At the core of our proposal are three cross-disciplinary themes, which have been identified as unifying aspects of systems operating far from equilibrium:

• Spontaneous development of structure and patterns: From the fractal structure of growing trees to the geometric regularity of snowflakes and crystals, pattern formation and spontaneous ordering are classic examples of emergent phenomena, occurring in both equilibrium and non-equilibrium settings. In this proposal, we consider examples as diverse as biological self-assembly1, community formation in social networks2, and zonal flows in plasma and atmospheric turbulence3. Emergence in equilibrium and non-equilibrium quantum systems has also drawn much interest of late. For instance, exotic quasi-particles emerge in topological phases of matter; magnetic monopole excitations govern magnetic relaxation properties in frustrated magnets; and novel phase transitions appear in driven quantum liquids. Similar theoretical methods are used in modelling this wide range of processes, and we aim to facilitate progress by combining expertise from different areas.
• Dynamics of large-scale failure: The sudden fracture of materials under strain is of clear industrial importance and has been studied extensively in mathematics, physics and materials science. Similar failure phenomena appear in many other contexts, ranging from earthquakes to the plasma collapses that hinder nuclear fusion research, and the stock-market crashes and epidemics that are predicted by statistical-mechanical analyses of human behaviour. In a very different but related context, sudden changes in the parameters of a strongly correlated quantum system (quantum quench) lead to non-equilibrium phenomena of great interest to quantum information and to the implementation of quantum computing, through the concepts of quantum coherence and entanglement. There are hints of common mathematical structures underlying these diverse phenomena, which we will explore, concentrating particularly on characteristic correlations that can be used to warn of impending systemic failure.
• Responses to strong driving forces and to shocks: A common setting for the appearance of complex emergent behaviour is that of systems with strong external forcing. Examples include turbulence in pipe flow at high Reynolds number, shear-banding in the non-linear rheology of soft-materials4, and non-equilibrium systems like the micro-maser5 and other driven quantum systems. A related issue concerns responses to sudden shocks. These may lead to unexpected behaviour, such as when physical or computational networks respond to breakage or failure of links. Near equilibrium, fluctuation-dissipation theory provides a generic framework for calculating responses; far-from-equilibrium generalisations of such a theory would be extremely valuable, but remain in their infancy.

We recognise that these are just initial topics, and the list above is not exhaustive. NetworkPlus activities (meetings and workshops) will be used to formulate a more complete list of topics. These initial topics, we hope, will bring some focus to our discussions of this very diverse scientific problem.