The complexity of wireless communication networks has grown considerably in recent years. This has been driven in part by academic research that has started to define the information theoretic boundaries and advantages of certain complex networking topologies and protocols. On the other hand, the demands from consumers and industry have pushed wireless networks towards more sophisticated architectures and solutions, primarily in order to ensure a broad range of services can be delivered using a common infrastructure. This is particularly true of 4/5G technologies, which many believe should support all things for all people, including voice, data, public safety, distributed sensing and monitoring, etc. However, similar beliefs and trends can be found in other sectors, such as smart grid networks and even satellite networks.

It is important that engineers understand the global properties of complex networks, and how these properties arise from local structure. Such information can be fed into models and optimisation routines so that practical networks can be designed to perform as well as possible. A common approach to tackling complex problems is to exploit randomness and statistical properties of the underlying system. Probabilistic approaches to network modelling are not without their difficulties, and some of the main problems that researchers have struggled with over the years arise from the fact that networks are finite entities with physical boundaries.

Recent research by the investigators has focused on the effects that boundaries have on connectivity when networks are embedded in some finite spatial domain. Analytic expressions for the overall connection probability have been obtained. These formulae quantify the intuitive phenomenon that nodes near the boundary are more likely to disconnect, and thus they explain how the network outage probability behaves at high node densities. This work has been extended considerably to explore notions of resilience (k-connectivity), the effects of node directivity, diversity and power scaling laws, complicated geometric bounding domains (both convex and non-convex), and even the interplay between higher layer trust protocols and the physical network set-up and spatial domain.

In this project, the probabilistic formalism alluded to above will be exploited further to study several key concepts that influence the structure of spatially embedded networks. The following four topics will be treated:

- continuum models of spatially embedded networks, including the investigation of spectral and centrality properties of random networks;
- mobility models in spatially embedded networks, including random waypoint and Levy flight processes;
- trust models in spatially embedded networks, including trust dynamics and protocol design;
- temporal models of spatially embedded networks, including dynamical node and link (edge) models.

The work will take a mathematical approach, but will always maintain a focus on practical implications and designs.

Potential Impact:
A number of parties and sectors will benefit from this research. These range from graduate students to industry leaders, and from private enterprise to spectrum regulators. Engagement with potential beneficiaries will largely take place through summer schools, workshops, and meetings with private industry.

Academic beneficiaries -- including Penrose (Bath), Win (MIT), Haenggi (Notre Dame) and many other academic institutions that are presently active in random graph/network research -- will no doubt create further scientific advances related to this research. New techniques and know how will be developed, and new results will of course be disseminated through scientific publications. Due to the timeliness of this research, its academic reach will be international.

Postdoctoral researchers and graduate students will benefit from skills training through an annual summer school and workshop programme. This training will strengthen the national pool of mathematically able engineers and problem solvers and attract others with similar skills to the UK for employment in the medium to long term. It is widely recognised that having a large number of skilled professionals such as these is of great importance to the UK in this age of information and technology, particularly with interest in electrical, electronic and computer engineering on the decline (Perkins' Review).

Industrial R&D facilities will feed research output into established businesses and markets by developing new products and services related to the Internet of Things and 5G cellular systems. They may also seek to create new businesses based on novel discoveries made in this project. R&D organisations are well represented on the panel of project partners (Toshiba, NEC, BT TSO), and thus it is envisaged that the time required for technology uptake will be minimal (2-3 years). In addition, due to the UK's reputation as a place for excellence in network research, which will be strengthened by this project, will bring further inward investment from companies like Toshiba and NEC.

Network operators and vendors will also utilise the output from this research. Network architectures will be able to be optimised by using techniques and theories developed in this project, thus enabling the delivery of new products that will enhance the end users' quality of life. Examples may include the exploitation of statistical connectivity or centrality information coupled with data analytics to offer users personalised services, or enable ad hoc device-to-device networks to form reliably.

Public sector organisations including health, police, and fire/rescue services will benefit from this research since it will aid with the development of enhanced public safety networks built on secure, ad hoc protocols and optimised network architecutres. The research will influence the design of truly intelligent transport systems (e.g., multihop vehicle-to-vehicle and vehicle-to-infrastructure), which will lead to safer UK road and rail networks.

Regulators are working hard to facilitate the creation of higher bandwidth next generation wireless services without allowing crippling levels of interference to affect transmissions. Interference is, however, a random quantity, being dependent upon the geographical locations of transmitting and receiving devices, the propagation medium and the likelihood that devices access the medium concurrently. Such random systems will be described by the proposed research, and thus the findings of this project will influence policy related to spectrum usage in the UK and further afield. In addition to these interference considerations avenues for influencing government policy may appear in other markets and technologies. Indeed, initial work by the investigators was used by Toshiba several years ago to inform and influence government policy on suitable communication network architectures for smart electricity meters.