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To better understand the flows of ideas or information through social andbiological systems, researchers develop maps that reveal important patterns innetwork flows. In practice, network flow models have implied memorylessfirst-order Markov chains, but recently researchers have introducedhigher-order Markov chain models with memory to capture patterns in multi-steppathways. Higher-order models are particularly important for effectivelyrevealing actual, overlapping community structure, but higher-order Markovchain models suffer from the curse of dimensionality: their vast parameterspaces require exponentially increasing data to avoid overfitting and thereforemake mapping inefficient already for moderate-sized systems. To overcome thisproblem, we introduce an efficient cross-validated mapping approach based onnetwork flows modeled by sparse Markov chains. To illustrate our approach, wepresent a map of citation flows in science with research fields that overlap inmultidisciplinary journals. Compared with currently used categories in scienceof science studies, the research fields form better units of analysis becausethe map more effectively captures how ideas flow through science.

Maps of sparse Markov chains efficiently reveal community structure in network flows with memory