有向圖形數據階層式因子模型

Abstract

圖論在現今成對關係分析中扮演關鍵角色，其普遍且廣泛地被應用到各大科學領域，用以描述成對關係中，各種複雜的交互影響關係。然而，至今為止，多數的相關研究僅提出探索群落結構的分群方法，並未提供其他有效方法深入分析其他多樣、複雜的結構。本論文延續先前提出處理無向網絡資料的因子模型，提出並改進能處理有向圖型資數據的階層式因子模型，其中我們認定每一個節點皆承載多種特徵，每一種特徵都以一介於0至1的實數值表示，用於分析各領域複雜的網絡數據。我們可以利用特定、相對應的鏈結函數定義兩節點之間的關係，再透由兩個不同節點各自的指入特徵、指出特徵，定義兩個節點之間的單向鏈接概率。我們提出的模型能夠自然結合不同類型的鏈接函數，用以產生不同類型的模塊。另外，偏差信息量準則(Deviance Information Criterion)被用於模型選擇，MCMC程序被用作模型特徵估計，k平均演算法被用以最後對於特徵數值的分群。這個模型使我們成功地分析網絡數據中，每個子群集的結構，並且找出每個子群集中的成員。這個模型目前僅基於各種最簡單的假設，未來，我們將嘗試選擇不同假設以提高準確率，尋找其他方法來提高運算之效率，並且更好地設定初始值。

Nowadays, graph theory has become a key role towards analyzing pairwise relations between objects. It is commonly and widely applied to describe complex interaction patterns in various scientific fields. Most of the researches explore community structure only. However, recent studies unveil more sophisticated modules; community is not the only structure we are interested in anymore. In this paper, we extend the previous undirected factor model and put forward a directed graph factor model, where every node carries several features, for analyzing various complex network data. Link functions map the features of each pair nodes to every one-direction edge probabilities, and a factor which can be determined by a specific kind of link function refers a channel for the one-way edge connection. This model naturally incorporates different kinds of link functions which yield distinct types of modules. DIC is used for our model selection, k-mean is for clustering, and MCMC procedures is for the inference of the model. We successfully analyze the structure of every sub-circle and the members in each sub-circle. In the future, we may choose different assumptions to improve the accurate rate, find other methods to improve the computation efficiency and to set better initial values.