论文标题
多路复用马尔可夫连锁店：对流周期和最佳性
Multiplex Markov Chains: Convection Cycles and Optimality
论文作者
论文摘要
多路复用网络是用于互连系统和多模式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。 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.一个主要发现是鉴定多重对流，从而固定分布表现出涉及多层的循环流。对流周期在流体中得到了充分的理解，但在网络上探讨了不足。我们的实验表明，对流的一种机制是（内部内）在不同层中（内部）节点的失衡。为了获得进一步的见解，我们采用光谱扰动理论来表征大小$ω$的限制的固定分布，并且我们表明，MMC固有地在其收敛速度和对流的程度方面对中级$ω$表现出最佳性。作为应用程序，我们对基于MMC的脑活动数据进行了分析，发现健康人与患有阿尔茨海默氏病患者之间的MMC不同。总体而言，我们的工作建议MMC和对流是与网络相关研究的两个重要新方向。
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.
