Abstract
Epistemic network analysis (ENA) has been used in more than 300 published studies to date. However, there is no work in publication that describes the transformations that constitute ENA in formal mathematical terms. This paper provides such a description, focusing on the mathematical formulations that lead to two key affordances of ENA that are not present in other network analysis tools or multivariate analyses: (1) summary statistics that can be used to compare the differences in the content rather than the structure of networks and (2) network visualizations that provide information that is mathematically consistent with those statistics. Specifically, we describe the mathematical transformations by which ENA constructs matrix representations of discourse data, uses those representations to generate networks for units of analysis, places those networks into a metric space, identifies meaningful dimensions in the space, and positions the nodes of network graphs within that space so as to enable interpretation of those dimensions in terms of the content of the networks. We conclude with a discussion of how the mathematical formalisms of ENA can be used to model networks more generally.
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Notes
- 1.
SVD is related to other commonly used dimension reduction/visualization techniques, such as principal component analysis (PCA) and factor analysis. SVD decomposes the original (non-symmetric) data matrix, Z, without centering, while PCA standardizes the columns of the original matrix and decomposes the data into a set of ordered, orthogonal components using the eigenvalues of the symmetric sample covariance matrix, thus centering the data prior to decomposition. Factor analysis derives a set of influential factors by decomposing the sample covariance matrix of a set of centered variables into two additive components, one of which is attributed to the set of common factors and the other is specific to each observation.
- 2.
Briefly, we prove orthogonality of V and \(\varvec{\mathbf {\mu }}\) by \(V^\prime \varvec{\mathbf {\mu }} = 0\). By Eqs. 3.17 and 3.18, and because \(\varvec{\mathbf {\mu }}\) is a unit vector \(\varvec{\mathbf {\mu }}^\prime \varvec{\mathbf {\mu }} = 1\),
$$\begin{aligned} V^\prime \varvec{\mathbf {\mu }}&= D^{-1}U^\prime (\mathbf{N}-\mathbf{N} \varvec{\mathbf {\mu }}\varvec{\mathbf {\mu }}^\prime )\varvec{\mathbf {\mu }}\\&= D^{-1}U^\prime \mathbf{N}\varvec{\mathbf {\mu }}-D^{-1}U^\prime \mathbf{N} \varvec{\mathbf {\mu }}\varvec{\mathbf {\mu }}^\prime \varvec{\mathbf {\mu }}\\&= D^{-1}U^\prime \mathbf{N}\varvec{\mathbf {\mu }}-D^{-1}U^\prime \mathbf{N} \varvec{\mathbf {\mu }}\\&= 0 \end{aligned}$$.
- 3.
We use \(\frac{1}{2}\) of the normalized weights because for any pair of codes \(a_i\) and \(a_j\), one half of the normalized weight of their connection is associated with each code. Thus the total normalized weights are preserved and \(\sum _i \tilde{\mathbf{w}}_i^k = \sum _i \mathbf{N}_i^k\).
- 4.
There is no intercept term included in Eq. 3.23. This is because the columns of \(\mathbf {\tilde{w_i}}\) (which function as the independent variables in the regression) are not linearly independent. In particular, by Eq. 3.20, \(\forall k, \sum _j \mathbf {\tilde{w}_{ij}^k} = 1\). Thus, adding an intercept term would make the equation ill-defined—that is, without a unique solution.
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Acknowledgements
This work was funded in part by the National Science Foundation (DRL-1661036, DRL-1713110), the Wisconsin Alumni Research Foundation, and the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin–Madison. The opinions, findings, and conclusions do not reflect the views of the funding agencies, cooperating institutions, or other individuals.
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Bowman, D. et al. (2021). The Mathematical Foundations of Epistemic Network Analysis. In: Ruis, A.R., Lee, S.B. (eds) Advances in Quantitative Ethnography. ICQE 2021. Communications in Computer and Information Science, vol 1312. Springer, Cham. https://doi.org/10.1007/978-3-030-67788-6_7
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