Structure Preserving Embedding (SPE) is an algorithm for embedding graphs in Euclidean space such that the embedding is lowdimensional and preserves the global topological properties of the input graph. Topology is preserved if a connectivity algorithm, such as k-nearest neighbors, can easily recover the edges of the input graph from only the coordinates of the nodes after embedding. SPE is formulated as a semidefinite program that learns a low-rank kernel matrix constrained by a set of linear inequalities which captures the connectivity structure of the input graph. Traditional graph embedding algorithms do not preserve structure according to our definition, and thus the resulting visualizations can be misleading or less informative. SPE provides significant improvements in terms of visualization and lossless compression of graphs, outperforming popular methods such as spectral embedding and Laplacian eigenmaps. We find that many classical graphs and networks can be properly embedded using only a few dimensions. Furthermore, introducing structure preserving constraints into dimensionality reduction algorithms produces more accurate representations of highdimensional data.

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Understanding the spatial networks formed by the trajectories of mobile users can be beneficial to applications ranging from epidemiology to local search. Despite the potential for impact in a number of fields, several aspects of human mobility networks remain largely unexplored due to the lack of large-scale data at a fine spatiotemporal resolution. Using a longitudinal dataset from the location-based service Foursquare, we perform an empirical analysis of the topological properties of place networks and note their resemblance to online social networks in terms of heavy-tailed degree distributions, triadic closure mechanisms and the small world property. Unlike social networks however, place networks present a mixture of connectivity trends in terms of assortativity that are surprisingly similar to those of the web graph. We take advantage of additional semantic information to interpret how nodes that take on functional roles such as 'travel hub', or 'food spot' behave in these networks. Finally, motivated by the large volume of new links appearing in place networks over time, we formulate the classic link prediction problem in this new domain. We propose a novel variant of gravity models that brings together three essential elements of inter-place connectivity in urban environments: network-level interactions, human mobility dynamics, and geographic distance. We evaluate this model and find it outperforms a number of baseline predictors and supervised learning algorithms on a task of predicting new links in a sample of one hundred popular cities.

In recent decades, the world has experienced rates of urban growth unparalleled in any other period of history and this growth is shaping the environment in which an increasing proportion of us live. In this paper, we use a longitudinal dataset from Foursquare, a location-based social network, to analyse urban growth across 100 major cities worldwide. Initially, we explore how urban growth differs in cities across the world. We show that there exists a strong spatial correlation, with nearby pairs of cities more likely to share similar growth profiles than remote pairs of cities. Subsequently, we investigate how growth varies inside cities and demonstrate that, given the existing local density of places, higher-than-expected growth is highly localized while lower-than-expected growth is more diffuse. Finally, we attempt to use the dataset to characterize competition between new and existing venues. By defining a measure based on the change in throughput of a venue before and after the opening of a new nearby venue, we demonstrate which venue types have a positive effect on venues of the same type and which have a negative effect. For example, our analysis confirms the hypothesis that there is large degree of competition between bookstores, in the sense that existing bookstores normally experience a notable drop in footfall after a new bookstore opens nearby. Other place types, such as museums, are shown to have a cooperative effect and their presence fosters higher traffic volumes to nearby places of the same type.

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