Covariate Information for Dynamic Network Embedding

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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.