Large Random Networks: Route and Information-Exchange Processes

该项目涉及大型随机网络数学模型的理论研究，重点是以下两个设置。 美国道路网的某些观测特征与它在5 - 500英里范围内的近似尺度不变是一致的。 提出研究尺度不变模型的数学性质。 这种研究提出了新的基础性和实质性问题。 第二，当前人们对社交网络中的信息流有很大的兴趣。 相关的抽象物理模型类似于以前在许多不同的科学和工程背景下研究的模型（流行病;计算机网络中的广播;统计物理中的相互作用粒子）。 建议划定一个特别广泛的类别，（“有限马尔可夫信息交换”），记录了已经知道的关于它们的知识，并制定和研究了许多新的问题。问题的制定是通过与有限马尔可夫链的混合时间周围的充分理解的理论类比来指导的。在研究层面上，从这项研究中设想的更广泛的影响是促进技术在不同研究团体之间的传播，并为大型网络的设计、分析和实验研究提供额外的工具。 在研究生培训阶段，第二个主题提供了一个开放性问题和不同的明显困难的肥沃的领域，其研究将使学生接触到广泛的技术。这两个主题也适合主要研究者在本科教学（“概率与真实的世界”课程）和监督本科研究项目中的活动。

The project concerns theoretical study of mathematical models of large randomnetworks, with a focus on the following two settings. Certain observed featuresof the U.S. road network are consistent with it being approximatelyscale-invariant over scales of 5 - 500 miles. It is proposed to studymathematical properties of models that are exactly scale-invariant. Such studyraises new foundational and substantive issues. Second, there is great currentinterest in information flow in social networks. Relevant abstract mathematicalmodels are similar to models previously studied in many different scientific andengineering contexts (epidemics; broadcasting in computer networks; interactingparticles in statistical physics). It is proposed to delineate a particularbroad class (``Finite Markov Information Exchange") of such models, record whatis already known about them, and formulate and investigate many new questions.The formulation of questions is guided by analogy with the well understood theorysurrounding mixing times for finite Markov chains.At the research level, broader impacts envisaged from this research arefacilitating the spread of techniques between different research communitiesand providing additional tools for the design, analysis and empiricalinvestigation of large networks. At the postgraduate training level thesecond topic provides a fertile field of open problems and varying apparentdifficulty, whose study will expose the student to a broad range of techniques.Both topics also fit well with the principal investigator's activities inundergraduate teaching (a "probability and the real world" course) andsupervising undergraduate research projects.