A systematic approach to the one-mode projection of bipartite graphs

Social Network Analysis and Mining volume 1, pages187–218(2011)Cite this article

Abstract

Bipartite graphs are common in many complex systems as they describe a relationship between two different kinds of actors, e.g., genes and proteins, metabolites and enzymes, authors and articles, or products and consumers. A common approach to analyze them is to build a graph between the nodes on one side depending on their relationships with nodes on the other side; this so-called one-mode projection is a crucial step for all further analysis but a systematic approach to it was lacking so far. Here, we present a systematic approach that evaluates the significance of the co-occurrence for each pair of nodes v, w, i.e., the number of common neighbors of v and w. It turns out that this can be seen as a special case of evaluating the interestingness of an association rule in data mining. Based on this connection we show that classic interestingness measures in data mining cannot be applied to evaluate most real-world product-consumer relationship data. We thus introduce generalized interestingness measures for both, one-mode projections of bipartite graphs and data mining and show their robustness and stability by example. We also provide theoretical results that show that the old method cannot even be used as an approximative method. In a last step we show that the new interestingness measures show stable and significant results that result in attractive one-mode projections of bipartite graphs.

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Notes

  1. 1.

    This part reproduces parts of the conference version published with IEEE (Zweig 2010). Reproduced with permission.

  2. 2.

    According to an interestingness measure called conviction.

  3. 3.

    This is given under the assumption that the occurrence of M in the random graph model is normally distributed.

  4. 4.

    Note that the actual number \(|{\fancyscript{G}}(L,R)|\) of graphs in \({\fancyscript{G}}(L,R)\) is not yet described by a closed formula (Greenhill and McKay 2008; Barvinok 2008).

  5. 5.

    As long as this does not lead to degree 0 or degree r + 1 (l + 1) in which case nothing is done.

  6. 6.

    Note that the order was chosen for displaying reasons—none of the data samples directly showed them in this order.

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Affiliations

  1. Interdisciplinary Center for Scientific Computing, University of Heidelberg, Speyerer Straße 6, 69115, Heidelberg, Germany

    Katharina Anna Zweig

  2. Wilhelm-Schickard-Institute, University of Tübingen, Sand 14, 72072, Tübingen, Germany

    Michael Kaufmann

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  1. Katharina Anna Zweig
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  2. Michael Kaufmann
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Correspondence to Katharina Anna Zweig.

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Zweig, K.A., Kaufmann, M. A systematic approach to the one-mode projection of bipartite graphs. Soc. Netw. Anal. Min. 1, 187–218 (2011). https://doi.org/10.1007/s13278-011-0021-0

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Keywords

  • Bipartite graphs
  • One-mode projection
  • Association rules
  • Interestingness measures