Maximally chaotic competition for attention in the cultural domain
Memory is a key determinant when cultural items compete for attention and, consequently, for success, as in the case of songs on a music chart. For modeling, one adds memory to Lotka-Volterra models, the reference for Markovian competitive processes. Here we treat memory in terms of an exponential moving average, finding that it leads to an extended region of winnerless chaos characterized by log-normal popularity statistics in trailing top-k charts. Importantly, the observed log-normal behavior collapses to a power-law distribution when the feedback dynamics is fast on the scale of the charting period. This result is in agreement with the observed statistics of real-world music charts (e.g., Billboard and Spotify). In a chaotic state, the size of the largest Lyapunov exponent is a measure of how unpredictable the system is. We find that the largest Lyapunov exponent varies strongly as a function of parameters in the phase where winnerless chaos is stable. Interestingly, the sets of parameters obtained by comparing simulations with real-world cultural-item dynamics extracted from Google Books and Google Trends, movies, Reddit, Wikipedia, Twitter, and scientific publications, are located close to the points in parameter space where the largest Lyapunov exponent reaches its local maximum, namely, close to the point of maximal unpredictability. This result suggests that cultural competitive processes are maximally chaotic when memory is a key determinant.