Tiago P. Peixoto
Phys. Rev. X 11, 021003

Community detection methods attempt to divide a network into groups of nodes that share similar properties, thus revealing its large-scale structure. A major challenge when employing such methods is that they are often degenerate, typically yielding a complex landscape of competing answers. As an attempt to extract understanding from a population of alternative solutions, many methods exist to establish a consensus among them in the form of a single partition “point estimate” that summarizes the whole distribution. Here, we show that it is, in general, not possible to obtain a consistent answer from such point estimates when the underlying distribution is too heterogeneous. As an alternative, we provide a comprehensive set of methods designed to characterize and summarize complex populations of partitions in a manner that captures not only the existing consensus but also the dissensus between elements of the population. Our approach is able to model mixed populations of partitions, where multiple consensuses can coexist, representing different competing hypotheses for the network structure. We also show how our methods can be used to compare pairs of partitions, how they can be generalized to hierarchical divisions, and how they can be used to perform statistical model selection between competing hypotheses.

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Markus Schläpfer, Lei Dong, Kevin O’Keeffe, Paolo Santi, Michael Szell, Hadrien Salat, Samuel Anklesaria, Mohammad Vazifeh, Carlo Ratti & Geoffrey B. West
Nature volume 593, pages 522–527 (2021)

Human mobility impacts many aspects of a city, from its spatial structure to its response to an epidemic. It is also ultimately key to social interactions, innovation and productivity. However, our quantitative understanding of the aggregate movements of individuals remains incomplete. Existing models—such as the gravity law or the radiation model—concentrate on the purely spatial dependence of mobility flows and do not capture the varying frequencies of recurrent visits to the same locations. Here we reveal a simple and robust scaling law that captures the temporal and spatial spectrum of population movement on the basis of large-scale mobility data from diverse cities around the globe. According to this law, the number of visitors to any location decreases as the inverse square of the product of their visiting frequency and travel distance. We further show that the spatio-temporal flows to different locations give rise to prominent spatial clusters with an area distribution that follows Zipf’s law. Finally, we build an individual mobility model based on exploration and preferential return to provide a mechanistic explanation for the discovered scaling law and the emerging spatial structure. Our findings corroborate long-standing conjectures in human geography (such as central place theory and Weber’s theory of emergent optimality) and allow for predictions of recurrent flows, providing a basis for applications in urban planning, traffic engineering and the mitigation of epidemic diseases.

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