Most theories of brain function depend on information processing and the manipulation of neural or cognitive representations. This information processing is thought to be efficient and manipulations are thought to update representations that are predictive of future needs. These ideas are formulated by theories of efficient coding and predictive coding. Efficient coding is transmitting maximal information while minimizing the use of limited resources. Predictive coding is transmitting maximal information about the future while minimizing the use of limited resources. Although these parsimonious theories have accumulated evidence at the cellular level and in sensory regions, different models and data are needed to test the theories at the macroscale and across the brain network. This dissertation investigates how we can generalize efficient and predictive coding to the brain network by drawing from network science, information theory, and control theory. Using these frameworks, we operationalize compression and control as two key processes underlying efficient and predictive coding. Data compression distills predictive from unpredictive information using limited metabolic resources. Optimal control governs how the brain network should distribute the control signals needed to transition to diverse future states according to feedback from structured representations of the world. We test the compression and control models with hypothesized features of an efficient and predictive code. We find relationships between our models and the dimensionality and timescales of brain activity, metabolic resource expenditure, myelin content, areal expansion, functional specialization, and behavioral speed and accuracy. These findings support the efficient and predictive coding hypotheses across the brain and open new avenues to investigate brain function and mental health.