Covariate Information for Dynamic Network Embedding

许多现实世界的大型数据集可以被认为是具有关联时间戳的对象之间的成对交互。这可以代表互联网上的计算机之间发送的数据包，或者世界各地机场之间的航班。这些交互可以表示为动态图，其中节点表示对象，边表示特定时间的交互。动态图嵌入技术在低维空间中产生网络中节点的表示，保留了原始结构的各个方面。基于谱分解的方法为每个节点提供了理想的轨迹嵌入，在离散时间[1]和连续时间[2]中都具有理想的稳定性。在许多应用中，我们还希望在动态嵌入中包含额外的协变量信息。这些协变量可以分为以下两类：节点协变量：图中的节点可能具有关联信息，这些信息可以用整数、实值或分类协变量表示。边缘协变量：图中的每条边都可能有相关的信息，这取决于两个端点和它发生的时间。同样，这些协变量可以是整数，实数或分类协变量。数值边缘数据可以表示为加权动态网络，而分类边缘数据可以表示为动态网络的复用。该项目的目标是开发可靠的方法，将边缘和节点协变量纳入动态网络的频谱嵌入。这可以通过修改邻接矩阵的展开来产生包含协变量信息的嵌入来实现，并且精确地理解这些嵌入的属性是本研究要回答的主要问题。当嵌入被用作后续统计分析的输入时，这一点特别有用，其中协变量信息对输出是有意义的。在动态网络嵌入中包含协变量信息的方法已经开发出来，但还没有用于谱方法，因为谱方法比非谱方法有更大的理论保证。许多动态网络具有协变量信息，涉及网络安全、交通运输、社会科学和生物等领域。具有协变量的网络数据固有地出现在这些领域中，更好地理解网络嵌入将为后续分析提供更多信息丰富的输入数据。本项目属于ESPRC数学科学研究领域伊恩·加拉格尔，安德鲁·琼斯，还有帕特里克·鲁宾·德兰奇。具有稳定性保证的动态网络的频谱嵌入。神经网络信息处理系统进展，34 (4):1058 -10170,2011 .[j] .[j]亚历山大·莫德尔、伊恩·加拉格尔、艾玛·切切里尼、尼克·怀特利和帕特里克·鲁宾-德兰奇。强度轮廓投影：动态网络连续时间表示学习的框架。[j] .中国科学院学报（自然科学版）：2306.06155,2023。

Many large real-world datasets can be considered as pairwise interactions between objects occurring with an associated timestamp. This could represent packets being sent between computers on the Internet, or flights between airports around the world. These interactions can be represented as a dynamic graph with nodes representing the objects and edges representing an interaction at a particular time.Dynamic graph embedding techniques produce a representation of the nodes in a network in a low-dimensional space that preserves aspects of the original structure. Approaches based on the spectral decomposition provide desirable trajectory embedding for each node with desirable stability properties in both discrete-time [1] and continuous time [2].In many applications, there is also extra covariate information that we wish to include in our dynamic embedding. These covariates can divided into one of two categories:Node covariate: Nodes in the graph may have associated information, which could be represented by an integer, real-valued or categorically covariate. Edge covariate: Every edge in the graph may have associated information, which will depend on the two endpoints and the time it occurred. Again, these could be integer, real-valued or categorically covariates. Numeric edge data could be represented as a weighted dynamic network, while categorical edge data could be represented as a multiplex of dynamic networks.The goal of the project is to develop reliable methods to incorporate edge and node covariates into the spectral embedding of dynamic networks.This can be achieved by modifying the unfolding of the adjacency matrices to produce embedding that includes the covariate information, and precisely understanding the properties of these embeddings is a major question to be answered by this research. This is particularly useful when the embedding is used as input for subsequent statistical analysis where the covariate information is meaningful for the output. Methods have been developed to include covariate information in dynamic network embedding, but not for spectral methods which have greater theoretical guarantees than non-spectral ones.Many dynamic networks have covariate information, ranging from cybersecurity, transportation, social science and biology. Network data with covariates inherently occurs in these fields and a better understanding of the network embeddings will provide more informative input data for subsequent analysis. This project falls within the ESPRC mathematical sciences research area.[1] Ian Gallagher, Andrew Jones, and Patrick Rubin-Delanchy. Spectral embedding for dynamic networks with stability guarantees. Advances in Neural Information Processing Systems, 34:10158-10170, 2021.[2] Alexander Modell, Ian Gallagher, Emma Ceccherini, Nick Whiteley, andPatrick Rubin-Delanchy. Intensity profile projection: a framework forcontinuous-time representation learning for dynamic networks. arXivpreprint arXiv:2306.06155, 2023.