In many real-world networks, the ability to withstand targeted or global attacks; extinctions; or shocks is vital to the survival of the network itself, and of dependent structures such as economies (for financial networks) or even the planet (for ecosystems). Previous attempts to characterise robustness include nestedness of mutualistic networks or exploration of degree distribution. In this work we present a new approach for characterising the stability and robustness of networks with all-positive interactions by studying the distribution of the k-shell of the underlying network. We find that high occupancy of nodes in the inner and outer k-shells and low occupancy in the middle shells of financial and ecological networks (yielding a "U-shape" in a histogram of k-shell occupancy) provide resilience against both local targeted and global attacks. Investigation of this highly-populated core gives insights into the nature of a network (such as sharp transitions in the core composition of the stock market from a mix of industries to domination by one or two in the mid-1990s) and allow predictions of future network stability, e.g., by monitoring populations of "core" species in an ecosystem or noting when stocks in the core-dominant sector begin to move in lock-step, presaging a dramatic move in the market. Moreover, this "U-shape" recalls core-periphery structure, seen in a wide range of networks including opinion and internet networks, suggesting that the "U-shaped" occupancy histogram and its implications for network health may indeed be universal. Whether one examines ecological networks 1-3 , financial networks 4-7 , social networks 8,9 , neural networks 10,11 , or beyond, the identification of the features of these networks that characterise their robustness and resilience against external shocks is an important problem in network science today. The relation between network structure and network stability is essential to understanding why some networks survive in the face of global and random local changes, and why others do not. There have been many attempts to characterise what defines robustness in a network based on the features of the network, and the effects of this robustness on network dynamics 9,12-17. Specifically, this has many applications in ecology 2,3,18-22 , where the robustness of an ecological network of species may determine the ability of that network to withstand environmental changes, and in finance 6,7 , where it is important to have a robust financial network in order that the economy does not collapse. Furthermore, by identifying the species or companies that are most integral to network robustness, we may discover markers of future network health. In the ecological case, these species are typically monitored to make sure that their numbers do not fall too low and thus endanger the health of the entire ecosystem; in the case of the stock market, the financial sectors composed of these most-important stocks are usually monitored for lock-step correlations in returns, which is a warning ...

Videos and commercials produced for large audiences can elicit mixed opinions. We wondered whether this diversity is also reflected in the way individuals watch the videos. To answer this question, we presented 65 commercials with high production value to 25 individuals while recording their eye movements, and asked them to provide preference ratings for each video. We find that gaze positions for the most popular videos are highly correlated. To explain the correlations of eye movements, we model them as “interactions” between individuals. A thermodynamic analysis of these interactions shows that they approach a “critical” point such that any stronger interaction would put all viewers into lock-step and any weaker interaction would fully randomise patterns. At this critical point, groups with similar collective behaviour in viewing patterns emerge while maintaining diversity between groups. Our results suggest that popularity of videos is already evident in the way we look at them, and that we maintain diversity in viewing behaviour even as distinct patterns of groups emerge. Our results can be used to predict popularity of videos and commercials at the population level from the collective behaviour of the eye movements of a few viewers.

We explain the structural origin of the jamming transition in jammed matter as the sudden appearance of k-cores at precise coordination numbers which are related not to the isostatic point, but to the emergence of the giant 3-and 4-cores as given by k-core percolation theory. At the transition, the k-core variables freeze and the k-core dominates the appearance of rigidity. Surprisingly, the 3-D simulation results can be explained with the result of mean-field k-core percolation in the Erdös-Rényi network. That is, the finite-dimensional transition seems to be explained by the infinite-dimensional k-core, implying that the structure of the jammed pack is compatible with a fully random network.