The impact of connection density on scale-free distribution in random networks

Preferential attachment is considered as a fundamental mechanism that contributes to the scale-free characteristics of random networks, which include growth and non-growth networks. There exist some situations of non-growth random networks, particularly for very sparse or dense networks, where preferential attachments cannot consequentially result in true scale-free features, but only in scale-free-like appearances. This phenomenon implies that, a close relationship exists between the connection density p and the scaling. In this study, we propose a self-organized model with constant network size to study the phenomenon. We show analytically and numerically that there exists a certain critical point pc. Only when p=pc, the random network evolves into steady scale-free state. Otherwise, the network exhibits a steady scale-free-like state. The closer the p approximates pc, the closer the scale-free-like distribution approximates the true scale-free distribution. Our results show that, in random network lack of growth, a preferential scheme does not necessarily lead to a scale-free state, and a formation of scale-free is a consequence of two mechanisms: (i) a preferential scheme and (ii) appropriate connection density.