网络本质：PageRank完备化与中心一致马尔可夫链
Network Essence: PageRank Completion and Centrality-Conforming Markov Chains
摘要：
马图谢克（1963-2015）在数学和算法设计方面有许多突破性的贡献。他的里程碑式的成果不仅意义深远，而且优雅。通过超越原来的对象-如欧几里德空间或线性程序- Jirka发现了具有挑战性的数学/算法问题的本质，以及对他来说很自然的美丽解决方案，但对该领域来说是令人惊讶的发现。在这篇简短的探索文章中，我将首先与读者分享我与Jirka的初次相遇，并讨论他在20世纪90年代初的一个基本几何结果。在社会和信息网络时代，我将把讨论从几何结构转向网络结构，试图向网络科学的圣杯迈出一小步，即理解所观察到的稀疏和多面网络数据背后的网络本质。我将讨论一个简单的结果，它总结了个性化PageRank矩阵的一些基本代数性质。与传统的二元关系传递闭包不同，个性化PageRank矩阵采用网络数据的“累积马尔可夫闭包”。其中一些代数性质在各种背景下都是已知的。但我希望在更广泛的背景下将它们放在一起，将有助于说明这种马尔可夫网络完备化的理想性质，并促进网络理论的系统发展，以理解大量和无处不在的多方面网络数据。
英文摘要：
Jiří Matoušek (1963-2015) had many breakthrough contributions in mathematics and algorithm design. His milestone results are not only profound but also elegant. By going beyond the original objects --- such as Euclidean spaces or linear programs --- Jirka found the essence of the challenging mathematical/algorithmic problems as well as beautiful solutions that were natural to him, but were surprising discoveries to the field. In this short exploration article, I will first share with readers my initial encounter with Jirka and discuss one of his fundamental geometric results from the early 1990s. In the age of social and information networks, I will then turn the discussion from geometric structures to network structures, attempting to take a humble step towards the holy grail of network science, that is to understand the network essence that underlies the observed sparse-and-multifaceted network data. I will discuss a simple result which summarizes some basic algebraic properties of personalized PageRank matrices. Unlike the traditional transitive closure of binary relations, the personalized PageRank matrices take "accumulated Markovian closure" of network data. Some of these algebraic properties are known in various contexts. But I hope featuring them together in a broader context will help to illustrate the desirable properties of this Markovian completion of networks, and motivate systematic developments of a network theory for understanding vast and ubiquitous multifaceted network data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
