有效的保阻有向图对称化
Effective Resistance Preserving Directed Graph Symmetrization
摘要：
本文提出了一种新的有向图对称化方法，即构造一个具有等价两有效电阻的无向图作为给定的有向图。因此，一个图形度量，有效电阻的平方根，被保留在有向图和它的对称化版本之间。它示出，该度量的保存允许在有向图的上下文中的对称化图的代数和谱性质的解释，由于有效电阻和拉普拉斯谱之间的关系。另外，利用李雅普诺夫理论证明了有向图的Laplacian矩阵可以分解为投影矩阵、斜对称矩阵和对称化图的Laplacian矩阵的乘积。在有向图的谱图划分和Kron约简的背景下，讨论了有效电阻保持图对称化的应用。
英文摘要：
This work presents a new method for symmetrization of directed graphs that constructs an undirected graph with equivalent pairwise effective resistances as a given directed graph. Consequently a graph metric, square root of effective resistance, is preserved between the directed graph and its symmetrized version. It is shown that the preservation of this metric allows for interpretation of algebraic and spectral properties of the symmetrized graph in the context of the directed graph, due to the relationship between effective resistance and the Laplacian spectrum. Additionally, Lyapunov theory is used to demonstrate that the Laplacian matrix of a directed graph can be decomposed into the product of a projection matrix, a skew symmetric matrix, and the Laplacian matrix of the symmetrized graph. The application of effective resistance preserving graph symmetrization is discussed in the context of spectral graph partitioning and Kron reduction of directed graphs.
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