University, Diss., 2003--Kiel.

Hypergraph partitioning lies at the heart of a number of problems in machine learning as well as other engineering disciplines. While partitioning uniform hypergraphs is often required in computer vision problems that involve multi-way similarities, non-uniform hypergraph partitioning has applications in database systems, circuit design etc. As in the case of graphs, it is known that for given objective and balance constraints, the problem of optimally partitioning a hypergraph is NP-hard. Yet, over the last two decades, several efficient heuristics have been studied in the literature and their empirical success is widely appreciated. In contrast to the extensive studies related to graph partitioning, the theoretical guarantees of hypergraph partitioning approaches have not received much attention in the literature. The purpose of this thesis is to establish the statistical error bounds for certain spectral algorithms for partitioning uniform as well as non-uniform hypergraphs. The mathematical framework considered in this thesis is the following. Let V be a set of n vertices, and ψ : V ->{1,…,k} be a (hidden) partition of V into k classes. A random hypergraph (V,E) is generated according to a planted partition model, i.e., subsets of V are independently added to the edge set E with probabilities depending on the class memberships of the participating vertices. Let ψ' be the partition of V obtained from a certain algorithm acting on a random realization of the hypergraph. We provide an upper bound on the number of disagreements between ψ and ψ'. To be precise, we show that under certain conditions, the asymptotic error is o(n) with probability (1-o(1)). In the existing literature, such error rates are only known in the case of graphs (Rohe et al., Ann. Statist., 2011; Lei \& Rinaldo, Ann. Statist., 2015), where the planted model coincides with the popular stochastic block model. Our results are based on matrix concentration inequalities and perturbation bounds, and the derived bounds can be used to comment on the consistency of spectral hypergraph partitioning algorithms. It is quite common in the literature to resort to a spectral approach when the quantity of interest is a matrix, for instance, the adjacency or Laplacian matrix for graph partitioning. This is certainly not true for hypergraph partitioning as the adjacency relations cannot be encoded into a symmetric matrix as in the case of graphs. However, if one restricts the problem to m-uniform hypergraphs for some m ≥ 2, then a symmetric tensor of order m can be used to express the multi-way interactions or adjacencies. Thus, the use of tensor spectral algorithms, based on the spectral theory of symmetric tensors, is a natural choice in this scenario. We observe that a wide variety of uniform hypergraph partitioning methods studied in the literature can be related to any one of two principle approaches: (1) solving a tensor trace maximization problem, or (2) use of the higher order singular value decomposition of tensors. We derive statistical error bounds to show that both these approaches lead to consistent partitioning algorithms. Our results also hold when the hypergraph under consideration allows weighted edges, a situation that is commonly encountered in computer vision applications such as motion segmentation, image registration etc. In spite of the theoretical guarantees, a tensor spectral approach is not preferable in this setting due to the time and space complexity of computing the weighted adjacency tensor. Keeping this practical scenario in mind, we prove that consistency can still be achieved by incorporating certain tensor sampling strategies. In particular, we show that if the edges are sampled according to certain distribution, then consistent partitioning can be achieved with only few sampled edges. Experiments on benchmark problems demonstrate that such sampled tensor spectral algorithms are indeed useful in practice. While vision tasks mostly involve uniform hypergraphs, in database and electronics applications, one often finds non-uniform hypergraphs with edges of varying sizes. These hypergraphs cannot be expressed in terms of adjacency matrices or tensors, and hence, use of a spectral approach is tricky in this context. The partitioning problem gets more challenging due to the fact that, in practice, these hypergraphs are quite sparse, and hence, provide less information about the partition. We consider spectral algorithms for partitioning clique and star expansions of hypergraphs, and study their consistency under a sparse planted partition model. The results of hypergraph partitioning can be further extended to address the well-known hypergraph vertex coloring problem, where the objective is to color the vertices such that no edge is monochromatic. The hardness of this problem is well established. In fact, even when a hypergraph is bipartite or 2-colorable, it is NP-hard to find a proper 2-coloring for it. We propose a spectral coloring algorithm, and show that if the non-monochromatic subsets of vertices are independently added to the edge set with certain probabilities, then with probability (1-o(1)), our algorithm succeeds in coloring bipartite hypergraphs with only two colors. To the best our knowledge, these are the first known results related to consistency of partitioning general hypergraphs.

Indiana University-Purdue University Indianapolis (IUPUI) / Brain connectome network visualization could help the neurologists inspect the brain structure easily and quickly. In the thesis, the model of the brain connectome network is visualized in both three dimensions (3D) environment and two dimensions (2D) environment. One is named “Brain Explorer for Connectomic Analysis” (BECA) developed by the previous research already. It could present the 3D model of brain structure with region of interests (ROIs) in different colors [5]. The other is mainly for the information visualization of brain connectome in 2D. It adopts the force-directed layout to visualize the network. However, the brain network visualization could not bring the user intuitively ideas about brain structure. Sometimes, with the increasing scales of ROIs (nodes), the visualization would bring more visual clutter for readers [3]. So, brain connectome network properties visualization becomes a useful complement to brain network visualization. For a better understanding of the effect of Alzheimer’s disease on the brain nerves, the thesis introduces several methods about the brain graph properties visualization. There are the five selected graph properties discussed in the thesis. The degree and closeness are node properties. The shortest path, maximum flow, and clique are edge properties. Except for clique, the other properties are visualized in both 3D and 2D. The clique is visualized only in 2D. For the clique, a new hypergraph visualization method is proposed with three different algorithms. Instead of using an extra node to present a clique, the thesis uses a “belt” to connect all nodes within the same clique. The methods of node connections are based on the traveling salesman problem (TSP) and Law of cosines. In addition, the thesis also applies the result of the clique to adjust the force-directed layout of brain graph in 2D to dramatically eliminate the visual clutter. Therefore, with the support of the graph properties visualization, the brain connectome network visualization tools become more flexible.

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京都大学 / 新制・課程博士 / 博士(情報学) / 甲第24039号 / 情博第795号 / 新制||情||134(附属図書館) / 京都大学大学院情報学研究科システム科学専攻 / (主査)教授 下平 英寿, 教授 田中 利幸, 教授 加納 学 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Time-series forecasting has long been studied and remains an important research task. In scenarios where multiple time series need to be forecast, approaches that exploit the mutual impact between time series results in more accurate forecasts. This has been demonstrated in various applications, including demand forecasting and traffic forecasting, among others. Hence, this dissertation focuses on graph-based models, which leverage the internode relations to forecast more efficiently and effectively by associating time series with nodes. This dissertation begins by introducing the notion of graph time-series models in a comprehensive survey of related models. The main contributions of this survey are: (1) A novel categorization is proposed to thoroughly analyze over 20 representative graph time-series models from various perspectives, including temporal components, propagation procedures, and graph construction methods, among others. (2) Similarities and differences among models are discussed to provide a fundamental understanding of decisive factors in graph time-series models. Model challenges and future directions are also discussed. Following the survey, this dissertation develops graph time-series models that utilize complex time-series interactions to yield context-aware, real-time, and probabilistic forecasting. The first method, Context Integrated Graph Neural Network (CIGNN), targets resource forecasting with contextual data. Previous solutions either neglect contextual data or only leverage static features, which fail to exploit contextual information. Its main contributions include: (1) Integrating multiple contextual graphs; and (2) Introducing and incorporating temporal, spatial, relational, and contextual dependencies; The second method, Evolving Super Graph Neural Network (ESGNN), targets large-scale time-series datasets through training on super graphs. Most graph time-series models let each node associate with a time series, potentially resulting in a high time cost. Its main contributions include: (1) Generating multiple super graphs to reflect node dynamics at different periods; and (2) Proposing an efficient super graph construction method based on K-Means and LSH; The third method, Probabilistic Hypergraph Recurrent Neural Network (PHRNN), targets datasets under the assumption that nodes interact in a simultaneous broadcasting manner. Previous hypergraph approaches leverage a static weight hypergraph, which fails to capture the interaction dynamics among nodes. Its main contributions include: (1) Learning a probabilistic hypergraph structure from the time series; and (2) Proposing the use of a KNN hypergraph for hypergraph initialization and regularization. The last method, Graph Deep Factors (GraphDF), aims at efficient and effective probabilistic forecasting. Previous probabilistic approaches neglect the interrelations between time series. Its main contributions include: (1) Proposing a framework that consists of a relational global component and a relational local component; (2) Conducting analysis in terms of accuracy, efficiency, scalability, and simulation with opportunistic scheduling. (3) Designing an algorithm for incremental online learning. / Doctor of Philosophy / Time-series forecasting has long been studied due to its usefulness in numerous applications, including demand forecasting, traffic forecasting, and workload forecasting, among others. In scenarios where multiple time series need to be forecast, approaches that exploit the mutual impact between time series results in more accurate forecasts. Hence, this dissertation focuses on a specific area of deep learning: graph time-series models. These models associate time series with a graph structure for more efficient and effective forecasting. This dissertation introduces the notion of graph time series through a comprehensive survey and analyzes representative graph time-series models to help readers gain a fundamental understanding of graph time series. Following the survey, this dissertation develops graph time-series models that utilize complex time-series interactions to yield context-aware, real-time, and probabilistic forecasting. The first method, Context Integrated Graph Neural Network (CIGNN), incorporates multiple contextual graph time series for resource time-series forecasting. The second method, Evolving Super Graph Neural Network (ESGNN), constructs dynamic super graphs for large-scale time-series forecasting. The third method, Probabilistic Hypergraph Recurrent Neural Network (PHRNN), designs a probabilistic hypergraph model that learns the interactions between nodes as distributions in a hypergraph structure. The last method, Graph Deep Factors (GraphDF), targets probabilistic time-series forecasting with a relational global component and a relational local model. These methods collectively covers various data characteristics and model structures, including graphs, super graph, and hypergraphs; a single graph, dual graphs, and multiple graphs; point forecasting and probabilistic forecasting; offline learning and online learning; and both small and large-scale datasets. This dissertation also highlights the similarities and differences between these methods. In the end, future directions in the area of graph time series are also provided.

Ensembles of graphs arise naturally in many applications, for example, the temporal evolution of social contacts or computer communications, tissue-specific protein interaction networks, annual citation or co-authorship networks in a field, or a family of high-likelihood Bayesian networks inferred from systems biology data. Several techniques have been developed to analyze such ensembles. A canonical problem is that of computing communities that are persistent across the ensemble. This problem is usually formulated as one of computing dense subgraphs (communities) that are frequent, i.e., appear in many graphs in the ensemble. In this thesis, we seek to find "unstable communities" which are the antithesis of frequent, dense subgraphs. Informally, an unstable community is a set of nodes that induces highly-varying subgraphs in the ensemble. In other words, the graphs in the ensemble disagree about the precise pairwise connections among these nodes. The primary contribution of this dissertation is to introduce the concept of unstable communities as a novel problem in the field of graph mining. Specifically, it presents three approaches to mathematically formulate the concept of unstable communities, devises algorithms for computing such communities in a given ensemble of networks, and shows the usefulness of this concept in a variety of settings. Our first definition of unstable community relies on two parameters: the first ensures that a node set induces several different subgraphs in the ensemble and the second guarantees that each of these subgraphs occurs in a large number of graphs in the ensemble. We present two algorithms to enumerate unstable communities that match this definition. The first approach, ClustMiner, is a heuristic that transforms the problem into one of computing dense subgraphs in a single graph that summarizes the ensemble. The second approach, UCMiner, is guaranteed to enumerate all maximal unstable communities correctly. We apply both approaches to systems biology datasets to demonstrate that UCMiner is superior to ClustMiner in the sense that ClustMiner's output contains node sets that are not unstable while also missing several communities computed by UCMiner. We find several node sets that capture the uncertain connectivity of genes in relevant protein complexes, suggesting that further experiments may be required to precisely discern their interaction patterns. Our second and third definitions of unstable community rely on a novel concept of (scaled) subgraph divergence, a formulation that uses the concept of relative entropy to measure the instability of a community. We propose another algorithm, SDMiner, that can exactly enumerate all maximal unstable communities with small (scaled) subgraph divergence. We perform extensive experiments on social network datasets to show that we can discover UCs that capture the main structural variations of the given set of networks and also provide us with interesting and relevant insights about these datasets. / Ph. D.

We make a first step towards adapting an existing approach for privacy-preserving publishing of linked data to Description Logic (DL) ontologies. We consider the case where both the knowledge about individuals and the privacy policies are expressed using concepts of the DL EL , which corresponds to the setting where the ontology is an EL instance store. We introduce the notions of compliance of a concept with a policy and of safety of a concept for a policy, and show how optimal compliant (safe) generalizations of a given EL concept can be computed. In addition, we investigate the complexity of the optimality problem.

在本篇論文，我們去找一個唯一著色的均勻混合超級圖的點數及邊數的下界。 我們證明為一著色的均勻混合超級圖的點數必須超過(l-1)(m-1)+1而且我們提出一個方法來建構為一著色的均勻混合超級圖。如果一個混合超圖是個D為空集合的r-均勻超級圖，當r=2則它是唯一著色的。否則，D為空集合的均勻超級圖不會是唯一著色的。我們介紹兩種有系統的方法建構唯一著色的均勻混合超級圖並且達到我們的邊界。首先，我們是著保持均勻混合超級圖的唯一著色下去減少D邊的個數。然後我們減少D邊的個數。我們考慮r均勻的C超圖和D超圖並找他們邊的個數的範圍。 / In this thesis, we find the lower bounds of number of vertices and edges of uniform mixed hypergraph which is uniquely colorable. We show that the size of vertex set of uniform mixed hypergraphs with unique coloring is more than (l-1)(m-1)+1 and we come up a way to construct uniquely colorable uniform mixed hypergraphs. If a mixed hypergraph is an r-uniform hypergraph with D empty, then it is uniquely colorable when r=2. Otherwise, an r-uniform hypergraph with D empty is not uniquely colorable. We will introduce two systematic ways to construct a uniform mixed hypergraph which is uniquely colorable and achieves our bounds. First,we reduce the number of C-edges such that uniform mixed hypergraphs keep being uniquely colorable. Then we reduce the number of D-edges. We consider r-uniform C-hypergraphs and D-hypergraphs and find the bounds on their number of edges.

The game of Subset Take-Away is an impartial combinatorial game posed by David Gale in 1974. The game can be played on various discrete structures, including but not limited to graphs, hypergraphs, polygonal complexes, and partially ordered sets. While a universal winning strategy has yet to be found, results have been found in certain cases. In 2003 R. Riehemann focused on Subset Take-Away on bipartite graphs and produced a complete game analysis by studying nim-values. In this work, we extend the notion of Take-Away on a bipartite graph to Take-Away on particular hypergraphs, namely oddly-uniform hypergraphs and evenly-uniform hypergraphs whose vertices satisfy a particular coloring condition. On both structures we provide a complete game analysis via nim-values. From here, we consider different discrete structures and slight variations of the rules for Take-Away to produce some interesting results. Under certain conditions, polygonal complexes exhibit a sequence of nim-values which are unbounded but have a well-behaved pattern. Under other conditions, the nim-value of a polygonal complex is bounded and predictable based on information about the complex itself. We introduce a Take-Away variant which we call “Take-As-Much-As-You-Want”, and we show that, again, nim-values can grow without bound, but fortunately they are very easily described for a given graph based on the total number of vertices and edges of the graph. Finally we consider Take-Away on a specific type of partially ordered set which we call a rank-complete poset. We have results, again via an analysis of nim-values, for Take-Away on a rank-complete poset for both ordinary play as well as misère play.