Connected Components in Random Graphs with Given Expected Degree Sequences

2002 ◽  
Vol 6 (2) ◽  
pp. 125-145 ◽  
Author(s):  
Fan Chung ◽  
Linyuan Lu
Keyword(s):  
Random Graphs ◽  
Connected Components ◽  
Degree Sequences

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 Degree sequences of random graphs

1981 ◽  
Vol 33 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Random Graphs ◽  
Degree Sequences

scholarly journals The degree sequences and spectra of scale-free random graphs

2006 ◽  
Vol 29 (2) ◽  
pp. 226-242 ◽  
Author(s):  
Jonathan Jordan
Keyword(s):  
Random Graphs ◽  
Degree Sequences ◽  
Scale Free

scholarly journals Erratum: Quasi-random graphs with given degree sequences

2008 ◽  
Vol 33 (4) ◽  
pp. 536-536
Author(s):  
Fan Chung ◽  
Ron Graham
Keyword(s):  
Random Graphs ◽  
Degree Sequences

scholarly journals Quasi-random graphs with given degree sequences

2007 ◽  
Vol 32 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Fan Chung ◽  
Ron Graham
Keyword(s):  
Random Graphs ◽  
Degree Sequences

scholarly journals On the Spectra of General Random Graphs

10.37236/702 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Fan Chung ◽  
Mary Radcliffe
Keyword(s):  
Independent Random Variable ◽  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Error Bounds ◽  
Adjacency Matrix ◽  
Random Variable ◽  
Degree Sequences ◽  
Chernoff Inequality

We consider random graphs such that each edge is determined by an independent random variable, where the probability of each edge is not assumed to be equal. We use a Chernoff inequality for matrices to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent upon the minimum and maximum expected degrees. In particular, we use these results to bound the spectra of random graphs with given expected degree sequences, including random power law graphs. Moreover, we prove a similar result giving concentration of the spectrum of a matrix martingale on its expectation.

scholarly journals Network dismantling

2016 ◽  
Vol 113 (44) ◽  
pp. 12368-12373 ◽  
Author(s):  
Alfredo Braunstein ◽  
Luca Dall’Asta ◽  
Guilhem Semerjian ◽  
Lenka Zdeborová
Keyword(s):  
Statistical Mechanics ◽  
Large Class ◽  
Random Graph ◽  
Random Graphs ◽  
Degree Distribution ◽  
Connected Components ◽  
Minimal Size ◽  
Epidemic Spreading ◽  
Minimal Set ◽  
Heavy Tailed

We study the network dismantling problem, which consists of determining a minimal set of vertices in which removal leaves the network broken into connected components of subextensive size. For a large class of random graphs, this problem is tightly connected to the decycling problem (the removal of vertices, leaving the graph acyclic). Exploiting this connection and recent works on epidemic spreading, we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed (light-tailed) degree distribution. Building on the statistical mechanics perspective, we propose a three-stage Min-Sum algorithm for efficiently dismantling networks, including heavy-tailed ones for which the dismantling and decycling problems are not equivalent. We also provide additional insights into the dismantling problem, concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well-performing nodes.

scholarly journals Limits of multiplicative inhomogeneous random graphs and Lévy trees: limit theorems

2021 ◽  
Author(s):  
Nicolas Broutin ◽  
Thomas Duquesne ◽  
Minmin Wang
Keyword(s):  
Random Graphs ◽  
Metric Spaces ◽  
Companion Paper ◽  
Explicit Construction ◽  
Convergence Condition ◽  
Asymptotic Distributions ◽  
Connected Components ◽  
Inhomogeneous Random Graphs ◽  
The Masses ◽  
The Continuum

AbstractWe consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs. arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), the asymptotic homogeneous case as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or the power-law case as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).

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