Mobile call graphs: beyond power-law and lognormal distributions

We analyze a massive social network, gathered from the records of a large mobile phone operator, with more than a million users and tens of millions of calls. We examine the distributions of the number of phone calls per customer; the total talk minutes per customer; and the distinct number of calling partners per customer. We find that these distributions are skewed, and that they significantly deviate from what would be expected by power-law and lognormal distributions. To analyze our observed distributions (of number of calls, distinct call partners, and total talk time), we propose PowerTrack , a method which fits a lesser known but more suitable distribution, namely the Double Pareto LogNormal (DPLN) distribution, to our data and track its parameters over time. Using PowerTrack , we find that our graph changes over time in a way consistent with a generative process that naturally results in the DPLN distributions we observe. Furthermore, we show that this generative process lends itself to a natural and appealing social wealth interpretation in the context of social networks such as ours. We discuss the application of those results to our model and to forecasting.