Evolving scale-free networks and node-based random edge deletion

We investigate a growing-network model that combines preferential and uniform attachment with two distinct mechanisms of edge deletion. In addition to the usual uniform random edge deletion, we introduce a novel node-based rule in which uniformly chosen non-isolated nodes lose one of their incident edges. This mechanism differs fundamentally from uniform edge deletion and leads to a nonlinear evolution for the stationary degree distribution due to the nonlinear dependence on the fraction of isolated nodes. We solve the general problem in the stationary regime and obtain closed-form expressions for the degree distribution in terms of hypergeometric and confluent hypergeometric functions. Depending on the balance between attachment and deletion rates, three asymptotic regimes for the degree distribution arise: power-law, exponential, and a critical regime characterized by a stretched-exponential decay. We show that the node-based edge deletion mechanism is less likely to disrupt the scale-free (power-law) regime than the uniform edge deletion. Moreover, we also demonstrate that the precise balance between preferential attachment and node-based deletion can transform a scale-free network into a critical one with stretched-exponential decay. Large-scale numerical simulations exhibit excellent agreement with the theoretical predictions.