2. Exploratory Graph Analysis: A (Very) Brief Overview

The origins of network models in psychology can be traced back to the seminal work of Cattell in the mid-60’s (Boker, Reference Boker2018; Cattell, Reference Cattell1965) and less explicitly to the proposition of image structural analysis by Guttman (Reference Guttman1953). It gained more traction, however, after the publication of the mutualism model of intelligence (Van Der Maas et al., Reference Van Der Maas, Dolan, Grasman, Wicherts, Huizenga and Raijmakers2006) and the proposition of the network perspective of psychopathological constructs (Borsboom, Reference Borsboom2008; Borsboom & Cramer, Reference Borsboom and Cramer2013; Cramer et al., Reference Cramer, Waldorp, Van Der Maas and Borsboom2010; Fried et al., Reference Fried, van Borkulo, Cramer, Boschloo, Schoevers and Borsboom2017). Network models have since been applied in clinical (Bork et al., Reference van Bork, van Borkulo, Waldorp, Cramer and Borsboom2018), cognitive (Golino & Demetriou, Reference Golino and Demetriou2017; Van Der Maas et al. Reference Van Der Maas, Kan, Marsman and Stevenson2017), social (Dalege et al., Reference Dalege, Borsboom, Harreveld, Waldorp and Maas2017), and many other areas of psychology (Epskamp et al., Reference Epskamp, Rhemtulla and Borsboom2017).

The rapid developments of network modeling in psychology spawned a new subfield of quantitative psychology termed network psychometrics (Epskamp, Reference Epskamp and Irwing Paul2018). In these models, nodes (circles) represent variables and edges (lines) represent associations between the nodes. Under this framework, Golino and Epskamp (Reference Golino and Epskamp2017) proposed the use of network psychometrics as a method for dimensionality assessment and termed this novel approach exploratory graph analysis. Unlike previous factor analytic methods, EGA produces a visual guide—network plot—that not only indicates the number of dimensions to retain, but also which nodes (e.g., items) cluster together and their level of association. Simulation studies have shown that EGA presents comparable or better accuracy than the state-of-the-art parallel analysis technique when estimating the number of simulated factors (Christensen, Reference Christensen2020; Golino & Epskamp, Reference Golino and Epskamp2017; Golino et al., Reference Golino, Shi, Garrido, Christensen, Nieto, Sadana and Martinez-Molina2020b).

The EGA approach currently uses two network estimation methods (for a review, see Golino et al., Reference Golino, Shi, Garrido, Christensen, Nieto, Sadana and Martinez-Molina2020b): graphical least absolute shrinkage and selection operator (GLASSO; Friedman et al., Reference Friedman, Hastie and Tibshirani2008) and triangulated maximally filtered graph (TMFG; Massara et al., Reference Massara, Di Matteo and Aste2016). After the network is estimated, an algorithm for community detection on weighted networks is used (Walktrap; Pons & Latapy, Reference Pons and Latapy2005). The next sections will briefly introduce the network estimation methods and the community detection algorithm used in EGA.

3. Dynamic Exploratory Graph Analysis

Dimensionality assessment is common in psychology, but also has a significant use in data mining, especially in the subfield of text mining. Text mining is a data-driven, exploratory method used to find patterns and trends in large sets of texts, enabling the transformation of unorganized text into succinct knowledge (Ananiadou & McNaught, Reference Ananiadou and McNaught2006). It is epistemologically compatible with content analysis, making it possible to collect, maintain, interpret, and discover relevant information hidden in texts in a systematic and efficient way (Singh, Hu, & Roehl, Reference Singh, Hu and Roehl2007). Recently, EGA was used in combination with text mining to estimate latent topics in texts, showing promising results (Kjellström & Golino, Reference Kjellström and Golino2019).

The current implementation of EGA, however, limits its application to data collected to a single time-point (i.e., cross-sectional data). Kjellström and Golino (Reference Kjellström and Golino2019), for example, used text data from single interviews made with multiple adults about their conceptions of health. To enable the identification of latent structures in texts from social media, the EGA technique needs to be expanded to accommodate short-term temporal dynamics. This, in our view, would provide a more valid way to estimate topics in texts that are produced in a series (such as posts on Twitter). People use words to communicate their ideas, thoughts, and feelings, with groupings of words indicating the underlying content of the text (i.e., topics). If the text data come from a single time-point, the topics can be estimated using the variance–covariance matrix of the words (Kjellström & Golino, Reference Kjellström and Golino2019). However, in the case of texts that are written on several different occasions, the temporal dynamics should be accounted for by the topic model, otherwise it may generate significant bias in the estimation of the underlying latent topics.

In texts published on Twitter, for example, people may use several words to talk about the topic of “violence” on one occasion (time or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} ). In the next occasion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t + 1)$$\end{document} , the same person may use a different set of words to express their feelings about “gun control” and yet in a subsequent occasion may use different words to communicate their views about the “mainstream media.” Across time, one person can write about several different topics, using words that may or may not be the same on each occasion. Instead of modeling the covariance of words without taking time into consideration (a single-measurement approach), a more ecologically valid way to understand topics published on Twitter is to model how words are varying together across time, capturing the short-term dynamics of the texts.

One way to address this problem is proposed as follows: A collection of texts (corpus) from one single individual (e.g., a Twitter account) over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} discrete time-points can be represented as a document-term matrix (DTM) in which each unique word is a column and each observation (e.g., a post on Twitter) is a row in the document-term matrix. The DTM is, therefore, a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \times K$$\end{document} matrix, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} is the number of time points and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K$$\end{document} is the number of unique words used in the entire collection of texts. The values of the DTM cells are the frequency of the words.

Since each column of the document-term matrix represents a time series of the word frequency, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U = \left\{ u_1, u_2,\ldots , u_N\right\} $$\end{document} , each time series can be transformed into a time-delay embedding matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {X}^{(n)}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} is the number of embedding dimensions. A time-delay embedding matrix is used to reconstruct the attractor of a dynamical system using a single sequence of observations (Takens, Reference Takens1981; Whitney, Reference Whitney1936). An attractor contains useful information about the dynamical system such as a series of values to which a system tends toward based on a set of starting conditions. In many empirical situations, however, the collection of possible system states (phase-space) and the equations governing the system are unknown. In such situations, attractor reconstruction techniques can be used as a means to reconstruct the phase-space dynamics using, for example, only a single time series with observable values.

In the time-delay embedding matrix, each row is a phase-space vector (Rosenstein et al., Reference Rosenstein, Collins and De Luca1993):

(1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X = \left[ X_1 X_2 \ldots X_M\right] ^\prime , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_t$$\end{document} is the state of the system at discrete time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} and is given by:

(2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X_t = \left[ x_t x_{t+\tau } \ldots x_{t + (n-1)\tau }\right] , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is the number of observations to offset successive embeddings (i.e., lag or reconstruction delay) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} is the embedding dimension. The time-delay embedding matrix is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \times n$$\end{document} matrix, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M = N - (n-1)\tau $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} is the number of observations.

Suppose that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_1$$\end{document} is a column in a given document-term matrix representing the time series (of the frequency) of the word gun, from time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = 1$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = 10$$\end{document} , so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_1 = \left\{ 5, 6, 7,\ldots , 14 \right\} $$\end{document} . The frequencies of the word gun in this example are way beyond what one finds in textual data, especially from social media platforms, but the goal of the example is to help describe how time-delay embedding works. Transforming the time series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_1$$\end{document} into a time-delay embedding matrix with five embedding dimensions and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau = 1$$\end{document} generates the following matrix:

(3) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {X}^{(5)} = \begin{bmatrix} 5 &{} 6 &{} 7 &{} 8 &{} 9 \\ 6 &{} 7 &{} 8 &{} 9 &{} 10 \\ 7 &{} 8 &{} 9 &{} 10 &{} 11\\ 8 &{} 9 &{} 10 &{} 11 &{} 12\\ 9 &{} 10 &{} 11 &{} 12 &{} 13\\ 10 &{} 11 &{} 12 &{} 13 &{} 14\\ \end{bmatrix} \quad \end{aligned}$$\end{document}

Once every time series of word frequency (columns) of the document-term matrix is transformed into a time-delay embedding matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {X}^{(n)}$$\end{document} , derivatives can be estimated using generalized local linear approximation (Boker et al., Reference Boker, Deboek, Edler, Keel, Chow, Ferrer and Hsieh2010; Deboeck et al., Reference Deboeck, Montpetit, Bergeman and Boker2009).

GLLA is a technique that can be used to estimate how a variable (e.g., the frequency of the word gun—word count per time point) changes as a function of time. The instantaneous change in one variable with respect to another variable is known as a derivative. The derivative can represent different aspects of change. The first derivative of a word’s time series, for example, estimates the rate of change of the word or the velocity at which the word’s frequency is changing over time. A negative first-order derivative indicates that a word is being used less often as a function of time, while a positive first-order derivative indicates that a word is being used more often as a function of time. The second derivative indicates the speed of the rate of change or the speed of how quickly a word’s frequency is changing (i.e., acceleration). A positive second-order derivative indicates an “acceleration” in the rate of change of a word’s frequency, while a negative second-order derivative indicates a deceleration in the rate of change of a word’s frequency.

In the GLLA framework (Boker et al., Reference Boker, Deboek, Edler, Keel, Chow, Ferrer and Hsieh2010; Deboeck et al., Reference Deboeck, Montpetit, Bergeman and Boker2009), the derivatives are estimated as:

(4) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {Y = \mathbf {X}L(L^{\prime }L)^{-1}}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Y}$$\end{document} is a matrix of derivative estimates, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {X}$$\end{document} is a time-delay embedding matrix (with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} embedding dimensions; to simplify the notation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {X}=\mathbf {X}^{(n)}$$\end{document} ), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {L}$$\end{document} is a matrix with the weights expressing the relationship between the embedding matrix and the derivative estimates. The weight matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {L}$$\end{document} is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times \alpha $$\end{document} matrix, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} is the number of embedding dimensions and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is the (maximum) order of the derivative. Each column of the weight matrix is estimated as follows, considering the order of the derivatives going from zero to k, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = [0, 1,\ldots , k]$$\end{document} :

(5) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {L}_{\alpha } = \frac{[\Delta _{t}(v-\bar{v})]^{\alpha }}{\alpha !}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{t}$$\end{document} is the time between successive observations in the time series, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document} is a vector from one to the number of embedded dimensions (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v = [1, 2,\ldots n]$$\end{document} ), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{v}$$\end{document} is the mean of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is the order of the derivative of interest, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha !$$\end{document} is the factorial of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} .

Continuing our example in which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_1$$\end{document} is a time series of the word gun’s frequency from time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = 1$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = 10$$\end{document} , we consider a time-delay embedding matrix with five dimensions, derivatives up to the second order (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = [0, 1, 2]$$\end{document} ), and a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{t}$$\end{document} of one. This weight matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {L}$$\end{document} is:

(6) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {L} = \begin{bmatrix} \frac{[1(v-\bar{v})]^{0}}{0!}, \frac{[1(v-\bar{v})]^{1}}{1!}, \frac{[1(v-\bar{v})]^{2}}{2!} \end{bmatrix} \quad = \begin{bmatrix} 1 &{} -2 &{} 2.0 \\ 1 &{} -1 &{} 0.5 \\ 1 &{} 0 &{} 0.0 \\ 1 &{} 1 &{} 0.5 \\ 1 &{} 2 &{} 2.0 \\ \end{bmatrix} \quad \end{aligned}$$\end{document}

Applying Eq. 4 to estimate the derivatives, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Y}$$\end{document} is:

(7) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {Y} = \begin{bmatrix} 6.5 &{} 1 &{} 0 \\ 7.5 &{} 1 &{} 0 \\ 8.5 &{} 1 &{} 0 \\ 9.5 &{} 1 &{} 0 \\ 10.5 &{} 1 &{} 0 \\ 11.5 &{} 1 &{} 0 \\ 12.5 &{} 1 &{} 0 \\ \end{bmatrix} \quad \end{aligned}$$\end{document}

where the first, second, and third columns represent the zeroth (observed values), first (rate of change or velocity), and second derivatives (speed of the rate of change or acceleration), respectively.

The process described above is repeated for each variable (e.g., each time series of word counts), and then, the resulting derivatives can be column bound to form a matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {D}$$\end{document} for each individual (e.g., each Twitter account). For instance, suppose text data from a Twitter handle ID1 is being analyzed and the final document-term matrix has only two words (president and america) and only a few observations. Using GLLA to compute the zeroth, first, and second derivatives per word and column biding them to form a matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {D}_{ ID01 }$$\end{document} would result in the following matrix (Table 1).

Table 1

Matrix D for individual ID1

Using GLLA preserves both linear and nonlinear dynamics for each individual and allows different levels of analysis to be implemented. If the goal is to investigate the population structure, then the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {D}$$\end{document} matrices can be stacked and EGA can be used to estimate the number of underlying dimensions using data from all individuals. If the data contain multiple groups of individuals (e.g., right- and left-leaning trolls), then the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {D}$$\end{document} matrices can be stacked by group, and EGA is applied separately in each resulting stacked matrix, generating one dimensionality estimation per group. Finally, if the goal of the analysis is in the intraindividual structure, then EGA can be used in each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {D}$$\end{document} . The result, in this case, will be a structure for each individual, separately. Irrespective of the level of analysis (population, group, or individual), the resulting clusters in the network correspond to variables (words) that are changing together. The main difference between EGA (Golino et al., Reference Golino, Shi, Garrido, Christensen, Nieto, Sadana and Martinez-Molina2020b) and DynEGA is that while the former uses the raw data to construct the network, the latter uses the n-order derivatives estimated via GLLA. This is, in summary, the general idea behind our Dynamic Exploratory Graph Analysis (DynEGA) approach.

One might argue that other network techniques, such as the graphical VAR network model (Epskamp et al., Reference Epskamp, Waldorp, Mõttus and Borsboom2018b), could potentially be used to model the same phenomena as the DynEGA technique. Standard graphical VAR statistical methods create both a contemporaneous network of relationships between observed time series and a temporal network of discrete linear mappings from variables at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} to the same set of variables at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t + L$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document} is some time lag. Nodes in these models represent variables at any given time, and the edges in these models represent either partial correlations (undirected) or average expected change between each variable at time lag \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document} (directed). What is important to note is that “change” in these models is represented solely in the estimation of the directed edges. The undirected edges do not contain information about dynamics. If, instead, nodes were created to represent how variables are changing over time (i.e., the derivatives of these variables), then the undirected edges in the contemporaneous network represent the average dynamics of change between variables and create a continuous map between each variable. Interpretations of edges between dynamic nodes in a network then represent consistent patterns of change between nodes, as opposed to value.

Undirected networks consisting of patterns of change versus value are not representing redundant information. Differences between edges in these two models may be stark. Consider two functions over time: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(t)=t^{(-1)}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(t)=t^2$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [1,2,\ldots ,20]$$\end{document} . The Pearson correlation between these two functions on this region is approximately \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-.56$$\end{document} . However, if we take the derivative of these functions, we get \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\prime (t)}=-t^{(-2)}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g^{\prime (t)}=2t$$\end{document} . The Pearson correlation between these two functions on this region is approximately \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$.52$$\end{document} , which is a change in sign from correlation of the original equations. More complex functional relationships (such as those observed in real data) may yield more differences. It is then highly likely that clustering networks based upon nodes representing change may yield valuable information above and beyond that of clustering networks based upon nodes representing values. Therefore, clustering methods based on undirected edges between raw variables (i.e., EGA) versus dynamic ones (dynEGA) can therefore lead to different findings since they are modeling different phenomena. It is important to emphasize that in the DynEGA technique, clusters of the dynamic nodes represent variables that are changing together.

The use of GLLA to estimate the derivatives in the DynEGA technique brings some challenges, especially in terms of the time-delay embedding assumption of equal time intervals between samples, and the need to set a hyperparameter in GLLA (e.g., the number of embedded dimensions) before computing the derivatives. Boker et al. (Reference Boker, Tiberio and Moulder2018) investigated the impact of sampling interval misspecification in time-delay embeddings and discovered that there was no significant difference between no correction and a sophisticated full information maximum likelihood method. On the other side, setting an optimal number of embedded dimensions should impact more directly the capacity of DynEGA to estimate the topic structure since this hyperparameter governs the size of the window (i.e., number of time points) that will be used to compute the derivatives in each row of the time-delay matrix. As a consequence, this hyperparameter should be tuned using a metric of quality or fit of a multidimensional partitioning. Recently, Golino et al. (Reference Golino, Moulder, Shi, Christensen, Garrido, Neto and Boker2020a) proposed a set of new fit metrics based on information theory especially designed for dimensionality assessment. In their simulation study, one of the metrics that presented the best performance (over more traditional metrics like CFI and RMSEA) was the total entropy fit index (TEFI), which assesses the degree of uncertainty of the partition of a multidimensional space into separate distinct categories (i.e., latent topics). Lower TEFI values indicate that a given dimensionality structure fits the data better than an alternative dimensionality solution with higher TEFI values, indicating that the former is more likely to represent the best organization of the variables. The TEFI index is calculated as follows:

(8) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} TEFI_{VN} = \left[ \frac{\sum _{i=1}^{N_{F}}{\mathcal {S}(\varvec{\rho }_{i})}}{N_{F}} - \mathcal {S}(\varvec{\rho })\right] + \left[ \left( \mathcal {S}(\varvec{\rho })-\sum _{i=1}^{N_{F}}{\mathcal {S}(\varvec{\rho }_{i})}\right) \times \sqrt{N_{F}}\right] , \end{aligned} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{F}$$\end{document} is the number of topics estimated by the Walktrap algorithm, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}(\varvec{\rho }_{i})$$\end{document} is the Von Neumann entropy for each individual topic, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}(\varvec{\rho })$$\end{document} is the total entropy of the system of variables. Golino et al. (Reference Golino, Moulder, Shi, Christensen, Garrido, Neto and Boker2020a) showed that the Von Neumann entropy can be approximately estimated in a correlation matrix by scaling it so that the trace of the matrix equals one (i.e., taking a correlation matrix and dividing all entries by the number of columns of the matrix). After scaling the correlation matrix to a so-called density matrix, an entropy-like metric can be obtained by the negative of the trace of the product of the density matrix by the log of elements of the density matrix (see Golino et al., Reference Golino, Moulder, Shi, Christensen, Garrido, Neto and Boker2020a). The TEFI index can be used to tune the number of embedded dimensions in the DynEGA technique, improving the overall accuracy of our technique to estimate the number of simulated topics. The impact of tuning the number of embedded dimensions using TEFI in the capacity of DynEGA to identify the number of topics is explored in our simulation study presented in the next few sections.

3.1. Extracting Latent Topic Trends with Topic Scores

3.1.1. Network Loadings

In these topics, certain terms contribute more information to topical trends than others. Measures to quantify the contribution of information at the nodal level (i.e., term-level) are called centrality measures. Centrality measures quantify the relative position of terms based on their connections to other nodes in the network. One of the most commonly used measures is called node strength, which corresponds to the (absolute) sum of a term’s connections in the network. In a series of simulation studies, Hallquist et al. (Reference Hallquist, Wright and Molenaar2019) demonstrated that node strength was roughly redundant with confirmatory factor analysis (CFA) loadings. They found, however, that a node’s strength represents a combination of dominant and cross-factor (cross-topic) loadings.

Considering this limitation of node strength, a more recent simulation study evaluated node strength when it’s split by dimensions (or topics) in the network (Christensen & Golino, Reference Christensen and Golino2021). In this study, Christensen and Golino (Reference Christensen and Golino2021) mathematically defined a measure called network loadings by splitting a node’s strength based on its connections within and between dimensions identified by EGA. This measure was then standardized to derive an equivalent measure to factor loadings. Their simulation study demonstrated that network loadings can effectively estimate the population (or true) loadings and are roughly equivalent with exploratory factor analysis (EFA) loadings. They also identified effect sizes that corresponded to 0.15 (small), 0.25 (moderate), and 0.35 (large). Notably, like CFA loadings, network loadings had zeros in the loading matrix from nodes in the network that were not connected. This places network loadings on a middle ground between a “saturated” (EFA) and simple structure (CFA). Below, we provide mathematical notation for how network loadings are computed. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {W}$$\end{document} represent a symmetric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \times m$$\end{document} weight matrix where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m$$\end{document} is the number of terms. Node strength is then defined as:

(9) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S_i = \sum _{j = 1}^n |\mathbf {W}_{ij}| \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mathbf {W}_{ij}|$$\end{document} is the absolute weight (e.g., partial correlation) between node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i$$\end{document} is the sum of the edge weights connected to node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document} across all nodes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} (i.e., node strength for node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document} ). Using the definition of node strength (9), we can define node strength split between the latent topics identified by DynEGA:

(10) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \ell _{ic} = \sum _{j \in c}^C |w_{ij}| \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{ic}$$\end{document} is the sum of edge weights in topic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document} that are connected to node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document} (i.e., node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document} ’s loading for topic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document} ), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C$$\end{document} is the number of topics (in the network). This measure can be standardized using the following formula:

(11) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \aleph _{ic} = \frac{\ell _{ic}}{\sqrt{\sum \ell _c}}, \end{aligned}$$\end{document}

Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{\sum \ell _c}$$\end{document} is equal to the square root of the sum of all the weights for nodes in topic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document} . As pointed by Christensen and Golino (Reference Christensen and Golino2021), the standardized network loadings ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph $$\end{document} ) are absolute weights with the signs being added after the loadings have been computed (following the same procedure as factor loadings; Comrey & Lee, Reference Comrey and Lee2016). In contrast to factor loadings, the network loadings are computed after the number of factors have been extracted from the network’s structure. Variables are assigned to specific factors via a community detection algorithm rather than the traditional factor analytic standard of their largest loading in the loading matrix. Therefore, some nodes may not have any connections to nodes in other topics in the network, leading some variables to have zeros for some topics in the network loading matrix. It is important to emphasize that network loadings are in the unit of association—that is, if the network consists of partial correlations, then the standardized network loadings are the partial correlation between each node and dimension.

3.1.2. Network Scores

Importantly, these network loadings form the foundation for computing network scores, which can be used to extract information about the topics in the network. Because the network loadings represent the middle ground between a “saturated” (EFA) and simple (CFA) structure, the network scores accommodate the inclusion of only the most important cross-loadings in their computation. This capitalizes on information often lost in typical CFA structures but reduces the cross-loadings of EFA structures to only the most important loadings. Below we detail the mathematical notation for computing network scores.

First, we take each topic and identify variables that do not have loadings on that topic equal to zero and divide them by the standard deviation of the corresponding variables in the data, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} :

(12) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varrho _{c} = \frac{\aleph _c}{\sqrt{\frac{\sum _{i=1}^{t \in c} (X_i - \bar{X})^2}{n - 1}}}, \end{aligned}$$\end{document}

where the denominator, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{\frac{\sum _{i=1}^{t \in c} (X_i - \bar{X_t})^2}{n - 1}}$$\end{document} , corresponds to the standard deviation of the variables (words) with nonzero network loadings in topic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varrho _{c}$$\end{document} is the weight for the nonzero loadings in topic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document} . These weights are then made into relative weights (or proportions) and multiplied by the corresponding variables in the data, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{i \in c}$$\end{document} , to obtain the topic score:

(13) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{\theta _c} = \sum \limits ^C_{c} X_{i \in c} \Bigg (\frac{\varrho _{t \in c}}{\sum ^C_c \varrho _{i \in c}}\Bigg ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta _c}$$\end{document} is the network score for topic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document} .

It is interesting to point that one of the first researchers to discover the equivalence between network models and factor models was Guttman (Reference Guttman1953). Although network models were not yet a specific area of research in psychology, Guttman (Reference Guttman1953) proposed a new psychometric technique termed “image structural analysis”, which is basically a network model with node-wise estimation (using multiple regression). Guttman demonstrated how his new psychometric technique relates to factor models and pointed that the former is a special case of the node-wise network model where the errors of the variables are orthogonalizedFootnote 1. Therefore, it should be expected that the network scores we define here should be directly related to traditional factor scores.

4. Checking the Feasibility of Dynamic Exploratory Graph Analysis to Estimate the Number of Underlying Factors/topics

Now that the DynEGA technique was described in detail, an important question must be addressed: how accurately does DynEGA estimate the number of latent topics (or latent factors). This section will briefly address this question.

A plausible underlying mechanism of latent topics can be represented as a direct autoregressive factor score model (DAFS; Engle & Watson, Reference Engle and Watson1981), which is characterized by the autoregressive structure of the latent dimensions (Nesselroade et al., Reference Nesselroade, McArdle, Aggen and Meyers2002). Since our paper focuses on modeling text data, we will adjust the nomenclature accordingly. In the DAFS framework, the observed variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {u}_{t}$$\end{document} at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t = 1, 2,\ldots , N)$$\end{document} are given by:

(14) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {u}_{t} = \varvec{\Lambda }\mathbf {f}_{t}+\mathbf {e}_{t} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Lambda }$$\end{document} is the topic (factor) loading matrix (a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times q$$\end{document} matrix), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {f}_{t}$$\end{document} is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \times 1$$\end{document} vector of topics at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {e}_{t}$$\end{document} is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times 1$$\end{document} vector with measurement errors following a multivariate normal distribution with mean zeros and covariance matrix Q (Nesselroade et al., Reference Nesselroade, McArdle, Aggen and Meyers2002; Zhang et al., Reference Zhang, Hamaker and Nesselroade2008).

The topic scores, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {f}_{t}$$\end{document} , are given by:

(15) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {f}_{t} = \sum _{l = 1}^{L}{\mathbf {B}_{l}\mathbf {f}_{t-l}+\mathbf {v}_{t}} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {B}_{l}$$\end{document} is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \times q$$\end{document} matrix of autoregressive and cross-regressive coefficients, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {f}_{t-l}$$\end{document} is a vector of topic score \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l$$\end{document} occasions prior to occasion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {v}_{t}$$\end{document} is a random shock vector (or innovation vector) following a multivariate normal distribution with mean zeros and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \times q$$\end{document} covariance matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }$$\end{document} (Nesselroade et al., Reference Nesselroade, McArdle, Aggen and Meyers2002; Zhang et al., Reference Zhang, Hamaker and Nesselroade2008). In the DAFS model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Lambda }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {B}_{l}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Q}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }$$\end{document} are invariant over time.

Data following the DAFS model can be simulated using the simDFM function of the EGAnet package (Golino & Christensen, Reference Golino and Christensen2019). Below, we present a brief simulation investigating how accurately DynEGA can recover the number of simulated topics. We also investigate the distribution of variables per topic and the correlation between the simulated and the estimated topic scores. Accuracy can be calculated as follows:

(16) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ACC = \frac{\Sigma _C}{N} \text{ for } C = \Bigg \{ \begin{array}{rcl} 1 \text{ if } \hat{\theta } = \theta \\ 0 \text{ if } \hat{\theta } \ne \theta \end{array} \Bigg \} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta }$$\end{document} is the estimated number of latent topics, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is the true number of latent topics used to simulate the data (i.e. ground truth), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} is the number of sample data simulated.

The distribution of the variables per topic can be checked using normalized mutual information (NMI; Horibe, Reference Horibe1985). NMI is used to compare the similarity between two vectors (of discrete variables) and assigns a value of zero where the two vectors are totally dissimilar, and a value of one where they are identical in an information theoretic perspective. For example, consider two vectors ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_2$$\end{document} ) representing the partition of a multidimensional space into two groups, so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1 = (1,1,1,1,1,2,2,2,2,2)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_2 = (2,2,2,2,2,1,1,1,1,1)$$\end{document} . The NMI of the two vectors equals one, since both vectors are presenting the same information (i.e., that the multidimensional space is grouped in two dimensions composed by the first five elements and the last five elements, respectively). If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_2$$\end{document} were to be compared to a third vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_3 = (1,1,1,2,2,2,2,3,3,3)$$\end{document} , the NMI between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_3$$\end{document} equals 0.38. Clearly, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_2$$\end{document} exhibits the same partitioning of the multidimensional space, different from the partitioning of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_3$$\end{document} .

The two vectors used to compute NMI are the vector of the assigned variables per topic—that generates the block diagonal matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Lambda }$$\end{document} Eq. 14—and the vector containing the estimated topic number per variable.

4.4. Results

4.4.1. Continuous Data

In the continuous data condition, the mean accuracy of DynEGA using GLASSO ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC_{glasso} =$$\end{document} 93.17%) and TMFG ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC_{TMFG} =$$\end{document} 95.78%) was very similar, as were the mean normalized mutual information ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NMI_{glasso} =$$\end{document} 96.62%, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NMI_{TMFG} =$$\end{document} 95.38%). Figures 1 and 2 show that the mean accuracy and normalized mutual information increase with the magnitude of the loadings, but decrease with the increase in the measurement error. It is interesting to note that although the TMFG network method is more accurate (see Fig. 1), the GLASSO approach gives the higher normalized mutual information, suggesting that the latter method more accurately allocates the variables into the correct latent topics. Both figures also show that the mean accuracy and normalized mutual information decrease with the increase in the measurement error.

Figure. 1

Mean accuracy per network method used in the DynEGA technique, magnitude of the loadings (x-axis), number of time points (vertical facets), and magnitude of measurement error (horizontal facets)

Figure. 2

Mean normalized mutual information per network method used in the DynEGA technique, magnitude of the loadings (x-axis), number of time points (vertical facets), and magnitude of measurement error (horizontal facets)

Figure. 3

Mean correlation between simulated and estimated topic scores per network method used in the DynEGA technique, magnitude of the loadings (x-axis), number of time points (vertical facets), and magnitude of measurement error (horizontal facets)

Figure 3 shows the mean correlation between simulated and estimated topic scores per network method used in the DynEGA technique, magnitude of the loadings, number of time points and magnitude of the measurement error. The DynEGA technique with the GLASSO network method presented a higher mean correlation (GLASSO) than the TMFG network method (TMFG). The differences are higher for factor loadings of 0.40 (see Fig. 3).

4.4.2. Ordered Categorical Data

In the ordered categorical data condition, the scenario is slightly different. The mean accuracy of DynEGA using the TMFG network method ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC_{TMFG} = 85.70\%$$\end{document} ) was higher than the mean accuracy of GLASSO ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC_{glasso} = 73.43\%$$\end{document} ). As with the continuous data type, DynEGA with the GLASSO network method presented a higher mean normalized mutual information than DynEGA with TMFG ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NMI_{glasso} = 88.20\%$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NMI_{TMFG} = 85.67\%$$\end{document} ).

The strategies used to select the optimal number of topics via LDA all presented a very low accuracy. The Arun algorithm (Arun et al., Reference Arun, Suresh, Veni Madhavan and Narasimha Murthy2010) had an accuracy of zero since it always selected the maximum number of topics compared (six) as the optimal number of topics. AIC and BIC presented a mean accuracy of 14.62% and 16.84%, respectively. Cao’s (Cao et al., Reference Cao, Xia, Li, Zhang and Tang2009) and Deveaud’s algorithm (Deveaud et al., Reference Deveaud, SanJuan and Bellot2014) presented the best mean accuracy: 28.60% and 31.24%, respectively. Figure 3 shows the mean accuracy per method in the ordered categorical data condition, per loadings (x-axis), number of time points (vertical facets), and magnitude of measurement error (horizontal facets). The DynEGA technique with the TMFG network method had the highest accuracy, followed by DynEGA with the GLASSO, and the Deveaud’s algorithm and Cao’s algorithm for LDA.

Figure. 4

Mean accuracy per method in the ordered categorical data condition. Magnitude of the loadings (x-axis), number of time points (vertical facets), and magnitude of measurement error (horizontal facets)

Figure 5 shows that the DynEGA with the GLASSO network method presented a slightly higher mean normalized mutual information than the DynEGA with the TMFG network method, specially with loadings of 0.40, 0.55 and 0.70.

Figure. 5

Mean normalized mutual information per network method used in the DynEGA technique, magnitude of the loadings (x-axis), number of time points (vertical facets), and magnitude of measurement error (horizontal facets)

Figure. 6

Mean correlation between simulated and estimated topic scores for the ordered categorical data condition per network method used in the DynEGA technique, magnitude of the loadings (x-axis), number of time points (vertical facets), and magnitude of measurement error (horizontal facets)

Figure 6 shows the mean correlation between simulated and estimated topic scores per network method used in the DynEGA technique, magnitude of the loadings, number of time points and magnitude of the measurement error. The DynEGA technique with the GLASSO network method presented a higher mean correlation between the simulated and estimated topic scores (GLASSO) than the TMFG network method (TMFG). The differences are higher for loadings of 0.40 (see Fig. 6).

4.4.3. Varying the number of embedded dimensions

As pointed before, the number of embedded dimensions used to compute the derivatives in the GLLA technique can impact the accuracy of DynEGA in the estimation of the number of simulated topics. We used a subset of the conditions implemented in the simulation above to investigate how setting the number of embedded dimensions to 3, 5, 7 and 9 impacted the NMI and accuracy of DynEGA in data simulated with 100 time points (length of the time series), 5 variables per topic, measurement error covariance matrix with diagonals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.15^2$$\end{document} , only three topics, and topic (factor) loadings with the following levels: .40, .55, .70, and 1.

Figure. 7

Mean accuracy per number of embedded dimensions, method, magnitude of the loadings (x-axis), number of embedded dimensions (vertical facets), and data type (horizontal facets)

Figure 7 shows the mean accuracy for each DynEGA network method, magnitude of the loadings, number of embedded dimensions and data type. It is interesting to note that the mean accuracy increases with the number of embedded dimensions, with an overall accuracy of 95.90% for three embedded dimensions, 99.10% for five, 99.78% for seven, and 99.90% for nine embedded dimensions. The impact of varying the number of embedded dimensions is higher for ordered categorical data, and for moderate and low factor/topic loadings. For example, the mean accuracy of DynEGA with the GLASSO network technique when the factor/topic loadings is 0.40 goes from 59.02% with three embedded dimensions to 91% with nine embedded dimensions for the ordered categorical data.

Figure. 8

Mean normalized mutual information per number of embedded dimensions, method, magnitude of the loadings (x-axis), number of embedded dimensions (vertical facets), and data type (horizontal facets)

A similar pattern happens with the normalized mutual information. Figure 8 shows the mean normalized mutual information for each DynEGA network method, magnitude of the loadings, number of embedded dimensions and data type. The mean NMI increases with the number of embedded dimensions, with an overall accuracy of 95.9% for three embedded dimensions, 99.1% for five, 99.78% for seven, and 99.9% for nine embedded dimensions. The impact of varying the number of embedded dimensions is higher for ordered categorical data, and for moderate and low factor/topic loadings. For example, the mean accuracy of DynEGA with the GLASSO network technique when the factor/topic loadings is 0.40 goes from 73.22% with three embedded dimensions to 94.40% with nine embedded dimensions for the ordered categorical data.

Using the total entropy fit index (TEFI) to tune the number of embedded dimensions allows the researcher to set this hyperparameter of the DynEGA technique in an objective way. The TEFI is a relative measure of fit, whose value is lower for structures that present a better fit to the data (Golino et al., Reference Golino, Moulder, Shi, Christensen, Garrido, Neto and Boker2020a). When the TEFI index is computed for the topic structures estimated using DynEGA with three, five, seven, and nine embedded dimensions, a decision can be made in terms of which number of embedded dimensions leads to the structure that best fits the data.

Figure. 9

Mean difference in accuracy and normalized mutual information between the structure with the lowest TEFI value and the maximum TEFI value per magnitude of the loadings (x-axis), network construction method (colors), and data type (horizontal facets)

Figure 9 shows the difference in accuracy and normalized mutual information between the structure with the lowest TEFI value and the structure with the maximum TEFI value per magnitude of the loadings (x-axis), network construction method (colors), and data type (horizontal facets). Therefore, the y-axis of Fig. 9 represents gain in accuracy and NMI obtained by tuning the number of embedded dimensions using the TEFI index, which is larger for the ordered categorical data than for the continuous data. A trend worth noting is that tuning the number of embedded dimensions leads to higher gains in conditions with more error due to lower topic (factor) loadings. In terms of gains in accuracy, DynEGA with the GLASSO network method benefited more than the TMFG network method, gaining 13.20% in the continuous data condition and 39.29% in the ordered categorical data condition for loadings of .40. The gains in accuracy and in NMI fade away with the increase in the topic (factor) loading since with higher loadings DynEGA is, in general, very accurate to recover the number and composition (i.e., correct placement of words per topic) of the simulated topic structure. Here, DynEGA with TMFG benefited more in terms of NMI than DynEGA with GLASSO.

5. Applying the Dynamic Exploratory Graph Analysis in the IRA-linked Twitter Data

The substantive problem this paper addresses is related to the use of social media, especially Twitter, by agencies or groups of people devoted to exploit social divisions in a society for political reasons. People are increasingly using online social platforms to communicate their ideas, and politicians and government bodies are taking advantage of these platforms to leverage their interaction with voters and citizens. Social media platforms are becoming more and more central in debates about relevant issues such as abortion, gun control, and other controversial topics (Rajadesingan & Liu, Reference Rajadesingan and Liu2014), sometimes leading to strongly polarized positions (Yardi & Boyd, Reference Yardi and Boyd2010).

In the USA, online activities were used to influence public opinion and voters during the 2016 campaign, with attacks occurring before and during the electoral process (Linvill & Warren, Reference Linvill and Warren2018). Twitter accounts linked to the Internet Research Agency (IRA), based in Russia, were used to try to divide voters in a wide range of issues (Linvill & Warren, Reference Linvill and Warren2018), and their information was released by the US congress after an investigation to discover Russian state-sponsored trolls (Zannettou, Caulfield, De Cristofaro, et al., Reference Zannettou, Caulfield, De Cristofaro, Sirivianos, Stringhini and Blackburn2019a). In this section, the DynEGA technique is applied to a large database of IRA-linked Twitter accounts extracted by Linvill and Warren (Reference Linvill and Warren2018) and posted online by the FiveThirtyEight team (Roeder, Reference Roeder2018). The goal is to investigate the strategies used by right- and left-wing Twitter accounts.

The original data contain almost 3 million Twitter posts, by 2843 unique accounts, starting in January 2013 to May 2018. Linvill and Warren (Reference Linvill and Warren2018) classified the accounts into five groups: right troll, left troll, news feed, hashtag gamer, and fearmonger. The first two types of accounts mimic right- or left-leaning people. The news feed accounts present themselves as local news aggregators, the hashtag gamer accounts specialize in playing hashtag games, and the fearmonger accounts spread news about a fake crisis (Roeder, Reference Roeder2018). In the current analysis, only accounts identified as right- and left-leaning trolls were used, with posts (not including retweets) from January 2016 to January 2017. Accounts with less than 50 posts were excluded, resulting in 276,752 posts (78.32% from left-leaning trolls) and 236 accounts. The 236 accounts included in the analysis can be considered influential, since they have a nonnegligible number of followers (median = 877, mean = 4838), with 12.37% having more than 10,000 followers. The high number of followers can help the trolls in pushing specific narratives to a much greater number of Twitter users (Zannettou et al., Reference Zannettou, Caulfield, De Cristofaro, Sirivianos, Stringhini and Blackburn2019a).

An omnibus analysis was implemented using data from both types of trolls (i.e., right and left-leaning trolls) that were pre-processed using the tm package from R (Feinerer et al., Reference Feinerer, Hornik and Meyer2008). URLs were removed from the text data as well as punctuation, numbers, and stop words. All characters were converted to lowercase, and the words were stemmed (i.e., reduced to their stem, base, or root using Porter’s algorithm; Porter, Reference Porter1980). The sparsity of the resulting document term matrix (i.e., a data frame where the columns are unique words and rows are different documents or posts) was decreased using a threshold of .9935, so that words with a sparsity above the threshold are removed from the document term matrix, resulting in 170 unique words for the right and 113 words for the left trolls. In text mining, deciding the number of words to use is done arbitrarily. In the current analysis, we decided to use between 100 and 200 words, so the final document-term matrix would have enough words to capture different types of topics, but not very specific topics (or niche content), which could happen if the number of words used increases (since the number of words used depends on the sparsity threshold, using more words means using words that are less frequently used).

To estimate the topics per account type, the DynEGA technique was used via the dynEGA function from the EGAnet package. Since the TMFG network method presented the highest accuracy in the estimation of the number of topics in the ordered categorical data, it was the network estimation method used in the current analysis. The optimum number of embedded dimensions was tuned using the procedure presented in our simulation study: different numbers of embedded dimensions were used (5, 10, 15, and 20), and the fit of the resulting topic structure estimated by DynEGA was verified using the total entropy fit index. The number of embedded dimensions associated with the lowest TEFI value was selected and referred to as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _n$$\end{document} , being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} the TEFI index and n the number of embedded dimensions. The TEFI values per number of embedded dimensions were as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(5)} = -108.00$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(10)} = -113.38$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(15)} = -107.03$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(20)} = -109.48$$\end{document} , and therefore, the number of embedded dimensions was set to 10. The other dynEGA arguments were set as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} (time-lag) was set to one and the time between successive observations for each time series was one (delta). The correlation between the first-order derivatives was computed using Pearson’s coefficient and was used as the input to estimate the network. The level of analysis used in this section was set to population, meaning that the derivatives are estimated per Twitter account and then row binded, creating a long data frame that is used to compute the correlation matrix. The estimated topics are, then, the “mean” structure of population.

Figure. 10

Network structure estimated using DynEGA of the right trolls document-term matrix showing eight topics (clusters). The nodes represent the words and the edges are the Pearson correlation of the words’ first-order derivatives

Figure 10 shows the network structure of the left- and right-leaning trolls Twitter data (170 words). The nodes represent the words and the edges are the Pearson correlation of the words’ first-order derivatives. Eleven topics were estimated. The first topic (topic A in Fig. 10) was one of the least interpretable topics, with words such as can’t, stop, believe. The second topic (topic B in Fig. 10) makes reference to money and Donald Trump’s campaign slogan (Make America Great Again or MAGA). The third topic (topic C in Fig. 10) makes reference to the Black Lives Matter movement, racism, and racial justice. The fourth topic (topic D in Fig. 10) is a small collection of slang words and cursing, and the fifth topic (topic E in Fig. 10) seems to demonstrate support for Trump running for president, make references to former President Obama and former Secretary of State Hillary Clinton (also running for President in 2016), and to the media. The sixth topic (topic F in Fig. 10) emphasizes the Black Twitter activities, while the seventh topic (topic G in Fig. 10) seems to invite people to follow users and/or topics (example of words: follow, thanks, now, twitter). The eighth topic (topic H in Fig. 10) has words as listen, video, show, now playing, watch, making references to live streaming of music and video content. The ninth topic (topic I in Fig. 10) makes reference to terrorism (using expressions as terrorism and the hashtag islamkill), refugees, the conservative movement in America (e.g., the abbreviations PJNET, CCOT and TCOT are referencing the Patriot Journalism Network, conservatives on Twitter, and Top Conservatives on Twitter) and the wake up America movement (a call for conservative principles endorsed by Donald Trump and others). The tenth topic (topic J in Fig. 10) makes reference to reports of police brutality and the eleventh topic (topic K in Fig. 10) talks about taking back the country, also a slogan used by pro-Trump supporters.

Figure. 11

Latent trends of the topics per date

To understand how each topic is expressed over time, the topic scores are estimated using Eq. 13. Figure 11 shows the mean topic score per date by type of trolls (left- and right-leaning trolls). Topics C (black lives matter), F (black twitter), H (references to music, videos and shows) and J (police brutality) are used more often by the left-leaning trolls, while topics E (supporting Trump for president), I (terrorism, refugees, conservatives) and K (take back the country) are more usually employed by the right-leaning trolls. Together, these topics (C, E, F, H, I, J and K) represent content more related to the political agenda of pro-Trump movements or more liberal, anti-racism groups in the USA. Some trends are interesting to mention. For example, topic E (supporting Trump for president) peaked three times, once in October 10th (the day after the second presidential debate, in which major news agencies started transcribing and analyzing all the controversial rhetoric of Donald Trump), once in October 20th (date of Trump’s rally in Delaware, in which he said that he would accept the results of the elections if he was the candidate elected) and once in November 2nd, 2016 (date in which he had multiple rallies in Florida for thousands of people). Topic I (Terrorism, Refugees, Conservatives) presented peaks on March 22nd (day of the bombings at Brussels airport and a metro station that killed 32 people) and in three days of February (02/10, 02/13, and 02/18), dates in which reports about the immigration crisis in Europe surfaced and were reported by major news outlet. Specifically on February 18, 2016, the Department of Homeland Security announced further travel restrictions for travelers that visited Libya, Somalia, and Yemen. The examples for topics E and I are presented in Fig. 12.

Figure. 12

Latent trends of topic E (supporting Trump for President) and topic I (Terrorism, Refugees, Conservatives)

The estimated topic scores can be used to estimate a second-order topic structure (similar to second-order factors), in which first-order topics can be grouped into second-order broad themes. To do that, DynEGA is used as before (i.e., tuning the number of embedded dimensions using the TEFI index), but instead of using the raw document-term matrix to estimate the first-order derivatives, the estimated topic scores can be used instead. Six values for the embedded dimensions were used (5, 10, 15, 20, 25, and 30), and the resulting TEFI values are: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(5)} = -5.02$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(10)} = -3.53$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(15)} = -3.59$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(20)} = -5.66$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(25)} = -5.63$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{(30)} = -5.63$$\end{document} . Therefore, the number of embedded dimensions was set to 20 (minimum TEFI value). The other dynEGA arguments were set as before, with a time-lag ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} ) and time between successive observations for each time series was both set to one (delta). Again, the correlation between the first-order derivatives was computed using Pearson’s coefficient and was used as the input to estimate the network, and the level of analysis was set to population (meaning that the derivatives are estimated per Twitter account and then row binded, creating a long data frame that is used to compute the correlation matrix). The resulting network of the first-order topics and the estimated second-order themes can be seen in Fig. 13. Three broad themes were estimated. One is a combination of political and social issues topics (C, E, F, H, I, J and K), while the second is a combination of general topics (B and K) and random ones (A and D). The third general theme is composed by topics inviting people to follow an account or twitter discussion (topic G) and mentioning music, videos, shows, and live streaming content (topic H). It is worth noting that within theme covering political and social issues content, the topics that are more common in the left-leaning trolls (topics C, F, and J) present a positive association among themselves and a negative association with topics E and I (more common in the right-leaning trolls). This negative association might be an indication that there is some underlying principle guiding the posts from left- and right-leaning trolls on Twitter.

Figure. 13

Network structure estimated using DynEGA of the first-order topics. The nodes represent the network scores for each topic, and the edges are the Pearson correlation of scores’ first-order derivatives

To check what type of trolls is leading (or forcing/driving) the posts of political/social issues themes on Twitter, we used convergent cross-mapping (Clark et al., Reference Clark, Ye, Isbell, Deyle, Cowles, Tilman and Sugihara2015; Sugihara et al., Reference Sugihara, May, Ye, Hsieh, Deyle, Fogarty and Munch2012)—a technique used to identify the coupling of time-series. If none types of trolls are driving the other, then convergent cross-mapping will lead to a nonsignificant “causal” signal, computed using the bootstrapped 95% confidence intervals for the estimated Pearson correlation coefficient of estimates of the time series of one group (time-series A) from the time series of the another group (time-series B) for each number observations tested (for more details, see Clark et al., Reference Clark, Ye, Isbell, Deyle, Cowles, Tilman and Sugihara2015). The convergent cross-mapping estimated a nonsignificant coupling in which the left trolls drive the right trolls ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p = 0.4254$$\end{document} ), and a small, but nonsignificant, p for the right-leaning trolls driving the left-leaning trolls ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p = 0.05$$\end{document} ). The small p value for the convergent cross-mapping in which the right-leaning trolls are driving the left-leaning trolls instigated us to explore the data in a more granular way, using the network scores from the first-order topics instead of the scores of the second-order theme with the political/social issues content. Since topics C (Black Lives Matter/Racial Justice) and I (Terrorism, Refugees and Conservatives) were among the two most frequent topics for the left- and the right-leaning trolls, respectively, we choose to focus on these two time series (i.e., the mean network score per date of topic C for the left and topic I for the right-leaning trolls). The convergent cross-mapping showed a nonsignificant coupling in which the topic C (Black Lives Matter/Racial Justice) for the left-leaning trolls drives topic I for the right-leaning trolls ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p = 0.074$$\end{document} ), but a significant coupling in which topic I for the right-leaning trolls drives or forces topic C for the left-leaning trolls ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p = 0.014$$\end{document} ). Convergent cross-mapping did not detect a significant coupling in neither direction for topics C and E (Supporting Trump for president).