Expression categories

We hypothesize that around the time of the 2020 U.S. presidential election, a new sub-identity emerged within the Make America Great Again (MAGA) movement. This sub-identity centered on beliefs in election fraud and frequently advocated extra‑institutional action; these discussions temporally coincided with the period leading up to the January 6, an event not necessarily endorsed by the broader MAGA movement.

We aim to investigate the development of this sub-identity, its dynamics over time, and the conditions that contributed to its emergence and influence within the larger movement. To do so, we employ an empirical approach that resembles that of Smith and coauthors [2,40]. Specifically, we focus on the social media posts that are relevant to the formation of this identity, classifying them via computer regular expressions into several categories of expression.

1. Grievance talk – expression of the relevant grievance (i.e., the voting and election fraud);
2. Action talk – advocacy for action that is meant to correct the grievance (e.g., an impactful gathering of people in protest of the fraud);
3. Efficacy talk – expression of belief in the efficacy of such action (e.g., that such a gathering is likely to overturn the election’s result);
4. Identity talk – expression of belonging to a collective identity that is associated with this grievance and this normative action (e.g., distinctive acronyms such as MAGA).
5. Validation talk – conversation that begins with expression that belongs to one of the above-mentioned four categories – grievance, identity, action, or efficacy – and is later followed by a response containing a validation-related regular expression (see the Supplementary Materials).

In the rest of the article, an italicized category name generally refers to the category itself or the number of posts in the category as identified by the regular expressions, versus a more abstract concept associated with the category. For example, identity can refer to the proportion of posts that we have obtained via this category’s filters. The un-italicized word “identity”, on the other hand, refers to the abstract concept of social identity that is the general topic of this article.

The regular expressions that we use to identify the above-mentioned categories of expression (see Materials and Methods, as well as the Supplementary Text in S1 File) are the first step in our operationalization of the existing social identity theories. In taking this step, we propose that the number of posts that these regular expressions select for each category reflects the collective mental state that corresponds to this category and that the theories seek to capture. We acknowledge that the correspondence between the theories and our operationalization may not be exact. For example, our identity regular expressions track a social identity that existed before the articulation of the relevant grievance and was subsequently co-opted for a sub-identity to mobilize collective action in response to the grievance. This is in contrast with the idea of an entirely new social identity that appears in response to the grievance.

Time series

We analyze ∼60 million posts in our “hate universe” database [41–43] (see also Materials and Methods), between Super Tuesday (March 3, 2020) and the Senate report on the Capitol attacks (June 8, 2021).

On each day between these two dates, we determine the fraction of posts that falls into one of the categories above. These time series are shown in Fig 1. We refer to the period before the election as Period 1, the period between the election and the attack as Period 2, and the period after the attack as Period 3. We scale each variable to have a mean of zero and standard deviation of one across all the time points in a given period (see Materials and Methods).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. Daily proportion of posts around the 2020 election, versus time.

Different colors correspond to different expression categories. The election day is November 3, 2020; it is marked as ‘11-03’. The day of the Capitol attacks is January 6, 2021; it is marked as ‘01-06’.

https://doi.org/10.1371/journal.pcsy.0000107.g001

The existing social identity theories imply dynamic interactions among collective mental states. Our scaled time series are thus a natural second step in operationalizing these theories, since they allow us to track such mental states over time.

Summary of the analysis

For each of the three periods, we perform the following analysis:

• As mentioned above, we scale each variable so that its mean is zero and its variance equals one.
• We determine whether each variable is stationary or trend-stationary via the Augmented Dickey-Fuller (ADF) test.
• We check for collinearity among the variables via the Variance Inflation Factor (VIF).
• We perform the dynamic vector auto-regression (VAR) analysis. The definition of VAR is in Materials and Methods and the VAR analysis implementation details are in the Supplementary Text in S1 File. This analysis consists of the following steps:
  • To produce the stationary versions of all the variables, we remove the trends from the trend-stationary variables and difference action in Period 1, where it is neither stationary nor trend-stationary.
  • We then fit the variables’ stationary versions to VAR models.
  • We choose the lag order for the VAR models according to the models’ information criteria and final prediction errors.
  • We identify the significant VAR model coefficients, which correspond to predictive interactions among the variables.
  • Toward a more detailed understanding of the interactions, we also calculate impulse response functions (IRF).

In addition, we perform a pathway analysis for Period 2, to test for long-term correlations across online communities during this period. See Supplementary Text in S1 File for all details of implementation.

Trends

Table 1 indicates the status of each variable with respect to trends and stationarity during each time period. In the cases where a variable is trend-stationary, this panel specifies whether the trend is increasing or decreasing.

Download:

• PNG
  larger image
• TIFF
  original image

Table 1. The changing stationarity properties in the 2020/2021 dataset.

https://doi.org/10.1371/journal.pcsy.0000107.t001

In order to better visualize the long-term trends in the trend-stationary and non-stationary time series, we smooth these time series by taking averages over 14-day rolling windows (see Fig 2).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. (a) Period 1 (b) Period 2 (c) Period 3 Rolling averages of scaled post proportions for the trend-stationary and non-stationary series in the 2020/2021 dataset.

Each point in a graph is labeled with the date that is at the center of the rolling window over which the average is calculated. Dashed lines indicate the regression models that we find significant; we subtract these models to obtain stationary, de-trended variables.

https://doi.org/10.1371/journal.pcsy.0000107.g002

In Period 1, there is a gradual rise of identity and validation, reflecting a steady growth of shared group identity and mutual reinforcement that prepare the ground for eventual actions. The dynamic of action shows two peaks that reflect specific mobilizing events. We hypothesize that the first peak reflects several events at the end of May 2020, including the George Floyd protests and President Trump’s influential warnings that mail-in voting would lead to election fraud. The second peak could be due to a combination of several events in mid-August, including the Democratic and Republican National Conventions, as well as additional spikes in the controversy around mail-in voting.

In Period 2, the declining trend in grievance might indicate a shift from expressing discontent to acting on it. Once grievance about election fraud is widely recognized, the focus shifts toward actionable responses. The corresponding increase in action could indicate an intensified mobilization leading up to the January 6 attacks. It underscores how accumulating grievance transforms into tangible calls for mobilization. This possible causal interpretation aligns with the results of the pathway analysis (see below).

In Period 3, fast quadratic declines in grievance, efficacy and validation may be due to a combination of disillusionment, fear of consequences, increased scrutiny, and legal repercussions. Identity is strongly correlated with the other variables during this period. In a multiple linear regression model with identity as the dependent variable and the remaining variables as independent, R² = 0.91. This suggests that the social identity has fully consolidated by this period, so that the identity variable is deeply embedded in the collective narrative and is no longer independently fluctuating.

Existing theories of social identity frequently posit a decisive influence of external factors on the development of identities that underlie collective actions. Here, we have split the time series into three periods, in accordance with the key external events that we expect to fundamentally affect the development of our focal identity. This allows us to operationalize the influence of these events by considering the way that they might affect the general time series trends. Additionally, the theories frequently speak of the social identity’s formation. We operationalize this latter idea as an increase in the correlation among our expression categories.

Overall, the time series trends support a coherent narrative where online collective identity and grievance initially fuel calls for action, and the subsequent mobilization efforts culminate dramatically on January 6. Following the latter event, participants either withdraw or entrench themselves more narrowly and cautiously within their communities.

Vector auto-regression

Here, we enter the stationary, non-collinear variants of the scaled variables (or, simply, the stationary variables – see Materials and Methods) into vector auto-regression (VAR) models in order to detect significant day-to-day interactions.

Fig 3 presents the self-effects and cross-effects that we infer from the fitted VAR models, in the form of the regression coefficients that are significant at the 5% level. These effects are the significant predictive interactions among the stationary portions of the scaled variables (and the differenced action in Period 1) on daily timescales. Impulse response functions in S2, S3 and S4 Figs show that the effects propagate over 3–4 day horizons. In the rest of this section, we provide an interpretation of Fig 3 as predictive associations consistent with prior theory.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. (a) Period 1 (b) Period 2 (c) Period 3 Self-effects and cross-effects in the VAR models for the 2020/2021 data.

The effects take the form of coefficients that are significant at the 5% level. The names of the stationary variables are underlined, those of the detrended variables are preceded by tildes, and the name of the differenced variable is preceded by . The self-lag coefficients are inside circles, the cross-lag coefficients are next to straight arrows. For each coefficient, the lag is one day.

https://doi.org/10.1371/journal.pcsy.0000107.g003

The self-lag coefficients (self-lags) of the stationary and detrended variables are comparable in magnitude and positive, except for differenced action in Period 1. Positive coefficients reflect each variable’s daily persistence or inertia. The negative coefficient for differenced action indicates that spikes in discussions about actions are followed by declines in discussion intensity. During Period 3, action exhibits the highest self-lag among all variables across periods, indicating a strong stabilization of this type of expression by this time – as well as stabilization of identity, which is highly correlated with action during this period (r = 0.92, also see the corresponding variance inflation factors and regression results in Materials and Methods).

We interpret the cross-lag coefficients (cross-lags, with values in parentheses) during Period 1 as follows. Grievance expression reliably predicts validation (0.14), meaning that airing grievances anticipates group affirmation, reinforcing their legitimacy. Validation promotes identity formation (0.15), as affirmation from peers solidifies collective identity. Expressing common identity strongly bolsters efficacy (0.32), increasing belief in the success of collective action. Efficacy maintains grievance expression (0.11), suggesting that higher efficacy sustains the airing of grievances.

Thus, Period 1 reveals a clear developmental sequence: grievance → validation → identity → efficacy → back to grievance, confirming psychological theories of social identity formation. This loop-like structure supports theories where group validation and identity formation reinforce each other, creating stable dynamics [30,40].

Our interpretation of the Period 2 cross-lags is as follows. Social identity strongly shapes beliefs about group efficacy (0.34). Efficacy expression increases grievance expression (0.15), suggesting that confidence in group power sustains the relevance of grievance. Efficacy also boosts validation (0.21), indicating that belief in group efficacy encourages mutual affirmation among members.

In sum, during this critical rallying phase, the collective identity strongly shapes collective confidence (i.e., efficacy beliefs), which then reinforces both grievance expression and validation. This sequence closely matches the social identity model of collective action [30], demonstrating that identity-driven efficacy is central to sustaining group cohesion and grievance expression. Action is not predicted by the other variables.

In Period 3, much like in Period 1, expression of grievance reliably prompts validation (0.22), meaning mutual affirmation remains important even after the mobilization peak. Increased efficacy in Period 3 reduces explicit calls for immediate action (−0.11), suggesting that confidence in group power may lower action urgency – potentially indicating a consolidation phase where efficacy beliefs persist without the accompanying calls to action.

Identity is not present explicitly in Period 3 due to its extremely high collinearity with the other variables (i.e., it is highly predictable by the other variables via linear regression; see above).

Overall, Period 3 marks a consolidation of the social identity: a strong, stable internal persistence across the variables, sustained validation of the grievance, and a notable reduction in immediate calls for action with increased efficacy. Identity, fully consolidated, no longer varies independently. This supports the idea of social identity becoming deeply internalized and self-reinforcing, which is consistent with the long-term stabilization that is predicted by theory [40,44].

On the whole, the VAR results reveal a predictive sequence that supports the following developmental trajectory for the sub-identity unified by shared beliefs in election fraud. This sub-identity emerges within the MAGA movement through a robust cycle from grievance through validation and identity expression toward efficacy. The identity then mobilizes its followers during Period 2, shaping the collective efficacy belief, as well as sustaining the expressions of grievance and validation. During this period, high daily self-lag coefficients for all five expression categories suggest strong collective readiness; for example, the amount of grievance expression on any given day strongly predicts the amount of this expression on the following day. Finally, the identity consolidates in Period 3, with the identity variable fully predicted by the other variables, grievance expression that continues to drive validation, and a robust equilibrium indicated by strong internal self-lags. Increased efficacy now reduces immediate calls for action, suggesting a post-event settling.

These findings align with those of Smith, van Zomeren and coauthors [30,40]. In particular, the sequence of grievance → validation → identity → efficacy → action is explicitly confirmed, consistent with the canonical developmental pathways in the literature. Here, mutual validation emerges explicitly as a critical factor stabilizing collective identity and grievance expression, aligning with the emphasis on validation in the work of Smith and her colleagues [40].

Furthermore, the inverse relationship between efficacy and explicit action calls in Period 3 provides an intriguing extension to the current theories: increased confidence in collective efficacy may reduce the immediate need or urgency for collective mobilization after major collective events, reflecting a psychological shift toward identity consolidation rather than continuing action.

Pathway analysis

As shown above, statistical analysis of the daily time scales via vector auto-regression reveals the predictive pathway grievance → validation → identity → efficacy on this time scale during Period 1. This pathway is a close match to the theory due to Smith and coauthors [40]. On longer time scales, we have identified increasing trends in validation and identity in Period 1, as well as decreasing trends in grievance, efficacy and validation in Period 3. We hypothesize that these trends are due to exogenous factors such as the impending election in Period 1 and the receding of the attacks into the past during Period 3. Additionally, according to our analysis of collinearity (see Materials and Methods), all five variables are strongly collinear in Period 3. Overall, our analysis of Periods 1 and 3 suggests the absence of long-term predictive interactions among the variables during these two periods.

The variables’ behavior is markedly different during Period 2. In particular, this period exhibits a decreasing trend in grievance and an increasing trend in action. In combination with the fact that this is the period between the factual or alleged realization of the grievance and the occurrence of the corrective action, these trends suggest the possibility that they are not entirely due to exogenous factors and that there is a causative relationship between them. Such a suggestion gains additional support when we examine the middle panel of S1 Fig and notice that, apart from the synchronous peak in all variables on December 12th, the temporal sequence of the highest points in four of the variables, grievance → validation → identity → action, closely follows the above-mentioned theoretical pathway from the literature.

In accordance with this qualitative observation, we conduct a pathway analysis, akin to that in a recent study by Smith and colleagues [3]. In particular, we sample across communities in the social media universe, similarly to the way Smith and coauthors [3] sample across individual social media users. We also distinguish among four non-intersecting time intervals in Period 2. Time 1 is an interval around the grievance peak at 11/06. Time 2 – an interval around the validation plateau between 11/08 and 11/10. Time 3 is an interval around the identity peak at 11/15. Time 4 is between 12/17 and 01/04, when action is prominently higher than the other variables.

For each community, we determine the number of grievance posts at time 1, the number of validation posts at time 2, the number of identity posts at time 3, and the number of action posts at time 4. The pathway analysis is a series of multiple regressions across the communities, where all the variables that precede a dependent variable in time are included as independent. In other words, we regress validation on grievance, identity on grievance and validation, and action on the other three. We assume that the residuals are normal, in accordance with common practice. To make sure our results do not depend on the exact boundaries between the four time intervals, we run the analysis with five different sets of intervals and compare the results (see Supplementary Text in S1 File). Fig 4 presents a summary of all the results; this figure is equivalent to Fig  2 in the article by Smith and colleagues [3]. Here, the arrows between different variables are pre-determined by the analysis design, whereas the significances, the signs and the magnitudes of the corresponding coefficients are due to the data. The coefficients that are predicted by Smith and coauthors [40] are positive and largely significant, offering support for their theory on the time scale of the entire second period. In addition to the predicted effects, we find a direct effect of grievance talk on identity expression and a direct effect of validation on calls to action.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. Results of the pathway analysis for Period 2 communities in the 2020/2021 data.

Five different sets of boundaries between the time intervals were used. The coefficients that are predicted by the psychological theory are in green. N.s.: not significant (P ≥ 0.05), one star: P < 0.05, two stars: P < 0.01, three stars: P < 0.001. For example, the text above the validation → identity arrow means that 4 out of 5 boundary sets produce significant coefficients, with a magnitude range between 0.20 and 0.24 and P < 0.05. The excluded community, 4Chan1, has thousands to tens of thousands of posts in each category.

https://doi.org/10.1371/journal.pcsy.0000107.g004

Social identity theories are based on the idea of logical connections between the collective mental states that reinforce each other with time. The above VAR and pathway analyses operationalize this idea in our time series, by seeing whether the expression in one of the categories at a given time reliably predicts expression in another category at a later time. In accordance with the above results, the signal that we detect generally aligns well with the existing theories, although the results also suggest theoretical refinements and additions.

2024 elections

We now apply our analytic methodology to the “hate universe” data around the 2024 presidential election, then compare and contrast our results between 2020 and 2024.

Time series.

The 2024 election occurred on November 5, 2024 (i.e., 11/05/24). To track the development of the focal social identity around this event, we analyze ∼30 million posts between 02/01/24 and 03/01/25. Fig 5 shows the frequencies of posts in different categories during this period of time. Here, the regular expressions for each expression category are the same as in our main analysis. Due to our dataset’s technical limitations (see Materials and Methods), we split the pre-election period in two, thus conducting our analysis separately on three periods: Period 1a, which is before the first grayed out interval in Fig 5; Period 1b, which is after this interval and before the election; and Period 2, which is after the election and includes the second grayed out interval. As in our analysis of the 2020 elections data, we scale the time series so that each has zero mean and unit variance in any given period. The scaled time series are shown in S6 Fig.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. Daily proportions of posts versus time, around the 2024 presidential election.

Different colors correspond to different expression categories. The election day is November 5, 2024; it is marked as ‘11-05’. The grayed out intervals are between August 17 and September 20 of 2024 and between December 11 and December 14 of 2024; during these intervals, only partial data is available due to instabilities in the 4Chan social media platform, which provides the majority of the data. Period 1a is up to the beginning of the first grayed out interval, Period 1b begins with the interval’s end and concludes with the election, Period 2 starts with the election and includes the second grayed out interval.

https://doi.org/10.1371/journal.pcsy.0000107.g005

We verified using an LLM that our regular expression filters capture very similar types of grievance and action between 2020 and 2024 (see Supplementary Text in S1 File). Apparently, for example, the grievance with regard to election fraud and the illegitimacy of the Biden administration remain topical and the calls for violent collective action remain popular in 2024. Nevertheless, as we show further in this section, the interactions among the different expression categories change dramatically between 2020 and 2024.

Trends.

Fig 6 illustrates the observed trends via 14-day rolling averages. In Period 1a, there is a statistically significant linear increase in identity (P < 0.05 for the ADF stationarity test with linear trend), accompanied by a quadratic increase in grievance, although the latter is significant only at the P = 0.1 level. A number of major events external to the social media discussion could be the causes of several local peaks in these variables. In particular, identity has a considerable peak around Donald Trump’s conviction in The People of the State of New York v. Donald J. Trump on May 30. Additionally, both identity and grievance show large peaks between the Biden-Trump debate on June 27 and Joe Biden’s withdrawal from the electoral race on July 21. The remaining variables are stationary during this period.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 6. (a) Period 1a (b) Period 1b (c) Period 2 Rolling averages of scaled post proportions for the trend-stationary and non-stationary series in the 2024/2025 dataset.

Stationarity is determined according to the Augmented Dickey-Fuller (ADF) test and, for the most part, the P < 0.05 significance criterion. Each point in a graph is labeled with the date that is at the center of the rolling window over which the average is calculated. Straight dashed lines indicate the linear regression models that we find significant; we subtract these models to obtain stationary, de-trended variables. Curved dashed lines indicate quadratic trends that the ADF test finds significant at the P < 0.10 but not the P < 0.05 level; we do not detrend the corresponding variables. Only partial data is available during the grayed out interval.

https://doi.org/10.1371/journal.pcsy.0000107.g006

During Period 1b, we observe statistically significant linear increases in identity and efficacy (P < 0.05), while quadratic increases in grievance and validation are present but significant only at P = 0.1. Action is stationary during this period. In Period 2, all variables except efficacy become stationary. Notably, around the Inauguration Day on January 20, efficacy’s running average exhibits values that are noticeably higher than during the rest of this period.

VAR.

Due to technical limitations of our data, we do not conduct a VAR analysis of Period 1b (see Materials and Methods). The results of the Period 1a and Period 2 VAR analyses are in Fig 7.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 7. (a) Period 1a (b) Period 2 VAR analysis of the 2024 data.

Self-effects and cross-effects inferred from the fitted models, in the form of the coefficients that are significant at the 5% level. Names of the stationary variables are underlined, those of the detrended variables are preceded by tildes, those of the differenced variables are preceded by . Self-lag coefficients are inside circles, cross-lag coefficients are next to straight arrows. For each coefficient, the lag is one day.

https://doi.org/10.1371/journal.pcsy.0000107.g007

During Period 1a, action (self-lag coefficient equal to 0.59), validation (0.51), and identity (0.49) exhibit strong persistence, indicating stable daily continuity within these categories. Differenced grievance has a negative self-effect (-0.24), indicating that expressed grievances on a given day likely decline on the next day, possibly due to a fluctuating discussion focus.

During Period 2, level grievance (0.70) is rather self-persistent, suggesting a day-to-day discussion of the focal grievance that is more consistent than in Period 1a. The self-lag coefficients for identity (0.58) and validation (0.47) are just as high during the post-election Period 2 as they are during the pre-election Period 1a, indicating that the election and its results do not have an effect on these variables’ daily continuity. Differenced efficacy shows strongly negative persistence (-0.42) during Period 2, indicating considerable volatility or fluctuations in collective efficacy beliefs post-election.

The negative effect of identity on grievance (cross-lag coefficient equal to -0.14) in Period 1a suggests that expression of belonging to the collective identity reduces the inclination to express grievance – a marked deviation from 2020, where this kind of grievance expression is foundational to identity formation. However, validation still drives action (0.15), maintaining a limited but important role in promoting the social identity’s normative action. Overall, unlike the 2020 pre-election period, no strong predictive pathways from grievance to action through identity and validation emerge. This suggests that activity related to the focal identity was driven primarily by internal persistence in 2024, rather than by external grievances, as in 2020.

In Period 2, identity significantly reinforces validation (0.19), highlighting a collective identity that fosters mutual affirmation. Increased action in this period slightly reduces grievance (-0.12), perhaps suggesting that once the need for mobilization is articulated, grievances may seem redundant or less immediate. Expressing these grievances strongly decreases collective efficacy (-0.27), potentially indicating that these grievances are no longer associated with the normative action, the way they are in 2020 and 2021.

To elaborate further, the periods immediately following the presidential elections show marked differences between 2020 and 2024. While the focal identity and the validation process still exist in 2024, calls to action no longer clearly stem from efficacy beliefs or validation processes, and the negative relationship between grievance and efficacy during this election cycle suggests a reduced group coherence. The peak in efficacy around the Inauguration Day suggests an increase in collective optimism or perceived group strength, likely influenced by the inauguration event itself.

2020 versus 2024

We split each election cycle into event-defined periods. For 2020, we analyze the pre-election period (Period 1), the post-election period through January 6 (Period 2), and the post-January 6 period (Period 3). For 2024, we analyze two pre-election sub-periods (Periods 1a and 1b) and a post-election period (Period 2). We find both similarities and differences between the cycles.

In the case of the 2020 cycle, the results are broadly consistent with the canonical models of collective action, in which grievance predicts validation, validation predicts identity, and identity predicts efficacy and action [2,30,31,40]. This canonical sequence is particularly salient during Periods 1 and 2; the combined evidence for all three periods suggests a progressive alignment among the theory components that culminates in a coordinated discourse around action by January 6.

The results for the 2024 cycle do not reliably support this sequence. Instead of a mutual reinforcement among the expression types, these are often weakly or negatively related. Grievance can be associated with weaker efficacy, action discourse can predict decreases in grievance, and identity predicts validation without consistently predicting efficacy beliefs or action talk. Rather than exhibiting a progressive alignment, the 2024 cycle presents a more fragmented configuration, suggesting that the mobilizing role of a focal grievance is context-dependent and may weaken when it is not widely validated or is inconsistent with the currently relevant political outcomes.

Operationalized theory components

Below, we connect our results to the established theories of collective action, focusing on the Social Identity Model of Collective Action (SIMCA) [30], the normative alignment model [31], and the identity-norm nexus (INN) [2,40]. We organize this discussion around the five theoretical components that we operationalize – grievance, validation, identity, action, and efficacy – and emphasize how the coupling that we observe between these components differs between 2020/2021 and 2024/2025.

Grievance. Many social-psychological theories of collective action treat the perceptions of grievance and injustice as foundational. SIMCA frames the injustice appraisals as one of the three antecedents to mobilization, and related accounts emphasize the pivotal role of the perceived gaps between the actual societal conditions and those that would be considered equitable [30,45]. Our 2020 results are consistent with this functional importance of grievance perception. Grievance expression regarding alleged voter fraud is relatively stable before the election and spikes immediately after the election. Across the three periods, VAR and regression analyses show grievance predicting increases in validation and identity expression. Here, the fraud claims appear sufficiently plausible within the studied communities to elicit agreement and to support the build-up of an associated in-group narrative.

In 2024, the grievance component behaves differently. Grievance expression does not consistently lead to validation or identity talk, and in the post-election period it can negatively predict efficacy while action discourse negatively predicts grievance. In other words, grievance is sometimes associated with a weakening, rather than a strengthening, of other mobilizing components. A plausible interpretation is contextual: in 2024, the Republican candidate both appears viable pre-election and wins the election. This likely reduces the perceived plausibility of fraud claims, relative to 2020, and weakens the capacity of grievance to generate validation and identity. More generally, these results suggest that grievance is not a uniform driver; its mobilizing potential depends on whether it is aligned with the perceived reality.

Validation. Validation corresponds to affirmation and agreement, and potentially leads to the emergence of shared norms through interaction. Accounts of inductive identity formation emphasize that consensus-building can be central to the emergence of a shared identity in discussion [28,29,46,47]. In our 2020 results, validation plays a bridging role. Validation is tightly linked to grievance and identity, and pathway analysis shows validation predicting identity and action in the period between the election and January 6. In this period, validation appears to function as the hinge by which grievance claims are transformed into a shared identity and coordinated action discourse.

In 2024, validation’s role shifts. In one pre-election sub-period, validation predicts action, suggesting that agreement can still energize mobilization. However, in the post-election period, validation is predicted by identity rather than by grievance, indicating a reversal of the expected sequence. In this configuration, validation may reflect the already established in-group positions, instead of producing a consensus around a newly salient injustice. This pattern reinforces the centrality of validation in interactive models, while showing that the direction of coupling can vary with context.

Identity. Social Identity Theory and Self-Categorization Theory describe identity as a basis for collective behavior [24,25]. The INN account further emphasizes a co-emergence of identity and action norms through discussion [2,40]. Our 2020 results support these interactive accounts. Identity expression increases across the pre-election period, is predicted by validation, and predicts efficacy beliefs and action talk. During the post-election period leading to January 6, identity is linked to action talk in pathway analysis, consistent with identity as a proximate motivator of collective action. In this cycle, identity is both a product of validation and a driver of efficacy and action discourse, consistent with a convergence on a corrective action norm under a perceived threat.

In 2024, identity remains active but functions differently. In the first pre-election sub-period, identity negatively predicts grievance, suggesting a decoupling from the perceptions of injustice. In the post-election period, identity predicts validation but not action. Thus, identity discourse can persist without aligning grievances and action. A contextual interpretation is again plausible: if the fraud claims are less credible, the perception of threat may be insufficient to produce a strong identity growth around the focal grievance or to translate identity into action norms. These results highlight a variability in identity’s role: identity can bridge grievance and action in some contexts (2020/2021), but in others (2024/2025) it may not produce a mobilizing alignment.

Action. Action discourse refers to the expressions of what should be done, ranging from calls to protest to demands for institutional change. Normative alignment theory emphasizes that action emerges when grievance, identity, and efficacy are aligned within a shared normative frame [30,31]. In 2020/2021, action discourse rises steadily between the election and January 6, and pathway analysis shows validation and identity predicting action in the run-up to the attack. These patterns support the view that action talk crystallizes when grievances are validated and linked to identity.

After January 6, we observe that efficacy can negatively predict action in VAR. One interpretation is that an increase in the perceived feasibility of collective action such as the Capitol attack due to the success of the attack, in combination with the attack’s observed consequences (including deaths and legal risk), reduces the subsequent advocacy for similar collective action, shifting the discourse away from mobilization.

In 2024/2025, action discourse is less connected to other expressions. In one pre-election sub-period, validation predicts action, but in the post-election period action negatively predicts grievance. Thus, rather than being a culmination of consensus around a focal injustice, action talk can sometimes undermine or displace grievance expression. This challenges a simple view of action as an endpoint of alignment and suggests that action discourse can operate independently of, or even in tension with, the central grievance narrative.

Efficacy. In SIMCA and related theories, efficacy beliefs mediate the relationship between identity and action [30,31]. Our 2020/2021 results show efficacy embedded in a broader dynamic system: identity predicts efficacy, efficacy is linked to grievance and validation, and efficacy can negatively predict action after January 6. These patterns suggest that efficacy is not a stand-alone belief but a discourse component shaped by identity that, in turn, shapes how grievances and action are discussed. The post-January 6 negative relationship between efficacy and action may reflect a decreased mobilization advocacy after a major collective event, consistent with the above-mentioned learning about both the feasibility and the costs of such events.

In 2024/2025, efficacy is both more fragile and less connected to other expressions. In the post-election period, grievance negatively predicts efficacy, suggesting that injustice expression can weaken the beliefs in collective capacity rather than strengthening them. This again underscores context sensitivity: grievances may mobilize only when they are validated and aligned with the identity and action norms; otherwise they may erode the collective confidence or reflect a resignation rather than the need for mobilization.

Confirmation, extension and refinement of theory

Taken together, the 2020/2021 results confirm several interactive accounts of collective action. Across multiple analyses, grievance predicts validation, validation predicts identity, and identity forecasts efficacy and action, consistent with SIMCA, the normative alignment theory, and the INN model [2,30,31,40]. In this cycle, validation appears central for translating grievance into shared identity and action norms, and efficacy serves as an important intermediary.

By contrast, the 2024/2025 results refine these accounts by showing that the canonical sequence can fail. The absence of consistent grievance → validation → identity → action coupling, alongside several negative relationships, suggests fragmentation rather than alignment. A practical implication is that models of mobilization may need to treat grievance validation and identity-action alignment as contingent on the context and the event structure rather than as the default. In particular, when a focal grievance is not credible or socially reinforced, the same discourse components may coexist without producing a shared pathway toward action.

The contrast between the two cycles also highlights the interplay of top-down events and bottom-up interaction. In 2020/2021, top-down events (the election outcome and January 6) coincide with sharp shifts in grievance expression and with bottom-up reinforcement via validation and identity discourse, consistent with accounts in which elite cues and salient events interact with grassroots consensus-building [1,38]. In 2024/2025, comparable top-down triggers that would reinforce grievance and align it with identity and action are weaker or absent, and bottom-up reinforcement is correspondingly weaker, producing a more fragmented system.

Methodologically, this study demonstrates that stationarity testing, VAR, and pathway analysis can be used to operationalize theory components in large-scale communication data and to evaluate theory-linked temporal predictions in naturalistic settings. In the same vein as prior digital-trace work on protest participation [34], we map theoretical constructs onto discourse, providing a replicable framework for testing and refining psychological theory with observational data.

Both theoretical work and empirical studies suggest that radicalization and mobilization can exhibit cascade-like dynamics and self-organized criticality (SOC) [13,18,48–50]. Our analysis does not show any signatures of SOC. In particular, we do not see any clear cascade-like precursors immediately before January 6, and the sharp post-election grievance increase is consistent with an external shock rather than a self-reinforcing communication cascade. Nevertheless, the alignment dynamics we observe in 2020/2021 may be foundational for the radicalization and offline mobilization toward the Capitol attack, particularly if additional processes (including higher-frequency cascades, network diffusion, or threshold dynamics) are present in this setting. A dedicated SOC analysis would require measures tailored to cascade processes (e.g., cascade-size distributions, branching-style reproduction metrics, or critical-transition diagnostics) and finer temporal resolution than our daily aggregates.

Qualitative features of the “hate universe” data, alongside our formal analysis, connect our results to broader accounts of online activism while also highlighting important differences between these accounts and our work. Firstly, our data suggest that salient events (e.g., election fraud claims or high-profile court cases) drive several of the largest discourse spikes, consistent with the idea that topic interest can be a strong predictor of participation [51]. Secondly, users frequently frame moderation or fact-checking as an attack on the movement, which can increase posting volume; this echoes broader “backfire” accounts in which perceived repression galvanizes engagement [52,53]. Thirdly, we observe substantial toxic out-group language, which appears to generate in-group engagement while discouraging cross-cutting conversation, consistent with evidence that out-group interactions can be more toxic and less engaging [54]. Fourth, the networks that we study include newcomers who may encounter approachable in-group arguments; this resonates with the work on allyship and legitimacy, where the engagement by advantaged-group allies can shape a movement’s dynamics [55]. Finally, although some literatures emphasize a mix of nonviolent and violent tactics in successful mobilization, discourse in our dataset often favors violent tactics and downplays peaceful alternatives [56]. Together, these comparisons suggest that right-wing mobilization around U.S. elections exhibits distinctive rhetorical cues and strategic adaptations that merit further study.

Consistent with recent work, conspiracy-oriented networks can validate grievances and circulate violent planning cues [35], and mobilization to violence can be associated with discourse about efficacy and action logistics [36,37]. Our results complement these studies by showing that, in 2020/2021, grievance validation and identity-action alignment are jointly present in the run-up to January 6, whereas in 2024/2025 similar components can persist without producing the same developmental pathway.

Study limitations

Our results may depend on the specific nature of the “hate universe” dataset and on the collective action context that we study. In particular, the combination of a selection for extremist content and our particular platform choice may bias both the prevalence estimates and the inferred pathways. Accordingly, any generalization should proceed with care: the results speak most directly to the discourse dynamics within the communities and time windows that we study, not necessarily to social media populations at large.

The analytic approach has additional limitations. First, the analyses are observational and assess predictability rather than causal effects; causal hypotheses require confirmation through experiments or further observational designs. Second, the measurement error from regular-expression filters can generate both false positives and false negatives. We estimate that the false positives have a limited impact on the qualitative conclusions if they do not introduce an appreciable bias, but time-varying error or category-specific error can bias the regression coefficients. Sarcasm in validation expression and retrospective mentions of past action in the supposed calls for future action can make such inaccuracies difficult to detect. Third, statistical testing in VAR raises multiple-comparisons concerns. For example, with P < 0.05 and five variables at lag order one, there are 20 coefficients and roughly one false positive expected by chance; future work may adopt stricter thresholds or correction procedures. Finally, our use of community distinctions is limited to a subset of analyses and does not incorporate network structure or platform-level diffusion.

Our statistical choices should not be seen as fixed. We keep the methods relatively simple because the overall analytic framework is novel, but several refinements to the statistical methodology may be natural in future work. One could handle collinearity via regression regularization; future research could analyze non-stationary series with co-integration tests and error-correction terms rather than differencing alone; and Bayesian VAR could stabilize the regression coefficient estimates, incorporate prior knowledge, and provide lag selection that is more robust. Likewise, rather than incorporating the major events via the splitting of the time series into periods, future models could incorporate exogenous time-varying covariates that represent the event effects, potentially allowing analysis over longer continuous time windows. Additional robustness checks are also important: repeating the pathway analyses under alternative normalizations (e.g., proportions rather than raw counts), varying the lag orders in VAR, and performing sensitivity analyses that inject plausible levels of measurement noise into each series would help quantify the dependence of specific regression coefficients on the modeling and preprocessing choices. Future analyses may also adopt multiple-testing adjustments (e.g., false discovery rate control) when interpreting sets of VAR coefficients. Finally, finer temporal sampling (e.g., hours rather than days) would allow for a better detection of short-lived surges, a better resolution of the responses to exogenous events, and a more sensitive identification of self-reinforcing cascades.

Future work can improve the robustness and external validity of our framework by applying it to other movements and to non-hateful speech, by incorporating additional data modalities (e.g., images and memes), and by using a finer temporal sampling to resolve faster dynamics. Accurate NLP classifiers (e.g., ones that measure emotions) and richer representations of platform and community structure (e.g., ones that include network ties, exogenous event covariates, or diffusion measures) can reduce the measurement error and better capture the mechanisms that link discourse to mobilization.

Ethics statement

The George Washington University Committee on Human Research determined that our study is exempt from Institutional Review Board (IRB) review. Our data consist only of anonymous, publicly available, community-level information.

“Hate Universe”

Our analysis is based on the data from the “hate universe” – a collection of communities on several social media platforms and these communities’ posts over an extended period of time. We build this dataset using the same methodology as prior publications [42,43,57,58], expanded to additional platforms. The platforms for the 2020–2021 dataset are Rumble, YouTube, Gab, 4Chan, Bitchute, VKontakte, Facebook, Telegram, Instagram, and Twitter. The platforms for the 2024–2025 dataset are the same, with the exception of Twitter and Instagram. Previously, we have utilized the 2020–2021 dataset to explore the changes in the hate universe’s network topology through the 2020 presidential election and the subsequent Capitol attacks [43].

The meaning of a “community” is platform-specific. For example, on the Russian state-controlled platform VKontakte, a community is called a Club. On Facebook, a community is called a Page; on Telegram, a Channel; on Gab, a Group. Each community contains anywhere from a few to a few million users and is unrelated to “community” as used in network community detection.

To identify hateful communities for inclusion in our data, subject matter experts (SMEs) examined several thousands of candidate communities on each platform. SMEs checked each community’s 20 most recent posts at some point during the year 2020. If at least 2 of these posts included hate speech, we included that community in our “hate universe” dataset. The SMEs were generally in very high agreement. Here, by “hate speech”, we refer to negative or hateful comments directed at a group or a class of persons on the basis of race, religion, skin color, sexual identity, gender identity, ethnicity, disability, or national origin [57,59].

We treated each community identified as a hate community as remaining so over time. Spot checks indicate that very few such communities stop posting hate speech in the long term, although their content is not always hateful.

Expression categories

We do not tailor the expression categories to our empirical context or our data set. Instead, we derive them from the existing theories of social identity formation. This ensures that the testing of said theories in our dataset is meaningful and does not yield a positive result via circular reasoning. For instance, a sense of shared grievance or injustice plays an important role in both SIMCA [30] and INN [2,40]. Mutual validation is a key mechanism that boosts the collective perception of shared grievance and translates it to action norms in both INN and the normative alignment model [31]. Shared efficacy belief plays important mediating roles in both the normative alignment model and SIMCA.

In contrast to the choice of categories, the specific regular expressions that represent them (see the Supplementary Text in S1 File) are rooted in our empirical context. We base these regular expressions on a prior experience with the “hate universe” database [41–43] (see also Materials and Methods), as well as a search through existing publications and other web-based sources.

A specific post can match more than one category of expression – for instance, both the grievance and the proposed action. In this case, we treat the contributions of such a post to the two types of expression as distinct. To put it another way, we assume that, for any given post, the probability of expression in one of the categories is independent of the probability of expression in another category. On the other hand,in designing our regular expression filters, we make an effort to ensure that the categories, as defined by these filters, are distinct – namely, that no two regular expressions from different categories match the same text (see the Supplementary Text in S1 File).

Our regular expressions could be missing some of the posts that fall in the categories that the posts intend to capture. Furthermore, according to a rather conservative, human-based verification procedure (see Supplementary Text in S1 File), about 30% of the posts that the regular expressions detect do not contain statements that fall in the corresponding category. Nevertheless, we posit that our time series generally track the expression categories that we want them to track, albeit with considerable noise. One qualitative piece of evidence in favor of this proposal is that the time series for the five categories appear to be punctuated by the election and the attack. In particular, maxima in the time series roughly correspond to these two events. Another piece of evidence is that the behaviors of the time series are distinctly different between the three time periods that are delimited by the events. The time series also appear to be different between categories, so that there is a possibility of a non-trivial effect of expression in one category on that in another, over time.

Time series

The magnitudes of post counts are substantially different between the different expression categories. For example, the posts with grievance expression are often several times more numerous than the posts that express belonging to the social identity. However, we wish to treat every category on an equal footing with every other. Specifically, we are interested in the way that proportional changes in one type of talk react to the proportional changes in another type of talk, not in the overall amounts of each type of talk. Accordingly, we scale each variable to have a mean of zero and a standard deviation of one across all the time points in a given period. The resulting time series, shown in S1 Fig, still differ qualitatively among the expression categories and among the three time periods. There, we observe that the propagated binomial error is smaller than the daily fluctuations in the scaled variables, even for efficacy, the expression category with the least numerous posts. In the remainder of this article, we work exclusively with the scaled versions of the five main variables.

Stationarity tests

We assess the stationarity of each variable during each time period via the Augmented Dickey-Fuller (ADF) test. This test is based on the following model of a single-valued time series:

(1)

where and p is the lag order. The null hypothesis for the test is non-stationarity, . The alternative hypothesis is stationarity, .

We implement the ADF test via the Python function adfuller in package statsmodels, with the function’s default option autolag = ‘AIC’. We set the regression option to ‘c’, ‘ct’, or ‘ctt’; these option values correspond to degrees 0, 1 and 2 of the time polynomial in the above equation. Degree d = 0 corresponds to , d = 1 corresponds to , and d = 2 – to a lack of any constraints on , or . Table 2 presents the P-values for the tests.

Download:

• PNG
  larger image
• TIFF
  original image

Table 2. Significance of the ADF test for stationarity in the 2020/2021 dataset.

https://doi.org/10.1371/journal.pcsy.0000107.t002

Whenever the ADF test is significant with d = 0, we say that the series is stationary. When it is not significant with d = 0 yet significant with either d = 1 or d = 2, we say that the series is trend-stationary. In this latter case, we “detrend” the variable, via the following procedure. We choose the smallest value of d where the series is trend-stationary. Subsequently, we build a regression of the focal variable on time with the appropriate degree – that is, a linear regression if d = 1 or a quadratic regression if d = 2. We fit the regression via the Python command statsmodels.api.OLS, with default options. We then subtract the regression equation from the variable, leaving only the residuals. These residuals constitute the detrended variable, which is stationary under d = 0 for all our originally trend-stationary variables. In the one case where no yields a stationary or a trend-stationary time series, we say that the variable is non-stationary. In this case, we difference the variable by substituting y_t with ; the latter is, once again, stationary under d = 0. The resulting detrended and differenced variables subsequently enter into the vector auto-regression (VAR) model, alongside the variables that are stationary without detrending or differencing.

Collinearity measure

We assess collinearity by the means of the variance inflation factor (VIF). This factor is equal to , where R² is the coefficient of determination for the linear regression of a focal variable on all the others. When VIF is above 5, this suggests that the focal variable is somewhat easy to predict via a linear regression from all the others. When VIF is above 10, the focal variable is very easy to predict from the others. High VIF indicates that the focal variable should be excluded in a dynamic model such as the vector auto-regression, where the goal is to establish temporal effects among independent components of the collective social identity. Specifically, when the lagged values of all five variables at some time are used to predict one of the variables at a later time in the context of a VAR model, collinearity makes it unclear which of the lagged variables contributes to the model’s prediction. To put it another way, there can be simultaneous changes to several of the corresponding coefficients that do not change the prediction. This can contribute to an uncertainty in the estimation of the coefficients.

To calculate the Variance Inflation Factor (VIF), we use the Python function variance_inflation_factor in package statsmodels.stats.outliers_influence. The VIF values for the original variables are in the upper panel of Table 3. In the lower panel of this table, the variables that are not stationary have been transformed via detrending or differencing, as described above. In Period 3, the VIF for the identity variable is particularly high in both panels. Correspondingly, during this period, identity is strongly correlated with the other variables. Quantitatively speaking, for the original versions of the variables, R² = 0.91 in a multiple linear regression model with identity as the dependent variable and the remaining variables as independent. Thus, more than 90% of the variance in identity is explained by the combined variance of the other variables. The regression coefficients are all significant at the 5% significance level. Their values are 0.66 for action, 0.36 for validation, 0.16 for efficacy and −0.14 for grievance. It appears that, in Period 3, the expression of belonging to a common action-focused identity is correlated with the expression of all the associated components except, interestingly, grievance – the component that initiates the identity’s development in the first place.

Download:

• PNG
  larger image
• TIFF
  original image

Table 3. Collinearity in the 2020/2021 dataset.

https://doi.org/10.1371/journal.pcsy.0000107.t003

Accordingly, we remove identity from the analysis in Period 3. Table 3 indicates that upon this removal, the VIF values of all the variables that enter into the vector auto-regression are below 5.

VAR and pathway analysis

Our statistical analysis of day-to-day influences among the different social identity components is based on the vector auto-regression (VAR) model. This model postulates that one can predict each variable’s value at a given time step from its value – and the values of the remaining variables – at previous times. The model’s formal definition is:

(2)

where Y_t is a k-dimensional column vector (or, simply, k-vector) of variables Y at time t. In our case, each component of Y is the scaled proportion of posts that falls in one of the above-mentioned categories. Eq (2) also contains the regression intercepts, which are the elements of the constant k-vector C, as well as the model’s residual errors at t, in the k-vector E_t. Index i is called lag and parameter p is the model’s maximum lag (otherwise known as its lag order). The lag order determines how far back in time one has to look to predict Y in the present. The k × k matrix contains the regression coefficients that determine the predictive interactions among the elements of Y at lag i. For instance, the diagonals of this matrix tell us how much the value of a given variable at time t − i contributes to the value of that same variable at time t, for any t. The VAR model can be seen as a special case of a multiple, multivariate linear regression, in which the design matrix consists of the variables’ lagged values. The model’s traditional form assumes time-invariant, normal error distributions and zero covariance among errors across the different components of Y.

The VAR model seeks to predict the variables’ current and future values from their values in the past. The model assumes that each variable is stationary – that is to say, that the variable’s statistical properties, such as its mean and its standard deviation, do not change with time. If the assumption of stationarity does not hold for at least one of the variables, then a constant matrix might not adequately describe the relationship among the variables at all times.

To fit the VAR model, we utilize the Stata command var.

To determine the optimal lag order of the VAR model for each period, we consider the final prediction error (FPE) and three information criteria versus lag. The information criteria are the Akaike information criterion (AIC), the Hannan-Quinn information criterion (HQIC), and Schwarz’s Bayesian information criterion (SBIC). We compute FPE, AIC, HQIC and SBIC via the Stata command varsoc. S7 Fig plots these criteria versus lag order for each period. Here, HQIC and SBIC both clearly indicate that the optimal lag order is p = 1 for all three periods. AIC and FPE concur with HQIC and SBIC for Period 2; however, AIC and FPE are ambiguous for Period 3 and disagree with HQIC and SBIC for Period 1.

According to the “Remarks and Examples” section of [60], FPE is not strictly speaking, an information criterion. The same source states that the BIC and the HQIC have a theoretical advantage over the AIC and the FPE: choosing the lag order to minimize the BIC and the HQIC provides consistent estimates of the true lag order. In contrast, minimizing the AIC or the FPE will overestimate the true lag order with positive probability, even with an infinite sample size. In accordance with all of the above observations and theoretical considerations, as well as toward consistency in the statistical procedure across the periods, we choose p = 1, as recommended by HQIC and SBIC for all three periods.

The variables that enter into VAR are the stationary variants of all the original, scaled variables. For the variables that are stationary to begin with, the stationary versions are the same as the original scaled variables. For the variables that are trend-stationary, the stationary variants are the stationary portions that remain after detrending. Since action during Period 1 is neither stationary nor trend-stationary, its stationary variant is the original variable’s differenced version. We also remove identity from the VAR analysis in Period 3, where this variable is highly collinear with the others.

In the pathway analysis of Period 2 in 2020/2021, one of the nodes, 4Chan’s “Politically Incorrect” board (hereafter referred to as “4Chan1”), is a clear, extreme outlier in the distributions of the number of posts. Its posts number thousands to tens of thousands for each category, whereas the numbers for other communities are in tens to hundreds. We exclude 4Chan1 from the pathway analysis, since including it would reduce the effective number of observations to one.

The proportions and counts of posts that enter into the VAR and pathway analyses are associated with the above-mentioned false positive error, which we conservatively estimate to be on the order of ∼30%. In the context of regression analyses, this error on individual points – either time points in the case of VAR or community points in the case of pathway analysis – translates to appreciably lower errors on the regression coefficients when the number of points is large. For example, in the case of Period 2 VAR analysis, each coefficient is estimated from n ≈ 60 points and the variance of the predictor, i.e., lagged, variable is . If the standard deviation due to the false positive error is , then the error in the regression slope is approximately . This error is no more than half the magnitudes of the smallest coefficients that we estimate as significant; these latter magnitudes are on the order of 0.1. For the majority of our coefficient estimates, the relative error due to the false positives in the regular expressions should be appreciably lower than this conservative estimate of about half the coefficient’s magnitude. Accordingly, if the statistical properties of the false positives are such that they do not introduce appreciable bias to our qualitative conclusions, we estimate that the potential effect of the false positives is, at most, to invalidate the significance of a few of the lowest-magnitude regression coefficients. We further point out that there is an inherent trade-off between the transparency and the perceived accuracy of natural language processing tools that propose to operationalize abstract psychological concepts in new data. Accordingly, our relatively simple regular expression procedure offers a methodology that is complementary to advanced tools such as large language models.

2024 election

The two grayed out intervals in Fig 5 are marked by instability in 4Chan, the social media platform that accounts for a large percentage of the total dataset. Due to this fact, the data during these intervals is incomplete. Furthermore, the data before the first grayed out interval was collected using a different sampling method than the data after this interval. Accordingly, we do not include the first grayed out interval in our analysis.

For the Period 1b VAR model, the data-to-parameter ratio is relatively low – only about 7.3 when the lag order p is 1; this ratio is even lower for higher lag orders. Accordingly, there is considerable uncertainty with regard to the optimal lag order, as evident in the middle panel of S8 Fig. Additionally, although three of the four information measures in this figure favor high lag orders by decreasing up to p = 5, Stata’s fits for p > 5 are not reliable. Thus, we conclude that the Period 1b data is insufficient to confidently fit the auto-regressive model. An implicit assumption of our analysis of the 2020 election cycle is that the rules of the system’s behavior are time-uniform before the election day. If we apply the same assumption to the 2024 cycle, then the absence of the data in the first grayed out interval of Fig 5 and the data in Period 1b only detracts from the statistical power of our conclusions regarding the pre-election behavior as inferred from Period 1a, without altering the qualitative nature of these conclusions.