Stability of degree distributions of social networks

Social network models formalize mechanisms of connection to provide a framework for understanding emerging topological properties of social relationships, interactions, and communications. Random and preferential attachment are widely-used mechanisms, which assume that as new social actors attach to a network, they establish a fixed number of new connections. Our work extends the class of random and preferential attachment models by considering scenarios in which the number of new connections may vary over time. For the original attachment mechanisms, we show that infinite-dimensional time-varying linear systems characterize the evolution of the cumulative degree distributions. Moreover, we show that the limit average degrees and the limit degree distributions are stable invariants. The stability of these sets implies that small perturbations to the network can only lead to small variations in the two topological measures at any point in time. Simulations illustrate how random and targeted perturbations impact the invariants. Finally, we present analogous results for preferential attachment models in which the number of new connections is defined by random variables obeying a binomial or power law probability function.