Structure of urban movements: polycentric activity and entangled hierarchical flows

Abstract

The spatial arrangement of urban hubs and centers and how individuals interact with these centers is a crucial problem with many applications ranging from urban planning to epidemiology. We utilize here in an unprecedented manner the large scale, real-time 'Oyster' card database of individual person movements in the London subway to reveal the structure and organization of the city. We show that patterns of intraurban movement are strongly heterogeneous in terms of volume, but not in terms of distance travelled, and that there is a polycentric structure composed of large flows organized around a limited number of activity centers. For smaller flows, the pattern of connections becomes richer and more complex and is not strictly hierarchical since it mixes different levels consisting of different orders of magnitude. This new understanding can shed light on the impact of new urban projects on the evolution of the polycentric configuration of a city and the dense structure of its centers and it provides an initial approach to modeling flows in an urban system.

Figure 1. Flow distribution.

Loglog plot of the histogram of the number of trips between two stations of the tube system. The line is a power law fit with exponent .

Figure 2. Ride distance distribution and propensity.

(a) Superimposition of the distance distribution of rides (circles) and of the distance distribution between stations (squares). The distribution of the observed rides can be fitted by a negative binomial law of parameters and , corresponding to a mean kms and standard deviation kms (solid line). This distribution is not a broad law (such as a Levy flight for example), in contrast to previous findings using indirect measures of movement , . (b) Ride distance propensity. Propensity of achieving a ride at a given distance with respect to a null-model of randomized rides.

Figure 3. Total flow distributions.

Zipf plot for the total inflows (red circles, below) and total outflows (blue squares, above) for morning peak hours (7am–10am). The inflow (outflow ) of a station () is defined as (). The straight lines are exponential fits of the form with for the inflow and for the outflow.

Figure 4. Hierarchical organization of the activity: Polycenters.

Breakdown of centers in terms of underlying stations and inflows. We gather stations by descending order of total inflow and we aggregate the stations to centers when taking into account more and more stations. In this process, all stations within meters of an already-defined center are aggregated to this main center. This yields the dendrogram shown here which highlights the hierarchical nature of the polycentric organization of this urban system. The bold names to the left of the aggregates — such as “West End” for the group of stations around Oxford Circus — are used throughout the paper as convenient labels to denote the polycenters.

Figure 5. The London subway (tube) system: polycenters and basins of attraction.

In the inset, we show the entire tube network while in the main figure, we zoom in on the central part of London. We represent the ten most important polycenters defined in the dendrogram of Figure 3, and show the corresponding propensity to anisotropy comparing actual flows with the null model defined in the text. A propensity of means that there is no deviation in a given direction with respect to the null model. Circles correspond to various levels of identical propensity values: the thicker circle in the middle corresponds to , inner circles correspond to propensities of and , and outer circles to and . The anisotropy is essentially in opposite directions from the center, thus showing a strong bias towards the suburbs for peripheral centers essentially, rather than for central centers. Moreover, most stations control their own regions and seem to have their own distinctive basins of attraction.

Figure 6. Structure of flows at and of the total flow.

When considering the most important flows from stations to centers such their sum represents of the total flow in the network, we observe sources (represented as squares) with outdegree such as London Bridge, Stratford, or Waterloo connecting to three different centers (represented as circles), as well as sources with (eg. Victoria) and (eg. Elephant and Castle). We also show how the pattern of flows is constructed iteratively when we go to larger fraction of the total flow (from shown in black to shown in red). We represent in red the new sources, centers and connections. The new sources connect to the older centers (eg. West End, City, etc) and the existing sources (eg. Victoria) connect to new centers (eg. Northern stations, Museums, and Parliament).

Figure 7. Most important links.

Proportion of links going from sources to centers of a certain group (I, II, III), considering links of decreasing importance for each given source, when raising (from the first link appearing, at left, to the last link, at right).

Figure 8. Transition matrix.

Typical form of the outdegree transition matrix , consisting essentially of a row vector (, inexistent sources before the transition) and an upper triangular matrix (made of a diagonal of sources having the same out-degree after the transition, and a submatrix of sources whose out-degree increases after the transition).

Figure 9. Evolution of the number of sources and their type.

(a) Number of new sources () versus the total flow . (b) Fraction of existing sources whose type is changing () when the total flow varies from to . Here .