Uncovering space-independent communities in spatial networks

Abstract

Many complex systems are organized in the form of a network embedded in space. Important examples include the physical Internet infrastucture, road networks, flight connections, brain functional networks, and social networks. The effect of space on network topology has recently come under the spotlight because of the emergence of pervasive technologies based on geolocalization, which constantly fill databases with people's movements and thus reveal their trajectories and spatial behavior. Extracting patterns and regularities from the resulting massive amount of human mobility data requires the development of appropriate tools for uncovering information in spatially embedded networks. In contrast with most works that tend to apply standard network metrics to any type of network, we argue in this paper for a careful treatment of the constraints imposed by space on network topology. In particular, we focus on the problem of community detection and propose a modularity function adapted to spatial networks. We show that it is possible to factor out the effect of space in order to reveal more clearly hidden structural similarities between the nodes. Methods are tested on a large mobile phone network and computer-generated benchmarks where the effect of space has been incorporated. complex networks | social systems U nderstanding the principles driving the organization of complex networks is crucial for a broad range of fields including information and social sciences, economics, biology, and neuroscience (1). In networks where nodes occupy positions in an Euclidian space, spatial constraints may have a strong effect on their connectivity patterns (2). Edges may either be spatially embedded, such as in roads or railway lines in transportation networks or cables in a power grid, or abstract entities, such as friendship relations in online and offline social networks or functional connectivity in brain networks. In either case, space plays a crucial role by affecting, directly or indirectly, network connectivity and making its architecture radically different from that of random networks (3). A crucial difference stems from the cost associated to long-distance links (4-12), which restricts the existence of hubs (i.e., high-degree nodes), and thus the observation of fat-tailed degree distributions in spatial networks.From a modeling viewpoint, gravity models (13-15) have long been used to model flows in spatial networks. These models focus on the intensity of interaction between locations i and j separated by a certain physical distance d ij . It has been shown for systems as diverse as the International Trade Market (16), human migration (17), traffic flows (18), or mobile communication between cities (19, 20) that the volume of interaction between distant locations is successfully modeled bywhere N i measures the importance of location i, e.g., its population, and the deterrence function f describes the influence of space. Eq. 1 emphasizes that the number of interactions between two locations is proportional to the numb...

Revisiting the Past: Replicating Fifty-Year-Old Flow Analysis Using Contemporary Taxi Flow Data

“…As we noted earlier, our own attempts employing cutting-edge link-clustering approaches required more than a month to complete. Unlike techniques adopted from network analysis (Expert et al 2011;Thomas et al 2012), therefore, PCA is able to cope with a highly skewed system of flows, as occurs in the AddLee data set, without producing a trivial partitioning in which most clusters contain a single observation and one cluster contains the rest of the data set. In the context of our data, this corresponds to a trivial coreperiphery regionalization in which the entire CBD is disaggregated into separate clusters before anything in the periphery is split into separate groups.…”

“…The second issue is more subtle because the exclusive association between zones (nodes) and regions (groups of nodes) can be a natural, and indeed powerful, approach to understanding human geography (Ratti et al 2010;Expert et al 2011). Where, for instance, though, there is evidence of polycentricism and of multiple business centers generating and receiving large numbers of trips across a wide area (Taylor, Evans, and Pain 2006), it is worth asking whether this type of exclusive partitioning is appropriate.…”

“…Flow data, however, have changed a great deal since the 1960s, becoming much more extensive through the development of sensors and systems able to capture data in real time and complementing or even supplanting the manual surveys that were the dominant mode of collection in the past. Reflecting the preoccupations of our "network society" (Castells 1996), methods derived from social network analysis are particularly popular (Demsar, Spatenkova, and Virrantaus 2007;Ratti et al 2010;Expert et al 2011). O'Sullivan and Manson (2015) also noted a rapid increase in physicists working on geographical topics, including flows and interactions-articulating and "solving the city" in these terms (Bettencourt and West 2010).…”

Space-Independent Community Structure Detection in United States Air Transportation

“…Most of them work on the principle of modularity (Newman, Girvan 2004) optimisation, aiming to maximise the modularity benefit function describing the quality of a network partition into communities. The more links that fall within a community compared to an ensemble of benchmark random networks with the same community structure, then the more bias there is for links to connect to nodes belonging to the same community, and therefore the higher the modularity Q in (1) (Expert et al 2011). In essence, modularity measures how sharply the modules are defined.…”

“…Therefore, standard community detection methods (typically based on the NG null model) will discover communities of nodes that are spatially close, as opposed to communities that have particularly strong internal interactions (Ball et al 2011, Estrada, Hatano 2009). To address this, (Expert et al 2011) proposed an alternative null model for P ij that takes into account the effect of space by favouring communities of nodes i and j that are more connected than expected, given the physical distance d ij between them:…”