E. Roberts

T. Coolen

Short loops (cycles) in real networks are a theoretical challenge for modeling. 

Networks observed in the real world often have many short loops. This violates the tree-like assumption
that underpins the majority of random graph models and most of the methods used for their analysis. In this
paper we sketch possible research routes to be explored in order to make progress on networks with many short
loops, involving old and new random graph models and ideas for novel mathematical methods. We do not present
conclusive solutions of problems, but aim to encourage and stimulate new activity and in what we believe to be
an important but under-exposed area of research. We discuss in more detail the Strauss model, which can be seen
as the ‘harmonic oscillator’ of ‘loopy’ random graphs, and a recent exactly solvable immunological model that
involves random graphs with extensively many cliques and short loops.

Random graph ensembles with many short loops

Next generation structures and materials are key to aerospace, the built environment and exploring space. Hierarchical materials, fractal structures, and polydisperse systems offer dramatic gains in efficiency. Advances in 3D printing and self-assembly mean that these novel technologies can be practically manufactured.

Extreme materials

Silicon-based electronic computing is hitting energetic and technological barriers, and quantum computing remains a theoretical challenge of the future. To maintain an exponential increase in computing power, we require alternative frameworks. Two which we are studying are memristors and photonic computing.

Low-energy computing

Information science underpins modern society. Quantum information science is creating the next generation of information technology, where effects like superposition and entanglement are exploited to envisage qualitatively new technologies. This field also gives an alternative perspective on physics, wherein information takes centre stage.

Quantum information

Innovation is to organizations what evolution is to organisms: it is how organisations adapt to changes in the environment and improve. New mathematical models of component recombination and the building blocks of economic complexity offer fundamental insights into how firms, countries and technologies develop.

Science of innovation

The global behaviour of local processes depends on the geometry of their underlying space. Studying processes on ideal graphs and on irregular networks derived from real data uncovers surprising behaviour in biological, social and financial systems. The tools of this field include spectral, replica, community and combinatoric techniques.

Graph theory

Institutional risk originates from ordinary economic uncertainties. But when these risks propagate across inter-connected institutions, internal amplification can lead to dramatic exposures. These systemic risks can be mitigated by applying ideas from cascading failures, diversification and extreme value theory.