Growth Scale-Free Networks by Various Generative Ways

In this article, the popularly discussed topic, i.e., how to construct available theoretical networked models that certainly capture some structural features popularly observed on realistic networks, is still our focus. Specifically, we first propose an evolving deterministic network \(N(t)\) using three types of growth ways. Then, we study some topological structural parameters including degree distribution, diameter, and clustering coefficient on network \(N(t)\) . The results demonstrate that the proposed network has scale-free feature and small-world property. In the meantime, we obtain an interesting finding, i.e., the first handshake between Fibonacci series and the “pure” preferential attachment mechanism. Next, we enumerate spanning trees on network \(N(t)\) and derive the closed-form solution of spanning trees number. Second, we introduce randomness into the growth process of network \(N(t)\) to further establish evolving stochastic networks \(\mathfrak{N}(t)\) that follow the same degree distribution as network \(N(t)\) and also determine some topological structural parameters so as to investigate effect of randomness on structural properties. We show analytically that such a randomization approach makes the resulting stochastic networks not only to greatly inherit some fundamental structural properties from deterministic network \(N(t)\) but also to considerably improve the robustness of network when encountering deliberate removal of edge. Lastly, we list out some open problems.