Magnetic Laplacian for Directed Networks

Numerous complex systems in nature and society can be modelled as networks where interacting agents are connected by edges. Examples include protein-protein interaction networks in biology, food-webs in ecology, and friendship networks in social sciences. Network science is a rapidly growing field that studies the network representations of these complex systems. Many of those networks are directed, and the edge directions encode essential information of the roles of the nodes. For example, in food-webs, edges direct from preys to predators; the input-output network in economics describes the chain of supply and purchase between industrial sectors. However, most traditional network algorithms do not consider edge directions.

The main objective of this project is to understand and visualize the structure of large-scale complex directed networks. A key question is how to assign physical positions to the nodes, in two dimensions or higher, in order to give insights about the underlying connectivity patterns. In particular, this could help unveil any hierarchical structure. We will tackle this problem with a novel spectral clustering method using the Magnetic Laplacian, which extends traditional methods by incorporating directional information. An important first step is to investigate the properties of the Magnetic Laplacian and understand its connection with random graph models. We will also investigate the choice of parameters and aim to develop an algorithm that automatically adapts these to the given network. Algorithmic developments will be tested and refined via computational experiments on artificial networks and also on real, public domain, data arising from online social interaction and from supermarket transactions.

MAC-MIGS develops computational modelling and its application to a range of economic sectors, including high-value manufacturing, energy, finance and healthcare. These fields contribute over £500 billion to the UK economy. The CDT involves collaborations with more than a dozen companies and organisations, including large corporations (AkzoNobel, IBM, Dassault, P&G, Aberdeen Standard Investments, Intel), mid-size firms, particularly in the engineering and power sectors (NM Group, which provides monitoring services to power grid operators in 30 countries, Artemis Intelligent Power, the world leader in digital displacement hydraulics, Leonardo, a provider of defense, security and aerospace services, and Oliver Wymans, a management consultancy firm) and startups such as Brainnwave, which develops data-modelling solutions, and Opengosim which designs state-of-the-art and massively parallel software for subsurface reservoir simulation. Government and other agencies involved will include the British Geological Survey, Forestry Commission, James Hutton Institute, and Scottish National Heritage. Engagement will be via internships, short projects and PhD projects. BIS has stated that "Organisations using computer generated modelling and simulations and Big Data analytics create better products, get greater insights, and gain competitive advantage over traditional development processes". Our partners share this vision and are keen to develop deeper collaborations with us over the duration of the CDT.

Our CDT will achieve the following:

- Produce 76 highly skilled mathematical scientists and professionals, ready to take up positions in academia or in companies such as our partners. The students will have exposure to projects, modelling camps and high-level international collaborations.

- Deliver economic and societal benefits through student research projects developed in close collaboration with our partners in industry, business and government and other agencies.

- Create pathways for impact on computer science, chemistry, physics and engineering by involving interdisciplinary partners from Heriot-Watt and Edinburgh Universities in the supervision and training of our students.

- Organise a large number of lectures and seminars which will be open to staff and students of the two universities. Such lectures will inform the wide university communities about the state-of-the-art in computational and mathematical modelling.

- Work with other CDTs both in Edinburgh and beyond to organise a series of workshops for undergraduates, intended to foster an increased uptake of PhD studentship places in technical areas by female students and those from ethnic minorities, with potential impact on the broader UK CDT landscape.

- Organise industrial sandpits and modelling camps which offer the possibility for our partners to present a challenge arising in their work, and to explore innovative ways to tackle that challenge, fully involving the CDT students. This will kick-start a change in the corporate mindset by exposing the relevant staff to new approaches.

- Develop a new course, "Entrepreneurship for Doctoral Students in the Mathematical Sciences" in conjunction with Converge Challenge (Scotland's largest entrepreneurial training programme) and UoE's School of Business. This and other support measures will develop an innovation culture and facilitate the translation of our students' ideas into commercial activities.