[论文解读] Small Lifts of Expander Graphs are Expanding.

Naman Agarwal, Alexandra Kolla+1 more|arXiv (Cornell University)|Nov 13, 2013

Graph theory and applications参考文献 21被引用 8

一句话总结

该论文证明了小规模随机 k-提升的 d-正则膨胀图继承了强膨胀性质：若基图的非平凡特征值在绝对值上以 λ 为界，则以高概率，提升图的所有非平凡特征值最多为 O(λ)，在适度膨胀条件下为 λ + O(√d)。该结果首次为有界 k 提供了谱界，表明拉马努金图的典型小规模提升图几乎是拉马努金图。

ABSTRACT

A k-lift of an n-vertex base-graph G is a graph H on n × k vertices, where each vertex of G is replaced by k vertices and each edge (u, v) in G is replaced by a matching representing a bijection piuv so that the edges of H are of the form (u, i), (v, piuv(i)). H is a (uniformly) random lift of G if for every edge (u, v) the bijection piuv is chosen uniformly and independently at random. The main motivation for studying lifts has been understanding Ramanujan expander graphs via two key questions: Is a “typical ” lift of an expander graph also an expander; and how can we (efficiently) construct Ramanujan expanders using lifts? Lately, there has been an increased interest in lifts and their close relation to the notorious Unique Games Conjecture [Kho02]. In this paper, we analyze the spectrum of random k-lifts of d-regular graphs. We show that, for random shift k-lifts, if all the nontrivial eigenvalues of the base graph G are at most λ in absolute value, then with high probability depending only on the number n of nodes of G (and not on k), the absolute value of every nontrivial eigenvalue of the lift is at most O(λ). Moreover, if G is moderately expanding, then this bound can be improved to λ +O(√d). While previous results on random lifts were asymptotically true with high probability in the degree of the lift k, our result is the first upperbound on spectra of lifts for bounded k. In particular, it implies that a typical small lift of a Ramanujan graph is almost Ramanujan, and we believe it will prove crucial in constructing large Ramanujan expanders of all degrees. We also establish a novel characterization of the spectrum of shift lifts by the spectrum of certain k symmetric matrices, that generalize the signed adjacency matrix (e.g. see [BL06]). We believe this characterization is of independent interest.

研究动机与目标

• 确定一个 d-正则膨胀图的典型小规模 k-提升图是否仍为良好的膨胀图。
• 解决有界 k 下随机 k-提升图的谱行为，克服了以往在 k 上的渐近性结果。
• 通过基于提升的方法，为构造大规模拉马努金膨胀图提供一条可行的构造路径。
• 通过 k×k 对称矩阵对平移提升的谱特性建立一种新颖的谱表征。

提出的方法

• 通过将邻接矩阵建模为由随机置换矩阵表示边匹配关系的分块矩阵，分析随机平移 k-提升图的谱。
• 使用谱分解技术，根据基图的谱间隙来界定提升图的特征值。
• 通过推广带符号邻接矩阵的 k 个对称矩阵的特征值，提出一种平移提升谱的全新表征方法。
• 应用概率方法，证明特征值以高概率集中在 O(λ) 附近，且该概率仅依赖于 n，而不依赖于 k。
• 实施谱交错论证，将提升图的特征值与基图的特征值关联起来。
• 利用提升图中随机置换矩阵的结构，推导出最大绝对值第二特征值的紧致界。

实验结果

研究问题

• RQ1 对于有界 k，d-正则膨胀图的典型小规模 k-提升图是否仍为良好的膨胀图？
• RQ2 当 k 较小且固定时，基图的谱特性是否能在其随机 k-提升图中得以保持？
• RQ3 随机 k-提升图的非平凡特征值的最佳可能上界，相对于基图的谱间隙，应如何表示？
• RQ4 除了带符号邻接矩阵框架外，如何从代数上表征平移提升的谱？
• RQ5 这种谱控制能否用于构造所有度数下的近乎拉马努金图？

主要发现

• 对于任意 d-正则基图 G，若其非平凡特征值在绝对值上以 λ 为界，则随机 k-提升图的所有非平凡特征值以高概率（仅依赖于 n）被限制在 O(λ) 以内。
• 若 G 具有适度膨胀性，该界可改进为 λ + O(√d)，从而提供更紧致的谱控制。
• 该结果适用于有界 k，是首个与 k 的增长无关的此类谱界。
• 平移提升的谱完全由 k 个推广带符号邻接矩阵的对称矩阵的特征值表征。
• 该方法表明，拉马努金图的典型小规模提升图几乎是拉马努金图，暗示了一条可行路径以构造大规模拉马努金膨胀图。
• 该谱表征提供了一种新的代数工具用于分析提升图，具有超越膨胀性的潜在应用。