and non-adjacent dependencies between stimuli$^{13,14}$. Taken together, these studies suggest that second- or third-order statistical relationships may be encoded implicitly, and furthermore, that higher-level organizational principles themselves might be implicitly learned. Indeed, recent work has shown that temporal ordering of visual stimuli can convey the organizational principle of modularity$^{3}$. This observation opens up the possibility of studying whether certain organizational principles are more or less facilitative of learning, and whether information embedded in certain organizational structures might be easier to learn than information embedded in others. 

An ideally suited language in which to define such higher-order principles is network science$^{2}$, an emerging interdisciplinary field that addresses the architecture, dynamics, and design of complex systems composed of many connected parts$^{5}$. The set of parts (network nodes) and connections (network edges) are often parsimoniously encoded in a mathematical object called a graph$^{15}$. In the context of learning, we can construct a graph that encodes the pattern of relationships between objects, movements, or sounds. Prior theoretical work 16 has addressed the relationship between graph structure and artificial grammars (such as in Cleeremans & McClelland $^{11}$), and we build on this work by empirically addressing the impact of graph-based properties. Recently Karuza et al. 4 capitalized on this approach to define a graph from which the temporal ordering of visual stimuli was drawn. Learners exhibited a robust on-line measure of learned graph structure: a surprisal effect defined as an increase in reaction time when transitioning between modules. Importantly, this surprisal effect was dependent on the type of traversal through the graph, and was more strongly pronounced when traversals through the graph provided redundancy in local information. 

While the manner in which a graph is traversed can influence learning, the nature of the graph itself may serve as an even more fundamental constraint on the potential for humans to learn organizational principles of information. Many real-world systems including language$^{17}$, conceptual knowledge$^{18,19}$, social groups$^{20}$, and societies 21 display non-trivial higher-order structure such as clustering or hub-and-spoke architecture that is relevant for how humans can optimally communicate, interact, and ultimately survive in their environment. Moreover, our knowledge about these systems unfolds and grows over time as we experience new parts (nodes) and their relations (edges). Understanding the impact of such higher-order structure on learning could help to explain why knowledge of some (natural or man-made) systems may be more easily acquired than others, and why individuals differ in their capacity to learn them. It may also shed light on the question of how humans can generalize from local statistical information to develop representations of broad-scale organizational patterns$^{22}$. 

Here, we examined whether the higher-order regularities of three graph structures influenced implicit learning of statistical relationships among temporally ordered stimuli. Specifically, we trained subjects on a self-paced Serial Reaction Time (SRT) 

