We propose a simple adaptive-network model describing recent swarming experiments. Exploiting an analogy with human decision making, we capture the dynamics of the model by a low-dimensional system of equations permitting analytical investigation. We find that the model reproduces several characteristic features of swarms, including spontaneous symmetry breaking, noise-and density-driven order-disorder transitions that can be of first or second order, and intermittency. Reproducing these experimental observations using a nonspatial model suggests that spatial geometry may have a lesser impact on collective motion than previously thought.

We consider voter dynamics on a directed adaptive network with fixed out-degree distribution. A transition between an active phase and a fragmented phase is observed. This transition is similar to the undirected case if the networks are sufficiently dense and have a narrow out-degree distribution. However, if a significant number of nodes with low out-degree is present, then fragmentation can occur even far below the estimated critical point due to the formation of self-stabilizing structures that nucleate fragmentation. This process may be relevant for fragmentation in current political opinion formation processes.

We consider the evolutionary dynamics of a cooperative game on an adaptive network, where the strategies of agents (cooperation or defection) feed back on their local interaction topology. While mutual cooperation is the social optimum, unilateral defection yields a higher payoff and undermines the evolution of cooperation. Although no a priori advantage is given to cooperators, an intrinsic dynamical mechanism can lead asymptotically to a state of full cooperation. In finite systems, this state is characterized by long periods of strong cooperation interrupted by sudden episodes of predominant defection, suggesting a possible mechanism for the systemic failure of cooperation in real-world systems.

Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Here we show that in high-dimensional systems, containing many variables, we frequently encounter an additional non-bifurcative saddletype mechanism leading to critical transitions. This generic class of transitions has been missed in the search for early-warnings up to now. In fact, the saddle-type mechanism also applies to lowdimensional systems with saddle-dynamics. Near a saddle a system moves slowly and the state may be perceived as stable over substantial time periods. We develop an early warning sign for the saddle-type transition. We illustrate our results in two network models and epidemiological data. This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition. In complex models and big data we anticipate that saddletransitions will be encountered frequently in the future.The low-dimensional systems that are commonly investigated in the context of critical transitions (or tipping points) 1,2 typically settle to stable, but not necessarily stationary, states 3 . The defining features of such states is that the system returns exponentially to the state after sufficiently small perturbations 4 . When environmental parameters change, a critical transition may occur when thresholds are crossed, where the system becomes unstable to certain perturbations. Low dimensional systems then typically depart quickly from the original state before approaching another, possibly distant, state. The thresholds at which such transitions occur are called bifurcation points 5 and the corresponding transitions are local bifurcations 3,6 . Warning signs for such bifurcation-induced transitions detect the loss of exponential restoring dynamics either through its impact on the statistics of noise-induced fluctuations 1,7,8 or by direct measurement of recovery rates 9,10

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