Tweeting #RamNavami: A Comparison of Approaches to Analyzing Bipartite Networks

The Data and the Network

Tweets that used the hashtag #RamNavami were collected using the Twitter API (Application Programming Interface) from midnight to 11am (Coordinated Universal Time) on 21 April 2021. This approach followed well-established procedures for gathering social network data from cyberspace (Steinert-Threlkeld, 2018). Initially, 1,319 tweets were assembled, excluding retweets and mentions. All hashtags (which are insensitive to capitalization) were extracted from the tweets to determine which were the most common hashtags related to #RamNavami. Hashtags could be written in any language, as long as they used the Latin/Roman alphabet. So hashtags written in Devanagari script (used for writing in Hindi) were not included in the analysis, a nontrivial limitation given the context.

From this initial set of tweets, two subsets were created: (a) a data set containing only hashtags that had been used two or more times (excluding #RamNavami, which was in all tweets by design), and (b) a data set containing only the top 30 hashtags. The first data set consists of 5,373 edges (i.e., sender-hashtag pairs), while the second consists of 3,128 edges.

Some appreciation for the nature of the data can be gleaned by considering the list of the top 30 hashtags listed in Table 1. Some of the most popular hashtags also made reference to Ram Navami but varied slightly from the specific hashtag that was used to identify the network, such as #ramnavami2021. Other hashtags made religious references, such as #navratri and #hindu. Still others alluded to the Covid pandemic sweeping India and the world at large, such as #staysafe and #stayhome. Hashtag coding was performed by graduate students fluent in Hindi and English.

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Table 1.

Top 30 Hashtags in the Ego-Network for #RamNavami on 21 April 2021

Hashtag	Count	Hashtag	Count
#ramnavami2021	468	#ramanavami2021	58
#jaishreeram	370	#ramanavami	56
#ramnavmi	224	#ramnavmi2021	56
#jaishriram	210	#festival	53
#ram	179	#wednesdaythought	53
#ayodhya	164	#rama	50
#india	117	#stayhome	49
#lordrama	116	#adipurush	47
#ramayana	108	#hinduism	47
#shriram	94	#happyramnavami2021	40
#hindu	92	#hanuman	38
#sitaram	79	#happyramnavami	38
#lordram	78	#ramayan	37
#navratri	70	#jayshreeram	35
#staysafe	67	#radhe	35

Source: https://developer.twitter.com/en/solutions/academic-research.

The network processes related to #RamNavami are evident in Figure 1. In this graph, white circles represent Twitter users, while the black squares represent the top 30 hashtags in the ego network for #RamNavami. This graph was generated using the spring-embedding algorithm in Netdraw 2.168 (Borgatti, 2002), which draws points closer to one another based on minimizing tension in the graph. Not all of the labels for hashtags are included in the figure due to visibility issues. Some structures apparent in the data include the approximate co-location of #staysafe and #stayhome, both references to the Covid pandemic. Similarly, the near co-location of religiously themed hashtags, #hinduism and #hanuman, provides evidence for the relevance to religion on the network structure. Finally, there appear to be structural holes (Burt, 1992) between the denser top part of the network and the sparser bottom part of the network.

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Figure 1.

Source: https://developer.twitter.com/en/solutions/academic-research.

These casual observations may (or may not) be of interest to the reader. Yet, if we wish to truly understand the network generating processes behind these data, we require formal statistical tools, which are considered in the next section.

The Analysis of Bipartite Networks

No single approach to two-mode networks should dominate any other. Rather, there are different situations in which one approach may be preferred to others. Dyadic Logit models have been applied to topics such as the study of international conflict and alliances, though these models have been the target of extensive criticism (see Cranmer et al., 2021). The substance of the criticism is that dyads are often unlikely to be independent on one another, as is assumed by the Logit model. For example, a (hypothetical) attack by Pakistan on India is unlikely to be independent of a subsequent (hypothetical) attack by the United States on Pakistan; the crisis fomented by the first attack conditions the strategic logic governing the second attack. Nonetheless, it may be helpful to set the Logit model as a baseline for comparison, as it is widely understood by scholars. Also, it is possible that in some situations, it may be reasonable to assume dyadic independence, in which case a Logit model may be preferred.

Despite our familiarity with Logit, the advantage of ERGMs is that they allow the user to specify the network dependencies that may be present in the data generating process. For example, system behaviour may exhibit reciprocity, transitivity, preferential attachment, homophily or other endogenous network tendencies (Lusher et al., 2013). Thus, ERGMs enable the specification and assessment of network processes in the data, which may both address dependencies and yield substantively relevant knowledge. However, ERGMs do face limitations. While restricting the data to discrete time periods may not be a problem in many applications, in other cases, it may miss crucial elements of the problem. In examining tweets, for example, the sequence may be of special interest, requiring a continuous-time model. Additionally, ERGMs tend to suffer from estimation difficulties resulting in degeneracies that make it impossible to estimate certain models. In these cases, it may be necessary to turn to other approaches to estimate network parameters.

The added value of REMs is that they incorporate continuous time into network models, provided that the assumption is made that no two events happen at the exact same time (Butts, 2008). Sequence statistics can be easily constructed in order to investigate how events relate to one another. However, current estimation procedures, such as those conducted using the informR package in R, limit the number of events (e.g., hashtags) that can be included in the estimation (Marcum & Butts, 2015). Further, including network attributes in REMs is not as straightforward as it is for ERGMs.

In light of these considerations, this article presents three simple models of the #RamNavami ego network using Logit, ERGM and REM estimators. In each case, variables are included for three hashtag characteristics: (a) direct reference to Ram Navami, (b) religious connotations (other than Ram Navami) and (c) reference to the Covid pandemic. These models are compared and then one extension is considered for each model.

Model Comparisons

Three models were estimated and are reported in Table 2. Model 1 used a Logit estimator including the three focus variables and a constant term (which is standard for this approach). Model 2 used an ERGM estimator with the three focus variables, an edges term (the analog to the constant term for ERGM) and an endogenous term for Twitter users that contributed at least two hashtags to the network. The inclusion of this endogenous term represents a minimal specification for the endogenous element of the network that is required to effectuate an ERGM. That is, without such a term, the ERGM would be equivalent to a Logit. Model 3 used a REM estimator that incorporated the unfolding of time during the 11 hours for which the data were collected. Sequence statistics were specified to approximate the variables in the Logit and ERGM models. The results of these exercises are reported in Table 2. The R code necessary to reproduce them is reported in the Appendix.

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Table 2.

Logit, ERGM, and REM Models for the #RamNavami Ego Network

Parameter	Model 1 Logit	Model 2 ERGM	Model 3 REM
		Coefficient (Standard Error)
Hashtag Reference to Ram Navami	–0.342 * (0.049)	–0.347 * (0.048)	1.980 * (0.159)
Hashtag Religious Connotation	0.560 * (0.047)	0.562 * (0.049)	0.080 (0.288)
Hashtag Covid Pandemic Reference	0.173 (0.089)	0.178 * (0.087)	3.514 * (0.532)
Constant / Edges	–2.879 * (0.042)	–2.888 * (0.044)
Endogenous Term for Two Tweets by Same User		–0.527 * (0.070)
Akaike information criterion (AIC)	21,333	21,270	15,818
Bayesian information criterion (BIC)	21,368	21,313	15,835

Source: https://developer.twitter.com/en/solutions/academic-research.

Note: * p ≤ 0.05.

Comparison of the estimates of Models 1 and 2 indicates a very close match. Both models report that the coefficient on Hashtag Reference to Ram Navami is significant and negative, documenting that these hashtags had a less than typical chance of being coupled with other hashtags. The coefficient on Hashtag Religious Connotation is positive and significant, thus showing that these hashtags had a greater than typical chance to be combined with other hashtags. Both models display borderline significant coefficients on Hashtag Covid Pandemic Reference, with Model 1 just above the significance threshold and Model 2 just below it. The endogenous term in Model 2 is significant and negative, demonstrating that two hashtags by the same sender was less common in this network than was the case for randomly generated networks with the same size and parameters. The significance on this term establishes that the hypothesis of dyadic independence should be rejected. Nonetheless, the estimates of the Logit (which assumes dyadic independence) and the ERGM (which assumes dyadic dependence) lead to substantively very similar conclusions.

The REM estimates, which take into account the temporal ordering of the data, suggest substantive conclusions that are virtually opposite to those derived from Logit and ERGM. They show a positive and significant coefficient on Hashtag Reference to Ram Navami, thus suggesting an effect in the opposite direction of the Logit and ERGM. Unlike the Logit and ERGM results, the coefficient on Hashtag Religious Connotation is insignificant. In contrast to Model 1 and 2, the coefficient on Hashtag Covid Pandemic Reference is positive and statistically significant. These coefficients demonstrate that viewing the data through a temporal lens makes a considerable difference.

One implication of these results is that, at least for this dataset, the Logit and ERGM approaches are virtual substitutes for one another. While the ERGM estimates do demonstrate that the data generating process is dyad dependent, this dependence does not have severe consequences for the parameter estimates. Thus, it may be reasonable to extend the Logit to a range of data not feasible for ERGM estimation. However, this conclusion should not be generalized because if dyadic dependence was stronger in the network, then the resulting Logit estimate could be far off the mark.

A second implication of these results is that the researcher must be careful in choosing between discrete-time and continuous-time specifications. In this case, at least, the two approaches produce drastically different results. Consequently, it is necessary to carefully theorize whether or not time is expected to be a critical element of the problem at hand. More specifically, a key question is whether the sequence of events is expected to matter in generating the outcomes of interest. If yes, then REM is the obvious choice. If no, then ERGM (or possibly Logit) is more suitable.

Model Extensions

The three models reported in Table 2 were specified to make them as similar as possible, thus centring the discussions on the methods of estimation. Having considered these models, it is possible to investigate extensions of each approach. As mentioned earlier, the ERGM approach frequently suffers from degeneracy such that some models are not estimable. This problem is present in a case at hand, as Model 2 did not converge when sender characteristics were added to the specification. Hence, all the models were paired down to omit these characteristics. Now, since the Logit model is not commonly plagued with degeneracy issues, it is possible to consider an extension of this model to incorporate sender characteristics.

Since the informR software has limitations on the number of events that it can model for two-mode data (Marcum & Butts, 2015), all models were paired down to only 30 hashtags. Since ERGM does not suffer from this limitation, it is possible to consider an extension to a dataset with all hashtags used two or more times. The reason for setting the limit at two is that any hashtag used only once merely adds noise to the network structure, while at the same time threatening to prevent statistical convergence.

Finally, it is standard for REMs to include intercepts for each event in the model. This feature was suppressed in Model 3, with the goal of making the three models as comparable as possible. Now it is possible to relax that restriction and add event intercepts to the model, which is typical for REMs.

Three model extensions were estimated. Model 4 is a Logit that incorporates parameters for sender characteristics. Model 5 is an ERGM estimated on all hashtags with two or more appearances in the data. Model 6 is a REM that adds event intercepts for all events, except for the base category, #ramnavami2021, which was the single most popular hashtag in the #RamNavami ego network. The results of this estimation are reported in Table 3.

The model extensions reported in Table 3 add insights to the network generating processes beyond what was evident in Table 2. The Logit results in Model 4 contain a negative and statistically significant coefficient on Sender Tweets, indicating that senders who are more active on Twitter were less likely to contribute edges to this network. The coefficients on Sender Accounts Followed and Sender Followers are numerically very small and statistically insignificant. A reasonable speculation is that the relatively miniscule magnitude for all three sender coefficients explains why the ERGM would not converge when they were inserted in the specification. It is relevant to note that neither the direction nor the significance of the parameters on the hashtag attributes change in Model 4 when compared to Model 1.

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Table 3.

Extensions of Logit, ERGM, and REM Models

Parameter	Model 4 Logit	Model 5 ERGM	Model 6 REM
		Coefficient (Standard Error)
Sender Accounts Followed	0.000
	(0.000)
Sender Followers	0.000
	(0.000)
Sender Tweets	–4 e–6 *
	(1e–6)
Hashtag Reference to Ram Navami	–0.343 * (0.048)	2.181 * (0.040)	0.929 * (0.169)
Hashtag Religious Connotation	0.560 * (0.047)	1.344 * (0.033)	–1.025 * (0.293)
Hashtag Covid Pandemic Reference	0.173 (0.089)	0.640 * (0.059)	4.017 * (0.542)
Constant / Edges	–2.842 * (0.042)	–5.996 * (0.028)
Endogenous Term for Two Tweets by Same User		–0.145 * (0.065)
#jaishreeram			–0.041
			(0.092)
#staysafe			–1.571 *
			(0.148)
#ramnavmi			–0.434 *
			(0.096)
#ramanavami2021			–2.735 *
			(0.250)
#ramnavmi2021			–1.887 *
			(0.172)
#stayhome			–1.794 *
			(0.161)
#ram			–0.436 *
			(0.102)
#jayshreeram			–2.717 *
			(0.251)
#wednesdaythought			–2.084 *
			(0.188)
#jaishiram			–0.601 *
			(0.105)
#hindu			–1.073 *
			(0.124)
#lordrama			–0.896 *
			(0.116)
#ayodhya			–0.896 *
			(0.116)
#adipurush			–2.116 *
			(0.190)
#rama			–1.766 *
			(0.163)
#lordram			–1.361 *
			(0.138)
#navratri			–1.507 *
			(0.147)
#shriram			–1.119 *
			(0.126)
#ramayan			–1.967 *
			(0.178)
#festival			–1.722 *
			(0.160)
#ramanavami			–1.700 *
			(0.159)
#happyramnavami			–2.842 *
			(0.266)
#india			–0.868 *
			(0.115)
#ramayana			–0.877 *
			(0.115)
#happyramnavami2021			–2.292 *
			(0.206)
#sitaram			–1.246 *
			(0.132)
#hanuman			–1.939 *
			(0.176)
#hinduism			–1.722 *
			(0.160)
#radhe			–2.459 *
			(0.222)
Akaike information criterion (AIC)	21,305	58,663	14,627
Bayesian information criterion (BIC)	21,366	58,722	14,812

Source: https://developer.twitter.com/en/solutions/academic-research.

Note: * p ≤ 0.05.

Model 5 reflected an expansion of the data examined in comparison to Model 2. The extant software for ERGM estimation was able to manage the vastly expanded number of hashtags (516 instead of only 30) more routinely than is possible with the extant software for REM estimation. This increased information is consequential for model estimation. Hashtag Reference to Ram Navami switches from a significant, negative coefficient to a significant, positive coefficient. The coefficient on Hashtag Covid Pandemic Reference appears now as positive and significant, whereas it was insignificant in Model 2. Other parameters in Model 5 did not change their significance and direction in comparison to Model 2.

Model 6 extended the REM to incorporate intercepts for 29 of the top 30 hashtags, consistent with typical REM specifications. Almost all (28 of 29) of the coefficients on these intercepts are significant and negative due to the fact that the base category is the most commonly used hashtag; all other hashtags are less likely in comparison. These new model terms do not affect our conclusions about the Hashtag Reference to Ram Navami or Hashtag Covid Pandemic Reference variables. However, the parameter on Hashtag Religious Connotation is significant and negative in Model 6, where it was insignificant in Model 3.

Conclusion

Social media helped people to transmit greetings related to the Hindu festival of Ram Navami, even during the siege of an unprecedented global plague. These messages were not disseminated randomly but travelled through the cognitive and social structures that mould the internet. Models of bipartite networks afford scholars tools to investigate how these processes work (or do not work).

Firm conclusions about the network processes around #RamNavami cannot be drawn from the statistical results at hand. The estimated models displayed sensitivity to data selection criteria, the time scale of the analysis, parameter specification and the statistical estimator chosen. Moreover, the scope of the example data on which the analysis relied was starkly narrow, linking primarily to the use of one hashtag on one day.

Nevertheless, this article highlights critical choices to be made in the research design in the study of two-mode networks like the #RamNavami ego network. First, it is necessary to decide if the dyads in the network can be assumed to be independent (as is the case with Logit) or if it is prudent to account for network dependence (as is possible with ERGM and REM). Second, time must be treated either in a discrete fashion (as is the case with Logit and ERGM) or handled as a continuous phenomenon (as is the case with REM). Third, decisions about the size of the data set and the variables to be included in the analysis have implications for model selection. While there are no hard and fast rules on this dimension, it is generally the case that Logit affords the widest berth, with ERGM allowing less flexibility and REM presenting even less latitude. These choices should be rooted in theoretical understanding of the case under study, as they are likely to make a difference in the resulting estimates.

The models discussed in this article are intentionally simple, though the ERGM and REM frameworks allow for more advanced applications. For example, Heaney and Leifeld (2018) demonstrates the utility of deploying structural zeros and structural ones in bipartite ERGMs to probe the intricacies of lobbying coalitions. Brandenberger (2018) illustrates how to test for reciprocity in congressional collaborations using REMs. These and other extensions are on the horizon in what is a rapidly evolving area of research.