在最近關於網路科學的研究中，指導教授與學長們研究出了一個專門用於分析無方向性網路和有方向性網路的機率框架。該框架的關鍵概念是取樣，透過對稱和不對稱的雙變量機率分布去對網路做取樣，接著使用取得的雙變量機率分布去定義各種觀念，包括重要性、相對重要性、社群和模塊化。
在這本論文中主要的目標就是把原本的定義延伸到特性網路上，並使用指數隨機圖模型取樣法去對雙變量機率分布做取樣。我們主要的貢獻就是我們找到一個方法去分析特性網路，而特性網路也包含我們常見的有號網路。利用取樣的方法，我們可以去定義在特性網路中有哪些重要性。影響力重要性和信任重要性可以在有號網路中去定義重要性。在特性網路中的點具有屬於自己的特性，而特定廣告影響力重要性可以幫我們完美的定義重要性。在真實世界數據實驗的結果可以看到在不同定義重要性的方法會看到不一樣的結果，且隨著溫度的改變結果也會不盡相同。還有更多的實驗是為了探討溫度的重要性。

In our recent works, we developed a probabilistic framework for structural analysis in undirected networks and directed networks. The key idea of that framework is to sample a network by a symmetric and asymmetric bivariate distribution and then use that bivariate distribution to formerly defining various notions, including centrality, relative centrality, community, and modularity. The main objective of this thesis is to extend the probabilistic definition to attributed networks, where sampling bivariate distributions by exponentially twisted sampling. Our main finding is that we find a way to deal with the sampling of the attributed network including signed network. By using the sampling method, we define the various centralities in attributed networks. The influence centralities and trust centralities correctly show that how to identify centralities in signed network. The advertisement-specific influence centralities also perfectly define centralities when the attributed networks that have node attribute. Experimental results on real-world dataset demonstrate the different centralities with changing the temperature. Further experiments are conducted to gain a deeper understanding of the importance of the temperature.

Contents 1
List of Figures 3
1 Introduction 4
2 Sampled graph 6
2.1 Review of the probabilistic framework of sampled graphs . . . . . . . . . 6
2.2 Exponentially twisted sampling . . . . . . . . . . . . . . . . . . . . . . . 10
3 Centralities 14
3.1 Centralities in signed networks . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 Influence centralities . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.2 Trust centralities . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Advertisement-specific influence centralities in networks with node attributes 18
4 Experiment 21
4.1 Experiment for influence centrality . . . . . . . . . . . . . . . . . . . . . 21
4.1.1 Experiment by _2=0 . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1.2 Experiment by _1 = 0:7 and _2 = 0:3 . . . . . . . . . . . . . . . . 26
4.2 Experiment for trust centrality . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Experiment for advertisement-specific influence centrality . . . . . . . . . 31
5 Conclusion 36

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