We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 - α , one imitates the opinion of the other; otherwise (i.e., with probability α ), the link between them is broken and one of them makes a new connection to an individual chosen at random ( i ) from those with the same opinion or ( ii ) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case ( i ), there is a critical value α _c which does not depend on u , with ρ ≈ u for α > α _c and ρ ≈ 0 for α < α _c . In case ( ii ), the transition point α _c ( u ) depends on the initial density u . For α > α _c ( u ), ρ ≈ u , but for α < α _c ( u ), we have ρ ( α , u ) = ρ ( α ,1/2). Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.

We investigate the degree to which impact in science and technology is associated with surprising breakthroughs, and how those breakthroughs arise. Identifying breakthroughs across science and technology requires models that distinguish surprising from expected advances at scale. Drawing on tens of millions of research papers and patents across the life sciences, physical sciences and patented inventions, and using a hypergraph model that predicts realized combinations of research contents (article keywords) and contexts (cited journals), here we show that surprise in terms of unexpected combinations of contents and contexts predicts outsized impact (within the top 10% of citations). These surprising advances emerge across, rather than within researchers or teams—most commonly when scientists from one field publish problem-solving results to an audience from a distant field. Our approach characterizes the frontier of science and technology as a complex hypergraph drawn from high-dimensional embeddings of research contents and contexts, and offers a measure of path-breaking surprise in science and technology.