An Evolving model of online bipartite networks

Understanding the structure and evolution of online bipartite networks is a significant task since they play a crucial role in various e-commerce services nowadays. Recently, various attempts have been tried to propose different models, resulting in either power-law or exponential degree distributions.However, many empirical results show that the user degree distribution actually follows a shifted power-law distribution, so-called \emph{Mandelbrot law}, which cannot be fully described by previous models. In this paper, we propose an evolving model, considering two different user behaviors: random and preferential attachment. Extensive empirical results on two real bipartite networks, \emph{Delicious} and \emph{CiteULike}, show that the theoretical model can well characterize the structure of real networks for both user and object degree distributions. In addition, we introduce a structural parameter $p$, to demonstrate that the hybrid user behavior leads to the shifted power-law degree distribution, and the region of power-law tail will increase with the increment of $p$. The proposed model might shed some lights in understanding the underlying laws governing the structure of real online bipartite networks.

💡 Research Summary

The paper addresses a fundamental gap in the modeling of online bipartite networks—systems where users interact with items (e.g., bookmarks, papers) and the resulting structure is a two‑mode graph. While many existing generative models focus exclusively on pure preferential attachment and consequently predict either a strict power‑law or an exponential degree distribution, empirical analyses of real platforms repeatedly reveal that user degree follows a shifted power‑law, also known as the Mandelbrot law. This discrepancy motivates the authors to propose a new evolving model that explicitly incorporates two distinct user behaviors: (1) random attachment, where a newly arriving user selects an existing item uniformly at random, and (2) preferential attachment, where the probability of selecting an item is proportional to its current degree.

A single scalar parameter (p) (0 ≤ p ≤ 1) governs the mixture of these behaviors. When (p=0) the model reduces to pure random linking; when (p=1) it collapses to the classic Barabási‑Albert style preferential attachment. At each discrete time step a new user and a new item are added, and each creates a single link according to the mixed rule. Using a master‑equation approach and continuous‑time approximation, the authors derive analytical expressions for the expected growth of user degree (k_u) and item degree (k_o). The resulting stationary distribution for user degree takes the form

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