結構性圖分群演算法的任務是將點分配給群集，使得同一群集中的點彼此緊密連結，而不同群集中的點則鬆散地連結。近幾十年來，圖分群演算法已經在理論研究和工業應用中被廣泛討論，例如社會網路、訊息網路和網絡搜索等。
　　由於訊息網路的快速發展，在本研究中我們進一步考慮了圖分群演算法的全動態模型，允許點的插入和刪除動作。Wen等人 [1]首先提出一種基於索引的結構性圖分群演算法，其中包含兩個階段：核心順序和鄰居順序。該演算法被發表在2018年的VLDB，是當前最新技術中最佳的演算法，同時參考效率和群集性能。然而，過去的研究只考慮邊的插入和刪除動作。為了更有效地解決全動態模型，意即減少連續動態邊的運算，我們使用不同的方式來考慮一群核心並更有效地建構群集。更準確來說，對於每個點的更新動作，我們證明最壞情況的時間複雜度可以從輸入圖的大小顯著改善到只需考慮群集的大小。除了理論分析之外，數值實驗也顯示所提出的全動態算法比現有的解決方案有更優越的性能表現。

Structural graph clustering is a task of assigning vertices to clusters such that vertices in the same cluster are densely connected to each other while vertices in different clusters are loosely connected. In recent decades, graph clustering has been widely discussed across theoretical studies and industrial applications, such as social networks, information networks, web search and so on.
Due to the rapid growth of information networks, in this study we further consider the fully dynamic model of graph clustering in which vertex insertion and deletion operations are allowed. Wen et al. [1] presented an index-based structural graph clustering algorithm, consisting of two phases: core orders and neighbor orders. The algorithm reported in the paper published in VLDB 2018, is the currently best algorithm among all the state-of-the-art algorithms in the literature, concerning efficiency and clustering performance. However, their study considered only edge insertion and deletion operations. In order to solve the fully dynamic model efficiently, that is, to reduce a sequence of consecutive dynamic edge operations, we use a different way to aggregately consider groups of cores and construct clusters more effectively. More precisely, for each vertex update operations, we show that the worst-case time complexity can be significantly improved from the order of the input graph to the size of a cluster. In addition to theoretical analysis, numerical experiments also demonstrate the superior performance of the proposed fully dynamic algorithm against existing solutions.

摘要 I
Abstract II
誌謝 III
Contents IV
List of Figures and Tables V
1. Introduction 1
1.1. Previous Results 2
1.2. Motivation 3
1.3. Our Strategy 4
1.4. Contributions 4
1.5. Outline 5
2. Preliminary 5
3. Index-Based Algorithms 8
4. Fully Dynamic Model 15
5. Experiments 21
6. Conclusion 31
References 32

[1] Dong Wen, Lu Qin, Ying Zhang, Lijun Chang and Xuemin Lin. Efficient Structural Graph Clustering: An Index-Based Approach. VLDB, (2018), 11(3), pp. 243-255.
[2] Lijun Chang, Wei Li, Xuemin Lin, Lu Qin and Wenjie Zhang. pSCAN: Fast and Exact Structural Graph Clustering. ICDE, (2016), pp. 253-264.
[3] H. Shiokawa, Y. Fujiwara, and M. Onizuka. Scan++: efficient algorithm for finding clusters, hubs and outliers on large-scale graphs. PVLDB, (2015), 8(11):1178–1189.
[4] X. Xu, N. Yuruk, Z. Feng, and T. A. J. Schweiger. Scan: A structural clustering algorithm for networks. KDD, (2007), pp. 824–833.
[5] A. Gibbons. Algorithmic Graph Theory. Cambridge University Press.
[6] S.E. Schaeffer. Graph clustering. Computer Science Review 1, (2007), pp. 27–64.
[7] X. Xu, N. Yuruk, Z. Feng, and T. A. J. Schweiger. Scan: A structural clustering algorithm for networks. KDD, (2007), pp. 824–833.