Stability properties of reciprocal networks

One of the aims of network formation models is to explain salient properties of empirical networks based on simple mechanisms for establishing links. Such mechanisms include random attachment (a generic abstraction of how new incoming nodes connect to a network), triadic closure (how the new nodes establish transitive relationships), and network response (how nodes react to new attachments). Our work analyzes the combined effect of the three mechanisms on various local and global network properties. In particular, we derive an expression for the asymptotic behavior of the local reciprocity coefficient as a function of the in-degree of a node. Furthermore, we show that the dynamics of the global reciprocity and the global clustering coefficients correspond to time-varying linear systems. Finally, we identify conditions under which the equilibria of both coefficients are asymptotically stable.