Phase transitions and natural scales

After community detection and smoothing for every percentile-scale, we are in a position to analyse the similarity between scales. More specifically, we are interested in seeing if there are well-defined ranges of scales that are sufficiently similar amongst each other and sufficiently distinct between ranges so that we can talk of natural scales, and reduce the 100 percentiles to a smaller number of scales.

Using a simple parameter-free breakpoint detection algorithm we are able to find phase transitions in scale space. We understand “phase transition” in a generic way, i.e. it corresponds to an abrupt change in the behaviour of partitions when slightly increasing the movement radius, going from a scale to the next ones. For the nine regions under study, our algorithm finds that the scale space is divided into no more than 2 or 3 well-differentiated intervals of scales characterised by very similar patterns. Aggregating increasingly long links while remaining below the upper bound of a given interval does not alter significantly the space partition typical of that interval. We call these intervals natural scales. Moreover, the breakpoints automatically found by our algorithm mostly match the visual intuition: in Fig. 1, we see that these phase transitions are also quite obvious simply by visual inspection.

Figure 1: Scale dissimilarity heat maps. Dissimilarity values are normalized per region to a [0, 1] scale. Lighter colors represent higher dissimilarity. Pure black (0.0) corresponds to a perfect match, bright yellow (1.0) to the maximum dissimilarity found for the region. Dashed blue lines indicate the discontinuities identified by the breakpoint detection algorithm and, accordingly, natural scales; green dots represent the prototypical scale for each natural scale interval. The mean absolute dissimilarity value per pair of intervals is shown. A value in a green background corresponds to an internal mean dissimilarity (the interval is being compared to itself); a black background indicates a mean dissimilarity between different intervals. Full size image

Multi-level partitions and prototypical scales

Given intervals of similar scales or natural scales, it is now desirable to have a method to visualise the boundaries defined by the partitions in those intervals. We propose a simple solution: identify the percentile that best represents the entire interval. We call this prototypical percentile a prototypical scale of the region under study. The prototypical scale of a given interval is the percentile of the interval with the corresponding partition that is the most similar to all other partitions in the interval. Prototypical scales found for each region are also represented in Fig. 1, along with natural scales. By construction, partitions the various scales of a given natural scale should thus roughly resemble the partition of the corresponding prototypical scale. In the following maps, prototypical scales are thus used as visual representations of natural scales.

Figure 2 uses the results for Belgium to illustrate how natural scales correspond to partitions in the map, and how the several natural scales can be combined in a single multiscale map, which provides richer information about the geographical patterns of the region than what is possible with more traditional methods. By using the full graph (percentile 100) and forcing the community detection algorithm to find the best partition in two communities, we present a bipartite division of the territory. As can be observed, the resulting partition matches almost perfectly the border between the two largest linguistic communities in Belgium. This is a well known result4,24 and it shows two things. On one hand, when we simplify our method this way, thus making it equivalent to previously published approaches, we obtain similar results, which provides some evidence of correctness. On the other hand, adopting these simplifications is most likely not the best way to unravel the structure of movement patterns in Belgium: to the contrary, for all scale phases, a bipartition does not achieve the best modularity, which usually corresponds to a larger number of geographical areas (Fig. 2).

Figure 2: Belgium borders at different scales. (a) Heat map extracted from Fig. 1. (b) Borders for the long distance scale. (c) Borders for the short distance scale. (d) Borders for the middle distance scale. (e) Multiscale borders. (f) Borders based on optimal two community partition of the full graph. (g) Language communities of Belgium. All maps except (g) were generated by the authors using the Basemap Matplotlib Toolkit ver. 1.0.8 (http://matplotlib.org/basemap/). Map in g) Wikimedia contributers Vascer and Knorck, licensed under CC BY-SA 3.0. The licence terms can be found on the following link: https://creativecommons.org/licenses/by-sa/3.0/. Full size image

Following our method, Belgium may be more precisely decomposed as an overlay of three territorial partitions of increasing fineness. The largest natural scale features a partition based on a small number of broad areas whose boundaries correspond to inter-urban mobility, as it only emerges when links longer than 81.5 km are included. By contrast, the smallest natural scale is such that most boundaries surround and enclose a local capital; it is based on links smaller than r ≤ 38.6 km. The middle natural scale appears when links between 38.6 and 81.5 km are considered. Interestingly, it diverges from the long-distance scale by only a few boundaries: for instance, while Hasselt is part of a broader Dutch-speaking cluster on the large-scale map, it belongs to the same cluster as Liege at the medium scale.

Finally, in Fig. 3, we further present the multi-scale maps for all nine regions. We also depict the absolute physical distances for all natural scale thresholds. Note that the absolute physical meaning of “large” or “small” scale is heavily region-dependent: for Paris, which is a quite dense metropolis extending over a comparably small area, the smallest natural scale typically covers a range of pedestrian movements (r ≤ 2.1 km). For Berlin, the switch between the small and large natural scales occurs at a radius of r ≤ 10.0 km which could correspond to “local” foot, bike or metro trips. For the largest regions such as Poland, Romania or Ukraine, they seem to correspond to a wider range of motorised inter-urban displacements, roughly around the order of magnitude of a hundred of kilometers.

Figure 3: Several multi-scale maps. Green corresponds to the smallest natural scale, blue to the middle (if it exists) and red to the largest. All maps were generated by the authors using the Basemap Matplotlib Toolkit ver. 1.0.8 (http://matplotlib.org/basemap/). Map tiles used in the background of the Berlin and Paris maps OpenStreetMap contributors, licensed under CC BY-SA (www.openstreetmap.org/copyright). The licence terms can be found on the following link: http://creativecommons.org/licenses/by-sa/2.0/. Full size image

A set of high-resolution maps for all natural scales as well as multi-scale representations of all regions may be found in the Supp. Info. Performing a thorough socio-geographic analysis of these maps is beyond the scope of this article, but we can identify some features that confirm folk knowledge about certain regions. In Portugal, large scale boundaries delineate the highly touristic beaches of Algarve in the south and fuzzily divide the country into north and south regions, while the short scales provide sensible local partitions, for example the dense city of Oporto and the socio-economic divide between the capital city of Lisbon and the neighbouring but more affluent Cascais/Estoril coastal area. The Benelux map enriches the previous insight on Belgium by providing a broader picture on potential cross-national interfaces – an achievement not possible with country-specific datasets traditionally used in the literature – here, the highest scale exhibits a mix of expected international borders (for instance between Belgium and the Netherlands) and fuzzy cross-national spaces (such as the wide commuting area surrounding Luxembourg, or the narrow strips adjacent to the French-Belgian border, e.g. around Lille), while leaving room for cross-border low-scale patterns. Paris features both the traditional east-west sociological partition of the city, while exhibiting more specific activity neighbourhoods at the lower level (Quartier Latin, Belleville, the governmental area).

Scale-dependent user behavior

Natural scales thus describe geographical areas and boundaries operating within a broad range of scale percentiles, though not beyond. In this respect, they correspond to a discrete spectrum of mobility behaviours which most likely unveil consistent yet distinct spatial practices of the underlying region. How are scales, boundaries and user behaviour related? For one, we observe on Fig. 3 that some regions such as Poland or Romania appear to exhibit a much higher proportion of smaller, lower-scale patterns than other regions such as Paris or Benelux. We find that these discrepancies have an interpretation in terms of user-level mobility behaviour: regions where movement distance distributions are broadest (i.e. where low and high percentiles correspond to markedly distinct physical distances) also exhibit a much larger amount of small-radius geographical patterns at the shortest scale (see Figs S3 and 4 of Supp. Info.). In other words, we show that the relative amount of patterns across the spectrum of natural scales corresponds to a relative spread of actual physical link distances across that same spectrum.

We further examine the relationship between natural scales and user-level behaviour by assigning to users the set of natural scales that they contributed to. We consider that users contribute to a natural scale if they perform at least one movement with a distance within the corresponding scale interval. Figure 4 shows a mixed picture. Overall, the proportion of users contributing exclusively to the highest scales is generally small, while the shortest scales are the most populated. At the same time, the most active users in terms of visited locations (as well as posted photos, see Supp. Info.) are those who span the most scales. From this we conclude that there exists a wide core of users active in all scales, which additionally always gathers a sizable proportion of all users (often the highest proportion). This hints at the fact that natural scales are based on scale-related behaviours rather than scale-related users.

Figure 4: Fraction of users contributing to each natural scale. The area of each slice/circle is proportional to the number of users active in the set of scales that it represents (for instance, “12” corresponds to users contributing exclusively to scales 1 and 2). User activity is represented by slice darkness, which is proportional to the number of visited locations relative to the maximal activity of a given region (100%): here, “123” users are always the darkest/most active slice, they consistently visit many more locations than other users. Full size image

Concluding remarks

By effectively distinguishing link scales and defining an increasing series of more and more global networks, we show that territories are automatically decomposable into a partially overlapping hierarchy of geographical partitions and, further, that this hierarchy exhibits a remarkably small number of natural scales. Besides, we fulfilled in the case of spatial mobility networks the ambition of finding natural phases in community partitions based on some notion of resolution (see ref. 25 for non-geographical scale-free networks). In contrast with the classical expectation that aggregate mobility data is essentially scale-free, we were able to uncover a discrete number of distance thresholds and radii configuring consistent movement patterns.

More broadly, understanding and breaking up mobility patterns as an overlay of a small number of endogenous scale-specific behaviours bears important consequences in diverse fields such as epidemiology, cultural contagion and public policy26,27,28 where the low-level modeling of displacements7,8,29,30,31 is pivotal: here, the introduction of boundary conditions based on a scaffolding of a small number of natural scales emerging endogenously from the data could prove to be particularly fruitful.