Adjacency spectral embedding beyond unweighted, undirected networks

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


There exists a rich variety of real-world networks beyond those with unweighted and undirected edges. This thesis considers the statistical analysis of adjacency spectral embedding beyond these standard networks. By embedding the nodes of these networks into low dimensional space in a consistent and meaningful manner, it is possible to make better inferences from the embedded data. This thesis extends the existing theory for unweighted, undirected networks with two major contributions.

First, we extend the generalised random dot product graph, a latent position graph model, to allow for weighted networks and provide new results regarding the asymptotic distribution of the adjacency spectral embedding of these networks. This opens up new possibilities as a network can be transformed using a different representation of the edge weights, for example, embedding an adjacency matrix of p-values or an adjacency matrix of log p-values. In the case of the weighted stochastic block model, we can compare the quality of the different embeddings using the size-adjusted Chernoff information and consider the optimal transformation for a network.

Second, we consider dynamic networks where individual nodes, communities or the entire graph may change over time. We show that unfolded adjacency spectral embedding produces an embedding for every node at every time point in a statistically consistent way. Up to noise, nodes behaving similarly at a given time are given the same embedding (cross-sectional stability), as is a single node behaving similarly across different times (longitudinal stability). We show that many common dynamic network embedding algorithms often lack one, or both, of these desirable properties.
Date of Award6 Dec 2022
Original languageEnglish
Awarding Institution
  • The University of Bristol
SupervisorFarhad Babaee (Supervisor) & Patrick Rubin-Delanchy (Supervisor)


  • Spectral embedding
  • Networks

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