AMERICAN ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE
Information Measures and Cognitive limits in multilayer navigation
By Riccardo Gallotti, Mason A. Porter & Marc Barthelemy
Cities and their transportation systems become increasingly complex and multimodal as they grow, and it is natural to wonder whether it is possible to quantitatively characterize our difficulty navigating in them and whether such navigation exceeds our cognitive limits. A transition between different search strategies for navigating in metropolitan maps has been observed for large, complex metropolitan networks. This evidence suggests the existence of a limit associated with cognitive overload and caused by a large amount of information that needs to be processed. In this light, we analyzed the world’s 15 largest metropolitan networks and estimated the information limit for determining a trip in a transportation system to be on the order of 8 bits. Similar to the “Dunbar number,” which represents a limit to the size of an individual’s friendship circle, our cognitive limit suggests that maps should not consist of more than 250 connection points to be easily readable. We also show that including connections with other transportation modes dramatically increases the information needed to navigate in multilayer transportation networks. In large cities such as New York, Paris, and Tokyo, more than 80% of the trips are above the 8-bit limit. Multimodal transportation systems in large cities have thus already exceeded human cognitive limits and, consequently, the traditional view of navigation in cities has to be revised substantially.
Introduction
The number of “megacities”—urban areas in which the human population is larger than 10 million—has tripled since 1990 (1). New York City (NYC), one of the first megacities, reached that level in the 1950s, and the world now has almost 30 megacities, which together include roughly half a billion inhabitants. The growth of such large urban areas usually also includes the development of transportation infrastructure and an increase in the number and use of different transportation modes (2, 3). For example, about 80% of the cities with human populations larger than 5 million have a subway system (4). This leads to a natural question: Is navigating transportation systems in very large cities too difficult for humans (5)? Moreover, how does one quantitatively characterize this difficulty?
It has long been recognized that humans have intrinsic cognitive limits for processing information (6). In particular, it has been suggested that an individual can maintain relationships only on the order of 150 stable relationships (7, 8). This “Dunbar number” was first proposed in the 1990s by the British anthropologist Robin Dunbar by extrapolating results about the correlations of brain sizes with the typical sizes of social groups for various primates. Although it is still controversial, these results have been supported by subsequent studies of, for example, traditional human societies (9) and microblogging (10).
When navigating for the first time between two unfamiliar places and having a transportation map as one’s only support, a traveler has to compare different path options to find an optimal route. In contrast to the schematization of partially familiar routes (11), here a traveler does not need to simultaneously visualize the whole route; it is sufficient to identify and keep track of the position of the connecting stations on a map. Therefore, a first important point to consider is that humans can track information for a maximum of about four objects in their visual working memory (12). This implies that a person can easily keep in mind the key locations (origin, destination, and connection points) for trips with no more than two connections (which corresponds exactly to four different points). In addition, recent studies on visual search strategies (13, 14) have shown a transition in search strategies between the simple cases of the Stuttgart and Hong Kong metropolitan networks and the case of Paris, which has one of the most complicated transportation networks in the world. The time needed to find a route in a transportation network grows with the complexity of its map, and the pattern of eye fixations also changes from following metro lines to a random scattering of eye focus all over the map (13). A similar transition from directional to isotropic random search has been observed for the visual search of hidden objects with an increasing number of distractors (15). The ability to manage complex “mental maps” is thus limited, and only extensive training on spatial navigation can push this limit with morphological changes in the hippocampus (16). Human-constructed environments have far exceeded these limits, and it is interesting to ask whether there is a navigation analog for the Dunbar number and whether there exists a cognitive limit to human navigation ability such that it becomes necessary to rely on artificial systems to navigate in transportation systems in large cities. If such a cognitive limit exists, what is it? In this paper, we answer this question using an information-processing perspective (6) to characterize the difficulty of navigating in urban transportation networks. We use a measure of “information search” associated with a trip that goes from one route to another (17). In most networks, many different paths connect a pair of nodes, and one generally seeks a fastest available path (which is not necessarily unique) that minimizes the total time to reach a destination. However, it tends to be more natural for most individuals to instead consider a “simplest path,” which has the minimal number of connections (see Fig. 1) (18).
Rosvall et al. (17) proposed a measure for the information needed to encode a shortest path from a route s to another route t. However, the amount of necessary information can depend strongly on the initial and final nodes, and we consider a trip from an origin node i in route s to a destination node j in route t. This trip is embedded in real space; among all of the possible simplest paths (17), we pick a fastest path p(i, s; j, t), which can differ from an actual fastest path between i and j (see Fig. 1A). In computing the travel time of a trip, we neglect the contribution of walking and waiting times (2). However, the choice of a simplest path already tends toward the minimization of such transfer costs, which strongly influence a traveler’s decisions (19).
![Fig. 1. Fastest and simplest paths in primal and dual networks. (A) In the primal network of the New York City (NYC) metropolitan system, a simplest path (light blue) from 125th Street on line 5 (dark green) to 121st Street on line J (brown) differs significantly from a fastest path (gray). There is only one connection for the above simplest path (Brooklyn Bridge–City Hall/Chambers Street) in Lower Manhattan. In contrast, the above fastest path needs three connections (5→F→E→J). We compute the duration of this path using travel times from the Metropolitan Transportation Authority (MTA) Data Feeds (see Materials and Methods). We neglect walking and waiting times. (B) In the dual space, nodes represent routes [where ACE, BDFM, and NQR are service names (49)], and edges represent connections. A “simplest path” in the primal space is defined as a shortest path with the minimal number of edges in the dual space (light-blue arrow). It has a length of C = 1 and occurs along the direct connection between line 5 (dark-green node) and line J (brown node). The above fastest path in the primal space has a length of C = 3 (gray arrows) in the dual space, as one has to change lines three times. [We extracted the schematic of the NYC metropolitan system from a map that is publicly available on Wikimedia Commons (45).]](https://www.infrastructureusa.org/wp-content/uploads/2016/03/transitsystems.jpg)
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Tags: AAAS, American Association for the Advancement of Science, Marc Barthelemy, Mason A. Porter, New York City, Riccardo Gallotti, Science Magazine
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