In urban systems, there is an interdependency between neighborhood roles and transportation patterns between neighborhoods. In this paper, we classify docking stations in bicycle-sharing networks to gain insight into the human mobility patterns of three major cities in the United States. We propose novel time-dependent stochastic block models, with degree-heterogeneous blocks and either mixed or discrete block membership, which classify nodes based on their time-dependent activity patterns. We apply these models to (1) detect the roles of bicycle-sharing stations and (2) describe the traffic within and between blocks of stations over the course of a day. Our models successfully uncover work blocks, home blocks, and other blocks; they also reveal activity patterns that are specific to each city. Our work gives insights for the design and maintenance of bicycle-sharing systems, and it contributes new methodology for community detection in temporal and multilayer networks with heterogeneous degrees.

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We consider decoupling for a fractal subset of the parabola. We reduce studying l 2 L p decoupling for a fractal subset on the parabola tpt, t 2 q : 0 ď t ď 1u to studying l 2 L p{3 decoupling for the projection of this subset to the interval r0, 1s. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when p{3 is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to r0, 1s. Our ideas are inspired by the recent work on ellipsephic sets by Biggs [1, 2] using nested efficient congruencing.

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