The concept of attractors is considered critical in the study of dynamical systems as they represent the set of states that a system gravitates toward. However, it is generally difficult to analyze attractors in complex systems due to multiple reasons including chaos, high-dimensionality, and stochasticity. This paper explores a novel approach to analyzing attractors in complex systems by utilizing networks to represent phase spaces. We accomplish this by discretizing phase space and defining node associations with attractors by finding sink strongly connected components (SSCCs) within these networks. Moreover, the network representation of phase space facilitates the use of well-established techniques of network analysis to study the phase space of a complex system. We show the latter by introducing a new node-based metric called attractivity which can be used in conjunction with the SSCC as they are highly correlated. We demonstrate the proposed method by applying it to several chaotic dynamical systems and a large-scale agent-based social simulation model.

The El Farol Bar problem highlights the issue of bounded rationality through a coordination problem where agents must decide individually whether or not to attend a bar without prior communication. Each agent is provided a set of attendance predictors (or decision-making strategies) and uses the previous bar attendances to guess bar attendance for a given week to determine if the bar is worth attending. We previously showed how the distribution of used strategies among the population settles into an attractor by using a spatial phase space. However, this approach was limited as it required N − 1 dimensions to fully visualize the phase space of the problem, where N is the number of strategies available. Here we propose a new approach to phase space visualization and analysis by converting the strategy dynamics into a state transition network centered on strategy distributions. The resulting weighted, directed network gives a clearer representation of the strategy dynamics once we define an attractor of the strategy phase space as a sink-strongly connected component. This enables us to study the resulting network to draw conclusions about the performance of the different strategies. We find that this approach not only is applicable to the El Farol Bar problem, but also addresses the dimensionality issue and is theoretically applicable to a wide variety of discretized complex systems.

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A slight error was found in reviewing the simulation code post publication of St. Luce and Sayama (2020) where the Last Incorrect strategy was performing incorrectly in three strategy sets, affecting the results in the Dynamic Analysis section for the Last Correct, Never, and Last Incorrect strategy set but unaffecting the final conclusions. By fixing this bug, we see that when more than 60% of the population are using the Last Correct strategy, then Last Correct will enter a cycle of which it is always wrong while Last Incorrect strategy will always be right and the Never strategy is right about half the time. This shows a synthesis of the dynamic results seen in the two-strategy sets in the original publication. The original paper and figures suggest that the simulations eventually shift away from the Last Incorrect strategy when the Best Switching method is used. In actuality, the Never strategy is eventually dismissed. The results from utilizing the best switching mechanic now eliminates the Never strategy entirely from consideration. Note that this result now matches the results from the original Last Correct and Last Incorrect dynamic results.