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

Author(s):  
Amin Coja-Oghlan ◽  
André Lanka
Keyword(s):  
2007 ◽  
Vol 16 (6) ◽  
pp. 923-946 ◽  
Author(s):  
AMIN COJA-OGHLAN

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.


2016 ◽  
Vol 50 (4) ◽  
pp. 584-611 ◽  
Author(s):  
Ronen Eldan ◽  
Miklós Z. Rácz ◽  
Tselil Schramm

10.37236/278 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
József Balogh ◽  
Béla Csaba ◽  
Martin Pei ◽  
Wojciech Samotij

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.


10.37236/227 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Amin Coja-Oghlan ◽  
André Lanka

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].


Author(s):  
V. F. Kolchin
Keyword(s):  

Author(s):  
A.C.C. Coolen ◽  
A. Annibale ◽  
E.S. Roberts

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.


Author(s):  
Mark Newman

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.


Sign in / Sign up

Export Citation Format

Share Document