网络专业化的光谱和动态后果
Spectral and Dynamic Consequences of Network Specialization
摘要：
真实网络的特征之一是，随着拓扑结构的发展，它们能够执行越来越复杂的任务。为了解释这一点，已经观察到，随着网络的增长，网络的某些子集开始专门化它们执行的功能。最近的网络增长模型的基础上，这个概念的专业化已经能够重现一些最知名的拓扑特征，发现在现实世界中的网络，包括右偏度分布，小世界属性，模块化以及层次拓扑结构等，在这里，我们描述了如何专业化在这个模型下也影响网络的频谱特性。这使我们能够给出网络在拓扑结构演变时能够保持其动态的条件。具体来说，我们证明，如果一个网络是内在稳定的，这是一个更强的版本的标准概念的全局稳定性，那么网络保持这种类型的动态网络的发展。这是统一对网络所表现出的两种动力学的严格研究的第一步。这些是\emph{dynamics of} a网络，这是网络结构的拓扑演化的研究，在这里通过网络专业化的过程建模，以及\emph{dynamics on} a网络，这是网络元素的变化状态，我们考虑的动态类型是全局稳定性。我们应用我们的结果的主要例子是递归神经网络，这是某些类型的机器学习算法的基础。
英文摘要：
One of the hallmarks of real networks is their ability to perform increasingly complex tasks as their topology evolves. To explain this, it has been observed that as a network grows certain subsets of the network begin to specialize the function(s) they perform. A recent model of network growth based on this notion of specialization has been able to reproduce some of the most well-known topological features found in real-world networks including right-skewed degree distributions, the small world property, modular as well as hierarchical topology, etc. Here we describe how specialization under this model also effects the spectral properties of a network. This allows us to give conditions under which a network is able to maintain its dynamics as its topology evolves. Specifically, we show that if a network is intrinsically stable, which is a stronger version of the standard notion of global stability, then the network maintains this type of dynamics as the network evolves. This is one of the first steps toward unifying the rigorous study of the two types of dynamics exhibited by networks. These are the \emph{dynamics of} a network, which is the study of the topological evolution of the network's structure, modeled here by the process of network specialization, and the \emph{dynamics on} a network, which is the changing state of the network elements, where the type of dynamics we consider is global stability. The main examples we apply our results to are recurrent neural networks, which are the basis of certain types of machine learning algorithms.
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