Humans excel at identifying abstract relationships between events, a skill crucial for abstract reasoning, pattern recognition, language processing, social understanding, and data segmentation. This ability involves detecting statistical regularities across various scales, going beyond the simple understanding of individual transition probabilities between stimuli to grasping higher-order structures like hierarchical patterns, temporal regularities, and semantic relationships. Network science provides a useful framework to represent these relationships, with stimuli as nodes and transitions as edges. Research using artificial grammars has shown that humans can infer the underlying network rules, and that network features like modularity (communities of densely connected nodes) impact learning and reaction times. However, the brain mechanisms underlying the learning of these abstract network features remain unclear. The prevailing intuition centers on either error-free Bayesian inference or computationally complex hierarchical learning algorithms. This paper proposes a different hypothesis: the brain maximizes accuracy while simultaneously minimizing computational complexity. This trade-off, formalized using the free energy principle, leads to a maximum entropy model of internal event representations. The model suggests that mental errors, rather than being detrimental, contribute to the emergence of higher-order network features in our mental representation, making the coarse-grained structure more apparent while finer details fade.

The paper draws upon a substantial body of work in cognitive science, statistical learning, and network science. Studies on infant language acquisition demonstrate the early emergence of statistical learning abilities (Saffran et al., 1996). Further research highlights human sensitivity to individual transition probabilities (Hyman, 1953; Sternberg, 1969; Fiser & Aslin, 2002) and higher-order structures (Newport & Aslin, 2004; Dehaene et al., 2015; Meyniel & Dehaene, 2017). The application of network science to model these relationships (Newman, 2003) has proven particularly fruitful in the study of artificial grammars (Cleeremans & McClelland, 1991; Gomez & Gerken, 1999) and the neural correlates of statistical learning (Schapiro et al., 2013; Kahn et al., 2018; Karuza et al., 2017, 2019). Prior research also points towards the importance of balancing accuracy and computational cost in cognitive processes (Tversky & Kahneman, 1974; Tononi et al., 1994; Cohen et al., 2007; Ortega & Braun, 2013; Friston et al., 2006), often implicitly or through related concepts like speed-accuracy tradeoffs (Wickelgren, 1977). The free energy principle (Friston et al., 2006) provides a formal framework to address the trade-off between accuracy and computational complexity, grounding the authors' model in existing neuroscientific and psychological theories.

The study employed two primary experimental paradigms: a probabilistic sequential motor task and an n-back memory task. The sequential motor task involved presenting subjects with sequences of visual stimuli representing nodes in a transition network. Each stimulus showed one or two highlighted squares out of five, and subjects responded by pressing corresponding keyboard keys (Fig. 1). The experiments used two network topologies with uniform transition probabilities: a modular graph with three densely connected communities and a lattice graph. Reaction times were measured to assess subjects' expectations about the transition structure. Data preprocessing involved removing initial trials, incorrect responses, and implausible reaction times (outside three standard deviations from the mean or below 100ms). Higher-order network effects were analyzed using linear mixed-effects models, regressing out effects of button combinations, trial number, and stimulus recency. A third experiment used a ring graph with interspersed transitions that violated the established network structure ('short' and 'long' violations based on topological distance). The n-back memory task involved presenting sequences of letters and asking subjects to identify letters matching the letter from n trials prior (n=1, 2, 3). For each correct response, the temporal distance (Δt) between the last occurrence of the stimulus and the target was measured, providing a direct measure of the memory distribution P(Δt). This distribution was analyzed to assess the form of mental errors.

The study revealed two key higher-order network effects on human reaction times in the sequential motor task. First, the *cross-cluster surprisal effect*: within-cluster transitions were faster than between-cluster transitions in the modular graph (35ms faster for random walks, 36ms for Hamiltonian walks). Second, the *modular-lattice effect*: reactions were faster in the modular graph compared to the lattice graph (23ms faster). These effects were robust even after controlling for stimulus recency. A maximum entropy model, derived from the free energy principle, successfully predicted these qualitative effects. The model incorporated mental errors in estimating transition probabilities using a Boltzmann distribution (Eq.5), where the inverse temperature β parameterizes the trade-off between accuracy and computational efficiency. By fitting the model to individual subjects' reaction times, the inverse temperature β was estimated (average β = 0.30 for random walks and 0.61 for Hamiltonian walks). The model's accuracy was compared to a hierarchy of competing models representing explicit learning of higher-order transition structures (Eq. 7), demonstrating that the maximum entropy model provided a superior fit (Figs. 4e,f, 4i,j). The n-back memory task confirmed the model's prediction of a monotonically decreasing exponential memory distribution P(Δt) (Fig. 5). The estimated β from the n-back task (0.32 ± 0.01) closely matched the average β from the reaction time data (0.30 for random walks), further supporting the model's assumptions. Finally, an experiment using a ring graph with topological violations (Fig. 6) showed that the magnitude of the surprise (increased reaction time) depended on the topological distance of the violation, aligning with the model's predictions.

The findings demonstrate that mental errors, rather than hindering abstract representation, play a constructive role. The maximum entropy model, incorporating these errors, accurately predicts higher-order network effects on human behavior. The model's ability to predict individual behavior based on estimated β highlights the significance of considering computational constraints in cognitive models. The close correspondence between β estimates from reaction time and memory tasks strongly supports the hypothesis that the same mental error mechanisms affect both learning and memory processes. This contrasts with alternative explanations like error-free Bayesian learning or advanced planning mechanisms. The results suggest that the structure of optimally learnable networks should possess a hierarchical community structure, a characteristic found in numerous real-world networks. This raises the intriguing possibility that these networks' structures have evolved to optimize human cognitive processing.

This study offers a novel perspective on abstract representation, demonstrating that mental errors are not mere noise but instead shape our understanding of the world in predictable ways. The maximum entropy model, grounded in the free energy principle, successfully captures higher-order network effects on human behavior. This framework has implications for the design of optimally learnable information sources, emphasizing the importance of hierarchical structure. Future research could investigate the influence of different network topologies on learning and memory, explore the interplay between recency effects and higher-order network structure, and further examine the relationship between mental errors and cognitive function.

The study's reliance on reaction times as a proxy for internal representations is a limitation. While reaction times are a reliable indicator of processing difficulty, they may not perfectly reflect the nuances of internal cognitive processes. The experiments focused on specific types of networks and stimuli; the generalizability of findings to other networks or cognitive domains warrants further investigation. The relatively simple linear model used to link anticipation and reaction times might benefit from refinement to incorporate more complex relationships. Finally, the cross-sectional nature of the study limits our ability to definitively conclude about causal relationships between mental errors and abstract representation.