Figure 2

Balance as upper bound of reciprocity in temporal networks. Distribution of fraction of reciprocations \(p(E_{rec})\) over nodes in dataset (box plot), and approximate upper bound \(p(E_{rec})= 2(1-b)\) (see Additional file 1 Section S6), both as function of balance b over nodes [20]. Balance is constrained to \(0.5 \leq b \leq 1\), with \(b=1\) an unbalanced, unidirectional relationship from one node to the other, and \(b=0.5\) perfect bidirectionality between two nodes. \(p(E_{rec})\) decreases as communication moves away from perfect balance, for both conversation (calls, sms, msg, email) and broadcasting (Twitter) channels. \(p(E_{rec})\) is highest for balanced conversations (\(b \in (0.5,0.6]\)), and smallest for severely unbalanced interactions between node pairs (\(b \in (0.9,1]\)). As an aggregate measure, balance does not convey the same information as temporal reciprocity, since \(p(E_{rec})\) varies greatly among the 6 datasets for the same interval of b; balance is a necessary but not sufficient condition for reciprocation. Outliers above the upper bound of balance are due to an approximation in its derivation (see Additional file 1 Section S6)