Dynamic networks occur in many fields of science, technology and medicine, as well as everyday life. Understanding their behaviour has important applications. For example, whether it is to uncover serious crime on the dark web, intrusions in a computer network, or hijacks at global internet scales, better network anomaly detection tools are desperately needed in cyber-security. Characterising the network structure of multiple EEG time series recorded at different locations in the brain is critical for understanding neurological disorders and therapeutics development. Modelling dynamic networks is of great interest in transport applications, such as for preventing accidents on highways and predicting the influence of bad weather on train networks. Systematically identifying, attributing, and preventing misinformation online requires realistic models of information flow in social networks. Whilst simple random networks theory is well-established in maths and computer science, the recent explosion of dynamic network data has exposed a large gap in our ability to process real-life networks. Classical network models have led to a body of beautiful mathematical theory, but do not always capture the rich structure and temporal dynamics seen in real data, nor are they geared to answer practitioners' typical questions, e.g. relating to forecasting, anomaly detection or data ethics issues. Our NeST programme will develop robust, principled, yet computationally feasible ways of modelling dynamically changing networks and the statistical processes on them. Some aspects of these problems, such as quantifying the influence of policy interventions on the spread of misinformation or disease, require advances in probability theory. Dynamic network data are also notoriously difficult to analyse. At a computational level, the datasets are often very large and/or only available "on the stream". At a statistical level, they often come with important collection biases and missing data. Often, even understanding the data and how they may relate to the analysis goal can be challenging. Therefore, to tackle these research questions in a systematic way we need to bring probabilists, statisticians and application domain experts together. NeST's six-year programme will see probabilists and statisticians with theoretical, computational, machine learning and data science expertise, collaborate across six world-class institutes to conduct leading and impactful research. In different overlapping groups, we will tackle questions such as: How do we model data to capture the complex features and dynamics we observe in practice? How should we conduct exploratory data analysis or, to quote a famous statistician, "Looking at the data to see what it seems to say" (Tukey, 1977)? How can we forecast network data, or detect anomalies, changes, trends? To ground techniques in practice, our research will be informed and driven by challenges in many key scientific disciplines through frequent interaction with industrial & government partners in energy, cyber-security, the environment, finance, logistics, statistics, telecoms, transport, and biology. A valuable output of work will be high-quality, curated, dynamic network datasets from a broad range of application domains, which we will make publicly available in a repository for benchmarking, testing & reproducibility (responsible innovation), partly as a vehicle to foster new collaborations. We also have a strategy to disseminate knowledge through a diverse range of scientific publication routes, high-quality free software (e.g. R packages, Python notebooks accompanying data releases), conferences, patents and outreach activities. NeST will also carefully nurture and develop the next generation of highly-trained and research-active people in our area, which will contribute strongly to satisfying the high demand for such people in industry, government and academia.

How do we understand whether epidemics will spread, or predict the likelihood of extreme meteorological events? How do we optimize nano-particles to make them efficient vehicles for targeted delivery of drugs in medical applications? Can we predict how a cell in a specific state will evolve in time, for example whether it will stay healthy or become diseased? How do we prevent over-heating in fast electronic devices, or understand how energy conversion in the next-generation of solar cells might work? Can insights into extreme events in the sciences be used to detect whether a network of financial institutions is close to a crash? These research challenges all relate to non-equilibrium systems. Such systems are typically irreversible, so that if a movie of the system was played backwards it would look very different. For equilibrium systems, on the other hand, a backwards movie would appear much like the original. This lack of an "arrow of time" makes equilibrium systems relatively easy to understand, and indeed most existing methods for predicting e.g. the behaviour of materials are based on the assumption of equilibrium. For non-equilibrium systems, on the other hand, we know much less. They can "age" towards an equilibrium state that is never reached, or exhibit extreme events. The latter are often the result of cascades of collective failure, as illustrated in phenomena ranging from stock market crashes to environmental disasters. As the examples in the first paragraph show, understanding, predicting and controlling non-equilibrium behaviour is an important challenge in many problems across the physical, mathematical, biological and environmental sciences. The starting point for the proposed interdisciplinary Centre for Doctoral Training is, therefore, that significant progress will require researchers that can exploit and strengthen such links between disciplines. They will need to be trained in how to analyse non-equilibrium systems theoretically, via mathematical models, how to study their behaviour via computer simulations, and how to extract information about them from possibly noisy or incomplete data. They will also need to be aware of the important non-equilibrium problems in different disciplines, to see connections, transfer methods and concepts from one application area to another, and develop new approaches. The Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES) will provide the training that such a new generation of researchers will need. In a substantial cohort of at least 10 PhD students per year it will make sure that ideas are exchanged systematically; research projects will be designed to build bridges between different disciplines employing similar methods, or to explore the connections between different approaches that are used to study non-equilibrium systems in the same area. Students will acquire transferable communication and presentation skills, and take part in outreach activities to increase the public understanding of non-equilibrium science. In `open questions sandpits', industry engagement events and dedicated careers events they will also obtain a solid understanding of the priorities of industrial partners working on non-equilibrium systems, and of attractive career paths outside of university. Overall, CANES researchers will emerge with a wide variety of skills that are highly sought after in academia and industry. CANES will also put UK research in non-equilibrium systems on the international map, helping the UK to compete against other countries like the U.S.A. where there is already a significant drive to strengthen research in this area.

Risk is the potential of experiencing a loss when a system does not operate as expected due to uncertainties. Its assessment requires the quantification of both the system failure potential and the multi-faceted failure consequences, which affect further systems. Modern industries (including the engineering and financial sectors) require increasingly large and complex models to quantify risks that are not confined to single disciplines but cross into possibly several other areas. Disasters such as hurricane Katrina, the Fukushima nuclear incident and the global financial crisis show how failures in technical and management systems cause consequences and further failures in technological, environmental, financial, and social systems, which are all inter-related. This requires a comprehensive multi-disciplinary understanding of all aspects of uncertainty and risk and measures for risk management, reduction, control and mitigation as well as skills in applying the necessary mathematical, modelling and computational tools for risk oriented decision-making. This complexity has to be considered in very early planning stages, for example, for the realisation of green energy or nuclear power concepts and systems, where benefits and risks have to be considered from various angles. The involved parties include engineering and energy companies, banks, insurance and re-insurance companies, state and local governments, environmental agencies, the society both locally and globally, construction companies, service and maintenance industries, emergency services, etc. The CDT is focussed on training a new generation of highly-skilled graduates in this particular area of engineering, mathematics and the environmental sciences based at the Liverpool Institute for Risk and Uncertainty. New challenges will be addressed using emerging probabilistic technologies together with generalised uncertainty models, simulation techniques, algorithms and large-scale computing power. Skills required will be centred in the application of mathematics in areas of engineering, economics, financial mathematics, and psychology/social science, to reflect the complexity and inter-relationship of real world systems. The CDT addresses these needs with multi-disciplinary training and skills development on a common mathematical platform with associated computational tools tailored to user requirements. The centre reflects this concept with three major components: (1) Development and enhancement of mathematical and computational skills; (2) Customisation and implementation of models, tools and techniques according to user requirements; and (3) Industrial and overseas university placements to ensure industrial and academic impact of the research. This will develop graduates with solid mathematical skills applied on a systems level, who can translate numerical results into languages of engineering and other disciplines to influence end-users including policy makers. Existing technologies for the quantification and management of uncertainties and risks have yet to achieve their significant potential benefit for industry. Industrial implementation is presently held back because of a lack of multidisciplinary training and application. The Centre addresses this problem directly to realise a significant step forward, producing a culture change in quantification and management of risk and uncertainty technically as well as educationally through the cohort approach to PGR training.