非负矩阵的边扩张与谱隙
组合数学
2019-09-30 v1 离散数学
摘要
经典的图论 Cheeger 不等式指出，若 M M M n × n n\times n n × n 1 − λ 2 ( M ) 2 ≤ ϕ ( M ) ≤ 2 ⋅ ( 1 − λ 2 ( M ) ) \frac{1-\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1-\lambda_{2}(M))} 2 1 − λ 2 ( M ) ≤ ϕ ( M ) ≤ 2 ⋅ ( 1 − λ 2 ( M )) ϕ ( M ) = min S ⊆ [ n ] , ∣ S ∣ ≤ n / 2 ( 1 ∣ S ∣ ∑ i ∈ S , j ∉ S M i , j ) \phi(M)=\min_{S\subseteq[n],|S|\leq n/2}\left(\frac{1}{|S|}\sum_{i\in S,j\not\in S}M_{i,j}\right) ϕ ( M ) = min S ⊆ [ n ] , ∣ S ∣ ≤ n /2 ( ∣ S ∣ 1 ∑ i ∈ S , j ∈ S M i , j ) M M M λ 2 ( M ) \lambda_{2}(M) λ 2 ( M ) M M M A A A ϕ ( A ) \phi(A) ϕ ( A ) 1 − Re λ 2 ( A ) 1-\text{Re}\lambda_{2}(A) 1 − Re λ 2 ( A ) λ 2 ( A ) \lambda_{2}(A) λ 2 ( A ) ϕ ( A ) \phi(A) ϕ ( A ) ϕ ( A ) ≤ 2 ⋅ ( 1 − Re λ 2 ( A ) ) \phi(A)\leq\sqrt{2\cdot(1-\text{Re}\lambda_{2}(A))} ϕ ( A ) ≤ 2 ⋅ ( 1 − Re λ 2 ( A )) ϕ ( A ) \phi(A) ϕ ( A ) ϕ ( A ) ∈ Θ ( 1 − Re λ 2 ( A ) log n ) , \phi(A)\in\Theta\left(\frac{1-\text{Re}\lambda_{2}(A)}{\log n}\right), ϕ ( A ) ∈ Θ ( log n 1 − Re λ 2 ( A ) ) , n n n ϕ ( A ) \phi(A) ϕ ( A ) n × n n\times n n × n A n A_{n} A n ϕ ( A n ) ≤ 1 − Re λ 2 ( A n ) n . \phi(A_{n})\leq\frac{1-\text{Re}\lambda_{2}(A_{n})}{\sqrt{n}}. ϕ ( A n ) ≤ n 1 − Re λ 2 ( A n ) . 0 0 0 A A A ϕ ( A ) ≥ 1 − Re λ 2 ( A ) 35 ⋅ n . \phi(A)\geq\frac{1-\text{Re}\lambda_{2}(A)}{35\cdot n}. ϕ ( A ) ≥ 35 ⋅ n 1 − Re λ 2 ( A ) . n n n R R R ϕ ( R ) \phi(R) ϕ ( R ) R R R
引用
@article{arxiv.1909.12497,
title = {Edge Expansion and Spectral Gap of Nonnegative Matrices},
author = {Jenish C. Mehta and Leonard J. Schulman},
journal= {arXiv preprint arXiv:1909.12497},
year = {2019}
}