu
Robust Spectral Detection of Global Structures in the Data by Learning a Regularization
[article]
2016
arXiv
pre-print
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Using extensive experiments, we illustrate that our method solves the localization problem and works down to the theoretical detectability limits in different kinds of synthetic data. ...
Using matrix perturbation analysis, we demonstrate that the learned regularizations suppress down the eigenvalues associated with localized eigenvectors and enable us to recover the informative eigenvectors ...
The eigenvalues of L X are also plotted in Fig. 5 where one can see clearly that there is a gap between the second largest eigenvalue and the third one. ...
arXiv:1609.02906v1
fatcat:cwkhymgfcfb2nawo2c22a6oxui
On Spectral Graph Drawing
[chapter]
2003
Lecture Notes in Computer Science
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The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. ...
We present a novel view of the spectral approach, which provides a direct link between eigenvectors and the aesthetic properties of the layout. ...
Thus, the series converges in the direction of the next dominant eigenvector, which is either u 2 , which has the largest positive eigenvalue, or u n , which possibly has the largest negative eigenvalue ...
doi:10.1007/3-540-45071-8_50
fatcat:vhxjoefvojb33fw7ccq5to2foa
First-principles multiway spectral partitioning of graphs
2014
Journal of Complex Networks
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defined by the leading eigenvectors of the graph Laplacian. ...
We consider the minimum-cut partitioning of a graph into more than two parts using spectral methods. ...
Acknowledgments The authors thank Raj Rao Nadakuditi for useful insights and suggestions. This work was funded in part by the National Science Foundation under grant DMS-1107796. ...
doi:10.1093/comnet/cnt021
fatcat:mdbiql3ry5d45mttkbtsbcpqiu
Steering Macro-Scale Network Community Structure by Micro-Scale Features
[article]
2016
arXiv
pre-print
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Similar to pointing and focusing a magnifying glass, the analysis can be directed to specific micro-scale structure, while the degree of interaction with the macro-scale community structure can be seamlessly ...
of the analysis (e.g., nodal features). ...
Spectral graph theory relates to the classical graph cut problem that consists of partitioning the graph into clusters of nodes such that the cut size is minimized; i.e., the total weight of the edges ...
arXiv:1603.05623v1
fatcat:s7hlfn52mbgolhi3cnrf567ab4
Efficient Filtering Of Graph Based Data Using Graph Partitioning
2017
Zenodo
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In this paper, we adapt the spectral graph partitioning method for partitioning of graphs and use it to reduce the computational cost of the filtering problem. ...
We use the example of denoising of the temperature data to illustrate the efficacy of the approach. ...
A limitation of the graph-Laplacian based approach is that it works only if the graph is undirected. ...
doi:10.5281/zenodo.1129853
fatcat:75phd6hjqvbi3ouhy4o44ni62e
Explainable Graph Theory-Based Identification of Meter-Transformer Mapping
[article]
2022
arXiv
pre-print
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all the k smallest eigenvalues of the voltage-based Laplacian matrix are smaller than the next smallest eigenvalue of the ideal Laplacian matrix. ...
Therefore, we propose to utilize the above observation via spectral embedding by using the property that inter-transformer meter consumptions are not the same and that the noise in data is limited so that ...
data is limited so that all the k smallest eigenvalues of the voltage-based Laplacian matrix are smaller than the (k + 1)-th smallest eigenvalue of the ideal Laplacian matrix. ...
arXiv:2205.09874v1
fatcat:wtumze2sijex7anasy5ltvij5u
Vertices cannot be hidden from quantum spatial search for almost all random graphs
2018
Quantum Information Processing
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The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the ·_∞ norm. ...
At the same time for p<(1-ε)(n)/n, the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight. ...
Here δ and ∆ denote respectively minimal and maximal degree of the graph. ...
doi:10.1007/s11128-018-1844-7
fatcat:y3b2pdan25a6rni5huqdjbu5uu
Minimally Redundant Laplacian Eigenmaps
2018
International Conference on Learning Representations
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In experiments on NORB and similarity-transformed faces we show that Minimally Redundant Laplacian Eigenmap (MR-LEM) significantly improves the quality of embedding vectors over Laplacian Eigenmaps, accurately ...
We present a simple extension of Laplacian Eigenmaps to fix this problem based on choosing embedding vectors which are both orthogonal and minimally redundant to other dimensions of the embedding. ...
This demonstrates the importance of choosing minimally predictable eigenvectors as the embedding. (a) Bottom eigenvectors of graph Laplacian on evenly spaced grid. ...
dblp:conf/iclr/PfauB18
fatcat:gtwyalwn7rdifdvdxzlvnuwbgi
A Tutorial on Spectral Clustering
[article]
2007
arXiv
pre-print
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We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. ...
The goal of this tutorial is to give some intuition on those questions. ...
Apart from applications of graph Laplacians to partitioning problems in the widest sense, graph Laplacians can also be used for completely different purposes, for example for graph drawing (Koren, 2005 ...
arXiv:0711.0189v1
fatcat:f3htfhjg3rhnvdcg2eron7w2gi
A tutorial on spectral clustering
2007
Statistics and computing
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We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. ...
The goal of this tutorial is to give some intuition on those questions. ...
Apart from applications of graph Laplacians to partitioning problems in the widest sense, graph Laplacians can also be used for completely different purposes, for example for graph drawing (Koren, 2005 ...
doi:10.1007/s11222-007-9033-z
fatcat:am7au6ssovadhelmisoqemt33m
RNA Graph Partitioning for the Discovery of RNA Modularity: A Novel Application of Graph Partition Algorithm to Biology
2014
PLoS ONE
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The graph partitioning algorithms utilize mathematical properties of the Laplacian eigenvector (m2) corresponding to the second eigenvalues (l2) associated with the topology matrix defining the graph: ...
The three types of algorithms, termed median, sign, and gap cuts, divide a graph by determining nodes of cut by median, zero, and largest gap of m29s components, respectively. ...
Funding This work is supported by the National Science Foundation (DMS-0201160, CCF-0727001) and the National Institute of Health (GM100469, GM081410). ...
doi:10.1371/journal.pone.0106074
pmid:25188578
pmcid:PMC4154854
fatcat:hj2whpdixrdpvhzacj6blhtf4a
Graph-Laplacian PCA: Closed-Form Solution and Robustness
2013
2013 IEEE Conference on Computer Vision and Pattern Recognition
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We propose a graph-Laplacian PCA (gLPCA) to learn a low dimensional representation of X that incorporates graph structures encoded in W . ...
In many applications, both vector data X and graph data W are available. Laplacian embedding is widely used for embedding graph data. ...
This work is supported by National Natural Science Foundation of China (No.61073116, 61211130309, 61202228) and U.S. National Science Foundation NSF-CCF-0917274, NSF-DMS-0915228. ...
doi:10.1109/cvpr.2013.448
dblp:conf/cvpr/JiangDLT13
fatcat:etrh4fkxwnacxlev6opa3iptkm
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