A gravity model for inter-city telephone communication networks

In this paper, we consider a network of mobile phone customers aggregated by geographical proximity. We analyze the anonymous communications patterns of 2.5 million customers of a Belgian mobile phone operator. Grouping customers by billing address, we obtain a two-layer network. The lower layer the microscopic network is built from individual human-to-human communications. The upper layer the macroscopic network is built from communications between 571 cities in Belgium. We show that inter-city communication intensity is characterized by a gravity model: the communication intensity between two cities is proportional to the product of the city population sizes divided by the square of their distance. PACS numbers: 89.75.Da, 89.75.Fb, 89.65.Ef ar X iv :0 90 5. 06 92 v1 [ ph ys ic s. so cph ] 5 M ay 2 00 9 A gravity model for inter-city telephone communication networks 2 Recent research has shown that different characteristics in cities grow in different ways with population size. While some characteristics are directly proportional to cities’ population sizes, other features such as productivity or energy consumption are not linear but exhibit instead superlinear or sublinear dependence to population size [1]. Interestingly, some of these features have strong similarities with those found in biological cells, and this observation has led to the metaphor of cities seen as living entities [2]. Particular interactions between cities, such as passenger transport flows and phone messages, have also been related to population and distance [3, 4]. In socioeconomical networks, interactions between entities such as cities or countries have led to models remembering Newton’s gravity law, where the sizes of the entities play the role of mass [5]. Road and airline networks between cities have also been studied recently [6, 7], and in the case of road networks, it appears that the strength of interaction also follows a gravity law. While these results have provided a better understanding of the way cities interact, a finer analysis at human level was until now difficult because of the lack of data. Recently, however, telephone communication data has opened a new way of analyzing cities at both a fine and aggregate level, and as Gottman already noted in 1957 [8]: “the density of the flow of telephone calls is a fairly good measure of the relationships binding together the economic interests of the region”. Several large datasets of email and phone calls have recently been available. By using these as proxy for social networks, they have enabled the study of humans connections and behaviors [9, 10, 11, 12, 13]. The use of geographical information makes it possible to go one step further in the study of individual and group interactions. For example, Lambiotte et al. use a mobile phone dataset to show that the probability for a call between two people decreases as the square of their distance [14]. However, while the structure of complex networks has already been widely studied [15, 16, 17, 18], to date, contributions have not yet analyzed large-scale features of social networks where people are aggregated based on their geographical proximity. In this work, we study anonymized mobile phone communications from a Belgian operator and derive a model of interaction between cities. Grouping customers together by billing address, we create a two-level network, containing both a microscopic network of human-to-human interactions, and a macroscopic network of interactions between cities. The data that we consider consists of the communications made by more than 3.3 million customers of a Belgian mobile phone operator over a period of 6 months in 2006 [14]. Every customer is identified by a surrogate key and to every customer we associate its corresponding billing address zip code. In order to construct the communication network, we have filtered out calls involving other operators (there are three main operators in Belgium), incoming or outgoing, and we have kept only those transactions in which both the calling and receiving individuals are customers of the mobile phone company. In order to eliminate “accidental calls”, we have kept links between two customers i and j only if there are at least six calls in both directions during the 6 A gravity model for inter-city telephone communication networks 3 months time interval. The resulting network is composed of 2.5 million nodes and of 38 million links. To the link between the customers i and j we associate a communication intensity by computing the total communication time in seconds lij between i and j. In order to analyze the relation between this social network and geographical positioning, we associate customers to cities based on their billing address zip code. Belgium is a country of approximately 10.5 million inhabitants, with a high population density of 344 inhab./km. The Belgian National Institute of Statistics (NIS) [19] divides this population in 571 cities (cities, towns and villages), whose sizes show an overall lognormal population distribution with approximate parameters μ = 4.05 and σ = 0.37. ‡ The analyzed communication network provides information for the operator’s Figure 1. Ranks of city population sizes (blue triangles) and number of customers (red squares) follow similar distributions. customers rather than for the entire population. However, the number of customers present in each city follows the same lognormal distribution as the total population and so this suggests that our dataset is not structurally biased by particular user-groups and market shares. This is also confirmed by the ranks of city population sizes, as shown in Fig. 1, they match with those of customers. In the rest of this article, with population of a city, we mean the number of customers living in this. By aggregating the individual communications at city level, we obtain a network of 571 cities in Belgium. We define the intensity of interaction between the cities A and B by (Fig. 2 (a)): LAB = ∑