This explorable illustrates beautiful dynamical patterns that can be generated by a simple game theoretic model on a lattice. The core of the model is the Prisoner’s Dilemma, a legendary game analyzed in game theory. In the game, two players can choose to cooperate or defect. Depending on their choice, they receive a pre-specified payoffs. The payoffs are chosen such that it seems difficult to make the right strategy choice.

This explorable illustrates one of the most famous and most fundamental models for the emergence of flocking, swarming and synchronized behavior in animal groups. The model was originally published in a 1995 paper by Tamás Vicsek and co-workers and is therefore called the Vicsek-Model. The model can explain why transitions to flocking behavior in groups of animals are often not gradual. Instead, one can expect a sudden emergence of flocking and synchronized movements if a critical density is crossed.

This explorable illustrates the geometric and dynamic properties of the physical process of diffusion and its intimate relation to a mathematical object known as a random walk. It also illustrates graphically the implications of the central limit theorem that explains why we so often (normally) observe Gaussian distributions in nature. In the context of random walks this means that in the long run and from a great distance the paths of different types of walks become statistically indistiguishable.

This explorable illustrates how strong heterogeneities, cluster-like structures and high variability in node connectivities can naturally emerge in growing networks. The mechanism explored here is very simple: When a new node is added to the network it picks another target node at random. Instead of linking to it, the new node establishes a link to one of the target’s neighbors.