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Social Ontologies as Generalized Nearly Acyclic Directed Graphs: A Quantitative Graph Model of Social Tagging

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Abstract

In this paper, we introduce a quantitative graph model of social ontologies as exemplified by the category system of Wikipedia. This is done to contrast structure formation in distributed cognition with classification schemes (by example of the DDC and MeSH), formal ontologies (by example of OpenCyc and SUMO), and terminological ontologies (as exemplified by WordNet). Our basic findings are that social ontologies have a characteristic topology that clearly separates them from other types of ontologies. In this context, we introduce the notion of a Zipfian bipartivity to analyze the relationship of categories and categorized units in distributed cognition.

MSC2000 Primary 05C75; Secondary 05C82, 68T50, 90B15, 91D30, 91F20.

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Notes

  1. 1.

    Note that within the social ontologies analyzed here, i.e. Wikipedia category systems, moderation may occur.

  2. 2.

    Such links are anchored within the body of an article, but not at the end of it where categorization links are located.

  3. 3.

    The order of a graph equals the number of its vertices [32].

  4. 4.

    Note that the documentation of the Wikipedia suggests the absence of cycles and that the category system of the Wikipedia spans DAGs (cf. http://en.wikipedia.org/wiki/Wikipedia:Categorization). We show more precisely the degree by which this is only approximately true.

  5. 5.

    This approach is in the line of [13] who develops information-theoretic indices of graphs and their topology. See also [87] for a related approach in the area of quantitative biology. A second impulse comes from [22] who calculates entropies of probability distributions of vertices in complex networks. Albeit this coincidence we deal with complex nearly DAG-like graphs apart from complex networks. In any event, it is our conviction that the analysis of graph structures can gain invaluable insights from these two approaches beyond of what has been done so far in complex network theory.

  6. 6.

    Note that the graphs in Fig. 10.5 denote different scenarios only schematically.

  7. 7.

    Note that we use the terms index and measure synonymously (cf. [41]).

  8. 8.

    Of course, every weakly connected component of the [ ]-variant of the SOC has at least one source. This information is already implicity explored by means of the connected component statistics. Thus, we focus on operating on the LCC of each SOG when calculating the multiplicity index.

  9. 9.

    Note that μ = . 514 and σ = . 3621.

  10. 10.

    This also means that if we disregard the multiplicity of sources we may say that SOGs tend to be tree-like.

  11. 11.

    By D − u we denote in the usual way the subdigraph of D induced by V ∖ { u}.

  12. 12.

    Note that [85] as well as [76] and [77] aim at improving shortest path algorithms by operating on nearly acyclic graphs but not on graph classification. Therefore, the deficiency we found is irrelevant for them.

  13. 13.

    At this point one might ask why we do not use a simpler notion of cyclicity [38] by counting, for example, the number of edges to be deleted in order to make a graph acyclic? The simple reason is that index (10.10) is more informative about the DAG-like structure of a SOG as it additionally includes the impact of multiple sources.

  14. 14.

    A synergetic model [2, 39] of this process could be a starting point to build an integrative model of all characteristics of structure formation considered here. However, at present this is out of reach.

  15. 15.

    At this point, we might also take the variance or mean of the feature vector q(x) as an aggregation function to measure the balance of x. See [52, 70] for such an approach.

  16. 16.

    Note that this main category is always unique.

  17. 17.

    An obvious alternative to this approach would be to analyze the distribution of imbalance values in a SOG – this will be one reference point for future work.

  18. 18.

    Botafogo et al. [14] already utilized depth as a reference quantity of imbalance. However, they unnecessarily use a recursive function for defining it and miss demonstrating its empirical significance.

  19. 19.

    Without following this line of research here, it may be interesting to consider this level of maximum order from the point of view of conceptual levels in prototype theory [74].

  20. 20.

    See [54] who have introduced this notion in the area of modeling web genres.

  21. 21.

    Cf. www.nlm.nih.gov/mesh/.

  22. 22.

    Cf. www.ontologyportal.org/.

  23. 23.

    Cf. http://wordnet.princeton.edu/man/wninput.5WN.html.

  24. 24.

    Thus, the reader should not confuse ℂ r with a random uniform distribution of the objects over the target classes.

  25. 25.

    In Table 10.9 we present the F-score of the best performing subset of features. This subset is reported in the 6th column of Fig. 10.22. We see that, by representing the target ontologies by this feature subset, the F-score is raised. Below we discuss this result in the context of testing the OSH.

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Acknowledgment

Financial support of the German Federal Ministry of Education (BMBF) through the research project Linguistic Networks, of the German Research Foundation (DFG) through the Excellence Cluster 277 Cognitive Interaction Technology (via the Project KnowCIT) and of the SFB 673 Alignment in Communication (via the Project A3 Dialogue Games and Group Dynamics and X1 Multimodal Alignment Corpora: Statistical Modeling and Information Management) is gratefully acknowledged. We also thank Dietmar Esch, Tobias Feith, and Roman Pustylnikov for the download of ontologies as well as Rüdiger Gleim, Olga Abramov, and Paul Warner for their fruitful hints which helped to reduce the number of errors in this chapter.

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    Alexander Mehler

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Mehler, A. (2011). Social Ontologies as Generalized Nearly Acyclic Directed Graphs: A Quantitative Graph Model of Social Tagging. In: Dehmer, M., Emmert-Streib, F., Mehler, A. (eds) Towards an Information Theory of Complex Networks. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4904-3_10

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