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On the Normalization and Visualization of Author Co-Citation Data Salton's Cosine versus the Jaccard Index

Loet Leydesdorff

arXiv:0911.1447v1physics.soc-phcs.DLcs.IR

TL;DR

When original citation data cannot be retrieved, researchers may have only a co-occurrence matrix and must choose how to normalize it. The paper compares similarity measures and concludes that diagonal-augmented Jaccard is the practical alternative in that setting, with empirical results similar to cosine-normalized citation matrices.

  • Problem

    In Web settings, retrieving original citation data is often infeasible, leaving researchers to choose a normalization for co-occurrence data without the underlying occurrence matrix.

    Source: §1 · pp. 1–2

  • Method

    The paper compares Jaccard, including citation-count diagonal values, with cosine and other measures across symmetrical co-occurrence and asymmetrical occurrence matrices.

    Source: §1 · p. 2, §2 · p. 4

  • Results

    When the occurrence matrix is unavailable, diagonal-augmented Jaccard is the recommended normalization and was as good as cosine-normalized citation matrices in the empirical examples, with ρ > 0.99 (p < 0.01) in the subset comparison.

    Source: §3 · p. 7, Conclusions · p. 8

  • Takeaways & Limitations

    Jaccard’s focus on co-occurrence strength rather than vector-distribution shape makes it suitable when only a co-occurrence matrix is available, and occurrence frequencies on the diagonal can improve normalization.

    Source: Conclusions · p. 8

    Which Jaccard normalization is preferable depends on the research question and data availability, and symmetrical matrices warrant caution as input to further statistical analysis.

    Source: Conclusions · p. 8

Abstract

from arXiv · show

The debate about which similarity measure one should use for the normalization in the case of Author Co-citation Analysis (ACA) is further complicated when one distinguishes between the symmetrical co-citation--or, more generally, co-occurrence--matrix and the underlying asymmetrical citation--occurrence--matrix. In the Web environment, the approach of retrieving original citation data is often not feasible. In that case, one should use the Jaccard index, but preferentially after adding the number of total citations (occurrences) on the main diagonal. Unlike Salton's cosine and the Pearson correlation, the Jaccard index abstracts from the shape of the distributions and focuses only on the intersection and the sum of the two sets. Since the correlations in the co-occurrence matrix may partially be spurious, this property of the Jaccard index can be considered as an advantage in this case.

1. Introduction

The paper separates the choice of similarity measure from the choice of matrix to normalize, focusing on cases where Web researchers can retrieve co-occurrence data but not original citations. It argues that Jaccard may be preferable in that setting and motivates the choice with a dataset whose group-level separation coexists with weaker cross-group author relations.

1. Introduction

  • Ahlgren et al. (2003) and Leydesdorff and Vaughan (2006) report different Schubert–Van Raan correlations when using co-citation versus underlying citation data.The reported values are r = +0.74 and r = −0.131 (p < 0.05), respectively.Source: §1 · p. 1
  • Web researchers may have only an index for constructing co-occurrence data, making retrieval of the original citation matrix often infeasible.The question is whether such co-occurrence data should be normalized with cosine, Pearson correlation, or another measure.Source: §1 · pp. 1–2
  • The paper argues that Jaccard may be preferred for co-occurrence-only data and compares it with cosine, Pearson correlation, and the probabilistic activity index.It distinguishes binary citation-matrix normalization with Jaccard from non-binary co-citation-matrix normalization with Tanimoto.Source: §1 · p. 2
  • The example dataset has positive correlations within two groups and negative correlations between them, yet individual authors retain weak cross-group relations.The paper frames an optimal representation as reflecting both the group separation and the weak overlap.Source: §1 · p. 2
  • Normalization choice and matrix type are distinct decisions, with cosine and Pearson correlation serving different visualization and statistical-analysis purposes.The passage notes that Pearson normalizes for the arithmetic mean, whereas cosine normalizes for the geometrical mean.Source: §1 · p. 2

2. The Jaccard index

This section defines Jaccard-style normalization through set overlap and contrasts it with vector-based measures and other co-occurrence normalizations. It also explains how citation totals on the diagonal or co-occurrence margintotals change the information used in normalization.

2. The Jaccard index

  • Jaccard normalizes the intersection of two sets by their union, while Tanimoto extends the approach to non-binary data.For co-citation, the intersection count is the co-citation frequency; relative co-citation frequency divides it by the union count.Source: §2 · pp. 2–3
  • Prior comparisons of Jaccard and cosine for co-occurrence data remain inconclusive, and their expected monotonicity does not make them identical for visualization.Cosine uses vector angles, whereas Jaccard uses the intersection relative to the union; denominator choices also differ between co-occurrence and occurrence matrices.Source: §2 · p. 3
  • For Schubert and Van Raan, non-binary Tanimoto using margintotals is 5 / (139 + 132 – 5) = 0.019, while binary Jaccard using citation counts is 5 / (60 + 50 – 5) = 0.048.These illustrate the two normalization options described for Table 1.Source: §2 · p. 3
  • An incidence matrix can yield two symmetrical matrices: one counting shared events among actors and the other shared actors among events.The latter is described as counting events simultaneously attended by a pair of actors.Source: §2 · p. 3
  • Adding total citation counts to the co-citation matrix’s main diagonal makes its Jaccard normalization correspond to Jaccard normalization of the asymmetrical occurrence matrix.The added diagonal values correspond to the asymmetrical matrix’s margintotals; without them, the co-occurrence matrix has lost information about cited-document counts.Source: §2 · pp. 3–4
  • The Jaccard index evaluates each cell against margintotals rather than the distributions across its row and column, unlike cosine and Pearson correlation.This can discard information in an asymmetrical matrix, while diagonal citation totals add information to the symmetrical matrix.Source: §2 · p. 4
  • The probabilistic activity index, like Jaccard and Tanimoto, can use the lower triangle of a symmetrical co-occurrence matrix, while cosine and Pearson use full vectors.The passage describes PAI as the ratio of observed to expected contingency-table values; cosine and Pearson thus use the symmetrical matrix’s information twice.Source: §2 · p. 4
  • Information measures can turn symmetrical co-occurrence data into an asymmetrical distance matrix and directed graph, where distance from A to B may differ from B to A.The passage contrasts this with the undirected graphs generated by the measures discussed in the paper.Source: §2 · p. 4

3. Results

The cosine-normalized asymmetrical occurrence matrix gives the best visualization of the underlying structure; when an occurrence matrix is unavailable, Jaccard normalization using total citations on the diagonal is a close alternative.

3. Results

  • The comparison uses Spearman rank correlations because proximity-matrix objects are dyadic and violate the independence assumption required for parametric significance tests.Table 2 reports the rank-order correlations among the lower triangles of the similarity matrices.Source: §3 · p. 5
  • The asymmetrical-matrix cosine and Jaccard rankings correlate perfectly (ρ = 1.00; p < 0.01), but their visualizations differ in fine structure.Figure 1 shows within-cluster detail to a greater extent than Figure 2.Source: §3 · pp. 5–6, Figure 1 · p. 5, Figure 2 · p. 6
  • The Jaccard index spans a smaller range than cosine, with variance 0.08 versus 0.21 for the cosine-based matrix, potentially limiting its dissolvent capacity in visualizations.Figure 3 presents the relation between the measures for the asymmetrical citation matrix.Source: §3 · p. 6, Figure 3 · p. 7
  • At a 0.05 threshold, only the Tijssen–Croft Jaccard relation remains, while cosine retains other relations including Price and Van Raan.The paper attributes the Tijssen–Croft result to both authors having relatively low main-diagonal values and thus low denominator values.Source: §3 · p. 7
  • The lower-triangle rankings correlate near unity with the symmetrical-matrix Tanimoto index (ρ = 0.998), while correlations with probabilistic affinity are slightly lower (ρ < 0.99).Despite relatively small differences among the lower triangles, the visualizations differ.Source: §3 · p. 7
  • The cosine-normalized asymmetrical occurrence matrix gives the best visualization; Jaccard normalization using total citations on the diagonal is the second-best alternative when an occurrence matrix is unavailable.For the twelve-scientometrician subset, the Jaccard-based results were very similar to the cosine-normalized occurrence matrix (ρ > 0.99; p < 0.01).Source: §3 · p. 7

Conclusions

For visualization, the cosine-normalized asymmetrical occurrence matrix is preferred; when it is unavailable, Jaccard normalization using total citations on the diagonal is the next-best alternative. The choice of Jaccard variant depends on the research question and data, and symmetrical co-occurrence input warrants caution for further statistical analysis.

Conclusions

  • For visualization, the cosine-normalized asymmetrical occurrence matrix is preferred; Jaccard with total citations on the diagonal is the next-best alternative when occurrence data are unavailable.The diagonal counts can be based on marginal totals from the asymmetrical matrix.Source: Conclusions · p. 8
  • Jaccard is preferred when only a co-occurrence matrix is available because it ignores distribution shape; adding occurrence frequencies on the diagonal can improve normalization.The preferred Jaccard option depends on the research question and data availability.Source: Conclusions · p. 8
  • Use caution with symmetrical co-occurrence matrices in further statistical analysis: multiplying the citation matrix by its transpose can change correlation size and potentially its sign.Using Jaccard with diagonal values from the asymmetrical matrix's marginal totals circumvents this problem.Source: Conclusions · p. 8
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