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Structural Recommendations in Networks

Abstract

The growth of various Web-enabled networks has enabled numerous models of recommendation. For example, the Web itself is a large and distributed repository of data, and a search engine such as Google can be considered a keyword-centric variation of the notion of recommendation. In fact, a major discourse in the recommendation literature is to distinguish between the notions of search and recommendations. While search technologies also recommend content to users, the results are often not personalized to the user at hand. This lack of personalization has traditionally been the case because of the historical difficulty in tracking large numbers of Web users. However, in recent years, many personalized notions of search have arisen, where the Web pages recommended to users are based on personal interests. Many search engine providers, such as Google, now provide the ability to determine personalized results. This problem is exactly equivalent to that of ranking nodes in networks with the use of personalized preferences.

Keywords

  • Collaborative Filter
  • Link Prediction
  • Undirected Network
  • User Node
  • PageRank Algorithm

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    A formal mathematical treatment characterizes this in terms of the ergodicity of the underlying Markov chains. In ergodic Markov chains, a necessary requirement is that it is possible to reach any state from any other state using a sequence of one or more transitions. This condition is referred to as strong connectivity. An informal description is provided here to facilitate understanding.

  2. 2.

    In some applications such as bibliographic networks, the edge (i, j) may have a weight denoted by w ij . The transition probability p ij is defined in such cases by \(\frac{w_{ij}} {\sum _{j\in Out(i)}w_{ij}}\).

  3. 3.

    An alternative way to achieve this goal is to modify G by multiplying existing edge-transition probabilities by the factor (1 −α) and then adding αn to the transition probability between each pair of nodes in G. As a result, G will become a directed clique with bidirectional edges between each pair of nodes. Such strongly connected Markov chains have unique steady-state probabilities. The resulting graph can then be treated as a Markov chain without having to separately account for the teleportation component. This model is equivalent to that discussed in the chapter.

  4. 4.

    The left eigenvector \(\overline{X}\) of P is a row vector satisfying \(\overline{X}P =\lambda \overline{X}\). The right eigenvector \(\overline{Y }\) is a column vector satisfying \(P\overline{Y } =\lambda \overline{Y }\). For asymmetric matrices, the left and right eigenvectors are not the same. However, the eigenvalues are always the same. The unqualified term “eigenvector” refers to the right eigenvector by default.

  5. 5.

    http://www.dmoz.org

  6. 6.

    It is possible to ameliorate this problem to some extent by making minor modifications such as adding self-loops to the graph. However, such methods are not a formal part of the original SimRank algorithm.

  7. 7.

    http://googleblog.blogspot.com/2009/12/personalized-search-for-everyone.html

  8. 8.

    An implicit assumption here is that the matrix A is positive semi-definite. However, by setting the (unobserved) diagonal entries of A to the node degrees, it can be shown that A is positive semi-definite. These unobserved diagonal entries do not affect the final solution because they are not a part of the optimization problem.

  9. 9.

    Sammy Sosa is a retired Major League baseball player.

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Aggarwal, C.C. (2016). Structural Recommendations in Networks. In: Recommender Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-29659-3_10

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  • DOI: https://doi.org/10.1007/978-3-319-29659-3_10

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