We describe a problem in complex networks we call the Node Vector Distance (NVD) problem, and we survey algorithms currently able to address it. Complex networks are a useful tool to map a non-trivial set of relationships among connected entities, or nodes. An agent—e.g., a disease—can occupy multiple nodes at the same time and can spread through the edges. The node vector distance problem is to estimate the distance traveled by the agent between two moments in time. This is closely related to the Optimal Transportation Problem (OTP), which has received attention in fields such as computer vision. OTP solutions can be used to solve the node vector distance problem, but they are not the only valid approaches. Here, we examine four classes of solutions, showing their differences and similarities both on synthetic networks and real world network data. The NVD problem has a much wider applicability than computer vision, being related to problems in economics, epidemiology, viral marketing, and sociology, to cite a few. We show how solutions to the NVD problem have a wide range of applications, and we provide a roadmap to general and computationally tractable solutions. We have implemented all methods presented in this article in a publicly available open source library, which can be used for result replication.

中文翻译：

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复杂网络中的节点向量距离问题

我们描述了复杂网络中的一个问题，我们称之为节点向量距离 (NVD) 问题，并且我们调查了目前能够解决它的算法。复杂网络是映射连接实体或节点之间的一组重要关系的有用工具。一个代理——例如，一种疾病——可以同时占据多个节点，并且可以通过边缘传播。节点向量距离问题是估计代理在两个时刻之间行进的距离。这与在计算机视觉等领域受到关注的最优运输问题（OTP）密切相关。OTP 解决方案可用于解决节点矢量距离问题，但它们不是唯一有效的方法。在这里，我们研究了四类解决方案，展示了它们在合成网络和现实世界网络数据上的差异和相似之处。NVD 问题比计算机视觉具有更广泛的适用性，与经济学、流行病学、病毒营销和社会学中的问题有关，仅举几例。我们展示了 NVD 问题的解决方案如何具有广泛的应用，并提供了通用和计算上易于处理的解决方案的路线图。我们已经在一个公开可用的开源库中实现了本文中介绍的所有方法，该库可用于结果复制。