Evolution Of Scale-free Networks Research Articles

Robustness of Controllability for Scale-free Networks Based on a Nonlinear Load-Capacity Model**This work is supported by the National Natural Science Foundation of China under Grant Nos. 61374053 and 61473016, HKU Conference and Research Council under Grant 201411159037, and the Fundamental Research Funds for the Central Universities YWF-15-ZDHXY-001.

Abstract Based on a nonlinear load-capacity model, the network evolution and controllability evolution of scale-free networks under the cascading failures triggered by removing the highest-load edge are simulated and discussed in this paper. It is shown by numerical simulations that the controllability evolution is consistent with the average degree evolution rather than the power law exponent evolution. Under the same network cost, it is found that the nonlinear load-capacity model exhibits the stronger robustness of controllability when the initial power law exponent of networks is small by comparing with the linear load-capacity model, while the linear load-capacity model is of stronger robustness of controllability when the initial power law exponent is large. Numerical results shows that high-load edges are becoming more critical to the robustness of controllability with the increase of initial power law exponent, and the nearly highest-load edges are becoming more critical with the increase of both initial average degree and power law exponent.

Time-Dependent Variation of the Centrality Measures of the Nodes during the Evolution of a Scale-Free Network

Scale-free networks are a type of complex networks in which the degree distribution of the nodes is according to the power-law. Centrality of the nodes is a quantitative measure of the importance of the nodes according to the topological structure of the network. The commonly used centrality measures are the degree-based degree centrality and eigenvector centrality and the shortest path-based closeness centrality and betweenness centrality. We use the widely studied Barabasi-Albert (BA) model to simulate the evolution of scale-free networks. The model works by adding new nodes to the network, one at a time, with the new node connected to m of the currently existing nodes. Accordingly, nodes that have been in the network for a longer time have greater chances of acquiring more links and hence a larger degree centrality. While the degree centrality of the nodes has been observed to show a concave down pattern of increase with time; but the time-dependent variation of the other centrality measures has not been analyzed until now. In this paper, we study the time-dependent variation of degree centrality, eigenvector centrality, closeness centrality and betweenness centrality of the nodes during the evolution of a scale-free network according to the BA model

The bounds of heavy-tailed return distributions in evolving complex networks

We consider the evolution of scale-free networks according to preferential attachment schemes and show the conditions for which the exponent characterizing the degree distribution is bounded by upper and lower values. Our framework is an agent model, presented in the context of economic networks of trades, which shows the emergence of critical behavior. Starting from a brief discussion about the main features of the evolving network of trades, we show that the logarithmic return distributions have bounded heavy-tails, and the corresponding bounding exponent values can be derived. Finally, we discuss these findings in the context of model risk.