Multiplex Markov Chains：对流循环与最优性
Multiplex Markov Chains: Convection Cycles and Optimality
摘要：
多重网络是互联系统和多模态数据的常见建模框架，但我们仍然缺乏对多重性如何影响随机过程的基本见解。我们引入了一种新的“马尔可夫链的马尔可夫链”模型，称为多重马尔可夫链（MMC），使可能的$(1-ω)\in [0,1]$随机步行者保持在同一层，并遵循（层特定的）层内马尔可夫链，而与概率$ω$，他们移动到不同的层以下（节点特定的）层间马尔可夫链。一个主要的发现是多重对流的识别，其中一个固定的分布表现出循环流动，涉及多个层。对流循环在流体中已被很好地理解，但在网络上还没有充分地探索。我们的实验表明，对流的一种机制是存在的不平衡（层内）度的节点在不同的层。为了获得进一步的了解，我们采用谱微扰理论来表征小的和大的$ω$的极限的平稳分布，我们表明，MMC固有地表现出最佳的中间$ω$在其收敛速度和对流的程度。作为一个应用程序，我们进行了基于MMC的脑活动数据的分析，发现MMC在健康人和阿尔茨海默病患者之间存在差异。总的来说，我们的工作表明MMC和对流作为两个重要的新方向，网络相关的研究。
英文摘要：
Multiplex networks are a common modeling framework for interconnected systems and multimodal data, yet we still lack fundamental insights for how multiplexity affects stochastic processes. We introduce a novel ``Markov chains of Markov chains'' model called multiplex Markov chains (MMCs) such that with probably $(1-ω)\in [0,1]$ random walkers remain in the same layer and follow (layer-specific) intralayer Markov chains, whereas with probability $ω$ they move to different layers following (node-specific) interlayer Markov chains. One main finding is the identification of multiplex convection, whereby a stationary distribution exhibits circulating flows that involve multiple layers. Convection cycles are well understood in fluids, but are insufficiently explored on networks. Our experiments reveal that one mechanism for convection is the existence of imbalances for the (intralayer) degrees of nodes in different layers. To gain further insight, we employ spectral perturbation theory to characterize the stationary distribution for the limits of small and large $ω$, and we show that MMCs inherently exhibit optimality for intermediate $ω$ in terms of their convergence rate and the extent of convection. As an application, we conduct an MMC-based analysis of brain-activity data, finding MMCs to differ between healthy persons and those with Alzheimer's disease. Overall, our work suggests MMCs and convection as two important new directions for network-related research.
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。
