A comparative analysis of multidimensional features of objects resembling sets of graphs
Introduction
Structured objects play an important role, e.g., for modeling and representing complex information processes. Generally, such objects are mathematically defined by relational algebras [1] and can often be represented by labeled graphs [2], [24], [27], [28]. The classification of structured objects by similarity relations is a still outstanding and challenging problem because the computational complexity of the underlying classical methods is often inadequate for processing large graph corpora [12], [10], [41], [45]. Also, it is a priori not clear which similarity aspects of the underlying objects should be addressed, e.g., structural similarity, semantic similarity or functional similarity [41]. In terms of determining the structural similarity of structured objects [6], [12], [10], [11], [13], [14], [15], classical methods are mostly based on defining graph metrics. Starting from a proper metric and based on an adequate spatial representation, one generally assumes that similar structured objects are geometrically closer than unsimilar ones. For example, a well-known class of graph metrics is based on graph or subgraph isomorphic relations [29], [35], [39], [40], [41], [46]. Furthermore, general metrical properties of graphs have been investigated by, e.g., Skorobogatov et al. [38].
In this paper, we introduce a novel method to compare abstract objects having characteristics that can be represented as graphs. In the following, we call these abstract objects styles to emphasize their embeddedness in some geometric space, meaning that pairs of abstract objects, or styles, can be compared meaningfully by some kind of quantitative measure. More precisely, we define a style as a multidimensional feature set. Starting from this definition, we further present a computational approach to quantify such styles based on a comparative analysis, i.e., we compare certain styles by comparing their corresponding feature sets. In Section 2, we introduce a special definition of a style whose features are represented by a set of graphs that can themselves be seen as structured objects. In order to perform the comparison of such styles, we map each feature set to a median graph [9], [30] and compare the resulting median graphs with a graph similarity measure, discussed in Section 4.3.
In order to distinguish our definition/method from already existing approaches, we briefly outline some known methods to measure the style between texts or other objects. The term style often appears in computational linguistics and semiotics, where semiotics denotes the study of signs, symbols, and sign systems [22]. Obviously, there do not exist general criteria to measure styles exactly. For example in quantitative linguistics, there exist known approaches for measuring relationships between text styles by defining measures based on certain quantitative text characteristics, e.g., length of words and sentences, type/token-index, and frequency distributions of words [32], [43], [44]. The goal of defining such measures is generally deciding to which text genre a given text belongs. Similar to the mentioned style analysis is the mathematical theory of aesthetics that has been originally developed by Birkhoff [4], [5]. The main problem of Birkhoff’s theory is to find all aesthetic characteristics of objects which belong to a certain class and then assign numerical values to each such characteristic. Then, the aesthetic feeling of an object during perception mainly depends on three characteristics: (i) the order denoted by O, (ii) the complexity denoted by C and the aesthetic measure denoted by M. Finally, Birkhoff [4], [5] defined the aesthetic measure as the ratio between O and C, i.e., M ≔ O/C, and he also applied this measure to network-like structures, e.g., compositions of certain polygons. A further development of Birkhoff’s theory has been mainly conducted by Fucks [20], [21], Shannon [37], and Gunzenhäuser [26] by using information-theoretical techniques, e.g., to define entropic measures. In contrast to all other approaches including Birkhoff’s, we do not aim to define a measure to quantify the style of single objects. Instead, we introduce a comparative approach of styles providing us to measure relational information between pairs of styles. Because our definition of a style is based on a multidimensional feature set, our measure to compare a pair of styles is based on the comparison of their underlying feature sets.
The paper is organized as follows. Starting from an introduction stated in Section 1, in Section 2 we mainly define the style of a set of structured objects and state some properties of such styles. In Section 3, we present a framework to quantify styles based on structural comparisons of the underlying structured objects. The graph similarity methods to perform such structural comparisons are introduced in Section 4. By applying certain graph similarity measures expressed in Section 4, we state in Section 5 some results of an experimental study to compare styles numerically. The paper finishes in Section 6 with a summary and conclusion.
Section snippets
Styles and properties of styles
In this section, we mainly state a definition of the style of structured objects. In this paper deal throughout with two-digit1 structured objects which are also called graphs [2], [24], [27], [28]. As a preliminary, we now repeat some known definitions to introduce basic graph-theoretical notations [2], [24], [27], [28]. In order to define abstract structured objects, we first give the definition of a general n-digit relation. Definition 2.1 Let be
Comparative analysis of styles
In this section, we present an approach to measure the similarity between styles. We solve this problem by applying similarity measures for measuring the structural similarity of graphs. By recalling the Definition 2.8, we notice that a style is a setIn the following, we briefly outline our novel method to measure the similarity between two given styles as follows:
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By applying graph similarity measures to the structured objects of a style , we generally compute the median
Measuring the similarity of structured objects
In Section 2, we defined the median of a certain style. Starting from such medians, we are now able to measure the similarity between arbitrary styles. We notice that the computation of a median and the comparison of medians mainly depends on a given similarity measure. In the following, we state some classical methods to measure the structural similarity between structured objects [6], [7], [8], [12], [29], [39], [40], [41], [46]. Furthermore, we express a graph similarity method of
Numerical results
In Section 4.3, we presented an approach to compare styles based on a local generalized tree decomposition, where the styles are sets of undirected structured objects. In order to apply this approach experimentally, we define the following styles:where denotes the edge density of . aij denotes the element in the ith row and jth
Summary and conclusion
In this paper, we introduced a definition of styles and presented a method to compare such styles computationally based on their underlying features. We now summarize the main contributions and arguments of our paper:
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In Section 1 we gave a short overview about the problem to measure the similarity of structured objects representing graphs. Furthermore, we briefly outlined some known approaches to measure text styles in semiotics and to define aesthetic measures [4], [5] for certain objects. In
Acknowledgments
Matthias Dehmer thanks the Max F. Perutz Laboratory for supporting parts of this work. Tanja Gesell was supported by ”GEN-AU project Bioinformatics Integration Network II”. Financial support from the Vienna Science and Technology Fond (WWTF) is also greatly appreciated.
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