在本論文中，我們考慮如何使用矩陣分解之方法來處理在時變網路中預測未來連結情形。傳統的嵌入方法會將每個時間點的網路各自嵌入成特徵矩陣，然而這會出現在不同時間點所嵌入而成的特徵不盡相同，如何對齊這些嵌入特徵就必須被考慮到。為了解決特徵對齊的問題，我們提出一個基於矩陣分解的方法稱之為點對時非負矩陣分解，使每一個點都各自嵌入到各自的嵌入空間。我們讓每一個點對應的點對時相似矩陣套用非負矩陣分解來獲得兩個特徵矩陣，我們假設其中一個特徵矩陣是靜態不變的，而另一個特徵矩陣是動態改變的。為了追蹤網路的演進，我們使用有限脈衝響應濾波器來達成，通過解脊回歸的問題，我們可以得到最佳的有限脈衝響應濾波器的參數來預測網路未來的情況。為了測試這些基於矩陣分解的方法的成效，我們利用人工生成的網路資料以及實際網路資料來進行實驗，實驗結果顯示點對時非負矩陣分解方法的準確率比其他基準方法來的高、經過嵌入之後損失較少的資訊以及需要比其他基於矩陣分解的方法還小的嵌入維度。

In this thesis, we consider the link prediction problem in temporal networks by matrix factorization based approaches. The traditional embedding methods embed each network individually, the meaning of the feature will vary with time. The issue of the alignment of embedding features might be considered. To address this issue, we propose a novel matrix factorization based approach, which is called Node-to-Time Nonnegative Matrix Factorization (NTNMF). It embeds each node into individual embedding space to prevent features from not be aligned. We apply the Nonnegative Matrix Factorization method on a node-to-time similarity matrix to obtain the latent feature matrices. We assume that one of the latent feature matrices is dynamic and the other is static. We then use the Finite Impulse Response (FIR) lter to track the evolution of the latent feature matrix. By solving a ridge regression problem, the best-estimated parameters of the FIR lter can be learned and used for predicting the latent features of the future. To evaluate the performance of the matrix factorization based approaches, we conduct our experiments on both synthetic datasets and real datasets. Our experimental results show that the matrix factorization based approaches are eective. The NTNMF is better than baseline approaches, loses lower information of the similarity matrix, and requires the lower dimension of the latent feature vector than two other matrix factorization based
approaches.

Contents 1
List of Figures 3
1 Introduction 4
2 Problem Definition 8
3 System Models 10
3.1 Similarity Approach (Sim) . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Symmetric Nonnegative Matrix Factorization for Temporal Networks (SNMF) 11
3.3 Node-to-Time Nonnegative Matrix Factorization (NTNMF) . . . . . . . 13
3.3.1 Step 0: Preprocessing Node-to-Time Matrices . . . . . . . . . . . 13
3.3.2 Step 1: Embedding Node-to-Time Matrices . . . . . . . . . . . . . 14
3.3.3 Step 2: Tracking Evolution . . . . . . . . . . . . . . . . . . . . . . 15
3.3.4 Step 3: Predicting the Similarity Matrix . . . . . . . . . . . . . . 15
3.4 Temporal Symmetric Nonnegative Matrix Factorization (TSNMF) . . . . 17
4 Experimental Results 20
4.1 Link Prediction and Performance Metric . . . . . . . . . . . . . . . . . . 20
4.2 Experiments on Synthetic Datasets . . . . . . . . . . . . . . . . . . . . . 21
4.2.1 Generating Synthetic Datasets . . . . . . . . . . . . . . . . . . . . 22
4.2.2 Link Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.3 Latent Features of The NTNMF Approach . . . . . . . . . . . . . 26
4.3 Experiments on Real Datasets . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1 Real Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.2 Link Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Conclusions 34
Appendices 37
A Temporal Deep Network Embedding (TDNE) 38
A.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A.2 Deep Network Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A.3 Tracking Network Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 41
A.4 Tricks for Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
A.5.1 Experiments on Synthetic Datasets . . . . . . . . . . . . . . . . . 43
A.5.2 Experiments on Real Datasets . . . . . . . . . . . . . . . . . . . . 43
A.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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