Bayesian multiple-network multi-layer exponential random graph models (MNML-ERGMs): developing efficient inference and application to neuroimaging

Statistical networks are mathematical representations of vertices connected by edges, which can be binary (present or absent) or weighted (associated with a weight). Studies often analyse topological structures of statistical networks using probabilistic modelling. However, most network models focus on describing a single binary network. The exponential random graph model (ERGM) belongs to such models, which characterise the distribution of networks through a set of network summary statistics. Each summary statistic provides a topological summary of a network, and ERGM is empowered by its flexibility to incorporate a wide range of summary statistics. Recent developments in ERGM have extended its applicability to model multiple binary networks and have been applied to neuroimaging data. However, many neuroimaging datasets are more effectively represented as weighted networks. Functional magnetic resonance imaging (fMRI) data are a key example and is the primary focus of our research application. To fill this gap, our study proposes an ERGM-based model to allow the joint probabilistic representation of multiple weighted networks. Furthermore, we use variational inference to enable efficient ERGM inference within a hierarchical Bayesian setup. In the first part of this thesis, we address the under-exploration of weighted ERGMs, for which most developed methods originate from diverse backgrounds. We systematically implement, review, and compare the existing weighted ERGM frameworks, focusing particularly on the fMRI context. We assess these models through a range of specific criteria. Our analysis concludes with the suitability of a multi-layer ERGM (ML-ERGM). ML-ERGMs target the dynamic topological evolution of weighted networks by probabilistically modelling the transition or the dissolution process between successive network layers. Building upon this, we propose to use a Bayesian hierarchical structure with the ML-ERGM so that we can jointly model a population of weighted networks, forming a multiple-network multi-layer ERGM (MNML-ERGM). MNML-ERGM pools the weighted information from individual networks into higher hierarchies, and therefore, can provide a description of the overall weighted dynamics of joint network topologies. We use synthetic data analysis to explore model properties and then implement it on real-world fMRI data from the Cambridge Centre for Ageing and Neuroscience (Cam-CAN) project, assessing the ageing effects on brain connectivity. In the final part of this thesis, we enable more efficient ERGM inference by using neural posterior estimation (NPE), which trains neural networks to directly estimate the posterior distributions. This largely improves the scalability of Bayesian inference for ERGM compared to traditional estimation approaches. We systematically assess NPE bias and provide ERGM-specific NPE implementation guidelines. This is the first formal NPE implementation for ERGM in the liter-ature. More importantly, we propose a novel NPE approach for Bayesian hierarchical structures called the Amortised Hierarchical Sequential Neural Posterior Estimation (AHS-NPE). AHS-NPE is directly applicable to multiple-network ERGMs. In comparison to other hierarchical NPE approaches in the literature, our model offers better scalability and flexibility. Crucially, our AHSNPE can accommodate a range of ERGM-specific adjustments for more robust ERGM estimation. We validate the accuracy of our AHS-NPE using Cam-CAN data and demonstrate its scalability by extending the MN-ERGM to model a significantly larger sample size.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2025

Author dc:creator

• Fan, Yefeng

Advisor dc:contributor.advisor

• White, Simon

• Exponential random graph model

Licence

• https://www.repository.cam.ac.uk/bitstreams/1eb7a70f-a0c8-4f54-bd01-1685aad68be5/download
• https://creativecommons.org/licenses/by/4.0/

Language dc:language

eng

Author Identifier

0009-0009-6039-2994

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/392654