Structural properties of random tree models and their applications in network flows, brain circulation networks and statistical physics

这项建议的主要目的是对计算机科学、生物学和统计物理中的应用产生的一些随机网络模型进行系统的数学研究，了解这些网络模型的动力学，并开发数学方法来从现实世界的网络中收集信息。利用分支过程嵌入和局部弱收敛技术，我们建议开发一组健壮的工具，可以用来分析在各种应用中出现的最重要的网络模型家族之一(依附族)。这些数学技巧将不仅给出局部泛函(如度分布)的渐近信息，而且还给出全局泛函(如(随机)邻接矩阵的最大度和谱分布)的信息。我们还利用连续时间分支过程技术分析了网络流模型和首次通过渗流模型，以了解无序对随机网络模型几何形状的影响以及流承载网络中拥塞的跨边传播。初步计算表明，对于一系列广泛的模型，随着网络规模的增加，宏观秩序从微观运输规则中出现，该项目将试图理解这一现象，并探索这些模型与稳定年龄分布理论和生物学中分支过程模型的马尔萨斯增长率之间的联系。我们还将研究由统计物理和生物学激发的新的随机树模型，其中使用随机行走构造和条件分枝过程，我们的目的是了解此类模型的比例极限。最后，我们建议开发数学方法来分析、理解和量化现实世界复杂网络结构中的变化来源，例如大脑中的血液循环网络等树木。过去几年，许多现实世界网络(包括社会网络、数据传输网络如互联网和各种生物网络)上的经验数据的可用性刺激了用于理解这些网络的一系列数学模型的爆炸性增长。许多领域的研究人员感兴趣的是了解这种网络的特性，这种网络随时间的演变和变化，以及这些网络上各种过程的动态，例如通过这些网络的流或流量的传输和这些网络上的流行病模型。对这些数学模型的行为的了解将使从业者能够收集关于现实世界中此类过程的重要信息和洞察，从设计更高效的网络，了解影响网络中流动过程或其他动态的拥塞扩散速度的因素，到有助于网络本身结构的实际出现的重要因素。对这类问题的数学分析导致了这些模型与数学概率的广泛领域之间有趣的联系，包括生物学中的分支过程模型和随机分形。这个项目的目的是开发数学方法来理解这种网络模型的性质，特别是理解当系统规模变大时会发生什么。与此同时，该项目还将开发技术，以准确理解各种生物网络产生的数据，如大脑中的血管网络，以及显著影响此类网络功能特性的因素。开发的技术将对广大研究人员有用，我们预计该项目将促进与国家和国际研究小组的跨学科合作项目，并促进对学生的培训，使他们接触到这一迅速崛起的研究领域。

The main aim of this proposal is a systematic mathematical study of a number of random network models arising from applications in computer science, biology and statistical physics, understanding dynamics on these network models, and developing mathematical methodology to glean information from real world networks. Using branching process embeddings and local weak convergence techniques, we propose to develop a set of robust tools that can be used to analyze one of the most important family of network models (the attachment family) which arises in a diverse range of applications. These mathematical techniques will give information on the asymptotics of not only local functionals such as degree distributions but global functionals such as the maximal degree and the spectral distribution of (random) adjacency matrices. Using continuous time branching process techniques we also propose to analyze models of network flow and first passage percolation, in order to understand the effect of disorder on the geometry of random network models and the propagation of congestion across edges in flow carrying networks. Preliminary computations suggest that for a wide array of models, macroscopic order emerges from microscopic rules of transport as the size of the network increases, and the project will attempt to understand this phenomenon and explore connections between these models and stable age distribution theory and the Malthusian rate of growth of branching process models in biology. New models of random trees motivated by statistical physics and biology will also be studied wherein using random walk constructions and conditioned branching processes, we aim to understand the scaling limits of such models. Finally we propose to develop mathematical methodology to analyze, understand and quantify sources of variation in the structure of real world complex networks such as trees arising as blood circulatory networks in the brain.Over the last few years the availability of empirical data on many real world networks including social networks, data transmission networks such as the Internet and various biological networks, has stimulated an explosion in the array of mathematical models proposed to understand these networks. Researchers in a wide array of fields are interested in understanding properties of such networks, the evolution and change of such networks over time, as well as the dynamics of various processes on these networks such as transporting flow or traffic through these networks and epidemic models on these networks. An understanding of the behavior of these mathematical models would allow practitioners to glean important information and insight about such processes in the real world, ranging from the design of more efficient networks, understanding the factors that influence the rate of spread of congestion of flow processes or other dynamics through the network, to the significant factors that contribute to the actual emergence of the structure of the network itself. A mathematical analysis of such problems leads to interesting connections between these models and wide areas of mathematical probability including branching process models in biology and random fractals. The aim of this project is to develop mathematical methodology to understand properties of such network models and in particular understand what happens when the system size grows large. At the same time the project will also develop techniques to accurately understand data arising from various biological networks such as vascular networks in the brain and the factors that significantly affect functional properties of such networks. The techniques developed will be of use to a wide community of researchers, and we anticipate that the project will foster interdisciplinary collaborative projects with both national and international research groups and facilitate the training of students and expose them to this rapidly emerging field of research.