Adaptation and Percolation in Complex Networks

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。许多重要的动态过程发生在网络上。例子包括流行病传播、基因调控、同步、信息传播等等。通常，这些动态过程对网络结构有修改作用。本项目将研究网络结构与网络过程的双向交互作用。作为一个重要且具有代表性的案例，我们将研究当网络链路和振荡器参数响应节点动力学时，网络耦合动力系统的同步问题。通过建立描述网络和振荡器参数演化的平均方程，将对网络和参数自适应进行数值和解析研究。网络空间的低维子集中可能的网络不动点、分支和吸引子将作为各种网络措施和自适应规则的函数进行研究。在一个相关的项目中，将研究非马尔可夫网络中的渗透。本文研究了马尔可夫网络和局部树状网络中网络结构对渗透阈值的影响。马尔可夫假设的有效性将在应用中出现的各种网络中进行测试。此外，将寻求一种在非树状的非马尔可夫网络中确定渗透阈值的方法。网络渗透与流行病传播模型、信息在网络中的传播或网络在攻击或随机故障下的鲁棒性有关。例如，在流行病的背景下，渗透阈值将疾病将消亡的网络与疾病将传播并感染相当一部分人口的网络分开。我们对网络结构（例如，流行病期间人们如何相互作用）如何决定这种转变的理论理解仅限于满足特定条件的网络。该项目的目标之一是直接测试在实践中发现的网络（如社交网络）是否满足这些要求。此外，现有的理论工具将扩展到不满足这些条件的网络。许多可以用网络来描述的过程，如无人机的通信网络、食物链网络和神经元网络，并不是发生在静态网络上，而是发生在响应节点动态变化的网络上。该项目的另一部分旨在增加我们对网络如何响应它们所调解的过程而变化的理解，以及它们如何被描述为动态对象。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Many important dynamical processes take place on networks. Examples include epidemic propagation, genetic regulation, synchronization, information propagation, and many more. Often, these dynamical processes have a modifying effect on the network structure. This project will study the bidirectional interaction of network structure and network processes. As an important and representative case, the synchronization of network-coupled dynamical systems will be studied when network links and oscillator parameters adapt in response to node dynamics. Network and parameter adaptation will be investigated numerically and analytically by developing averaged equations that describe the evolution of the network and oscillator parameters. Possible network fixed points, bifurcations, and attractors in low-dimensional subsets of the space of networks will be studied as a function of various network measures and adaption rules. In a related project, percolation in non-Markovian networks will be studied. The effect of network structure on the percolation threshold has been studied for Markovian networks and for locally tree-like networks. The validity of the Markovian assumption will be tested for various networks arising in applications. Additionally, a way to determine the percolation threshold in non-Markovian networks that are not tree-like will be sought.Network percolation is related to models of epidemic propagation, the propagation of information in a network, or the robustness of networks under attack or random failures. For example, in the epidemic context, the percolation threshold separates networks in which a disease will die out from those in which it will propagate to infect a significant fraction of the population. Our theoretical understanding of how network structure (for example, how people interact with each other during an epidemic) determines this transition is restricted to networks satisfying specific conditions. One of the goals of this project is to directly test whether or not networks found in practice, such as social networks, satisfy them. In addition, existing theoretical tools will be extended to networks that do not satisfy these conditions. Many processes that can be described in terms of networks, such as communication networks of unmanned aerial vehicles, food-chain networks, and neuron networks, do not take place on a static network, but on a network that changes in response to node dynamics. The other part of the project seeks to increase our understanding of how networks change in response to the processes that they mediate, and how they can be described as dynamical objects.