Snowballing Effects in Preferential Attachment: The Impact of the Initial Links

This paper studies the node degree snowballing effects (i.e., degree growth effects) in the age-sensitive preferential attachment model, where nodes are iteratively added one by one to a growing network. Upon entering the network, each new node connects to a suitably chosen set of existing nodes, while the attachment probability for an existing node to get connected depends on both its node degree and age difference. We are interested in accelerating the node degree snowballing effects through the impact of the initial links. If a new node enters the growing network with more initial links (a larger degree), it could attract many more links from the later nodes, and thus, its degree snowballs faster. We find that the initial links are only impactful when neither the node degree nor the age difference dominates the attachment probability. In that case, the relationship between the ratio of the additional initial link and the gain ratio of the eventual node degree is shown to include two stages (linear stage and diminishing return stage). Applications of our work involve citation networks and online social networks. For example, in citation networks, we answer the question that whether an author can attract additional citations through self-citations. Finally, real data-driven experiments verify the accuracies of our results, which cast some new light in real-world growing networks.