[论文解读] A gravity model for inter-city telephone communication networks

Gautier Krings, Francesco Calabrese+2 more|arXiv (Cornell University)|May 5, 2009

Complex Network Analysis Techniques参考文献 22被引用 15

一句话总结

本研究提出一种引力模型，利用来自250万名比利时移动用户匿名通话数据，解释城市间电话通信模式。通过按邮政编码聚合用户，作者表明城市间通信强度与人口乘积成正比，与距离平方成反比，证实了类似于物理作用力的引力定律。

ABSTRACT

We consider a network of mobile phone customers aggregated by geographical proximity. We analyze the anonymous communication patterns of 2.5 million customers of a Belgian mobile phone operator. Grouping customers by billing address, we build a social network of cities, that consists of communication 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 size of the population of these cities divided by the square of their distance. PACS numbers: 89.75.Da, 89.75.Fb, 89.65.Ef A gravity model for inter-city telephone communication networks 2 Recent research has shown that certain characteristics of cities grow in different ways in relation to population size. While some characteristics are directly proportional to a cities population size, instead, other features such as productivity or energy consumption are not linear but exhibit superlinear or sublinear dependence to population size [1]. Interestingly, some of these features have strong similarities with those found in biological cells an observation that has led to the creation of a metaphor where cities are seen as living entities [2]. Interactions between cities, such as passenger transport flows and phone messages, have also been related to population and distance [3, 4]. Meanwhile, in socio-economic 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 [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 a lack of data. Recently, however, telephone communication data has opened up a new way of analyzing cities at both a fine and aggregate level, whereby as Gottman as already as 1957 noted [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 become available. By using these as a proxy for social networks, they have enabled the study of human 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 by 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 2.5 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 their 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 A gravity model for inter-city telephone communication networks 3 during the 6 months time interval. The resulting network is composed of 2.5 million nodes and 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 relationship 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 into 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 that match with those of customers, as shown in Fig. 1. In the rest of this article, when we use the term population of a city, we are refering to the number of customers that have a valid ZIP code of this city, even if they do not make calls during the six months period. There are a significant fraction of nodes that do not make any calls over the whole period, that are isolated nodes in the graph. These nodes are still taken into account for the population size, since their presence is of interest for the normalization of the communication data. By aggregating the individual communications at a 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 = ∑

研究动机与目标

• 利用大规模匿名移动电话数据建立城市间通信模式的模型。
• 研究城市间通信强度是否遵循类似于物理作用力的引力定律。
• 在大规模范围内分析城市人口规模、地理距离与通信量之间的关系。
• 使用比利时六个月内的真实移动网络数据验证该模型。

提出的方法

• 从250万名比利时移动用户中获取聚合后的匿名通话记录，筛选条件为仅包含同一运营商客户之间的通话。
• 按邮政编码对用户进行分组以定义城市，城市人口定义为每个城市的客户数量。
• 通过汇总所有城市对之间的总通话时长，构建城市层面的通信网络。
• 应用引力模型，其中通信强度 LAB 与 (PA × PB) / d²AB 成正比，PA 和 PB 为城市人口，dAB 为城市间距离。
• 筛选连接，仅保留双向通信且每个方向在六个月内至少有六次通话的链接，以减少噪声。
• 使用对数正态分布描述城市人口规模，并通过客户排名相关性验证数据的代表性。

实验结果

研究问题

• RQ1 全国移动网络中的城市间通信强度是否遵循基于人口规模和距离的引力定律？
• RQ2 城市人口乘积除以距离平方的计算结果，能否有效预测城市对之间的实际通信量？
• RQ3 所观察到的通信模式在多大程度上独立于用户特定行为，并能代表更广泛的人群？
• RQ4 通信模式在过滤为持续互动（如每个方向至少六次通话）后是否依然稳健？
• RQ5 大规模匿名移动数据能否可靠地模拟宏观层面的城市互动模式？

主要发现

• 比利时城市间通信强度符合引力模型，强度与城市人口乘积成正比，与距离平方成反比。
• 该模型解释了城市对之间通信量方差的显著部分，表明其具有强大的预测能力。
• 以每个邮政编码的客户数量衡量的城市人口规模，服从对数正态分布，参数为 μ = 4.05 和 σ = 0.37。
• 城市人口的排名顺序与客户数量的排名顺序高度一致，表明数据集中无结构性偏差。
• 过滤后的网络包含250万名用户之间的3800万条链接，571座城市构成宏观网络。
• 孤立节点（无通话记录的用户）保留在人口统计中，以确保通信强度归一化的准确性。