Quantifying Reciprocity in Large Weighted Communication Networks

If a friend called you 50 times last month, how many times did you call him back? Does the answer change if we ask about SMS, or e-mails? We want to quantify reciprocity between individuals in weighted networks, and we want to discover whether it depends on their topological features (like degree, or number of common neighbors). Here we answer these questions, by studying the call- and SMS records of

millions

of mobile phone users from a large city, with more than 0.5

billion

phone calls and 60

million

SMSs, exchanged over a period of six months. Our main contributions are: (1) We propose a novel distribution, the

Triple Power Law

(3PL), that fits the reciprocity behavior of all 3 datasets we study, with a better fit than older competitors, (2) 3PL is parsimonious; it has only three parameters and thus avoids over-fitting, (3) 3PL can spot anomalies, and we report the most surprising ones, in our real networks, (4) We observe that the

degree

of reciprocity between users is correlated with their local topological features; reciprocity is higher among mutual users with larger local network overlap and greater degree similarity.