The First Eigenvalue of (&lt;i&gt;c&lt;/i&gt;,&lt;i&gt;d&lt;/i&gt;)-Regular Graph

Nakagawa, Kotaro; Yamaguchi, Hiroki

doi:10.1587/transinf.e96.d.433

Cited by 1 publication

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References 11 publications

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“…The expression of ( 27) was also obtained recently in a mathematically rigorous manner for ∆ = 1 [37]. We will term solutions of this type defect solutions because of their physical implications shown by the following naive analysis [11].…”

Section: Defect Modelmentioning

confidence: 88%

First eigenvalue/eigenvector in sparse random symmetric matrices: influences of degree fluctuation

Kabashima,

Takahashi

2012

Preprint

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The properties of the first (largest) eigenvalue and its eigenvector (first eigenvector) are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions. In principle, one should be able to accurately calculate them by solving a functional equation concerning auxiliary fields which come out in an analysis based on replica/cavity methods. However, the difficulty in analytically solving this equation makes an accurate calculation infeasible in practice. To overcome this problem, we develop approximation schemes on the basis of two exceptionally solvable examples. The schemes are reasonably consistent with numerical experiments when the statistical bias of positive matrix entries is sufficiently large, and they qualitatively explain why considerably large finite size effects of the first eigenvalue can be observed when the bias is relatively small.

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Section: Defect Modelmentioning

confidence: 88%