scholarly journals The Spectral Gap of Random Graphs with Given Expected Degrees

2006 ◽  
pp. 15-26 ◽  
Author(s):  
Amin Coja-Oghlan ◽  
André Lanka
Keyword(s):  
Random Graphs ◽  
Spectral Gap

On the Laplacian Eigenvalues of Gn,p

2007 ◽  
Vol 16 (6) ◽  
pp. 923-946 ◽  
Author(s):  
AMIN COJA-OGHLAN
Keyword(s):  
Random Graphs ◽  
Spectral Gap ◽  
Degree Sequences ◽  
Laplacian Eigenvalues ◽  
Combinatorial Laplacian ◽  
Laplacian Spectra

We investigate the Laplacian eigenvalues of sparse random graphs Gnp. We show that in the case that the expected degree d = (n-1)p is bounded, the spectral gap of the normalized Laplacian $\LL(\gnp)$ is o(1). Nonetheless, w.h.p. G = Gnp has a large subgraph core(G) such that the spectral gap of $\LL(\core(G))$ is as large as 1-O (d−1/2). We derive similar results regarding the spectrum of the combinatorial Laplacian L(Gnp). The present paper complements the work of Chung, Lu and Vu [8] on the Laplacian spectra of random graphs with given expected degree sequences. Applied to Gnp, their results imply that in the ‘dense’ case d ≥ ln2n the spectral gap of $\LL(\gnp)$ is 1-O (d−1/2) w.h.p.

scholarly journals Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

2016 ◽  
Vol 50 (4) ◽  
pp. 584-611 ◽  
Author(s):  
Ronen Eldan ◽  
Miklós Z. Rácz ◽  
Tselil Schramm
Keyword(s):  
Random Graphs ◽  
Spectral Gap ◽  
Braess’S Paradox

scholarly journals Large Bounded Degree Trees in Expanding Graphs

10.37236/278 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
József Balogh ◽  
Béla Csaba ◽  
Martin Pei ◽  
Wojciech Samotij
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Spectral Gap ◽  
Sufficient Condition ◽  
Regular Graphs ◽  
Bounded Degree ◽  
Large Trees ◽  
Remarkable Result ◽  
Bounded Degree Trees ◽  
Expanding Graphs

A remarkable result of Friedman and Pippenger gives a sufficient condition on the expansion properties of a graph to contain all small trees with bounded maximum degree. Haxell showed that under slightly stronger assumptions on the expansion rate, their technique allows one to find arbitrarily large trees with bounded maximum degree. Using a slightly weaker version of Haxell's result we prove that a certain family of expanding graphs, which includes very sparse random graphs and regular graphs with large enough spectral gap, contains all almost spanning bounded degree trees. This improves two recent tree-embedding results of Alon, Krivelevich and Sudakov.

scholarly journals The Spectral Gap of Random Graphs with Given Expected Degrees

10.37236/227 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Amin Coja-Oghlan ◽  
André Lanka
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Degree Distribution ◽  
High Probability ◽  
Spectral Gap ◽  
Sparse Graphs ◽  
Laplacian Eigenvalues

We investigate the Laplacian eigenvalues of a random graph $G(n,\vec d)$ with a given expected degree distribution $\vec d$. The main result is that w.h.p. $G(n,\vec d)$ has a large subgraph core$(G(n,\vec d))$ such that the spectral gap of the normalized Laplacian of core$(G(n,\vec d))$ is $\geq1-c_0\bar d_{\min}^{-1/2}$ with high probability; here $c_0>0$ is a constant, and $\bar d_{\min}$ signifies the minimum expected degree. The result in particular applies to sparse graphs with $\bar d_{\min}=O(1)$ as $n\rightarrow\infty$. The present paper complements the work of Chung, Lu, and Vu [Internet Mathematics 1, 2003].

Random Graphs

1998 ◽  
Author(s):  
V. F. Kolchin
Keyword(s):  
Random Graphs

Applications of random graphs

2017 ◽  
Author(s):  
A.C.C. Coolen ◽  
A. Annibale ◽  
E.S. Roberts
Keyword(s):  
Social Networks ◽  
Random Graphs ◽  
Interaction Network ◽  
Power Grids ◽  
Protein Protein Interaction ◽  
Topological Features ◽  
First Case ◽  
The Stability ◽  
Protein Protein Interaction Network

This chapter reviews graph generation techniques in the context of applications. The first case study is power grids, where proposed strategies to prevent blackouts have been tested on tailored random graphs. The second case study is in social networks. Applications of random graphs to social networks are extremely wide ranging – the particular aspect looked at here is modelling the spread of disease on a social network – and how a particular construction based on projecting from a bipartite graph successfully captures some of the clustering observed in real social networks. The third case study is on null models of food webs, discussing the specific constraints relevant to this application, and the topological features which may contribute to the stability of an ecosystem. The final case study is taken from molecular biology, discussing the importance of unbiased graph sampling when considering if motifs are over-represented in a protein–protein interaction network.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

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