key: cord-254513-7d10vd86
authors: Phillips, B. C.; Bauch, C.
title: Echo chambers as early warning signals of widespread vaccine refusal in social-epidemiological networks
date: 2020-10-20
journal: nan
DOI: 10.1101/2020.10.17.20214312
sha: 
doc_id: 254513
cord_uid: 7d10vd86

Sudden shifts in population health and vaccination rates occur as the dynamics of some epidemiological models go through a critical point; literature shows that this is sometimes foreshadowed by early warning signals (EWS). We investigate different structural measures of a network as candidate EWS of infectious disease outbreaks and changes in popular vaccine sentiment. We construct a multiplex disease model coupling infectious disease spread and social contact dynamics. We find that the number and mean size of echo chambers predict transitions in the infection dynamics, as do opinion-based communities. Graph modularity also gives early warnings, though the clustering coefficient shows no significant pre-outbreak changes. Change point tests applied to the EWS show decreasing efficacy as social norms strengthen. Therefore, many measures of social network connectivity can predict approaching critical changes in vaccine uptake and aggregate health, thereby providing valuable tools for improving public health.

Low vaccine rates stemming from vaccine refusal result in outbreaks of vaccine-preventable diseases in some populations [1] , and the high costs of intervention and treatment incurred by public health systems [2] motivate us to find tools warning of epidemics. The connection between social network activ- 5 ity and health issues in populations has long been exploited by researchers [3] , especially relating to disease spread [4] , with the assertion that firm understanding of social network structure is important to the implementation of effective change is driven primarily by exposure to news sources and contrasting views from neighbours, communities can support the reinforcement of sentiments already held [17] .

Echo chambers, described as well connected groups of people promoting and reinforcing the same bias [18] , have recently come under media scrutiny since 30 these groups can facilitate vaccine scares [18] , support political candidates [12] , lead to skewed evaluation of objective fact and decreased accuracy of opinion [19] . Furthermore, some studies indicate that in some cases these homogeneous sub-networks may reinforce bias [20] , in some part due to avoidance of cognitive dissonance [21] . Given that interaction between dislike agents in a network can 35 sometimes correct false beliefs [22] , these echo chambers may be seen as drivers of polarisation [23] . This is especially since much anti-vaccine content is shared without thought of its veracity [24] , with Facebook anti-vaccine groups serving the dual purpose of opinion-reinforcing echo chamber and "fake news" source in a time where a large number of people draw on social media sites for their health information [25] .

In the same vein, much work has focused on the modularity of social networks. Modularity is a graph theoretic measure of the segregation of a graph [26] ; a high degree of modularity may indicate increasing segregation of a network into clusters [27] , with other work showing that modularity is not a "direct nodes, where effective communication between disagreeing persons can lead to change of opinion [12] .

The occurrence of opinion change within opinion-based communities has seen much attention in political studies, with some work asserting the ability of the 'wisdom of the crowd' to overcome bias [33] , while other work shows that group phenomena reinforce the opinions held [19] . Others have argued that a 60 commonly held belief within a community can still become more accurate even as the homogeneity of the group increases [34] . This leads to our interest in the rate of opinion change in the network as yet another potential indicator of dynamical regime change. Systems moving from one polarised state to another undergo phase transition through a sole critical point [35] . Called critical transitions, they sometimes result in a demonstration of characteristic system behaviours such as critical slowing down [36] . These events give us easily recognisable 'hints' of approaching transitions called early warning signals [37] .

We show that trends in all of the measurements described above (modularity, global clustering coefficient, census and sizes of communities and echo 70 chambers) provide early warning signals of epidemic and vaccine crisis events for a coupled disease-behaviour model of childhood disease. We use a binary vaccine opinion dynamic and an SIRV p disease process occurring on a random network to model a childhood infectious disease. By quantifying and comparing their performance, we find that trends in the sizes of anti-vaccine communities 75 were the best-performing signals, with the modularity and clustering coefficients of communities also performing well. All in all, we verify that changes to fundamental graph structure driven solely by opinion dynamics are good predictors of disease events and aggregate sentiment towards vaccination. This paper is organised as follows: in Sec. 2, we will describe the disease-80 behaviour model used in the study and a description of the warning signals used. Section 3 will show the change in trends of the signals with respect to perceived risk of adverse vaccine effects and the social pressure of an injunctive norm, as well as a quantification of the warning provided by each measure through the use of the Standard Normal Homogeneity change point test (SNHT). We 85 will elaborate on the limitations of the measures used and the implication of the results in Sec. 4. Information showing the results obtained through the application of other change point tests are given in the Appendix.

For simulation, we use an ABM identical to that described in [38] , shown in Fig. 1 . A perfectly effective vaccine with no effect on mortality is immediately available to susceptible agents upon gaining pro-vaccine status S → V p . As shown in Fig. 1a , the disease follows the SIRV p model; each agent physically interacts with all their neighbours per time step. If a susceptible agent becomes

(a) Schematic of the infection dynamics of the model. Effective contacts occur between susceptible S and infected I agents with probability p per time step (1 week). Upon deciding to vaccinate (with probability Pn(N → Vs)), a susceptible agent n becomes physically vaccinated (S → Vp). Infection lasts � = 2 weeks after which agents recover (I → R). Upon death (with probability µ per week), an agent is "rebirthed" with either vaccinated (probability α · µ) or susceptible (probability (1 − α) · µ) status.

Representation of the opinion dynamics of the model. Per time step, each agent switches between pro-(Vs) and anti-vaccine (N ) opinion with probabilities Pn(N → Vs) and Pn(Vs → N ) respectively upon interaction with a dissenting neighbour. α gives the probability of being birthed with pro-vaccine opinion Vs. become both ill and infectious for � = 2 time steps (each time step represents a week); recovery I → R and vaccination S → V p are both permanent. Injunctive social norms 0 ≤ σ ≤ 2 and a cost of −0.4 ≤ κ ≤ 0.2 represent peer pressure 100 and adverse vaccine effects respectively [39] . Figure 1 features a binary social (opinion) dynamic, where agents demonstrate either pro-vaccine (V s ) or anti-vaccine (N) sentiment. Change of sentiment occurs through imitation of neighbours, where a randomly chosen neighbour is sampled each time step; an effective interaction with a disagreeing neighbour (with different sentiment than the agent sampling) prompts a reevaluation and change of sentiment with probability P n (N → V s ) for anti-vaccine agents and P n (V s → N ) for pro-vaccine agents. Any susceptible agent that adopts pro-vaccine sentiment (N → V s ) is immediately vaccinated (S → V p ); for agent n, these changes in sentiment depend on the perceived vaccine risk kappa and 4 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted October 20, 2020. . https://doi.org/10.1101/2020.10.17.20214312 doi: medRxiv preprint neighbours I n :

Indices U N →Vs n and U Vs→N n in equation (1) are utility functions defined as

where d * n represents the number of neighbours of n with sentiment * , with d n representing the total number of neighbours.

Four of the six EWS explored in this paper are related to the detection of community structure in networks; the global clustering coefficient, echo chamber, opinion-based community and modularity score are all important tools that enable biological modelling [26] . The topological phenomenon of community refers not to a single central construct, but rather a general notion of variation in connection density. Communities in social networks are vaguely defined in the 110 literature as groups with the following basic property: members of the community are more connected to each other than with non-members [40] . Vagueness in science usually leads to artistic licence and the specific treatment of context and purpose; by the above definition, communities can then be alternately conceptualised in different areas and studies as modules, clusters, groups and so on 115 [41] . Such refinements then lead to tighter and more technical definitions.

Here, we conceptualise topological communities as (connected) components on the network. Components in undirected networks are maximal disjoint groups of agents such that there is a path between every pair of agents in the group. [42] . Even more specifically, the concept of a giant connected com-120 ponent (GCC) describes a component that contains a "significant fraction of all the nodes" [43] . The role of components (both non-giant and giant) in spreading processes can be conceptualised as such: where some infection can be spread from person to person, GCCs are formed by historical person-to-person contacts (as opposed to current contacts) [44] . Practical indirect analysis and 125 exploitation of this component structure in policy design and epidemiological intervention is facilitated through contact tracing [43] .

Recent political upheaval has thrust the phenomenon of the echo chamber into social consciousness and modern parlance, with many lay articles arguing for and against their existence , though many articles do not directly define the 130 concept before exploiting it. There exist different definitions of echo chambers in 5 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted October 20, 2020. . literature; also called tribes [45] , they can be described either as a community where at least some percentage of the members hold a particular sentiment [46] , or else a subset of community members overwhelmingly likely to restrict their neighbourhood communication to contacts with shared opinion [47] . As 135 such, echo chambers are usually conceptualised as closed subsystems of social networks containing members with a single orientation [17] , and are therefore considered synonymous with homophily and strongly associated with the quick spread of misinformation [48] , polarisation and the insulation and reinforcement of belief despite their veracity [17] . One example is the concept of the 'filter 140 bubble', detailing the biased filtering of information unfortunately created by personalisation algorithms [49] .

Given the described ubiquity on social networks and their importance in information diffusion (and therefore decision making) [48] , we test whether observations of the size and number of communities Z * and echo chambers J * give 145 early warnings of approaching vaccine scares and crises, as well as falls in vaccination rates. We retrieve the echo chambers J * by first listing the agents in the network with the desired vaccine opinion; the members of the network are then sorted into two primary classes. The peripheral members maintain links with at least one disagreeing neighbour and core members only maintain links 150 with agreeing neighbours (these are direct analogies of the boundary and interior of a topological space respectively). We then characterise echo chambers as communities of core members.

Clustering has been shown to greatly facilitate the spread of such phenomena as political extremism [50] and infectious disease [51] . Specifically, clustering 155 refers to the propensity of connection between two persons if they have a mutual friend [52] . As an indicator of this organisation, the global clustering coefficient indicates the prevalence of clusters, dense highly-connected groups of nodes in a graph [53] . This done by finding the density of triplets on the network. An open triplet is a group of three nodes connected by 2 edges, while a closed triplet 160 is a group of three nodes joined by three unique edges (also called triangles for this reason). The global clustering coefficient (GCC) is then calculated as The modularity measure is similar to the global clustering coefficient as a measure of organisation; highly modular networks possess many modules, which 170 feature dense interconnectivity between nodes similar in some way and sparse connectivity between dislike nodes. Specifically, this measures the correlation between the probability of connection of two nodes and their membership of 6 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 20, 2020. . the same module [55] . The modularity score is calculated as follows [56] ; let an undirected network be divided into two disjoint groups Λ and Ξ, with each node n given the score

with an adjacency matrix A of the network, so that A ij gives the number of edges between nodes i and j. Let k * represent the degree of node * , so that

gives the number of edges in the network and the expected number of edges between nodes i and j is

The network modularity (Q) is then given as

In the model investigated, vaccination occurs only through the first adoption of pro-vaccine opinion N → V s (Sec. 2.1); as such, the number of changes of opinion Θ * undertaken by agents of either opinion can conceivably describe both the social and infection dynamics of the model and thereby yield warning signals 185 of sudden transitions. Moreover, equation (2) shows that the probability of switching sentiment is sensitive to the number of infected agents in the individual neighbourhood; this dependence makes it highly likely that the probability of having an infected neighbour Γ * is correlated not only with vaccine coverage both also with the rate of change of opinion throughout the length of each 190 simulation.

Both (infection and social) layers of the network have an Erdős-Rényi random network structure G (10000, 0.003); network size N = 10000 and mean degree � d n � = 30 were chosen for alignment with studies of similar coupled 195 behaviour-infection models [57, 38] , as well as for computational tractability. Each simulation starts with proportion α = 0.25 of vaccinated pro-vaccine agents (states represented by pair (V s , V p )), with all others being susceptible anti-vaccine agents (pairs (N, S)). The probability of death per time step is µ, so that α · µ gives the probability of reset with initial state V s and (1 − α) · µ the 200 probability of reset with initial state S. ξ = 1 × 10 −4 represents the probability of switching sentiment randomly. Noise parameter ξ = 0.0001 represents unsystematic fluctuations commonly seen in empirical studies [58] . The birth/death probability µ = 2.4 × 10 −4 affords an average life span of 80 years to each agent 7 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 20, 2020. . https://doi.org/10.1101/2020.10.17.20214312 doi: medRxiv preprint [38] . This is due to the permanence of the vaccine; an agent can change their vaccine opinion throughout their lifetime, but they cannot become 'unvaccinated' should their 220 stances change.

The chosen parameter space (κ, σ) ∈ [−1, 1] × [0, 3] was sufficiently broad to capture transitions in both the infection (Fig. A.8a) and social (Fig. A.8b) dynamics. The contours in each panel of Fig. (A.8) show the obvious correspondence between social (K s ) and infection (K p ) transitions and substantial 225 changes in the clustering coefficient of the pro-vaccine sub-network � C Vs � (Fig.  A.8c ) and the mean size of anti-vaccine communities � |Z N | � (Fig. A.8d ).

8 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 20, 2020. .

We again define the infection transition K p as the value of perceived vaccine risk κ value at which the mean number of recovered agents in the model 230 surpasses that of the vaccinated agents and vice versa, so that 

this is marked by the dotted vertical line in Fig. 3a showing the approximate intersection of

(a) Changes in trend in the κ-series of output variables The distance between the two transitions (i.e. K p − K s ) is called the intertransition distance [38] ; as with similar models, we find that this intertransition distance decreases with increasing strength of the social norm σ. This is ex-240 plicitly demonstrated by Fig 3a, where an increase in the social norm from σ = 0 (Fig. 3a, left) to σ = 0.5 (Fig. 3a, right) brings the two vertical lines (representing transitions K s and K p respectively) together. Figure 3b gives a full picture of the intertransition distance over the investigated parameter range 0 ≤ σ ≤ 2.2; its decreasing trend with strengthening 245 social norm σ is shown by the purple curve, with the inset panel showing the locations of the social K s (blue) and infection K p (red) transitions. The positivity of the graph tells us that K s < K p for all strengths of the social norm σ ≤ 2.5, so that the social transition always precedes the disease transition. Since physical vaccination is driven primarily by changes in sentiment, this is to 250 be expected; strengthening social norms increases the alignment of behaviour and vaccine uptake, resulting in decreasing intertransition distance K p − K s .

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 20, 2020. . https://doi.org/10.1101/2020.10.17.20214312 doi: medRxiv preprint (a) Mean community size;

(b) Mean number of communities; #Z N (anti-vaccine), #Z V (pro-vaccine).

(c) Mean echo chamber size; Warning signals can potentially indicate both transitions, or maybe only one of the two; this subtlety is lost with shrinking intertransition distance and so may impact the predictive power of any early warning signals tested. Figure 4 shows the trends in the means of the numbers and sizes of communities and echo chambers (both pro-and anti-vaccine) at equilibrium. As for the number of echo chambers � #J * � (Fig. 4d) , there is not much resemblance to the trend of the mean number of connected components � #Z * � (Fig. 4b) ; this 260 is partly due to the low occurrence of echo chambers in this network. For σ = 0 (Fig 4d, left) , there is no warning given by the mean number of pro-vaccine echo chambers � #J V � ≈ 0, whereas the mean number of anti-vaccine echo chambers � #J N � increases before K s , reaching a maximum between the two transitions 10 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 20, 2020. . https://doi.org/10.1101/2020.10.17.20214312 doi: medRxiv preprint and reaching zero value approaching K p . At first glance, the change in trend as 265 the social norm strengthens to σ = 0.5 (Fig 4d, right) � (Fig. 5a) , the number of pro-vaccine communities � #Z V � (Fig.  5c ) and the number of pro-vaccine echo chambers � #J V � (Fig. 5c) ; they all give very little warning of the social transition K s for all strengths of the social norm σ, as well as giving the best warning (of all the EWS) for only ≈ 9% of the tested range of the social norm σ. They all demonstrate significantly lower 295 lead distances than those of the other EWS, giving the highest lead distance of all EWS for only 9% of σ values (as compared to ≥ 30% for other EWS shown in Fig. 5 ). The performance of these EWS under different change point detection tests is shown in Figs. (C.11-C.13). As social norm σ increases, pressure on each agent to conform to surrounding opinion becomes the main 300 driver of self-organisation of the social dynamics. Since it also increases the 11 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 20, 2020. speed of this transition between these opposing organised states, lead distances K s − SNHT{ * } will decrease as the social morn σ increases. 

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The copyright holder for this preprint this version posted October 20, 2020. . This behaviour disappears when social norm σ → 0.5 (Fig. B.9c, right) due to the shrinking intertransition distance

for σ = 0 and σ = 0.5 are similar (0.95 and 0.98 respectively), showing that the distances between 320 the warnings and the transitions K p are similar in both cases. As σ increases to 0.5, K s 'moves closer' to K p ; this suggests that any warning of K s occurs incidentally, rather than being directly caused by the model dynamics. This is intuitive; any measure of the probability of having an infected neighbour is an observation of the infection dynamics, therefore an assumption of some direct For all the EWS in Fig. (B.10) , we can see that lead distance decreases with strengthening social norm σ, with all the EWS eventually failing (giving negative lead distances, so that warnings follow K s -these are useless); modularity scores for the sub-networks formed by pro-vaccine � Q V � (Fig. B.10a ) and anti-vaccine � Q N � (Fig. B.10a ) agents give useful warnings for all social norms 340 σ ≤ 2.40625, while both the number of opinion changes � Θ * � (Fig. B.10b ) and the probability of having an infected neighbour SNHT �� Γ * �� give useful signals for σ ≤ 2.90625. As stated, the global clustering coefficient of the entire social network SNHT{ � C Σ � } (Fig. B.10d ) was undefined for most σ and negative for quite a few others, resulting in the worst performance of all the EWS 345 tested and giving the highest lead distance for only 4% of the total range of the social norm σ. The sub-networks generated by pro-vaccine agents gave often unsubstantial though positive leads SNHT{ � C V � } (Fig. B.10d) , while the sub-networks formed by anti-vaccine agents (Fig. B.10a ) gave very little lead distance over most of the range of σ. The performances of these EWS under 350 different change point detection tests are shown in Appendix C.

We compare the EWS by finding the proportion of σ values for which each EWS gives the largest warning; this is shown in Fig. 6 . We specify a good warning as one that gives the highest lead distance of all warnings for a single σ 355 value and a bad warning as one that gives either a minimal, negative or undefined lead distance. For many σ values, the largest and smallest lead distances were not unique, so that the ratios on neither side sum to 100%.

Comparison of the panels of Fig. 6 gives a notion of 'dependability'; were a restricted set of EWS to be employed, we would prefer good EWS (ones that 360 13 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 20, 2020. . 14 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 20, 2020. . give the largest lead distances) that aren't also bad (giving the smallest lead distances). Some of the foremost EWS best satisfying this criterion are the counts of all different opinion changes � Θ * � and the mean size of anti-vaccine communities � |Z N | � (30% best, 15% worst). The best performers are the modularity scores of the opinion sub-networks � Q V � , being the best EWS for 32% of 365 σ values, with 18% bad warnings. Conversely, Fig. 6 suggests that the numbers of echo chambers � #J * � both bring up the rear, providing good warnings for only 1% of σ values while giving bad warnings for 98% of social norm σ values tested.

Overall, all observations of the sizes of anti -vaccine echo chambers |J N | on 370 the network are poor EWS. Conversely, the sizes of pro-vaccine echo chambers |J V | perform generally well (≥ 26% best, 15% worst). Also, the mean and maximum sizes of anti-vaccine communities ( � |Z N | � and � max(|Z N |) � respectively) give two of the highest ratios of best warnings (both 30%) and not many bad warnings (≤ 15%), but observations of the minimum size � min(|Z N |) � give only 375 1% good warnings and 52% bad warnings. The global clustering coefficient � C * � performed particularly badly as an EWS, with ≤ 4% best and 35% − 80% worst warnings. Another observation made from Fig. 6 would be the relationship between the percentages of good and bad warnings; they are actually strongly anti-380 correlated, with a coefficient of −0.77. The previously shown behaviours of the lead distances eliminate the possibility of an EWS that densely alternates between good and bad warnings, but some EWS do neither; for example, the total modularity of the network � Q Σ � is trivially neither good (0%) nor bad (0%) by virtue of being everywhere undefined. Nontrivially, the clustering coefficient 385 of the anti-vaccine sub-network � C N � gives intermediate warnings (neither good nor bad) for 60% of social norm strengths σ. Therefore, no choice need be made between minimising the percentage of bad warnings and maximising the percentage of good warnings; both strategies yield largely identical results. Figure 7 show the performance of each EWS per σ value. Red tiles show 390 where the EWS gave the smallest positive lead distance of all its peers, while green tiles represent the σ values for which the EWS gave the largest lead distance. Yellow columns show where all EWS gave equal lead distances and black tiles represent failed warnings (either undefined or negative lead distances). Therefore, the relative length of an EWS' red bar in Fig. 6 represents the per-395 centage of that EWS' red tiles in its row in Fig. 7 ; the same correspondence holds between the length of an EWS' green bar in Fig. 6 and the percentage of green tiles in the related row in Fig. 7 . The main insight provided by Fig. 7 deals with patterns of performance; for instance, the overwhelming red colouration of the rows corresponding to 400 the mean number of pro-vaccine communities and anti-vaccine echo chambers ( � #Z V � , � #J N � respectively) and all EWS related to anti-vaccine echo chambers J N show that these EWS are quantitatively the worst of the group. Also, there is no detectable pattern in performance visible on the grid; in other words, the effectiveness of the EWS cannot be broken down by ranges of σ value. For 405 

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The copyright holder for this preprint this version posted October 20, 2020. . higher values of the social norm σ ≥ 2.625, the prevalence of yellow columns provides the observation that performance seems not to vary as much among the EWS as it does for smaller σ values, but nothing else is immediately apparent.

For social norms σ = 2.90625 and σ = 3, none of the EWS give valid warnings; lead distances are all negative, except for the total clustering coef-410 ficient � C Σ � , the number of opinion changes by anti-vaccine agents � Θ N � and the network's total modularity score � Q Σ � which are undefined. This confirms behaviour seen for large σ in Figs. 5 and (B.10); indeed, � C Σ � is undefined for most social norms σ because of the disconnection in the social network. Total modularity � Q Σ � is everywhere undefined (row of white tiles) for this reason. 

In this paper, we tested the use and effectiveness of different network measures as early warning signals (EWS) of sudden transitions in the social and infection dynamics of a multiplex model of disease. For the parameter values used, we found that observations of the mean and maximum sizes of anti-vaccine 420 communities appear to be the most effective EWS of all tested, unlike the size of the smallest anti-vaccine community (though it does give warnings signals); trends in the global clustering coefficient of the sub networks formed by pro-and anti-vaccine agents respectively also marked these events, as well as the number of both respective communities and echo chambers preceding both transitions.

This reflects the breakup of connected components on a network preceding a 16 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted October 20, 2020. . critical transition, an observation well supported by literature on percolation thresholds in random graphs.

A phenomenon of particular interest in this study was the formation and breakup of pro-and anti-vaccine echo chambers; we found that all observations of the sizes of pro-vaccine echo chambers (maximum, minimum and mean) performed well as warning signals, while observations of the sizes of anti -vaccine echo chambers performed poorly compared to other EWS. The modularity measure of the social network and the rate of opinion changes also warn of transitions of the social and disease transitions, representing changes in aggregate vaccine 435 opinion and vaccine uptake crises respectively. As a direct observation of the infection dynamics of the model, the probability of having an infected neighbour (for both pro-and anti-vaccine agents) performed well as an EWS of vaccine crisis.

Through our proposal and study of effective graph connectivity measures, 440 this study complements others in the field of early warning signals. A potential limitation to the study is our strict definition of an echo chamber; it remains to be seen whether different descriptions such as those featured in other studies of social media networks will result in a more effective or dependable EWS. Also, the graph connectivity measures seem suitable for tracking the dynamics of an 445 evolving network; the inclusion of preferential link formation in the dynamics, as well as social and 'on the ground' interventions for different strengths of the social norm, present other interesting avenues of research. Finally, the inclusion of directionality of communication in the network may render the model more realistic [61, 62, 63, 64] .

Together with other markers of spatial correlation and aggregation, the graph connectivity measures presented here contribute to the set of tools allowing us to leverage the ubiquity of social media involvement and the resulting data sets in the pursuit of adaptive strategies for maintaining public health.

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The copyright holder for this preprint this version posted October 20, 2020. . https://doi.org/10.1101/2020.10.17.20214312 doi: medRxiv preprint Cambridge ¡England¿ ;, 1994.

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The copyright holder for this preprint this version posted October 20, 2020. 

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Tweeting from left to right: Is online political communication more than an echo chamber?

Mitigating the spread of fake news by identifying and disrupting echo chambers

Users Enjoy the Opportunities for Political Debate and Engagement that Social Media Facilitates, But Many More Express Resignation, Frustration Over the Tone and Content of Social Platforms

Echo chamber? what echo chamber? reviewing the evidence

Comparing extremism propagation patterns in continuous opinion models

Dynamics and control of diseases in networks with 610 community structure

Clustering and preferential attachment in growing networks

Clustering in weighted networks

Collective dynamics of 'small-world' networks

Detecting complex network modularity by dynamical clustering

Finding and evaluating community structure in networks

Spatial autocorrelation as an early warning signal of regime shifts in a multiplex disease-model behaviour network

Stochastic formulation of ecological models and their applications

Table 39-10-0007-01 life expectancy and other elements of the life table

trend: Non-Parametric Trend Tests and Change-Point Detection

Social Media Mining: An Introduction

What is twitter

Interest communities and flow roles in directed networks: the twitter network of the uk riots

Information network or social network? the structure of the twitter follow graph