  To better understand the flows of ideas or information through social and
biological systems, researchers develop maps that reveal important patterns in
network flows. In practice, network flow models have implied memoryless
first-order Markov chains, but recently researchers have introduced
higher-order Markov chain models with memory to capture patterns in multi-step
pathways. Higher-order models are particularly important for effectively
revealing actual, overlapping community structure, but higher-order Markov
chain models suffer from the curse of dimensionality: their vast parameter
spaces require exponentially increasing data to avoid overfitting and therefore
make mapping inefficient already for moderate-sized systems. To overcome this
problem, we introduce an efficient cross-validated mapping approach based on
network flows modeled by sparse Markov chains. To illustrate our approach, we
present a map of citation flows in science with research fields that overlap in
multidisciplinary journals. Compared with currently used categories in science
of science studies, the research fields form better units of analysis because
the map more effectively captures how ideas flow through science.
