Covariate Information for Dynamic Network Embedding

许多真实世界的大型数据集可以被认为是对象之间的成对交互，这些对象具有关联的时间戳。这可能代表在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。这些交互可以被表示为动态图，其中节点表示对象，边表示特定时间的交互。动态图嵌入技术产生了网络中节点在低维空间中的表示，该表示保留了原始结构的各个方面。基于谱分解的方法在离散时间[1]和连续时间[2]中为每个节点提供了期望的轨迹嵌入，并且具有期望的稳定性。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两类：节点协变量：图中的节点可能有关联信息，这些信息可以用一个整数、实数或绝对协变量来表示。边协变量：图中的每条边可能都有关联信息，这取决于两个端点及其发生的时间。同样，这些变量可以是整数、实数或绝对协变量。数字边缘数据可以表示为加权的动态网络，而分类边缘数据可以表示为多个动态网络，该项目的目标是开发可靠的方法将边和结点协变量合并到动态网络的谱嵌入中，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，而准确理解这些嵌入的性质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，这是特别有用的，其中协变量信息对输出有意义。许多动态网络都含有协变量信息，包括网络安全、交通、社会科学和生物学等领域的协变量信息。带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性的输入数据。该项目属于ESPRC数学科学研究领域。[1]Ian Gallagher、Andrew Jones和Patrick Rubin-Delancy。具有稳定性保证的动态网络的谱嵌入。神经信息处理系统进展，34：10158-10170,2021年。[2]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：动态网络的连续时间表示学习框架。Arxiv预印本arxiv：2306.06155,2023年。

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.