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

本提案的主要目的是对计算机科学、生物学和统计物理学应用中产生的一些随机网络模型进行系统的数学研究，理解这些网络模型的动力学，并开发从现实世界网络中收集信息的数学方法。使用分支过程嵌入和局部弱收敛技术，我们建议开发一套健壮的工具，可用于分析在各种应用中出现的最重要的网络模型家族之一（附件家族）。这些数学技术不仅提供了局部泛函（如度分布）的渐近性信息，还提供了全局泛函（如（随机）邻接矩阵的最大度和谱分布）的渐近性信息。我们还建议使用连续时间分支过程技术来分析网络流和第一通道渗流模型，以了解无序对随机网络模型几何形状的影响以及流量承载网络中沿边缘的拥塞传播。初步的计算表明，对于大量的模型，随着网络规模的增加，宏观秩序从微观的运输规则中浮现出来，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。由统计物理和生物学驱动的随机树新模型也将被研究，其中使用随机行走结构和条件分支过程，我们的目标是了解这些模型的缩放限制。最后，我们建议发展数学方法来分析、理解和量化现实世界复杂网络（如大脑血液循环网络中的树）结构中的变异来源。在过去的几年里，许多现实世界网络的经验数据的可用性，包括社交网络，数据传输网络，如互联网和各种生物网络，刺激了一系列数学模型的爆炸式增长，这些模型被提出来理解这些网络。许多领域的研究人员都有兴趣了解这些网络的特性，这些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运输流量或流量以及这些网络上的流行病模型。理解这些数学模型的行为将使实践者能够收集有关现实世界中此类过程的重要信息和见解，从设计更高效的网络，了解影响流量过程或网络中其他动态的拥塞传播速度的因素，到有助于网络本身结构实际出现的重要因素。对这些问题的数学分析导致这些模型与广泛的数学概率领域之间的有趣联系，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属性，特别是理解当系统规模变大时会发生什么。同时，该项目还将开发技术，以准确理解各种生物网络（如大脑中的血管网络）产生的数据，以及显著影响这些网络功能特性的因素。所开发的技术将被广泛的研究人员使用，我们预计该项目将促进与国内和国际研究小组的跨学科合作项目，并促进学生的培训，使他们接触到这一迅速兴起的研究领域。

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.