熵波、锯齿图积和新的常度
Entropy waves, the zig-zag graph product, and new constant-degree
摘要：
这项工作的主要贡献是一种新型的图产品，我们称之为{ \it zig-zag产品}。取一个大图和一个小图的乘积，得到的图（大致）继承了大图的大小，小图的度，以及两者的扩展属性！迭代产生简单的显式结构的常数度扩展器的任意大小，从一个常数大小的扩展器。对我们直觉（和简单分析）这个图形产品的性质至关重要的是，将扩展器视为充当"熵波”传播器的函数--它们将熵集中在一个区域的概率分布转换为这种集中被耗散的分布。在这些术语中，图形乘积提供了两个这样的波的相长干涉。随后的工作[ALW 01]，[MW 01]将图的Z字形积与群的标准半直积联系起来，得到了扩充Cayley图的新结果和构造.
英文摘要：
The main contribution of this work is a new type of graph product, which we call the {\it zig-zag product}. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of arbitrary size, starting from one constant-size expander. Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as ``entropy wave" propagators -- they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated. In these terms, the graph products affords the constructive interference of two such waves. Subsequent work [ALW01], [MW01] relates the zig-zag product of graphs to the standard semidirect product of groups, leading to new results and constructions on expanding Cayley graphs.
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