Accessibility percolation

Accessibility percolation was introduced by Nowak and Krug as a model for evolution. In this model, a graph represents possible genotypes or phenotypes, with each vertex assigned a fitness value. The objective is to identify paths of vertices whose fitness values increase, signifying viable evolutionary pathways. In the 'House of Cards' model, fitness values are independently and identically distributed. In the 'Rough Mount Fuji' model, fitness values exhibit some form of drift as well as an independent and identically distributed component. The primary aim is to obtain theoretical insights into the asymptotic behaviour of the House of Cards and Rough Mount Fuji models across various settings, including on trees, the hypercube, random graphs, or even the integer lattice.

Our first priority will be to investigate trees, since the lack of cycles reduces dependencies between different parts of the graph. In this case much is already known for the House of Cards model, so we will concentrate on the Rough Mount Fuji model. We can use a coupling with Bernoulli percolation, introduced by Hegarty and Martinsson on the hypercube but equally applicable to trees, to show that there is accessibility percolation for the RMF model on regular trees when the drift parameter is sufficiently large. The aim then is to show that there is no accessibility percolation when the drift parameter is small; we have an argument to do this by splitting paths into a fixed number of segments of equal length, and using the negative correlation of the segments. Next we aim to show that the critical value of the drift parameter, when the probability of percolation goes from zero to something strictly positive, is of order 1/n, where n is the number of children of each vertex of the tree. A hands-on combinatorial argument, where we bound the probability of labels being ordered by the probability that 0, 1, 2 or more i.i.d. random variables are out of order but still within close proximity, appears promising.

Once we have established this result for the tree we aim to generalise it to the hypercube, which is a more complicated graph but can be viewed to a certain extent like a pair of non-regular trees glued together.

The idea for Erdos-Rényi graphs is to use the second-moment method, similar to how it was applied to regular trees in the HoC setting scenario. Instead of only focusing on paths above the diagonal as in Roberts and Zhao paper, the analysis will now include paths between two diagonals. This is because in Erdos-Rényi graphs, there is added complexity where paths can repeatedly join and split. To address this, the approach is to consider paths within two diagonals, by doing so, we not only eliminate k-forks kind of paths, but we also account for situations where paths were initially separate and then joined at generation k'. There is also the possibility of paths can repeatedly join and split multiple times but, we expect that this type of increasing path is rare.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.