Modelling and Predicting Online Vaccination Views using Bow-tie Decomposition

TL;DR

The paper applies bow-tie decomposition to temporal, directed online vaccination networks to reveal stance-specific information flow. By analyzing within-group (vaccination groups) and across-group (Infomap communities) bow-tie structures, it finds pro-vaccination pages tend to form large SCCs while anti-vaccination pages exhibit large OUT components, with neutral pages showing more variable patterns. It demonstrates that incorporating bow-tie structure improves predictions of fan-count dynamics via supervised ML and agent-based SIR simulations, offering interpretable links between structure and information cascades. The framework is generalizable to other multi-stance temporal networks and can inform interventions to mitigate misinformation or tailor information diffusion strategies.

Abstract

Social media has become increasingly important in shaping public vaccination views, especially since the COVID-19 outbreak. This paper uses bow-tie structure to analyse a temporal dataset of directed online social networks that represent the information exchange among anti-vaccination, pro-vaccination, and neutral Facebook pages. Bow-tie structure decomposes a network into seven components, with two components "SCC" and "OUT" emphasised in this paper: SCC is the largest strongly connected component, acting as an "information magnifier", and OUT contains all nodes with a directed path from a node in SCC, acting as an "information creator". We consistently observe statistically significant bow-tie structures with different dominant components for each vaccination group over time. In particular, the anti-vaccination group has a large OUT, and the pro-vaccination group has a large SCC. We further investigate changes in opinions over time, as measured by fan count variations, using agent-based simulations and machine learning models. Across both methods, accounting for bow-tie decomposition better reflects information flow differences among vaccination groups and improves our opinion dynamics prediction results. The modelling frameworks we consider can be applied to any multi-stance temporal network and could form a basis for exploring opinion dynamics using bow-tie structure in a wide range of applications.

Figures (13)

• Figure 1: Bow-tie structures in online social networks. The arrows highlight that in this paper, an edge from node A to B in online social networks (solid arrow) represents an interaction from page A to B (e.g., page A recommends page B to its members), while the direction of information flow (dashed arrow) goes in the opposite direction (e.g., content about page B is presented or "flows" to page A). a, primitive bow-tie structure. This panel illustrates a bow-tie structure that divides a toy example network into three components: SCC, IN, and OUT. This decomposition establishes pairwise relations between these components, assigning distinct roles to each in terms of information flow: IN - "listeners", SCC - "magnifiers", and OUT - "creators". b, extended bow-tie structure. This panel expands on the bow-tie structure in panel a by introducing additional components: TUBES, INTENDRILS, OUTTENDRILS, and OTHERS. In this structure, IN not only "listens" to SCC but also to INTENDRILS and TUBES, while OUT not only "creates" information that is delivered to SCC but also to TUBES and OUTTENDRILS. c, recursive bow-tie structure. This panel displays an example of a recursive bow-tie structure, where the entire graph is partitioned into subgraphs, and bow-tie decomposition is applied to each of them. Note that edges across partitioned subgraphs are disregarded in this case.
• Figure 2: Online recommendation networks about vaccination views. a, February network. This is a snapshot of the largest weakly connected subgraph in February 2019, reproduced from Johnson et al.'s paper Johnson-2020. It includes over 94% of nodes and 99% of edges from the entire network. Each node represents a page. Its node colour depicts page polarity: red for anti, green for neutral, and blue for pro. Its node size is proportional to its page fan size. The node layout follows ForceAtlas2 in Gephi. The edge colour follows the colour of its source node (where the edge starts from). b, node-level data. It describes the total number of nodes and the total fan size for each vaccination group. By observation, the neutral vaccination group dominates with the largest number of pages and fans. The pro-vaccination group has fewer pages but a stronger fan base than the anti-vaccination group, mainly due to three pages with over a million fans. The anti-vaccination group has no pages with over a million fans in February and October but experienced the largest percentage increase in fans from February to October. c, d, edge-level data. It describes the edge number (panel c) and the average edge weight (panel d) within and across vaccination groups. Every edge is directed from A to B (i.e., A recommends B). It can be observed that the direction and weight of recommendations are important. Despite that, there are a larger number of edges within vaccination groups than across vaccination groups (except for pro- pages), and the highest edge weights flow from anti- to neutral and pro- to neutral groups (possibly due to the neutral group's high activity in interacting with both groups). Additionally, the anti- and pro-vaccination groups had minor interaction in February but interestingly experienced drastic increases in both edge number and average weight from February to October.
• Figure 3: Within-group and across-group bow-tie structures in the February and October 2019 online recommendation-based networks. The figure displays within-group bow-tie structures in the upper part and across-group bow-tie structures in the lower part for networks at both timestamps. Each part includes an explanatory diagram of the decomposition scheme, where the February network is divided into subgraphs, and the bow-tie structure within each subgraph is revealed using an organised layout of node and arrow, maintaining consistency with Figure \ref{['fig:intro']}b. Nodes are colour-coded by vaccination group and proportionally sized based on fan size, consistent with Figure \ref{['fig:data']}a. Note that, the across-group diagram shows the largest five communities, labelled as $C_i$, ranked by node counts, with larger communities having smaller indices, which collectively represent $49.4\%$ of all pages. While all five largest communities are primarily composed of neutral pages, communities $C_1$ and $C_2$ stand out with nearly half of their pages anti- and pro-vaccination, respectively. Three Sankey diagrams in each part illustrate the bow-tie structures for pages with different vaccination views. Each diagram has two columns representing bow-tie roles for February and October, with the flow indicating role variations. The stability, indicated beneath each diagram, quantifies the percentage of pages that maintain the same bow-tie roles at both timestamps. Overall, these results indicate that the pro-vaccination group exhibits a large SCC in bow-tie decomposition for both choice of discursive communities, while the anti-vaccination group is comparatively dominated by OUT component. Moreover, these structures are stable over time. In contrast, the neutral group yields inconsistent bow-tie structures and exhibits less temporal stability.
• Figure 4: Distinctions among the influence of information pieces initialised from different bow-tie components. Each bow-tie component generates $1000$ information pieces, with each page inside a component holding an equal probability of being the initial source for a single information piece, and the SIR epidemic process for each information piece is set at $\beta = 0.5$ and $\gamma = 0.3$. The violin plots depict the distribution of information within-group (across-group) influence, based on initialisation from different within-group (across-group) bow-tie components. We observe that the hierarchy of influence, both within-group and across-group, adheres to the following ranking: SCC > OUT > IN. This aligns with the roles of these components (i.e., SCC - "magnifiers", OUT - "creators", and IN - "listeners") formed by the bow-tie decomposition. Additionally, a large quantity of information pieces remain confined within-group and have limited dissemination across-group, mirroring the real-world dynamics.
• Figure 5: Distinctions among the influence of pages in different bow-tie components when varying their probability of generating information pieces. Each heatmap pixel represents the correlation coefficient (CC) between the page influence and the page fan count fluctuations during the specified SIR epidemic process. We generate $N = 3000$ pieces of information for each pixel with $\beta = 0.5$ and $\gamma = 0.3$ (aligning with Figure \ref{['fig:sir1']}), and customise the probability of generating information $\mathds{P}^{\text{initialiser}}$ for pages in different bow-tie components. Specifically, in each heatmap, the x and y axes represent the information generation probabilities for pages in the respective bow-tie components, divided by the information generation probabilities for pages in other components that are not involved in either axis. Our results indicate that both within-group and across-group page influence appear more correlated with fan count variations of expanding pages, in contrast to non-expanding pages. The higher CCs of across-group page influence than within-group potentially suggest that the increase in fan counts for anti- and pro- pages may be more strongly influenced by their interactions with neutral pages instead of similar-minded ones. In light of these two points, the upper-left red corner in the W-BT:OUT and W-BT:SCC heatmap illustrates that within-group OUT pages ("creators") are likely to produce more information pieces that recruit fans of pages sharing the same vaccination stance, compared with SCC pages. This result is also supported by other within-group heatmaps. Conversely, across-group SCC pages ("magnifiers") tend to generate more information pieces that possibly recruit neutral pages' fans compared with OUT pages. Both across-group and within-group IN pages ("listeners") exhibit a limited trend in generating information pieces contributing to fan size variations, though surprisingly having a good performance when $x = 10$ in the heatmap related to W-BT:SCC and W-BT:IN. Moreover, the CCs of within-group and across-group page influence can maintain relatively high values compared with other numeric features in Table \ref{['tab:ml2']}. This demonstrates their high potential in aiding the prediction of fan count increases for anti- and pro- pages.