Cognitive function is driven by dynamic interactions between large-scale neural circuits or networks, enabling behaviour. However, fundamental principles constraining these dynamic network processes have remained elusive. Here we use tools from control and network theories to offer a mechanistic explanation for how the brain moves between cognitive states drawn from the network organization of white matter microstructure. Our results suggest that densely connected areas, particularly in the default mode system, facilitate the movement of the brain to many easily reachable states. Weakly connected areas, particularly in cognitive control systems, facilitate the movement of the brain to difficult-to-reach states. Areas located on the boundary between network communities, particularly in attentional control systems, facilitate the integration or segregation of diverse cognitive systems. Our results suggest that structural network differences between cognitive circuits dictate their distinct roles in controlling trajectories of brain network function.

Small-world networks, according to Watts and Strogatz, are a class of networks that are ''highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.'' These characteristics result in networks with unique properties of regional specialization with efficient information transfer. Social networks are intuitive examples of this organization, in which cliques or clusters of friends being interconnected but each person is really only five or six people away from anyone else. Although this qualitative definition has prevailed in network science theory, in application, the standard quantitative application is to compare path length (a surrogate measure of distributed processing) and clustering (a surrogate measure of regional specialization) to an equivalent random network. It is demonstrated here that comparing network clustering to that of a random network can result in aberrant findings and that networks once thought to exhibit small-world properties may not. We propose a new small-world metric, x (omega), which compares network clustering to an equivalent lattice network and path length to a random network, as Watts and Strogatz originally described. Example networks are presented that would be interpreted as small-world when clustering is compared to a random network but are not small-world according to x. These findings have important implications in network science because small-world networks have unique topological properties, and it is critical to accurately distinguish them from networks without simultaneous high clustering and short path length.

Literature has shown that exercise is beneficial for cognitive function in older adults and that aerobic fitness is associated with increased hippocampal tissue and blood volumes. The current study used novel network science methods to shed light on the neurophysiological implications of exercise-induced changes in the hippocampus of older adults. Participants represented a volunteer subgroup of older adults that were part of either the exercise training (ET) or healthy aging educational control (HAC) treatment arms from the Seniors Health and Activity Research Program Pilot (SHARP-P) trial. Following the 4-month interventions, MRI measures of resting brain blood flow and connectivity were performed. The ET group's hippocampal cerebral blood flow (CBF) exhibited statistically significant increases compared to the HAC group. Novel whole-brain network connectivity analyses showed greater connectivity in the hippocampi of the ET participants compared to HAC. Furthermore, the hippocampus was consistently shown to be within the same network neighborhood (module) as the anterior cingulate cortex only within the ET group. Thus, within the ET group, the hippocampus and anterior cingulate were highly interconnected and localized to the same network neighborhood. This project shows the power of network science to investigate potential mechanisms for exercise-induced benefits to the brain in older adults. We show a link between neurological network features and CBF, and it is possible that this alteration of functional brain networks may lead to the known improvement in cognitive function among older adults following exercise.

Although graph theory has been around since the 18th century, the field of network science is more recent and continues to gain popularity, particularly in the field of neuroimaging. The field was propelled forward when Watts and Strogatz introduced their small-world network model, which described a network that provided regional specialization with efficient global information transfer. This model is appealing to the study of brain connectivity, as the brain can be viewed as a system with various interacting regions that produce complex behaviors. In practice, graph metrics such as clustering coefficient, path length, and efficiency measures are often used to characterize system properties. Centrality metrics such as degree, betweenness, closeness, and eigenvector centrality determine critical areas within the network. Community structure is also essential for understanding network organization and topology. Network science has led to a paradigm shift in the neuroscientific community, but it should be viewed as more than a simple ''tool du jour.'' To fully appreciate the utility of network science, a greater understanding of how network models apply to the brain is needed. An integrated appraisal of multiple network analyses should be performed to better understand network structure rather than focusing on univariate comparisons to find significant group differences; indeed, such comparisons, popular with traditional functional magnetic resonance imaging analyses, are arguably no longer relevant with graph-theory based approaches. These methods necessitate a philosophical shift toward complexity science. In this context, when correctly applied and interpreted, network scientific methods have a chance to revolutionize the understanding of brain function.