網路研究是一門非常重要的領域。網路可以描述我們世界中許多不同的系統，例如社會、生物、及科技網路等。如何找出最佳化的網路成長的模型是我們研究的目的。我們建構了許多不同的能量及限制條件的二維成長網路模型，運用 Metropolis 方法及 Simulated Annealing（模擬降火法）來執行模擬的工作。我們的目標是得到一個網路具有最小材料的消耗（例如網路的長度及能量）而同時間能夠得到最大的訊息內容。在這篇論文中，我們藉由不同的的能量及限制條件得到了許多不同特徵的網路，且在某些網路中觀察到了類似小世界網路及訊息內容的相變現象。我們對模擬出來的網路模型分析了許多網路性質，且進一步和一些真實的網路做比較。最後我們得到一些系統相圖，且發現到了一些和真實網路比較的有趣現象。 Network research is important because complex networks can describe a wide variety of systems in our world. How to find a optimized growing networks is the purpose of this research. We construct models with different Hamiltonians and constraints for the growth of some two dimensional networks, and employ the Metropolis algorithm and Simulated Annealing to perform simulation. Our proposed model aimed at minimizing the material cost while at the same time maximizing the information content. In this thesis, we obtain different kinds of networks display small world characteristics. We also observe a possible phase transition phenomenon in the information content. We analyze many network properties of our results and go further to compare our model networks with some realistic networks. Finally, several "phase diagrams" for our model system are presented and it appears some interesting phenomena can also be found in realistic networks.
