Source-linked AI summary
A comparison of two techniques for bibliometric mapping: Multidimensional scaling and VOS
Nees Jan van Eck, Ludo Waltman, Rommert Dekker, Jan van den Berg
TL;DR
The paper addresses the lack of an extensive comparison between multidimensional scaling and VOS for bibliometric mapping. It combines theoretical and experimental analyses and finds that VOS generally represents data more satisfactorily because common MDS approaches suffer from artifacts.
Problem
An extensive comparison of multidimensional scaling and VOS for constructing bibliometric maps was previously lacking.
Method
The paper analyzes the mathematical relationship between MDS and VOS and experimentally compares two MDS approaches with one VOS approach across author, journal, and keyword data sets.
Results
VOS maps provide a more satisfactory representation of the underlying data set than maps produced by the two MDS approaches, which suffer from serious artifacts.
Takeaways & Limitations
VOS can serve as an alternative to MDS for constructing bibliometric maps.
Takeaways & Limitations
Ordinal or interval MDS applied to association-strength similarities is not completely satisfactory because the similarity ratios do not correspond straightforwardly to expected map distances.
Abstract
from arXiv · showhide
VOS is a new mapping technique that can serve as an alternative to the well-known technique of multidimensional scaling. We present an extensive comparison between the use of multidimensional scaling and the use of VOS for constructing bibliometric maps. In our theoretical analysis, we show the mathematical relation between the two techniques. In our experimental analysis, we use the techniques for constructing maps of authors, journals, and keywords. Two commonly used approaches to bibliometric mapping, both based on multidimensional scaling, turn out to produce maps that suffer from artifacts. Maps constructed using VOS turn out not to have this problem. We conclude that in general maps constructed using VOS provide a more satisfactory representation of a data set than maps constructed using well-known multidimensional scaling approaches.
Introduction
Bibliometric maps represent relationships among scientific entities using bibliographic data, with multidimensional scaling historically dominant and VOS introduced as an alternative. This paper compares MDS- and VOS-based approaches theoretically and experimentally.
- Bibliometric maps represent relations among authors, documents, journals, or keywords using citation, co-citation, bibliographic coupling, or keyword co-occurrence data.
- MDS has been widely used to construct maps of authors, documents, journals, and keywords.
- VOS is a mapping technique introduced as an alternative to MDS.
- The paper addresses the lack of an extensive comparison between MDS and VOS through theoretical and experimental analyses.
- The experimental analysis compares three mapping approaches across author, journal, and keyword co-occurrence data sets.
Multidimensional Scaling
MDS constructs low-dimensional maps from similarities or dissimilarities by optimizing distances between items. Its bibliometric use involves transforming co-occurrence data into similarities and selecting a stress-minimization variant.
- Similarity Measures: Co-occurrence frequencies are transformed into similarities because raw frequencies generally do not properly reflect similarity between items.
- Similarity Measures: Direct similarity measures normalize pairwise co-occurrence frequencies, whereas indirect measures compare vectors of co-occurrence frequencies.
- Similarity Measures: Association strength normalizes co-occurrence frequencies and is identified as an appropriate direct similarity measure for bibliometric mapping.It is proportional to observed co-occurrences divided by expected co-occurrences under statistical independence.
- The Technique of Multidimensional Scaling: MDS locates items in a low-dimensional space so distances reflect similarities as accurately as possible, with stronger relations represented by smaller distances.
- The Technique of Multidimensional Scaling: MDS minimizes a weighted sum of squared differences between transformed proximities and map distances.
- The Technique of Multidimensional Scaling: Ratio, interval, and ordinal MDS differ in how the transformation function treats proximity scales; bibliometric applications presumably use ordinal MDS most often.
VOS
VOS pursues the same mapping aim as MDS but uses a similarity-weighted objective function. It typically combines nonnegative association-strength similarities with a constraint preventing all items from collapsing to one location.
- VOS aims to place items in a low-dimensional space where distances reflect similarity or relatedness, like MDS.
- VOS uses nonnegative similarities as input and typically calculates them with association strength.
- VOS minimizes a weighted sum of squared pairwise distances, with each distance weighted by the similarity between items.
- The average distance between items is constrained to equal one to avoid the trivial solution in which all items share the same location.
- VOS maps are produced by minimizing the objective function subject to the distance constraint, using a SMACOF variant in available implementations.
- An item’s ideal location is defined as a similarity-weighted average of the locations of all other items.
Relationship Between Multidimensional Scaling and VOS
The paper shows that MDS and VOS are closely related under certain conditions, while arguing that common ordinal and interval MDS approaches inadequately represent ratio-scale similarities. VOS and an alternative MDS formulation address this issue through specific transformations and weighting choices.
- Limitations of common MDS approaches: Ordinal and interval MDS are not completely satisfactory for association-strength similarities because their transformation functions have too much freedom.For ratio-scale similarities, this freedom can treat maps with substantially different distance ratios as equally perfect representations.
- Alternative MDS formulation: An alternative MDS approach uses the identity transformation, preserving similarities directly rather than freely transforming them.Because the relevant objective function requires dissimilarities, similarities must first be converted, here using d_ij = 1 / s_ij.
- VOS weighting: For VOS-related weighting, setting w_ij = s_ij avoids division by zero when items have no co-occurrences.The weights increase linearly with similarity; faster growth would remove the penalty for placing completely dissimilar items close together.
- Mathematical relationship: The formal proposition states that a globally optimal solution to either the VOS or corresponding weighted-MDS problem can be rescaled by a positive constant to solve the other.This establishes the techniques’ close mathematical relationship without making them identical in every formulation.
- Mathematical relationship: Under certain conditions, VOS can be regarded as weighted MDS with specially chosen proximities and weights.The paper formally establishes a two-way correspondence between globally optimal solutions, up to multiplication by a positive constant.
Experimental Comparison
The experiment compares two MDS approaches with VOS across author, journal, and keyword data sets. Across these maps, VOS avoids the prominent circularity and centralization artifacts observed in the MDS maps, making subfield separation and item relatedness easier to interpret.
- Experimental Design: The study compares MDS-AS, MDS-COS, and VOS using author co-citations, journal co-citations, and keyword co-occurrences from Web of Science.The authors, journals, and keywords data sets contain 405 authors, 2079 journals, and 831 keywords, respectively.
- Experimental Design: Each data set is represented by three two-dimensional maps, with MDS using ordinal scaling and VOS using association-strength similarities.MDS was run with PROXSCAL; the two MDS approaches differ in their direct versus indirect similarity measures.
- Global Map Structure: MDS-AS maps tend toward almost perfect circles, while MDS-COS maps show weaker circularity and both MDS approaches place important items centrally and less important items peripherally.These effects are especially clear for the authors and keywords data sets.
- Global Map Structure: The observed map differences are robust across ordinal versus interval MDS, alternative direct similarity measures, and numerous data sets, except that near-perfect circularity is absent for data sets with fewer than 100 items.The authors report similar results across many data sets, while noting this size-related boundary for the circular structure.
- Authors Data Set: In the authors data set, all three approaches place similar authors near one another, but only VOS clearly displays the separation between the ISR and informetrics subfields.The MDS maps scatter ISR authors and tend to place prominent authors at the center, obscuring subfield structure.
- Interpretation: MDS-AS and MDS-COS can impose artificial structure that makes distances misleading, whereas VOS does not seem to impose such artifacts across the authors, journals, and keywords maps.The circularity of MDS-AS is linked to the large proportion of zero co-occurrences, which makes many zero-similarity distances difficult to represent in low dimensions.
Conclusions
The paper concludes that VOS offers an alternative to MDS, with a mathematically related formulation and maps that better represent the underlying data. VOS can also be accessed through freely available VOSviewer software for visualization and interactive examination.
- VOS can be regarded as a kind of weighted MDS with specially chosen proximities and weights.
- Maps constructed using VOS provide a more satisfactory representation of the underlying data set than maps constructed using either MDS approach.The MDS approaches suffer from central-placement and circular-structure artifacts.
- The VOS approach is available through freely accessible VOSviewer software with a graphical interface for mapping and interactive map examination.
Appendix
The appendix proves both parts of Proposition 1 by contradiction, relating globally optimal solutions of the two optimization problems through scaling transformations.
- Both parts of Proposition 1 are proved by contradiction.
- For part (i), globally optimal solutions are related by defining scaled solutions U = cX and V = Y / c.The argument shows that U satisfies the constraint in Equation 6 and derives a contradiction from assuming it is not globally optimal.
- For part (ii), the proof similarly defines U = cX and V = Y / c and uses the constraint to derive a contradiction.The contradiction establishes global optimality for the corresponding solution and completes the proposition.