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Keith Smith


Physics and Mathematics

Staff Group(s)
Physics and Mathematics


Keith Smith is a Lecturer in Mathematics. His overarching research interest is to understand more about the mathematical and statistical rules and limitations for complex biological systems. His work to date involves network science, complex systems science, brain structure in health and disease, biological networks, and biomedical signal processing.

For 2020/21 Keith is the module leader for Statistical Data Analysis and Visualisation for the MSc in Computer Science. He also teaches for Statistical Applications to Data Analysis.

Career overview

  • UKRI Innovation Fellow in Health Data Science, University of Edinburgh, Edinburgh, UK & Health Data Research UK, London, UK (Feb 2018-Jan 2021)
  • PhD studentship in Biomedical Signal Processing, University of Edinburgh, Edinburgh, UK (Nov 2014-Feb 2018)
  • MSc Actuarial Science, Heriot-Watt University, Edinburgh, UK (2011-2012)
  • BSc Mathematics, University of Glasgow, Glasgow, UK (2007-2011)


  • Attendee at Complex Systems Summer School, Santa Fe Institute, New Mexico, USA (Jun 2019)
  • Visiting PhD student in Graph Signal Processing, EPFL, Lausanne, Switzerland (Sep 2015 - Dec 2015)

Research areas

Most major common forms of disease today— including sporadic Alzheimer’s disease, Parkinson’s disease, diabetes, stroke and heart disease—are known as complex trait diseases. Such diseases are not known to be caused by any single deleterious genetic mutation (such as in sickle cell disease or familial Alzheimer’s disease) nor any particular environmental condition. Rather, they appear to be caused by different kinds of combinations of such factors with high levels of co-occurrence (multimorbidity)— perhaps as broad outcomes of the complex human system failing in certain ways. The complexity and volume of factors at a biological level which can contribute to common human diseases makes them not only great societal and economic problems, but some of the most challenging problems in all of science.

While standard reductionist strategies have struggled to make the long hoped-for breakthroughs for complex trait diseases, new systems-level paradigms and the development of powerful and biologically-informed computational methods are gaining traction in enabling more general, holistic perspectives of these problems. Particularly, network science is being realized as a key tool to help us understand the interconnected nature of biological systems and their dynamic organizational principles. In this vein, I pursue two main branches of research:

  1. Networks of brain structure and function in health and disease
  2. Theory of complex biological networks

i) Networks of brain structure and function in health and disease

During my PhD, I introduced a new paradigm for complex networks called hierarchical complexity (HC). While complexity has typically been characterized as arising in the spectrum between ordered and random systems, our characterization allowed us to single out complex networks (involving diverse functionality and generative processes) from ordered and random networks (both involving simple, global generative processes). This was enabled by using neighbourhood degree sequences to encode connectivity patterns (a network tool which I also introduced).  Applying this idea to brain networks, we have found EEG functional connectivity and dMRI structural connectivity of the human brain to have particularly strong HC. Concurrently, I have found that HC is not a general characteristic of real-world networks, with tentative evidence that it is not even a general characteristic of animal brain networks. This interestingly points to HC as a particular characteristic of the human brain.

Recent work with collaborators on HC of adult structural connectomes opens up new ways of understanding and interpreting brain structure and function. Indeed, evidence is gathering of the importance of this characteristic for healthy brain networks and its disruption in a number of clinical conditions. We have found global disruptions to HC in EEG functional connectivity in Alzheimer’s disease; disruptions to the HC of hub regions in lupus; and decreased HC in sensorimotor and heteromodal regions in preterm-born infants.

Still, important fundamental questions remain. We have some understanding of the differences in HC in brain structure between neonates and adults, but I want to go further to understand how HC develops throughout human life. Important clinical considerations include the systemic principles which underlie this development, and how the failure of such principles could be associated with neurodegenerative conditions. With a slightly more ambitious outlook, I want to understand how HC presents in other animals across the tree of life.

Complementary to this work in brain networks, I have a keen interest in implementation and interpretations of graph machine learning approaches to human biological data. To this end, I am primary supervisor of a PhD student focused on applying graph machine learning algorithms to structural features and connectomes from the UK Biobank with applications to major depressive disorder.

ii) Theory of complex biological networks

Due to my underlying interests in exploring the finer biology of the brain, I have helped to substantiate two projects in the area of protein interaction networks. One is an international collaboration with evolutionary biologists (including colleagues at the Santa Fe Institute and Network Science Institute), looking at how the distribution of genetic expression can explain the evolution and particularly the resilience of protein networks, another is  with health data scientists at the Usher Institute, looking for new genes of relevance to cerebral small vessel disease.

While I have focused on applications of signal processing and mathematics to the neurosciences and biology, I have at the same time been driven to developing the theoretical aspects of my work. More recently, I have been developing a new first principles theory of networks. Backed by the main overarching themes within the literature, it poses two components as underlying factors of network emergence (a) the proclivity for individual nodes to make connections (node fitness), which I argue best modelled by a log-normal distribution, and (b) pairwise (dyadic) nodal similarity, where the probability that two nodes interact is determined by latent variables describing their similarity/compatibility, which I argue can be effectively modelled by a high-dimensional Euclidean space where each dimension describes one such latent variable. Modelling using this theory shows striking improvements of topological accuracy on popular competing theories and provides a reconciliatory solution to the outstanding public disagreement on the nature and prevalence of the scale-free degree distributions in complex networks.

Log-normal distributions have yet to receive any significant attention in the network science literature to date. Yet, they are common in nature (such as in gene expression) as the asymptotic distribution of the product of independent variables (i.e. the central limit theorem in the log domain) as well as being incredibly versatile statistical objects. Importantly, there are many open questions as to the composition of complex biological networks which systems-level statistical modelling can help to address. I want to understand how general properties of the distributions expected in nature play a role in defining biological networks and how the influence of biological noise plays a role in the adaptive properties necessary to enable the development of stable biological systems.

Sponsors and collaborators

My recent collaborators include:

EEG & Signal processing:

  • Javier Escudero, IDCOM, University of Edinburgh
  • Mario Parra, Psychology, University of Strathclyde

Structural MRI:

  • Simon Cox, Lothian Birth Cohort Studies, University of Edinburgh
  • Maria Valdes-Hernandez, Centre for Clinical Brain Studies, University of Edinburgh
  • Mark Bastin, Centre for Clinical Brain Studies, University of Edinburgh
  • Heather Whalley, Psychiatry, University of Edinburgh
  • Stewart Wiseman, Centre for Clinical Brain Studies, University of Edinburgh
  • Saturnino Luz, Usher Institute, University of Edinburgh

Neonatal Connectomes:

  • Manuel Blesa, MRC Centre for Reproductive Health, University of Edinburgh
  • Paola Galdi, MRC Centre for Reproductive Health, University of Edinburgh
  • James Boardman, MRC Centre for Reproductive Health, University of Edinburgh

Protein networks:

  • April Kleppe, Department of Molecular Medicine, Aarhus University
  • Honghan Wu, Institute of Health Informatics, UCL
  • Kristiina Rannikmae, Usher Institute, University of Edinburgh


Course(s) I teach on

  • Students using laptops
    Postgraduate taught / Professional | Full-time / Part-time / Sandwich | 2021