Mobile Phone Graph Evolution: Findings, Model and Interpretation

What are the features of mobile phone graph along the time? How to model these features? What are the interpretation for the evolutional graph generation process? To answer the above challenging problems, we analyze a massive who-call-whom networks as long as a year, gathered from records of two large mobile phone communication networks both with 2 million users and 2 billion of calls. We examine the calling behavior distribution at multiple time scales (e.g., day, week, month and quarter), and find that the distribution is not only skewed with a heavy tail, but also changing at different time scales. How to model the changing behavior, and whether there exists a distribution fitting the multi-scale data well? In this paper, first, we define a delta-stable distribution and a Multi-scale Distribution Fitting (MsDF) problem. Second, to analyze our observed distributions at different time scales, we propose a framework, Scale Power, which not only fits the multi-scale data distribution very well, but also works as a convolutional distribution mixture to explain the generation mechanism of the multi-scale distribution changing behavior. Third, Scale Power can conduct a fitting approximation from a small time scale data to a large time scale. Furthermore, we illustrate the interesting and appealing findings from our Scale Power model and large scale real life data sets.