Social groups are common among animals and humans [1]–[5]. In humans, they reflect friendship, kinship, and work relationships, and can also be seen as social networks. From an evolutionary and historical perspective, the formation of such network groups – consisting of agglomerations of dyadic interactions – has been constrained by geography. In contrast, larger social units, enabled by modern technology and political organization, offer drastically different opportunities for social interactions and for group assembly over larger geographic ranges. This raises two sorts of questions. First, is the structure of “old-fashioned” groups similar to the large-scale groups possible in modern society? And second, what role does geography play in group formation?

If we represent the social relationships among a population of people as a network, then groups can be seen as “communities” within the population that consist of sets of nodes that are relatively densely connected to each other but sparsely connected to other nodes in the network [6], [7]. While social communities have been studied for a long time [8], it has recently become feasible, with mobile phone data, to monitor the social interactions and geographic positions of millions of individuals [9], [10], and to apply algorithmic detection of communities on a large scale [6], [7]. The structure of dyadic social interactions is known to depend on geography, for example, as shown by the decay of friendship probability with distance, based on voluntary self-reports of hometown and US state, in a blog community [11], and the decrease in communication probability with distance based on the zip codes of cell phone billing addresses [12]. In addition, a previous study has shown that smaller communities are more homogeneous with respect to the billing postal codes of their members [13], while another presented evidence that this persists across a hierarchy of communities [14]. However, there are no prior large-scale studies of the way in which community structure depends on geography, where the actual communication locations are used and where geographical properties of communities themselves are examined (see Fig. 1).

With respect to group formation, geography can be seen as a kind of constraint. That is, social connections not only face network constraints and opportunities (we tend to form ties with others who are the friends of our friends), but also, quite obviously, geographic constraints and opportunities. What is unclear, however, is the way in which such geographic constraints and opportunities affect and shape network communities above and beyond their effect on dyadic interactions.

Our findings on the geographic decay of ties differ from those obtained for a network constructed from the customers of a Belgian mobile operator. Using zip codes provided for billing purposes to compute distances between individuals, Lambiotte et al. showed that the probability for two individuals to be connected decays as , which led them to suggest that the decay follows a so-called gravity model [12]. Our result, essentially showing that , differs for various possible reasons: we used the maximum-likelihood technique to estimate the value of the exponent [15]; our range of distances is larger ( km vs. km), allowing for more statistical power; we used the location of maximal phone use as opposed to the location of the billing address (which is often not reliable); and the population density in our target country is significantly lower than that of Belgium. Our result that tie strength does not vary with distance is complementary to the finding of Lambiotte et al., who report that the average duration of phone calls increases with distance, reaching a plateau around 40 km. Therefore, while the number of calls made to long-distance individuals friends is slightly smaller than those made to short distance friends, the average duration may be twice as long [12].

Communities appear to have particular properties in relationship to geography, properties that are distinct from the underlying interactions between pairs of individuals. Geography constrains group formation in important ways that nevertheless differ from the way it constrains dyadic interactions. On the one hand, comparison of topological and geographical centrality of nodes within communities demonstrated that the two are essentially uncorrelated. On the other hand, we find that the geographic shapes of social groups, measured in terms of geographic span and spatial clustering, vary in regular ways with the size of the group. For small communities, as their size increases, their expected geographic span increases smoothly at first, but then experiences a sudden bounce as the community size reaches about 30 members. To exemplify this behavior, an increase in community size from 10 to 20 members is associated with an increase in span by about 40%, whereas, in contrast, an increase from 30 to 40 members leads to an increase of about 100% in geographic span. This suggests that the tendency of human groups to remain geographically cohesive gradually gives in as the group size exceeds 30. Similarly, the number of clusters within a single topological group also increases with community size. Intriguingly, the number 30 is also close to the optimal group size for which cooperation in social dilemma situations, modeled, for example, by the public goods game, is maximized [20], [21].

Just as the structures of observed social interactions may be compared to randomized networks, the observed localities of individuals in communities may be compared to randomized locations. Indeed, if social ties could be formed without consideration for the underlying geography, we would expect the tie probability to be independent of distance, and the geographic span of groups to follow the proposed null models closely. However, we observe neither of the two. This demonstrates that network ties and network communities, in this context, do not behave as if they were in well mixed populations, suggesting that geography continues to maintain its power as a compartmentalizing factor. Thus, the assumption of perfect mixing of individuals, sometimes made in the study of infectious disease or technology diffusion in humans, does not then appear to hold either at the topological or at the geographic level.

The extent to which a spreading process follows the assumptions of well-mixed populations often depends on a number of conditions, including the nature of the spreading process. For example, network models can better account for the spread of diseases that spread via the formation of a physical tie (such as STD's) than those that spread by simple proximity (like the common cold). This is illustrated in a mobile phone context by Wang et al. [22], who find that the nature of the spreading process, and its dependence on proximity, clearly affect the dynamics of the spreading. Similarly, localization of interacting proteins within the geography of the cell can explain certain disease associations [23]. In addition, other work has suggested that the diffusion-like movement of people alone can often explain how a pathogen spreads, such as the plague in medieval Europe [24]. On the other hand, with the onset of air travel, pathogens are not constrained in the same way, as the epidemics of SARS and H1N1 documented [25], [26].

Ideally, models of the flow of pathogens or information through human populations would account for the simultaneous roles of geographic and network constraints, and our work helps shed light on the intersecting relationship between the two. Future work will explore the complex interrelationship between network topology and geography and their joint importance in understanding how phenomena spread through populations.

All networks were constructed from four weeks of anonymized mobile phone call and text messaging data from an operator based in an unnamed European country. Only interactions that took place between customers of the operator were considered, and only individuals who made at least two calls were included as nodes. To filter out sporadic calls and texts that are unlikely to correspond to meaningful social interactions, we required there to be a minimal level of reciprocation for a tie to be included in the network; each person had to initiate at least one interaction, where the initial transaction could be either a call or a text, and this could be reciprocated by either a call or a text.

We detected topological communities using the popular method of modularity maximization [6], [7], [16], [17] in the following manner. We first converted the original network consisting of directed voice calls and text messages into a symmetric unweighted network, effectively combining the two modes of interaction. We then proceeded to maximize modularity defined as (2)

where the adjacency matrix element denotes the presence () or absence () of a connection between nodes and , is the degree of node , the total weight of the edges in the network, the community assignment of node , and is the Kronecker delta function, which is unity if and only if , otherwise it is zero. Modularity measures the difference between the total fraction of edges that fall within groups versus the fraction one would expect by chance. A common null model, sometimes called the Newman-Girvan null model, is codified by the term, and it takes degree heterogeneity into account by preserving the expected degree distribution. High values of indicate network partitions in which more of the edges fall within groups than expected by chance. While maximizing modularity is known to be an NP-hard problem [27], there are numerous computational heuristics available [6], [7]. Since we are dealing with networks consisting of millions of nodes, we chose the Louvain method for its computational efficiency [28].