A model, applicable to a range of innovation diffusion applications with a strong peer-to-peer component, is developed and studied, along with methods for its investigation and analysis. A particular application is to individual households deciding whether to install an energy efficiency measure in their home. The model represents these individuals as nodes on a network, each with a variable representing their current state of adoption of the innovation. The motivation to adopt is composed of three terms, representing personal preference, an average of each individual's network neighbors' states, and a system average, which is a measure of the current social trend. The adoption state of a node changes if a weighted linear combination of these factors exceeds some threshold. Numerical simulations have been carried out, computing the average uptake after a sufficient number of time-steps over many randomizations at all model parameter values, on various network topologies, including random (Erdös--Rényi), small world (Watts--Strogatz), and (Newman) highly clustered, community-based networks. An analytical and probabilistic approach has been developed to account for the observed behavior, which explains the results of the numerical calculations.

Networks of interactions between competing species are used to model many complex systems, such as in genetics, evolutionary biology or sociology and knowledge of the patterns of activity they can exhibit is important for understanding their behaviour. The emergence of patterns on complex networks with reaction-diffusion dynamics is studied here, where node dynamics interact via diffusion via the network edges. Through the application of a generalisation of dynamical systems analysis this work reveals a fundamental connection between small-scale modes of activity on networks and localised pattern formation seen throughout science, such as solitons, breathers and localised buckling. The connection between solutions with a single and small numbers of activated nodes and the fully developed system-scale patterns are investigated computationally using numerical continuation methods. These techniques are also used to help reveal a much larger portion of of the full number of solutions that exist in the system at different parameter values. The importance of network structure is also highlighted, with a key role being played by nodes with a certain so-called optimal degree, on which the interaction between the reaction kinetics and the network structure organise the behaviour of the system.

Heat waves are one of the deadliest of natural hazards and their frequency and intensity will likely increase as the climate continues to warm. A challenge in studying these phenomena is the lack of a universally accepted quantitative definition that captures both temperature anomalies and associated mortality. We test the hypothesis that social media mining can be used to identify heat wave mortality. Applying the approach to India, we find that the number of heat-related tweets correlates with heat-related mortality much better than traditional climate-based indicators, especially at larger scales, which identify many heat wave days that do not lead to excess mortality. We conclude that social media based heat wave identification can complement climatic data and can be used to: (1) study heat wave impacts at large scales or in developing countries, where mortality data are difficult to obtain and uncertain, and (2) to track dangerous heat wave events in real time.