複雜網路是一項十分重要的研究領域，在現實世界裡有很多系 統可以被網路所描述。我們透過將網路的連結對應到 Ising Model 的自旋來找出最佳化網路(最低成本)的解，除此之外，基於平均場 理論，我們還發展出一個演算法來有效地去計算最佳解。在之前的 研究裡，藉由不同的連結以及節點本身的分布，我們觀察到許多網 路結構的相變現象。在這篇論文裡，我們研究最佳化網路的一些網 路性質，像是聚集係數、最短路徑、度分布等等。我們假設真實的 網路往往是長時間的演化而形成的，這種演化結果必然是通過某種 最佳化的方法，我們的目標是想知道網路當中節點和連結的微觀性 質，跟最佳化的網路性質之間的關係。透過假設網路本身不同的微 觀性質，我們發現到在特定條件下，我們產生的最佳化網路具有類 似現實網路的一些現象，例如小世界網路和無尺度網路。 ;The network growth model is designed as a problem of finding the minimal wiring cost while achieving maximal connections. By mapping to Ising spin models, two kinds of models were investigated and they show different phase transition behaviors for different wiring weight distributions and node weight distributions. Previously, the network properties of undirected network have been investigated. In this research, we focus on the network properties of our optimized directed network, such as cluster coefficients, in and out degree dis- tributions, minimal path length and so on. Based on the mean-field theory, an- alytical results are also derived. These optimized network properties are sim- ulated by the efficient algorithm which was developed in our previous work and fit well with the analytical results. For some specific edge weight and node weight distributions, the growth of optimized networks behave like scale-free networks, which are found in many networks in biological system and social networks. Besides, motifs (sub-graphs) are measured in the optimized network, which help us to gain insight on the structure of networks.
