Temporal dynamics of scale-free networks

Many social, biological, and technological networks display substantial non-trivial topological features. One well-known and much studied feature of such networks is the scale-free power-law distribution of nodes' degrees. Several works further suggest models for generating complex networks which comply with one or more of these topological features. For example, the known Barabasi-Albert "preferential attachment" model tells us how to create scale-free networks. Since the main focus of these generative models is in capturing one or more of the static topological features of complex networks, they are very limited in capturing the temporal dynamic properties of the networks' evolvement. Therefore, when studying real-world networks, the following question arises: what is the mechanism that governs changes in the network over time? In order to shed some light on this topic, we study two years of data that we received from eToro: the world's largest social financial trading company. We discover three key findings. First, we demonstrate how the network topology may change significantly along time. More specifically, we illustrate how popular nodes may become extremely less popular, and emerging new nodes may become extremely popular, in a very short time. Then, we show that although the network may change significantly over time, the degrees of its nodes obey the power-law model at any given time. Finally, we observe that the magnitude of change between consecutive states of the network also presents a power-law effect.