The degree sequences and spectra of scale-free random graphs

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.

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Degree sequences of random graphs

Discrete Mathematics ◽

10.1016/0012-365x(81)90253-3 ◽

1981 ◽

Vol 33 (1) ◽

pp. 1-19 ◽

Cited By ~ 57

Author(s):

Béla Bollobás

Keyword(s):

Random Graphs ◽

Degree Sequences

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Robustness and Vulnerability of Scale-Free Random Graphs

Internet Mathematics ◽

10.1080/15427951.2004.10129080 ◽

2004 ◽

Vol 1 (1) ◽

pp. 1-35 ◽

Cited By ~ 121

Author(s):

Béla Bollobás ◽

Oliver Riordan

Keyword(s):

Random Graphs ◽

Scale Free

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Optimal shattering of complex networks

Applied Network Science ◽

10.1007/s41109-019-0205-5 ◽

2019 ◽

Vol 4 (1) ◽

Author(s):

Nicole Balashov ◽

Reuven Cohen ◽

Avieli Haber ◽

Michael Krivelevich ◽

Simi Haber

Keyword(s):

Complex Networks ◽

Random Graphs ◽

Polynomial Time ◽

Optimal Performance ◽

Regular Graphs ◽

Scale Free ◽

Scale Free Networks ◽

Minimum Number

Abstract We consider optimal attacks or immunization schemes on different models of random graphs. We derive bounds for the minimum number of nodes needed to be removed from a network such that all remaining components are fragments of negligible size.We obtain bounds for different regimes of random regular graphs, Erdős-Rényi random graphs, and scale free networks, some of which are tight. We show that the performance of attacks by degree is bounded away from optimality.Finally we present a polynomial time attack algorithm and prove its optimal performance in certain cases.

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