Asymptotically Consistent Estimation of Preferential Attachments in Growing Networks

Preferential attachment, also referred to as the "rich-get-richer" mechanism, characterizes the ability of already existing nodes in an evolutionary network to acquire new connections from newly-coming nodes throughout the growing process of the network. This mechanism is responsible for the emergence of some critical structures in many real networks, such as the Saccharomyces cerevisiae protein-protein interaction network, scientific collaboration network, and bitcoin transaction network. Clearly, it is very important to accurately estimate the preferential attachment mechanism for a growing network in interest. Although it has long been aware that the existing widely-used methods yield significantly biased estimates of preferential attachments, the underlying mechanism is not completely understood and not well explained. By rewriting preferential attachment from a deterministic formulation to a stochastic one, this paper reveals two major problems existing in the current methodologies, thereby well explaining why traditional methods introduce biases to the estimates of preferential attachments. To avoid the negative effects of these two problems, a new method is proposed based on the Poisson Pseudo Maximum Likelihood method for estimating preferential attachments, which provides asymptotically consistent estimation to preferential attachments. More importantly, even with insufficient information, the new method can still provide an estimate with high precision, showing a significant advantage in measuring preferential attachments based on small-sized samples. The new method is also robust against network parameter changes. The findings may shed some lights onto deeper understanding of the evolutionary processes of real networks.