Markus Hagenbuchner (University of Wollongong, Australia), Chung Tsoi (Macau University of Science and Technology, China), Shu Jia Zhang (University of Wollongong, Australia) and Milly Kc (University of Wollongong, Australia)

DOI: 10.4018/978-1-61350-356-0.ch010

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TopThe traditional approach to processing data is by using numeric vectors of some fixed dimension as a means of representing the data, where the vector elements describe features of an associated data item (Haykin, 1994). Such representation is suitable when the data items are independent or when any existing dependencies are not relevant for the solving of a given problem. A common situation in which data items are dependent is in time series information processing (Haykin, 1994). In this case, a data entry may depend on another data entry which occurred in a prior time instance, and such dependency may be relevant for a given problem (Haykin, 1994). Data items which are sequentially organized are referred to as *sequences*. For example, in financial forecasting or in speech recognition, it is important that such dependency can be represented and encoded (Haykin, 1994). Note that unlike data vectors which are often of the same dimension, in the cases where they are not of the same dimension, then the technique of “zero padding” is used to ensure that all data vectors are of the same dimension (Haykin, 1994), there is not normally a restriction to the length of sequences. When data items are “related” to several other data items, then such dependencies are popularly described by a *graph*, hence such instances are referred to as *graph data structures*. Note that here, the data items are “related” in that the data objects, represented by nodes of a graph, are connected by links, directed or un-directed, cyclic, or acyclic, which express the relationships among the data objects. A very large number of practical problems can be represented as a graph data structure. For example, problems in molecular chemistry, image processing, computer security systems, weather forecasting, etc. can be addressed by processing the dependencies which are represented in graph structured data. Since sequences are a special case of graphs (graphs with an out-degree of 1), and since vectors are also a special case of graphs (a graph with a single node), graph data structures provide the most generic means for data representation. Any approach capable of encoding graphs can also solve time series problems and problems in which the data can be represented in vector spaces.

Simple graphs consist of two basic elements, namely, *nodes* and the binary relations called *links*. The nodes represent atomic entities of a domain. For example, a document may be represented by a node. A node may be labeled to describe the associated entity. For example, a node label may contain descriptive features of the associated document. These features could be the number of words in the document, the occurrence of certain words in the document, the frequency with which certain words occurred in the document, etc. A link exists between any two nodes if these nodes are “related” in some way. For example, a document may contain a hyperlink to another document. In this case, there is a directed link from the node representing the source document to the node which represents the target document of the hyperlink. A link may also be labeled in order to describe its properties. For example, a label associated with a link may indicate the strength of a link. Links may be directed (such as in the hyperlink case) or undirected (such as the links indicating atomic bindings of a molecule). The total number of all outgoing links of a node is called the *outdegree* of the node, while the total number of all incoming links is called the *indegree* of a node. In the case of undirected links, this is referred to as the *degree* of a node. The size of a graph *G* is the total number of nodes in *G* and is denoted by *n*=|*G*|. If there exists a path along the links in *G* from one node to another then these nodes are said to be *connected*, otherwise they are *disconnected*. The shortest path along the link structure between a pair of nodes in *G* is called the *distance* between these nodes.^{1}

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