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THe largest eigenvalue of sparse random graphs
[article]

2001
*
arXiv
*
pre-print

We prove that for all values

arXiv:math/0106066v1
fatcat:io5mbidx7fhxbd7grfwwhswnzm
*of**the*edge probability p(n)*the**largest**eigenvalue**of*a*random**graph*G(n,p) satisfies almost surely: λ_1(G)=(1+o(1))max√(Δ),np, where Δ is a maximal degree*of*G, and*the*o ... Here we are to find*the*asymptotic value*of**the**largest**eigenvalue**of**sparse**random**graphs*. ...*The*subject*of*this paper is asymptotic behavior*of**the**largest**eigenvalue*λ 1 (G(n, p))*of**random**graphs*. ...##
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The Largest Eigenvalue of Sparse Random Graphs

2003
*
Combinatorics, probability & computing
*

We prove that, for all values

doi:10.1017/s0963548302005424
fatcat:ske7l3usbvhjde5rra5qjqvusa
*of**the*edge probability p(n),*the**largest**eigenvalue**of**the**random**graph*G(n, p) satisfies almost surely λ 1 (G) = (1 + o(1)) max{ √ ∆, np}, where ∆ is*the*maximum degree ...*of*G, and*the*o(1) term tends to zero as max{ √ ∆, np} tends to infinity. ... Concluding remarks In this paper we have found*the*asymptotic value*of**the**largest**eigenvalue**of**the**random**graph*G(n, p), or*the*spectral radius*of**the*corresponding*random*real symmetric matrix. ...##
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Spectra of "real-world" graphs: Beyond the semicircle law

2001
*
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
*

While

doi:10.1103/physreve.64.026704
pmid:11497741
fatcat:g4rh4k6ne5h4bh7kkomzujvzue
*the*semi-circle law is known to describe*the*spectral density*of*uncorrelated*random**graphs*, much less is known about*the**eigenvalues**of*real-world*graphs*, describing such complex systems as*the*... Many natural and social systems develop complex networks, that are usually modelled as*random**graphs*.*The**eigenvalue*spectrum*of*these*graphs*provides information about their structural properties. ... Section IV contains our results concerning*the*spectra and special*eigenvalues**of**the*three main types*of**random**graph*models:*sparse*uncorrelated*random**graphs*in Sec. ...##
###
Discrepancy and eigenvalues of Cayley graphs

2016
*
Czechoslovak Mathematical Journal
*

This positively answers a question

doi:10.1007/s10587-016-0302-x
fatcat:tw7ohim7hbcj5a2tborfaijrmu
*of*Chung and Graham ["*Sparse*quasi-*random**graphs*", Combinatorica 22 (2002), no. 2, 217-244] for*the*particular case*of*Cayley*graphs**of*abelian groups, while in general ... ., small discrepancy) and having large*eigenvalue*gap are equivalent properties for such Cayley*graphs*, even if they are*sparse*. ... Statement*of**the*main result We use*the*following notation. If G = (V, E) is a*graph*, we write e(G) for*the*number*of*edges |E| in G. ...##
###
Optimal Laplacian regularization for sparse spectral community detection
[article]

2020
*
arXiv
*
pre-print

In this paper we formally determine a proper regularization which is intimately related to alternative state-

arXiv:1912.01419v2
fatcat:asj5lyjb6fhkvj2mvgq7tvkk7y
*of*-*the*-art spectral techniques for*sparse**graphs*. ... Regularization*of**the*classical Laplacian matrices was empirically shown to improve spectral clustering in*sparse*networks. ... ACKNOWLEDGEMENTS Couillet's work is supported by*the*IDEX GSTATS DataScience Chair and*the*MIAI LargeDATA Chair at University Grenoble Alpes. ...##
###
Spectral redemption in clustering sparse networks

2013
*
Proceedings of the National Academy of Sciences of the United States of America
*

*The*spectrum

*of*this operator is much better-behaved than that

*of*

*the*adjacency matrix or other commonly used matrices, maintaining a strong separation between

*the*bulk

*eigenvalues*and

*the*

*eigenvalues*... Here we introduce a new class

*of*spectral algorithms based on a non-backtracking walk on

*the*directed edges

*of*

*the*

*graph*. ...

*of*

*random*regular

*graphs*. ...

##
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Localized eigenvectors of the non-backtracking matrix
[article]

2016
*
arXiv
*
pre-print

In

arXiv:1505.07543v3
fatcat:v6fnjoympzahpkrd36hsupsb2e
*the*case*of**graph*partitioning,*the*emergence*of*localized eigenvectors can cause*the*standard spectral method to fail. ... However, we show that localized eigenvectors*of**the*non-backtracking matrix can exist outside*the*spectral band, which may lead to deterioration in*the*performance*of**graph*partitioning. ... This work was supported by JSPS KAKENHI No. 26011023 and*the*JSPS Core-to-Core Program "Non-equilibrium dynamics*of*soft matter and information." ...##
###
Spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities

2009
*
Physical Review E
*

We also find a characteristic pattern

doi:10.1103/physreve.80.026123
pmid:19792216
fatcat:qwfc5mpchvgojhki6mn4i73x4m
*of*periodic localization along*the*chains on*the*tree for*the*eigenvector components associated with*the**largest**eigenvalue*equal 2*of**the*Laplacian. ... We corroborate*the*results with simulations*of**the**random*walk on several types*of*networks. ...*The*simulations*of**random*walks on trees and on*sparse*modular*graphs*with minimal connectivity M Ն 2 is presented in Sec. V. ...##
###
Percolation on Sparse Networks

2014
*
Physical Review Letters
*

By considering

doi:10.1103/physrevlett.113.208702
pmid:25432059
fatcat:rulcxe2xh5eknaq7j5yovecqwm
*the*fixed points*of**the*message passing process, we also show that*the*percolation threshold on a network with few loops is given by*the*inverse*of**the*leading*eigenvalue**of**the*so-called ...*The*calculations are exact for*sparse*networks when*the*number*of*short loops in*the*network is small, but even on networks with many short loops we find them to be highly accurate when compared with direct ... After this work was completed we learned*of*concurrent work by Hamilton and Pryadko [24] in which a similar result for*the*percolation threshold is derived. ...##
###
Similarity-Aware Spectral Sparsification by Edge Filtering
[article]

2018
*
arXiv
*
pre-print

*The*proposed method has been validated using various kinds

*of*

*graphs*obtained from public domain

*sparse*matrix collections relevant to VLSI CAD, finite element analysis, as well as social and data networks ... Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from

*the*original

*graph*to form

*the*backbone

*of*

*the*sparsifier, and then recover small portions

*of*spectrally-critical ... to preserve

*the*original

*graph*spectrum within ultra-

*sparse*subgraphs (

*graph*sparsifiers), which allows preserving not only cuts in

*the*

*graph*but also

*eigenvalues*and eigenvectors

*of*

*the*original

*graph*...

##
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An iterative Jacobi-like algorithm to compute a few sparse eigenvalue-eigenvector pairs
[article]

2021
*
arXiv
*
pre-print

We show

arXiv:2105.14642v2
fatcat:chbypx6i7jdpdonolr4hy6rhbm
*the*effectiveness*of**the*method for*sparse*low-rank approximations and show applications to*random*symmetric matrices,*graph*Fourier transforms, and with*the**sparse*principal component analysis ...*The*method is also particularly well suited for*the*computation*of**sparse*eigenspaces. ... We generate*random*community*graphs**of*n = 256 nodes with*the**Graph*Signal Processing Toolbox 2 for which we decompose*the**sparse*positive semidefinite Laplacians as L = UΛU T and recover*the*eigenvectors ...##
###
Balancing sparse matrices for computing eigenvalues

2000
*
Linear Algebra and its Applications
*

Results are given comparing

doi:10.1016/s0024-3795(00)00014-8
fatcat:45p3qspjg5eozmdz2d3ng2rcoi
*the*Krylov-based algorithms to each other and to*the**sparse*and dense direct balancing algorithms, looking at norm reduction, running times, and*the*accuracy*of**eigenvalues*... We first discuss our*sparse*implementation*of**the*dense algorithm; our code is faster than*the*dense algorithm when*the*density*of**the*matrix is no more than approximately .5, and is much faster for large ... Acknowledgements We would like to thank Zhaojun Bai for useful discussions, Beresford Parlett for reading an earlier version*of*this paper, and Weihua Shen for helping to write*the*...##
###
Top Eigenpair Statistics for Weighted Sparse Graphs
[article]

2019
*
arXiv
*
pre-print

*The*analytical results are in perfect agreement with numerical diagonalisation

*of*large (weighted) adjacency matrices, and are further cross-checked on

*the*cases

*of*

*random*regular

*graphs*and

*sparse*Markov ... We develop a formalism to compute

*the*statistics

*of*

*the*top eigenpair

*of*weighted

*sparse*

*graphs*with finite mean connectivity and bounded maximal degree. ... Acknowledgments

*The*authors acknowledge funding by

*the*Engineering and Physical Sciences Research Council (EPSRC) through

*the*Centre for Doctoral Training in Cross Disciplinary Approaches to Non-Equilibrium ...

##
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Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations
[article]

2022
*
arXiv
*
pre-print

We analyse

arXiv:2009.11211v4
fatcat:s3xvh5oo6vgs7o756fd25ssxfi
*the*stability*of*linear dynamical systems defined on*sparse*,*random**graphs*with predator-prey, competitive, and mutualistic interactions. ... We develop an exact theory for*the*spectral distribution and*the*leading*eigenvalue**of**the*corresponding*sparse*Jacobian matrices. ... Kühn who contributed in*the*initial stages*of**the*project and for insightful discussions on*the*replica method. We thank J-P. Bouchaud, F.L. Metz, T. Galla, G. Torrisi, and V.A.R. ...##
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GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis
[article]

2020
*
arXiv
*
pre-print

Spectral

arXiv:1911.04382v3
fatcat:rpwnqsymhfbq7m4ykfnqqowmne
*graph*sparsification aims to find ultra-*sparse*subgraphs whose Laplacian matrix can well approximate*the*original Laplacian*eigenvalues*and eigenvectors. ... Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from*the*original*graph*to form*the*backbone*of**the*sparsifier, and then recover small portions*of*spectrally-critical ... We implement*the*accelerated spectral*graph*partitioning algorithm, and test it with*sparse*matrices in [8] and several 2D mesh*graphs*synthesized with*random*edge weights. ...
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