Academic literature on the topic 'Hypergraphe binaire'

Relationship is the core building block of a network, and today's world advances through the complex networks. Graph theory deals with such problems more efficiently. But whenever vagueness or imprecision arises in such relationships, fuzzy graph theory helps. However, fuzzy hypergraphs are more advanced generalization of fuzzy graphs. Whenever there is a need to define multiary relationship rather than binary relationship, one can use fuzzy hypergraphs. In this chapter, interval-valued fuzzy hypergraph is discussed which is a generalization of fuzzy hypergraph. Several approaches to find shortest path between two given nodes in an interval-valued fuzzy graphs is described here. Many researchers have focused on fuzzy shortest path problem in a network due to its importance to many applications such as communications, routing, transportation, etc.

A basic task of machine learning and data mining is to automatically uncover <b>patterns</b> that reflect regularities in a data set. When dealing with a large database, especially when domain knowledge is not available or very weak, this can be a challenging task. The purpose of <b>pattern discovery</b> is to find non-random relations among events from data sets. For example, the “exclusive OR” (XOR) problem concerns 3 binary variables, A, B and C=A<img src="http://resources.igi-global.com/Marketing/Preface_Figures/x_symbol.png">B, i.e. C is true when either A or B, but not both, is true. Suppose not knowing that it is the XOR problem, we would like to check whether or not the occurrence of the compound event [A=T, B=T, C=F] is just a random happening. If we could estimate its frequency of occurrences under the random assumption, then we know that it is not random if the observed frequency deviates significantly from that assumption. We refer to such a compound event as an event association pattern, or simply a <b>pattern</b>, if its frequency of occurrences significantly deviates from the default random assumption in the statistical sense. For instance, suppose that an XOR database contains 1000 samples and each primary event (e.g. [A=T]) occurs 500 times. The expected frequency of occurrences of the compound event [A=T, B=T, C=F] under the independence assumption is 0.5×0.5×0.5×1000 = 125. Suppose that its observed frequency is 250, we would like to see whether or not the difference between the observed and expected frequencies (i.e. 250 – 125) is significant enough to indicate that the compound event is not a random happening.<div><br></div><div>In statistics, to test the correlation between random variables, <b>contingency table</b> with chi-squared statistic (Mills, 1955) is widely used. Instead of investigating variable correlations, pattern discovery shifts the traditional correlation analysis in statistics at the variable level to association analysis at the event level, offering an effective method to detect statistical association among events.</div><div><br></div><div>In the early 90’s, this approach was established for second order event associations (Chan & Wong, 1990). A higher order <b>pattern discovery</b> algorithm was devised in the mid 90’s for discrete-valued data sets (Wong & Yang, 1997). In our methods, patterns inherent in data are defined as statistically significant associations of two or more primary events of different attributes if they pass a statistical test for deviation significance based on <b>residual analysis</b>. The discovered high order patterns can then be used for classification (Wang & Wong, 2003). With continuous data, events are defined as Borel sets and the pattern discovery process is formulated as an optimization problem which recursively partitions the sample space for the best set of significant events (patterns) in the form of high dimension intervals from which probability density can be estimated by Gaussian kernel fit (Chau & Wong, 1999). Classification can then be achieved using Bayesian classifiers. For data with a mixture of discrete and continuous data (Wong & Yang, 2003), the latter is categorized based on a global optimization discretization algorithm (Liu, Wong & Yang, 2004). As demonstrated in numerous real-world and commercial applications (Yang, 2002), pattern discovery is an ideal tool to uncover subtle and useful patterns in a database. </div><div><br></div><div>In pattern discovery, three open problems are addressed. The first concerns learning where noise and uncertainty are present. In our method, noise is taken as inconsistent samples against statistically significant patterns. Missing attribute values are also considered as noise. Using a standard statistical <b>hypothesis testing</b> to confirm statistical patterns from the candidates, this method is a less ad hoc approach to discover patterns than most of its contemporaries. The second problem concerns the detection of polythetic patterns without relying on exhaustive search. Efficient systems for detecting monothetic patterns between two attributes exist (e.g. Chan & Wong, 1990). However, for detecting polythetic patterns, an exhaustive search is required (Han, 2001). In many problem domains, polythetic assessments of feature combinations (or higher order relationship detection) are imperative for robust learning. Our method resolves this problem by directly constructing polythetic concepts while screening out non-informative pattern candidates, using statisticsbased heuristics in the discovery process. The third problem concerns the representation of the detected patterns. Traditionally, if-then rules and graphs, including networks and trees, are the most popular ones. However, they have shortcomings when dealing with multilevel and multiple order patterns due to the non-exhaustive and unpredictable hierarchical nature of the inherent patterns. We adopt <b>attributed hypergraph</b> (AHG) (Wang & Wong, 1996) as the representation of the detected patterns. It is a data structure general enough to encode information at many levels of abstraction, yet simple enough to quantify the information content of its organized structure. It is able to encode both the qualitative and the quantitative characteristics and relations inherent in the data set.<br></div>

Knowledge graphs store facts using relations between two entities. In this work, we address the question of link prediction in knowledge hypergraphs where relations are defined on any number of entities. While techniques exist (such as reification) that convert non-binary relations into binary ones, we show that current embedding-based methods for knowledge graph completion do not work well out of the box for knowledge graphs obtained through these techniques. To overcome this, we introduce HSimplE and HypE, two embedding-based methods that work directly with knowledge hypergraphs. In both mod