Visualizing Bibliometric Networks

Abstract

This chapter provides an introduction to the topic of visualizing bibliometric networks. First, the most commonly studied types of bibliometric networks (i.e., citation, co-citation, bibliographic coupling, keyword co-occurrence, and coauthorship networks) are discussed, and three popular visualization approaches (i.e., distance-based, graph-based, and timeline-based approaches) are distinguished. Next, an overview is given of a number of software tools that can be used for visualizing bibliometric networks. In the second part of the chapter, the focus is specifically on two software tools: VOSviewer and CitNetExplorer. The techniques used by these tools to construct, analyze, and visualize bibliometric networks are discussed. In addition, tutorials are offered that demonstrate in a step-by-step manner how both tools can be used. Finally, the chapter concludes with a discussion of the limitations and the proper use of bibliometric network visualizations and with a summary of some ongoing and future developments.

Appendix: Normalization, Mapping, and Clustering Techniques Used by VOSviewer

In this appendix, we provide a more detailed description of the normalization, mapping, and clustering techniques used by VOSviewer.

1.1 Normalization

We first discuss the association strength normalization (Van Eck & Waltman, 2009) used by VOSviewer to normalize for differences between nodes in the number of edges they have to other nodes. Let a _ij denote the weight of the edge between nodes i and j, where a _ij  = 0 if there is no edge between the two nodes. Since VOSviewer treats all networks as undirected, we always have a _ij  = a _ji . The association strength normalization constructs a normalized network in which the weight of the edge between nodes i and j is given by

$$ {s}_{ij}=\frac{2m{a}_{ij}}{k_i{k}_j}, $$

(1)

where k _i (k _j ) denotes the total weight of all edges of node i (node j) and m denotes the total weight of all edges in the network. In mathematical terms,

$$ {k}_i={\displaystyle \sum_j{a}_{ij}}\kern1em \mathrm{and}\kern1em m=\frac{1}{2}{\displaystyle \sum_i{k}_i}. $$

(2)

We sometimes refer to s _ij as the similarity of nodes i and j. For an extensive discussion of the rationale of the association strength normalization, we refer to Van Eck and Waltman (2009).

1.2 Mapping

We now consider the VOS mapping technique used by VOSviewer to position the nodes in the network in a two-dimensional space. The VOS mapping technique minimizes the function

$$ V\left({\mathbf{x}}_1,\dots, {\mathbf{x}}_n\right)={\displaystyle \sum_{i<j}{s}_{ij}{\left\Vert {\mathbf{x}}_i-{\mathbf{x}}_j\right\Vert}^2} $$

(3)

subject to the constraint

$$ \frac{2}{n\left(n-1\right)}{\displaystyle \sum_{i<j}\left\Vert {\mathbf{x}}_i-{\mathbf{x}}_j\right\Vert }=1, $$

(4)

where n denotes the number of nodes in a network, x _i denotes the location of node i in a two-dimensional space, and ||x _i  − x _j || denotes the Euclidean distances between nodes i and j. VOSviewer uses a variant of the SMACOF algorithm (e.g., Borg & Groenen, 2005) to minimize (3) subject to (4). We refer to Van Eck et al. (2010) for a more extensive discussion of the VOS mapping technique, including a comparison with multidimensional scaling.

1.3 Clustering

Finally, we discuss the clustering technique used by VOSviewer. Nodes are assigned to clusters by maximizing the function

$$ V\left({c}_1,\dots, {c}_n\right)={\displaystyle \sum_{i<j}\delta \left({c}_i,{c}_j\right)\left({s}_{ij}-\gamma \right)}, $$

(5)

where c _i denotes the cluster to which node i is assigned, δ(c _i , c _j ) denotes a function that equals 1 if c _i  = c _j and 0 otherwise, and γ denotes a resolution parameter that determines the level of detail of the clustering. The higher the value of γ, the larger the number of clusters that will be obtained. The function in (5) is a variant of the modularity function introduced by Newman and Girvan (2004) and Newman (2004) for clustering the nodes in a network. There is also an interesting mathematical relationship between on the one hand the problem of minimizing (3) subject to (4) and on the other hand the problem of maximizing (5). Because of this relationship, the mapping and clustering techniques used by VOSviewer constitute a unified approach to mapping and clustering the nodes in a network. We refer to Waltman et al. (2010) for more details. We further note that VOSviewer uses the recently introduced smart local moving algorithm (Waltman & Van Eck, 2013) to maximize (5).