  Many classes of network growth models have been proposed in the literature
for capturing real-world complex networks. Existing research primarily focuses
on global characteristics of these models, e.g., degree distribution. We aim to
shift the focus towards studying the network growth dynamics from the
perspective of individual nodes. In this paper, we study how a metric for node
influence in network growth models behaves over time as the network evolves.
This metric, which we call node visibility, captures the probability of the
node to form new connections. First, we conduct an investigation on three
popular network growth models -- preferential attachment, additive, and
multiplicative fitness models; and primarily look into the "influential nodes"
or "leaders" to understand how their visibility evolves over time.
Subsequently, we consider a generic fitness model and observe that the
multiplicative model strikes a balance between allowing influential nodes to
maintain their visibility, while at the same time making it possible for new
nodes to gain visibility in the network. Finally, we observe that a spatial
growth model with multiplicative fitness can curtail the global reach of
influential nodes, thereby allowing the emergence of a multiplicity of "local
leaders" in the network.
