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THe largest eigenvalue of sparse random graphs [article]

Michael Krivelevich, Benny Sudakov
2001 arXiv   pre-print
We prove that for all values 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.  ...

The Largest Eigenvalue of Sparse Random Graphs

MICHAEL KRIVELEVICH, BENNY SUDAKOV
2003 Combinatorics, probability & computing
We prove that, for all values 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.  ...

Spectra of "real-world" graphs: Beyond the semicircle law

Illés J. Farkas, Imre Derényi, Albert-László Barabási, Tamás Vicsek
2001 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
While 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

Yoshiharu Kohayakawa, Vojtěch Rödl, Mathias Schacht
2016 Czechoslovak Mathematical Journal
This positively answers a question 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]

Lorenzo Dall'Amico, Romain Couillet, Nicolas Tremblay
2020 arXiv   pre-print
In this paper we formally determine a proper regularization which is intimately related to alternative state-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

F. Krzakala, C. Moore, E. Mossel, J. Neeman, A. Sly, L. Zdeborova, P. Zhang
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.  ...

Localized eigenvectors of the non-backtracking matrix [article]

Tatsuro Kawamoto
2016 arXiv   pre-print
In 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 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

Brian Karrer, M. E. J. Newman, Lenka Zdeborová
2014 Physical Review Letters
By considering 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]

Zhuo Feng
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  ...

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

Cristian Rusu
2021 arXiv   pre-print
We show 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

Tzu-Yi Chen, James W. Demmel
2000 Linear Algebra and its Applications
Results are given comparing 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]

Vito Antonio Rocco Susca, Pierpaolo Vivo, Reimer Kuehn
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  ...

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

Andrea Marcello Mambuca, Chiara Cammarota, Izaak Neri
2022 arXiv   pre-print
We analyse 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.  ...

GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis [article]

Zhuo Feng
2020 arXiv   pre-print
Spectral 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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