2021

1]Wegener Center for Climate and Global Change, University of Graz,
Brandhofgasse 5, Graz, 8010, Styria, Austria

2]Complexity Science Hub Vienna, Josefstädter Straße 39, Vienna, 1080, Vienna,
Austria

# Guided rewiring of social networks reduces polarization and accelerates
collective action

Lilli Frei<EMAIL_ADDRESS>Jordan Everall<EMAIL_ADDRESS>Andrew K. Ringsmuth<EMAIL_ADDRESS>[  [

###### Abstract

Global challenges like climate change may be considered as collective action
problems that require sufficient cooperation with pro-mitigation norms, soon
enough to be effective. Socio-political polarization is a barrier to
collective action. Prior agent-based models of behavioural change on
structured networks in a shared socio-political environment have shown that
polarization emerges naturally in such systems and that the speed of consensus
formation is limited by the rate at which polarized clusters can be dissolved.
Here we study how guided social link rewiring affects the speed of network
depolarization. We investigate rewiring algorithms representing random
meetings, introduction by mutual acquaintances, and bridging between socially
distant communities. We find that building lasting links between polarized
individuals and communities can accelerate consensus formation when the
sociopolitical environment is favourable. This strengthens the evidence that
promoting connection between polarized communities could accelerate collective
action on urgent global challenges.

###### keywords:

social network, common-pool resource, collective action, polarization, link
rewiring

##  1 Introduction

Many societal challenges are social dilemmas in which individual rationality
leads to irrational large-scale behaviour (kollock_social_1998, ). Examples
include common-pool resource (CPR) management challenges such as
deforestation, the depletion of carbon budgets and loss of biodiversity.
Evidence of successful bottom-up CPR management shows that individuals can
overcome myopic strategies that lead to resource depletion and limit their
resource usage to a sustainable level (jager_using_2002, ;
janssen_evolution_2008, ). Such bottom-up management is more likely to succeed
for small-scale CPRs that have clear boundaries and enforceable access rules
but becomes more difficult for resources that are large relative to the scale
of resource extraction by individuals (ostrom_beyond_2010, ). Especially for
global challenges like climate change, it is difficult to define and monitor
clear boundaries, such as national or per-capita emissions permits
(Williges_fairness_2022, ). The mismatch of scales between the large CPR and
comparatively small extractions by individuals may make it difficult for
individuals to perceive the resource’s response to their actions. Without
obvious feedback, individuals might not realize the need for restraint until
critical thresholds have been crossed (Scheffer2012, ). While formal
institutions such as state governments or supranational institutions have the
power to alter structures such that conservation strategies become
individually rational, they depend on social support to ensure successful
implementation and avoid backlash Brownstein2022 . Without individual
behaviour obviously threatening the resource, however, finding social support
for conservation policies is challenging (griskevicius_evolutionary_2012, ).

Although the urgency of global socio-ecological challenges such as climate
action and pandemic containment has become clearer (ipcc_climate_2022, ),
governments struggle to enforce policies aimed at tackling them through
collective action (ringsmuth_2022, ; vasconcelos_segregation_2021, ;
pawloff_contrarians_2014, ). Rather, the opposite is the case; research shows
that political and social polarization around such global challenges is high
(carothers_democracies_2019, ; gladston_social_2019, ;
mccarty_polarization_2019, ). Empirical studies have found that high levels of
political and social polarization can significantly inhibit collective action
(vasconcelos_segregation_2021, ; carothers_democracies_2019, ;
axelrod_preventing_2021, ) and suggest that dissolving polarization could
promote efficient policy-making and restore a functioning democracy
(vasconcelos_segregation_2021, ; pawloff_contrarians_2014, ;
axelrod_preventing_2021, ). In general, a polity is polarized when dominated
by two opposite or contradictory tendencies that lack significant common
ground (stavrakakis_paradoxes_2018, ). This usually also involves increased
support for extreme political views relative to more moderate ones
(mccarty_polarization_2019, ). Since individuals’ opinions are highly impacted
by their social context (janssen_evolution_2008, ; gowdy_behavioral_2008, ;
tavoni_survival_2012, ), polarization is catalyzed by social structures that
reinforce ideological separation by excluding external information
(gladston_social_2019, ; williams_network_2015, ; nguyen_echo_2020, ). It may
be that information does not enter polarized clusters because of filters that
impair its flow. Such filtering may be individuals’ natural reaction to
exposure to more information than they can process (nguyen_echo_2020, ). It
may also be externally imposed through media control, systematic censorship or
algorithmic filtering (williams_network_2015, ; nguyen_echo_2020, ). Even if
information does reach polarized clusters, it may be discredited through
targeted manipulation of trust (nguyen_echo_2020, ). In both cases, social
restructuring can help to provide members of polarized clusters with relevant
information or to restore chains of trust to external information sources
(nguyen_echo_2020, ; lee_homophily_2019, ). In the latter case, building trust
in social, political, and scientific institutions is crucial, as simply
confronting members with outside information will not help to dissolve the
polarized cluster and may even reinforce the value system under attack
(axelrod_preventing_2021, ; nguyen_echo_2020, ).

These empirical results underscore the importance of careful engagement with
different views. Consistent with this, previous modeling work analyzing
opinion dynamics has shown that the rate of depolarization is essentially
determined by the frequency of interactions between different opinion clusters
and the response assumed when opinions are exchanged (axelrod_preventing_2021,
; fotouhi_conjoining_2018, ; andersson_dynamics_2021, ). These findings
suggest that increasing opportunities for interaction between agents with
dissimilar views through social rewiring might speed up depolarization and,
accordingly, collective action. In this paper we test this hypothesis by
extending a pre-existing agent-based model (ABM) on a social network. The ABM
was introduced by Andersson et al. (andersson_dynamics_2021, ) and simulates
the spread of a pro-conservation norm on a clustered scale-free social network
embedded in a shared sociopolitical environment. Andersson et al.
(andersson_dynamics_2021, ) used this framework to explore factors affecting
the speed of collective action to conserve a large CPR. We extend this by
adding link rewiring, which allows agents to alter their social environment by
breaking links to neighbours and forming new links to non-neighbours. This
allows an agent inside an opinion cluster to form links to agents outside the
cluster and become exposed to dissimilar views.

We aim to identify any dependence of network depolarization rate on properties
of link rewiring algorithms, and how this might be used to accelerate
depolarization relative to static networks. We test three rewiring algorithms.
First, purely random rewiring as a baseline scenario to analyse the effect of
a variable network structure without further assumptions. Second, a ‘local
rewiring’ algorithm based on current understanding of the processes by which
social networks usually change: individuals are most likely to form new
relationships with people who are similar and topologically close to them
(williams_network_2015, ; santos_link_2021, ; dunbar_anatomy_2018, ). Finally,
‘bridge rewiring’ that incorporates recommendations for dissolving
polarization from the literature and establishes links between differently
thinking agents to increase their exposure to differing views
(gladston_social_2019, ; williams_network_2015, ; nguyen_echo_2020, ).

##  2 Results

To learn how rewiring algorithms affect depolarization we study the time
evolution of the average agent cooperativity (hereafter simply
‘cooperativity’) and polarization. Parameters are defined in the Methods
section. We quantify polarization by calculating the standard deviation of the
cooperativity in each network realization and averaging these to obtain the
polarization shown in the plots.

###  2.1 Representative dynamics

Figure 1: Time evolution of cooperativity and polarization for a static and
dynamic networks. The standard deviation of cooperativity represents
polarization. A The static network scenario serves as a benchmark.
Cooperativity initially declines rapidly due to the predominance of defectors
at $t=0$. The slightly positive (pro-cooperation) external field drives the
system towards full cooperation through a slower process of polarized cluster
dissolution. B Representative dynamics under random rewiring. The legend
refers to different probabilities for establishing and breaking links.
Convergence is faster compared to the static network, this is matched by a
faster decline in polarization. Different rewiring probabilities have minimal
impact on convergence. C Representative dynamics under local rewiring.
Convergence is slower relative to the static network. D Representative
dynamics under bridge rewiring. Convergence is faster relative to the static
network which is matched by an earlier decline in polarization.

In figure 1A-D we see that the qualitative stages of the dynamics originally
observed by Andersson et al. for a static network (andersson_dynamics_2021, )
are maintained under dynamic network structures. In the first stage, the
slight defector majority that is assumed to exist at $t=0$ dominates and
cooperativity declines. Agents with strong opinions spread them to their
neighbours, resulting in growing opinion clusters that rapidly polarize the
network. This is quantified by the increasing standard deviation of the
cooperativity. Any interactions within the opinion clusters serve to reinforce
opinions and, thus, polarization. Even without assuming bounded confidence,
this cluster formation stage ends in a fully polarized state in which only
extreme opinions are present. This corresponds to the maximum standard
deviation. The next stage of the dynamics is a slower process of cluster
dissolution (depolarization), where meaningful interactions take place only at
the cluster perimeters, between agents holding opposing opinions. Here the
assumption of a slightly positive external field, representing a socio-
political environment favorable to resource conservation, is pivotal, as it
slightly biases the interactions such that defector-to-cooperator conversion
is more likely than cooperator-to-defector conversion. We refer to the time
derivative of the cooperativity during the depolarization stage as the
convergence rate.

The depolarization stage can itself be divided into two substages. In the
first, cooperativity grows approximately exponentially. In the second, the
convergence rate slows, likely due to the shrinking clusters’ finite sizes in
combination with fluctuations (andersson_dynamics_2021, ). As social change
depends more on achieving a cooperative majority than winning over the last
few holdouts (andersson_dynamics_2021, ; otto_social_2020, ;
gardiner_perfect_2011, ), we focus on the convergence rate within the
exponential stage of cluster dissolution.

###  2.2 Random rewiring

We first study the effects of random (unguided) rewiring as a baseline for
introducing dynamic network structure. Figure 1B compares the cooperativity
and polarization dynamics under random rewiring with the dynamics in a static
network. During the initial cluster formation stage, the two systems behave
almost identically until the standard deviation reaches roughly half of its
eventual maximum value. This is unsurprising since, in an unpolarized network,
random rewiring does not on average introduce agents to opinions that are
different from those in their local environment. However, once polarized
clusters have formed, random rewiring allows agents inside opinion clusters to
link with agents holding opposite opinions, effectively lengthening the
cluster perimeters along which opinion change is possible. Since we assume a
slightly positive external field, more frequent interactions between
differently thinking agents cause a slightly earlier peak in polarization and
then accelerate convergence towards a fully cooperative network. Concurrently,
the rate of depolarization is also accelerated relative to the static network
case. Random rewiring, however, cannot counteract the finite size effects that
slow down convergence as full cooperation approaches. The similar effect of
different rewiring probabilities suggests that convergence rates are
insensitive to the exact rewiring probability chosen.

###  2.3 Local rewiring

We analyse how social rewiring dynamics commonly seen in real social networks
affect the convergence rate through modified, local rewiring that limits
potential new neighbours to those within two network steps. We further enforce
a high level of homophily (see Methods). Figure 1C shows that this slows the
convergence rate relative to random rewiring. Under local rewiring, links are
established only with agents holding the same opinion, while links are broken
with uniform probability across all neighbours. This increases the link
density within opinion clusters and reduces inter-cluster links. The resulting
decrease in interactions between agents holding different opinions lowers the
chance of opinion change, which slows convergence.

###  2.4 Bridge rewiring

As recommended by (gladston_social_2019, ; williams_network_2015, ;
fotouhi_conjoining_2018, ), we implement an algorithm that guides the
establishment of links between differently thinking agents. We limit the
potential new neighbours to those outside agent  $\ignorespaces i$ ’s own
cluster and enforce a high level of heterophily (see Methods). Figure 1D shows
that this reduces the initial dip in cooperation and then accelerates
convergence relative to random rewiring. This is likely due to a reduction in
link density between similar opinion clusters and concurrent increase in link
density between polarized clusters resulting from agent  $\ignorespaces i$
rewiring only to agents holding the opposite opinion while links are broken
with uniform probability across all neighbours. Together with the assumed
slightly positive external field, the resulting increased likelihood of
cooperator-defector interaction reduces opinion reinforcement and
extremization during the polarization stage and subsequently accelerates
depolarization and convergence.

###  2.5 Effects of toplogical constraints and opinion preferences

Figure 2: Isolating the effects of topological constraint and opinion
preference. Rewiring is topologically limited to agents within 2 network steps
(local) or outside of agent  $\ignorespaces i$ ’s own cluster (bridge). In the
(similar) algorithms a link is established only if the agents are of a similar
opinion (homophily). In (opposite) rewiring, links are established only if the
agents are of opposite opinion (heterophily). A and B compare the role of the
limitation set on the pool of potential neighbours for (similar) and
(opposite) rewiring. We observe that the role of the topological constraint on
potential neighbours is small. C and D compare the role of homophily and
heterophily when establishing links. The effect on convergence rate is
significant. Homophily (i.e. (similar) rewiring) leads to reduced convergence
rates while heterophily (i.e. (opposite) rewiring) increases convergence rates

Our local and bridge rewiring algorithms differ on two dimensions: a
topological constraint (local or bridge) and an opinion preference determining
whether agents link to agents with similar or opposite opinions
(homo-/heterophily). To better understand the effects of each, we study the
cooperativity and polarization dynamics under changes in each alone (see
Methods). Figure 2A and B show that the effect of changing the topological
constraint alone is relatively small, with both local(similar) and
bridge(similar) rewiring leading to lower convergence rates than on a static
network. Establishing links between agents with similar opinions leads to a
decline in links between defectors and cooperators, thereby reducing the
probability of meaningful interaction. We see that the initial drop in
cooperation is slightly larger with the local(similar) algorithm. This is
likely because local(similar) rewiring increases the link density within
existing opinion clusters and thereby facilitates the initial spread of
strongly defective opinions.
While the convergence rates under both local(similar) and bridge(similar)
rewiring are lower than that achieved on the static network, the rate under
bridge(similar) rewiring approaches the static network rate, particularly when
the cooperativity exceeds 0.75. This may be because bridge(similar) rewiring
creates links that connect different clusters of the same opinion. Since we
include a clustering step in the link-building process (see Methods), this
creates a ‘super cluster’, making it increasingly unlikely that a non-
neighbour outside of agent  $\ignorespaces i$ ’s cluster is of the same
opinion. The number of new connections made therefore decreases until all
like-minded agents belong to the same cluster and no new connections are made.
The rate of convergence is then equal to that which occurs on the static
network.

We see quite different dynamics in panel B. Links are established only between
agents holding opposite opinions, which increases the likelihood of
interactions between them. Since those interactions most likely lead to more
cooperative opinions due to the influence of the external field, the
convergence rates are higher than those in the static network. Interestingly,
the convergence rates of local(opposite) rewiring and bridge(opposite)
rewiring are virtually the same, suggesting that the topological constraint
determining the pool of potential new neighbours does not significantly affect
the convergence rate.

Panels C and D isolate the effects of homo- or heterophily when establishing
links and we clearly see that these are significantly larger than the changes
in convergence rate caused by limitations on the pool of potential neighbours.
Moreover, it is not surprising that the assumption of homophily (i.e.
local(similar) and bridge(similar)) leads to a lower convergence rate than the
assumption of heterophily (i.e. local(opposite) and bridge(opposite)).
Notably, while the initial dip in cooperation levels can be reduced by
establishing links only between agents of opposite opinions, a corresponding
reduction in the maximum polarization is not seen. However, the significantly
narrower peaks in polarization indicated that the increased numbers of
interactions between cooperators and defectors in the ‘opposite’ algorithms
significantly shortens the polarization stage relative to those when no, or
‘similar’, rewiring are applied.

###  2.6 Comparing all effects

Figure 3: Comparing network evolutions for all rewiring algorithms. Rewiring
leads to accelerated convergence rates unless homophily is enforced when
establishing links (i.e. (similar) rewiring). Bridge(opposite) and
local(opposite) rewiring achieve the highest convergence rates. Local(similar)
leads to the lowest convergence rate.

Figure 3 compares the effects of all rewiring algorithms and their variations
(see Methods). We see that enabling a variable network structure accelerates
convergence rates when random rewiring is applied and when heterophily is
enforced while establishing links. This is true for both topological
constraints, which generally do not significantly impact the convergence rate.
Thus, to promote depolarization it is important to foster meaningful
interactions between cooperators and defectors. However, it is unimportant
whether the actors are located in different regions of the network or along
cluster boundaries. The algorithms assuming homophily for link establishment
result in lower convergence rates than those observed on the static network
structure. The convergence rate of the bridge(similar) algorithm approaches
the rate observed without rewiring.

###  2.7 External field

Figure 4: Effect of the external field on the convergence rate with random
rewiring. A stronger external field accelerates the convergence rate when
random rewiring is applied. The same holds true for local and bridge rewiring.
The strength of the effect diminishes with increasing field strength.

As shown in figure 4, the strength of the external field is a deciding factor
in convergence rate as well as the final state. As on the static network
(andersson_dynamics_2021, ), a stronger influence from the external field
accelerates the convergence rate for all rewiring algorithms. However, the
response of the convergence rate diminishes as the field strength increases.

###  2.8 Stubbornness

The response of the agent affected in each interaction is determined by the
external field, stochastic external factors as well as the level of
stubbornness and susceptibility, which are drawn from a normal distribution
for every agent and link (see Methods). While Andersson et al.
(andersson_dynamics_2021, ) found that effects of varying intervals for
stochastic external factors are only marginal, they also found a more
interesting dependence on stubbornness and susceptibility. Since the
implementations of stubbornness and susceptibility in the agent interaction
are similar (see Methods), (andersson_dynamics_2021, ) found that the
parameter dependence is qualitatively comparable. We therefore limit the
sensitivity analysis to stubbornness. Figure 5A shows that the convergence
rate peaks when stubbornness is assumed to be around 0.55; varying from this
value in either direction leads to lower convergence rates. The heat map in
figure 5B shows that for stubbornness levels outside of the interval [0.3,0.8]
it becomes increasingly unlikely that a final state of full cooperation is
reached. If the level of stubbornness is very low, individuals’ opinions are
volatile and the system becomes unpredictable. Coupled with stochasticity in
model inputs, this leads to a high variance in the distribution of final
states. The more stubborn agents are, the less likely they are to change their
opinion. Since the model starts with a majority of defectors, the likelihood
of defective end states is high when high stubbornness is assumed. Only at
intermediate levels of stubbornness will agents go with the dominant flow. If
this tends towards cooperation, as when a positive external field is applied,
a cooperative end state is reached.

Figure 5: Effects of stubbornness on convergence rate and final state with
random rewiring. A illustrates the convergence rates corresponding to
different levels of stubbornness. There is a peak in convergence rate when the
level of stubbornness is assumed to be 0.55. The convergence is slower the
further stubbornness departs from this. The heatmap in B depicts the final
states corresponding to the different levels of stubbornness. Final states are
volatile for levels of stubbornness below 0.3. For intermediate levels of
stubbornness, the system reaches a final state with full cooperation when a
positive external field is assumed. For high levels of stubbornness, it
becomes increasingly unlikely that the initial majority of defectors can be
persuaded, and the final state is more defective. The parameter dependence
does not change significantly when local or bridge rewiring are applied. To
show the small probability of values the saturation point of the color map is
set at 3 percent of probability.

##  3 Discussion

Our results show that variable network structure leads to faster
depolarization when links are established to agents holding different
opinions. If instead agents are homophilic when rewiring, depolarization is
slower than in a static network. This is true whether the potential neighbours
are limited to friends of friends or outside an agent’s local cluster.
Convergence to full cooperation is further accelerated by a strongly pro-
conservation sociopolitical environment but this does not qualitatively affect
our findings with respect to the topological constraints and opinion
preferences. Therefore, we find that reducing negative sentiment between
agents of opposite opinions, such that they are able to form persistent links,
can be a powerful lever for accelerating large-scale collective action.
This is consistent with previous findings by Axelrod et al.
(axelrod_preventing_2021, ) and Nguyen (nguyen_echo_2020, ), that interactions
between polarized clusters can act to maintain polarization when opposite
opinions are actively discredited through manipulations of trust. To dissolve
such echo chambers, trust relationships to agents and information sources
outside echo chambers need to be carefully rebuilt (nguyen_echo_2020, ), to
increase tolerance of alternative views (axelrod_preventing_2021, ). Centola
(Centola_2013, ), on the other hand, found that such heterophily can prevent a
society from achieving a social tipping point and thus collective action.
However, whereas Centola (Centola_2013, ) assumed that agents can perceive the
linear response of a collective good to their and others’ behaviour, in our
model it is assumed that such perception is impossible due to the common-pool
resource’s large size. As a result, interactions between agents play a vital
role in spreading opinion changes beyond cluster boundaries, and this happens
faster when agents link preferentially to those of opposite opinion.
It is natural to ask how our findings can inform measures to foster
depolarization in the real world. We propose that the current role of social
media in causing and maintaining polarization could potentially be limited and
even reversed if algorithms that determine displayed content and recommend
social links were modified to promote and maintain links between groups
holding opposite opinions (i.e out-groups) (santos_link_2021, ). Perhaps the
key challenge in this process would be to reduce negative emotions and instead
foster lasting empathy between the polarized groups. In an analysis of Twitter
data, Williams et al. (williams_network_2015, ) found a negative correlation
between the frequency of interaction between agents with opposite opinions and
the negative sentiment expressed towards them. This suggested that either
users with negative feelings towards those of opposite opinions are less
likely to interact with them repeatedly, or that users who frequently interact
with differently thinking users become less negative toward them over time (or
both). If the latter inference holds, more interaction between opinion
clusters could help to reduce negative sentiments and build empathy between
polarized groups. To do this effectively, however, algorithms may need to be
redesigned, away from simply maximising user engagement to also identifying
and promoting common ground between individuals who disagree on some
dimensions but may still agree on others (pawloff_contrarians_2014, ;
gladston_social_2019, ; williams_network_2015, ). According to Nguyen
(nguyen_echo_2020, ), for offline communities, restoring trust in political
and scientific institutions is central to fostering lasting connections
between actors who hold opposing views. This is because doing so could help to
weaken the influence of structures such as echo chambers that could manipulate
information from the outside (nguyen_echo_2020, ).
Our results give qualitative insight into the effects of different rewiring
strategies on the rates of convergence and depolarization in a social network.
However, we caution that the simplicity of our model algorithms limits the
translation of our findings to the real world. Crucially, we assumed that
newly established connections are stable whether or not they connect agents
holding similar or opposite opinions. This is a strong assumption, contrary to
the evidence that disagreeing individuals tend to avoid contact
(dunbar_anatomy_2018, ). This assumption could be relaxed in a future
extension of our work by adding link decay rates that depend on the agents’
states.
Our analysis reveals that both opinion preference and stubbornness can
strongly affect social dynamics in the presence of rewiring. However, both are
treated quite simply. Opinion preference is a binary parameter (homo- or
heterophily). Although stubbornness is chosen from a continuous spectrum, both
it and opinion preference are assumed to be uniform across all agents.
Generalising opinion preference to a wider range of discrete values or
continuous spectrum, and/or assuming heterogeneity in agents’ opinion
preferences and stubbornness values, would be other interesting model
extensions. A particularly interesting question is whether consensus time
might depend nonlinearly on the distribution of opinion preferences such that
only partial heterophily could accelerate consensus similarly to pure
heterophily as seen in our model. This might make our findings easier to apply
in real social networks. Moreover, varying the stubbornness between agents
would imply coexistence of easily persuaded and intransigent agents. Based on
the findings of Andersson et al. andersson_dynamics_2021  we might expect the
former to cluster around the latter, who are effectively ‘influencers’ with a
range of opinions, thus reinforcing polarization. An interesting question is
therefore whether and how strongly the depolarization dynamics depends on the
stubbornness distribution. Beyond simply expanding the parameter space, a next
step could be to convert parameters to coupled dynamical variables and analyze
feedback between agents’ opinion states, opinion preferences and stubbornness.
Nguyen suggests that the closer an opinion is to an extreme end of the
spectrum and the longer it has been since the opinion changed, the greater the
opinion holder’s stubbornness (nguyen_echo_2020, ). This could be captured by
making stubbornness dependent on agents’ current opinion state as well as its
trajectory over time. A higher level of stubbornness might also entail a
higher threshold that discourages actors from engaging with opposite opinions
(homophily). Opinion preference, in turn, affects stubbornness as it shapes
agents’ future contact networks, and thus the possible future encounters that
might alter agents’ opinions and break through their stubborn adherence to
their own opinion, creating a reinforcing feedback loop. Finally, including
feedback from agents’ opinion states to the sociopolitical environment could
effectively introduce institution formation into the model. In modeling such a
feedback loop, the sociopolitical environment could be made to depend on the
average opinion level of the network.

Another way to more realistically portray agents’ social environment would be
to extend the current model of simple contagion, which involves an opinion
update after one interaction, to one that represents complex contagion in
which opinions are updated only after repeated interactions
(centola_spread_2010, ; guilbeault_topological_2021, ). Furthermore, expanding
the current single variable agent opinion state into a multidimensional vector
would allow us to represent multiple character and cultural traits (Pham2021,
; Axelrod1997, ). Such a representation could be used to analyse how social
media algorithms could be designed to focus on peoples commonalities rather
than highlighting their differences, which could be a crucial step towards
building and maintaining empathy in online social networks
(dunbar_anatomy_2018, ; gladston_social_2019, ).

##  4 Methods

###  4.1 ABM model and setup

To gain qualitative insights into what qualities in link rewiring algorithms
accelerate the dissolution of polarized clusters, we extend an ABM on a
clustered scale-free network by Andersson et al. (andersson_dynamics_2021, ).
In contrast to the previous work, we include a variable network topology,
which is realised through link rewiring. The ABM is similar to those used in
statistical physics to model opinion dynamics without bounded confidence in
the interaction step (andersson_dynamics_2021, ; castellano_statistical_2007,
). We analyse polarization surrounding a collective action problem around a
large-scale CPR that shows no obvious feedback to individual actions. Hence,
the resource is not modelled explicitly, but implicitly by analysing the
spread of cooperation with a pro-conservation behavioural norm
(andersson_dynamics_2021, ; andersson_simple_2020, ). As a means of
implementation we rely on the networkx 2.3 package for Python 3.7.2. We use
the Holme-Kim clustered scale-free network growth algorithm
(holme_growing_2002, ) with an average degree of  $\ignorespaces\langle
k\rangle=8$ to establish a network of 1089 agents. Agents that are connected
by a link are referred to as neighbours or friends interchangeably. An agent’s
level of cooperation is represented by a continuous real valued variable
$\ignorespaces a\in[-1,1]$ , and is also referred to as the opinion of an
agent. An opinion of  $\ignorespaces-1$ signifies total defection,
$\ignorespaces 1$ signifies total cooperation. We refer to the agents as
defectors and cooperators accordingly. We differentiate between topological
clusters, detected by the Louvain algorithm (blondel_fast_2008, ), and opinion
clusters, which are made up of nodes sharing the same opinion. For each
scenario, 80 independent iterations are run. Results are obtained by computing
average trajectories over these iterations. Since we are concerned with
building collective action from a state of inaction, our simulations begin
with network states in which most agents defect.

The simulation iteratively follows eight steps:

(1) | Randomly select an agent  $\ignorespaces i$
---|---
(2) | Randomly select a neighbour  $\ignorespaces j$ of  $\ignorespaces i$
(3) | Perform interaction between agent  $\ignorespaces i$ and  $\ignorespaces j$ as described in 4.2
(4) | Update  $\ignorespaces i$ ’s opinion state
(5) | Select a non-neighbour  $\ignorespaces k$ of  $\ignorespaces i$ following one of the rewiring algorithms
(6) | poss. establish a new link between  $\ignorespaces i$ and  $\ignorespaces k$ as described below
(7) | Randomly select a neighbour  $\ignorespaces l$ of  $\ignorespaces i$
(8) | poss. break a link between  $\ignorespaces i$ and  $\ignorespaces l$ as described below

This process is iterated until the network converges to a steady state. While
the probability of being chosen as agent  $\ignorespaces i$ is independent of
an agent’s degree, the likelihood of being selected as a neighbor
$\ignorespaces j$ for interaction increases with rising degree. Updating i’s
behaviour, rather than  $\ignorespaces j$ ’s, ensures that agents with more
connections are more influential as their opinion state changes more slowly
over time.

###  4.2 Agent interaction

The change in the opinion  $\ignorespaces a$ of an agent  $\ignorespaces i$
when interacting with a neighbouring agent  $\ignorespaces j$ is given by

$\Delta a_{i}=\lvert a_{i}-a_{j}\rvert[w^{-}(1-a_{i})-w^{+}(1+a_{i})]$ (1)

Where  $\ignorespaces w^{+}$ and  $\ignorespaces w^{-}$ are the weights for an
agent becoming more or less cooperative respectively. To simplify we set
$\ignorespaces w^{+}=f(x_{ij},r,\xi$ ) and  $\ignorespaces w^{-}=1-w^{+}$ and
obtain

$\Delta a_{i}=\lvert a_{i}-a_{j}\rvert[2f(x_{ij},r,\xi)-1-a_{i}]$ (2)

where,

$f(x_{ij},r,\xi)=\begin{cases}0&\text{if $x_{ij}+\xi<-r$}\\\
\frac{1}{2r}(x_{ij}+r-\xi)&\text{if $x_{ij}+\xi\in[-r,r]$}\\\ 1&\text{if
$x_{ij}+\xi>r$}\end{cases}$ (3)

and,

$x_{ij}=w_{i}a_{i}+w_{ij}a_{j}+\phi$ (4)

The function  $\ignorespaces f(x_{ij},r,\xi)$ captures the response of the
affected agent. The parameters  $\ignorespaces w_{i}$ and  $\ignorespaces
w_{ij}$ represent the self-weight and the weight between agents
$\ignorespaces i$ and  $\ignorespaces j$ . This captures the local mechanisms
of stubbornness as well as the susceptibility, which determines the effect of
an agent on another.  $\ignorespaces\xi$ is a random noise term that is
distributed over the interval  $\ignorespaces\pm r$ and represents stochastic
external factors that may affect the interaction, such as the mood the agent
was in.  $\ignorespaces\phi$ captures global effects that influence all agents
at all times and can be referred to as the external field. This simulates
channels shared by all actors such as relevant political policies, media
reporting or the moral hegemony of society(andersson_simple_2020, ).

The parameters are chosen according to Andersson et al.
(andersson_dynamics_2021, ) and are summarized in table 1.

Parameter/Variable |  Represents |  Value
---|---|---
External field (  $\ignorespaces\phi$ ) |  Global factors:
Policy regulation, mass media, morals, etc |  0.05
Self weight (  $\ignorespaces w_{i}$ ) |  Resistance to influence:
Stubbornness, ideological commitment, etc |  0.6
Agent-agent weight (  $\ignorespaces w_{ij}$ ) |  Susceptibility:
Friendship, animosity, charisma, etc |  0.5
Agent-agent weight STD |  Susceptibility variance |  0.15
Initial cooperation ratio |  Initial average cooperativity |  -0.25
Initial cooperation SD |  Variance in initial cooperativity |  0.15
Randomness (  $\ignorespaces r$ ) |  Noise intensity in agent interactions |  0.1
Average degree |  Degree of nodes added during network growth |  8
Clustering |  Relative level of clustering in networks |  0.5
Table 1: Summary of model parameters and variables to provide an overview.

The external field is taken to be  $\ignorespaces\phi=0.05$ . It is chosen to
be positive in order to reflect a society that incentivizes cooperative
behaviour. The agent-to-agent weights  $\ignorespaces w_{ij}$ are drawn from a
normal distribution with mean  $\ignorespaces\mu=0.5$ and standard deviation
$\ignorespaces\sigma=0.15$ . We choose the self-weight to be  $\ignorespaces
w_{i}=0.6$ as it is assumed that people are more likely to keep their own
opinion than change it, but are quite prone to opinion change, nevertheless
(mallinson_effects_2018, ). The random noise term  $\ignorespaces\xi$ is
uncorrelated with the interaction and hence assumed to be uniformly rather
than normally distributed. The width of the noise is set to  $\ignorespaces
r=0.1$ . For a more detailed discussion of parameters we refer to
(andersson_dynamics_2021, ).

###  4.3 Link rewiring

By adding a link rewiring step after interaction, we allow for a variable
network structure where the neighbours of agent  $\ignorespaces i$ can change
over time. This potentially increases the variety of opinions an agent can be
exposed to and may help to dissolve polarized clusters faster
(andersson_dynamics_2021, ; lee_homophily_2019, ; fotouhi_conjoining_2018, ).
We formulate three rewiring algorithms to determine which properties of
rewiring algorithms accelerate the dissolution of polarized clusters. The
algorithms represent limiting cases on extreme ends of probability spectra and
are not accurate representations of real-life interaction. They are meant to
give first qualitative insights into how different ways of restructuring one’s
immediate social surroundings can affect the dissolution of polarization.
Since the rewiring algorithms make two additional assumptions compared to the
static network approach, we introduce algorithm variants, each differing by
only one assumption, to isolate the effects of each algorithm assumption.

Figure 6: Link establishing for different rewiring algorithms. The color of a
node depicts its level of cooperation. Orange nodes are cooperators and blue
nodes defectors. We focus on individual  $\ignorespaces i$ and mark the agents
available for establishing a link to with dashed circles. In A, B local
rewiring,  $\ignorespaces i$ can choose among agents within 2 network steps
(i.e friends of friends). The dashed lines show examples of nodes to whom a
link is actually established. In A local(similar) rewiring  $\ignorespaces i$
only establishes a link if the agents are of the same general opinion. In B
local(opposite) a link is established only if  $\ignorespaces i$ and the
chosen friend-of-friend are of opposite opinions. In C, D bridge rewiring
$\ignorespaces i$ can establish links to agents outside its own topological
cluster. In C bridge(opposite) a link is established only if the agents
opinions differ. In D bridge(similar)  $\ignorespaces i$ only establishes a
link if it agrees with the chosen agent.

####  4.3.1 Random rewiring

Random rewiring is taken as a baselLine scenario in order to evaluate the role
of unguided network change. Agent  $\ignorespaces i$ establishes a link to a
random non-neighbour  $\ignorespaces k$ with a probability  $\ignorespaces p$
. With a probability  $\ignorespaces q$ a random agent  $\ignorespaces l$ from
the pool of neighbours of agent  $\ignorespaces i$ is chosen and the link
connecting the two is cut.  $\ignorespaces q$ does not change with the number
of neighbours. The finding that cognitive capacity and available time limit
the number of friends an individual is able to hold (dunbar_anatomy_2018, ) is
included only indirectly, as an agent with high degree has a higher likelihood
of being chosen as the neighbour with whom a link is cut. To maintain a stable
average degree we assume  $\ignorespaces p=q$ .

####  4.3.2 Local rewiring

The local rewiring algorithm includes two tendencies found in empirical
studies to simulate a more realistic, yet highly simplified way of
establishing friendships. Firstly, an individual’s social surroundings
influence the pool of people they are likely to meet, making them more likely
to engage with people who are topologically close (santos_link_2021, ;
nasiri_new_2021, ; dunbar_anatomy_2018, ). Secondly, individuals’ engagements
are subject to homophily, meaning they will most likely show preferential
attachment to individuals with whom they have a lot in common
(williams_network_2015, ; dunbar_anatomy_2018, ; kozma_consensus_2008, ).
For implementation we use an approach similar to Crawford et al.
(crawford_resisting_2018, ). Figure 6A illustrates our assumptions. The color
of a node depicts its opinion. We focus on agent  $\ignorespaces i$ and limit
the pool of available agents  $\ignorespaces k$ for rewiring to those non-
neighbours that are within  $\ignorespaces d$ network steps. These are marked
with dashed circles. We set  $\ignorespaces d=2$ (i.e. agent  $\ignorespaces
i$ can only choose among friends of friends). This is a practical choice as
$\ignorespaces d=3$ (  $\ignorespaces\langle k\rangle^{3}=512$ ) would allow
agent  $\ignorespaces i$ to establish links with almost half of our chosen
network. We introduce homophily by assuming that an agent  $\ignorespaces i$
only establishes a link to selected agent  $\ignorespaces k$ if they are of
the same general opinion, where the general opinion is either above or below
0. I.e. a link is established if and only if  $\ignorespaces a_{i}\wedge
a_{k}<0$ or  $\ignorespaces a_{i}\wedge a_{k}\geq 0$ . We include preferential
attachment to maintain the network’s small world characteristics. For breaking
a link, a random neighbour  $\ignorespaces l$ of agent  $\ignorespaces i$ is
selected. To maintain a constant average degree, a link is broken if and only
if a link is established.

####  4.3.3 Bridge rewiring

In the bridge rewiring algorithm we incorporate recommendations for dissolving
polarization. These mainly focus on the role of inter-community interactions
and highlight their importance as information channels to increase
individuals’ exposure to differing views (andersson_dynamics_2021, ;
gladston_social_2019, ; williams_network_2015, ; lee_homophily_2019, ;
fotouhi_conjoining_2018, ). We implement bridge rewiring as a targeted
mechanism that deliberately aims to increase connectivity between disparate
communities. As figure 6C illustrates, we limit the potential new neighbours
of agent  $\ignorespaces i$ to the non-neighbours outside of  $\ignorespaces
i$ ’s own cluster. Clusters are identified by the Louvain algorithm
(blondel_fast_2008, ). Nevertheless it is likely that multiple communities of
defectors separated by ones of cooperators exist within a network. To ensure
that the newly established link is connecting agents of opposite opinion we
assume that agents are subject to extreme heterophily. Agent  $\ignorespaces
i$ establishes a link to agent  $\ignorespaces k$ if and only if their general
opinion is opposite, i.e. if  $\ignorespaces a_{i}\geq 0\wedge a_{k}<0$ or
$\ignorespaces a_{i}<0\wedge a_{k}\geq 0$ . We use the term, out-group to
refer to non-neighbors outside agent  $\ignorespaces i$ ’s cluster who hold an
opposite opinion. As with the local rewiring algorithm, preferential
attachment is included. In order to keep the average degree stable, agent
$\ignorespaces i$ breaks a link with a random neighbour  $\ignorespaces l$ if
and only if a new link was established.

####  4.3.4 Rewiring variations

Compared to random rewiring the effects of two additional assumptions are
tested in the local and bridge rewiring algorithm. The topological constraint
places a limitation on the pool of potential neighbours. The opinion
preference determines whether agents are constrained by homo- or heterophily
in their decision whether or not to establish a link. In order to isolate the
effects of each assumption, additional variations that differ by only one
assumption are tested. We keep the terms local and bridge rewiring to refer to
the topological constraint. A notation of (similar) and (opposite) is added to
indicate whether links are established only if agents are of the same opinion
(i.e homophily) or only if they are of a different opinion (i.e heterophily).
Local(similar) therefore refers to the originally formulated local rewiring
algorithm, in which agent  $\ignorespaces i$ chooses a random agent
$\ignorespaces k$ within two network steps and establishes a link if they are
of the same general opinion (see figure 6A). Local(opposite) describes a
variation of the algorithm in which agent  $\ignorespaces i$ establishes a
link to agent  $\ignorespaces k$ , who is selected within two network steps,
if and only if they disagree in their general opinions (see figure 6B).
Bridge(similar) (see figure 6D) and bridge(opposite) (see figure 6C) both
limit the potential neighbours of agent  $\ignorespaces i$ to non-neighbours
outside of  $\ignorespaces i$ ’s own cluster. However, a link is established
only if the agents agree in the former, and only if they disagree in the
latter.

###  4.4 Code availability

The Python3 scripts used for the simulations and evaluation are provided
online at https://github.com/lifrei/It-s_how_we_talk_that_matters.git.

Acknowledgments We thank D. Andersson, who provided us with the initial code
and significantly helped during the first phase of the project.

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$f:{\mathfrak{h}}\rightarrow{\mathfrak{h}}\otimes x{\mathbb{C}}[[x]]$ in
Proposition 5.7 is assumed to be a linear map
$f:{\mathfrak{h}}\rightarrow{\mathfrak{h}}\otimes(x{\mathbb{C}}[[x]]\cap{\mathbb{C}}_{\phi}((x)))$,
then using Lemma 4.19 (a technical result) we can show that the $B_{L}$-module
structure $Y_{M}^{f}(\cdot,x)$ on $V_{L}$ is also $\phi$-compatible.

Combining Theorems 2.25 and 3.6, we have:

######  Theorem 5.9.

Let $f:{\mathfrak{h}}\rightarrow{\mathfrak{h}}\otimes x{\mathbb{C}}[[x]]$ be
any linear map. Then there exists a quantum vertex algebra structure on
$V_{L}$, whose vertex operator map, denoted by $Y^{f}(\cdot,x)$, is uniquely
determined by

(5.29) $\displaystyle
Y^{f}(e_{\alpha},x)=Y(e_{\alpha},x)\exp(\Phi(f(\alpha,x)))\quad\textnormal{{for
}}\alpha\in L,$ (5.30) $\displaystyle
Y^{f}(h,x)=Y(h,x)+\Phi(f^{\prime}(h,x))\quad\textnormal{{for
}}h\in{\mathfrak{h}}.$

Denote this quantum vertex algebra by $V_{L}^{f}$. Then $V_{L}^{f}$ is an
irreducible $V_{L}^{f}$-module and in particular, $V_{L}^{f}$ is non-
degenerate. Furthermore, the following relations hold:

$\displaystyle[Y^{f}(a,x),Y^{f}(b,z)]$ (5.31) $\displaystyle=$
$\displaystyle{\langle}a,b{\rangle}\frac{\partial}{\partial
z}x^{-1}\delta\left(\frac{z}{x}\right)-{\langle}f^{\prime\prime}(a,x-z),b{\rangle}+{\langle}f^{\prime\prime}(b,z-x),a{\rangle},$
$\displaystyle[Y^{f}(a,x),Y^{f}(e_{\beta},z)]={\langle}a,\beta{\rangle}Y^{f}(e_{\beta},z)x^{-1}\delta\left(\frac{z}{x}\right)$
(5.32) $\displaystyle-$ $\displaystyle
Y^{f}(e_{\beta},z)({\langle}f^{\prime}(\beta,z-x),a{\rangle}+{\langle}f^{\prime}(a,x-z),\beta{\rangle}),$
(5.33)
$\displaystyle\textnormal{{Res}}_{x}x^{-1}Y^{f}(\alpha,x)e_{\alpha}=\mathcal{D}e_{\alpha}+{\langle}f^{\prime}(\alpha,0),\alpha{\rangle}e_{\alpha},$
$\displaystyle(x-z)^{-{\langle}\alpha,\beta{\rangle}-1}Y^{f}(e_{\alpha},x)Y^{f}(e_{\beta},z)$
$\displaystyle\ \ \ \
-(-z+x)^{-{\langle}\alpha,\beta{\rangle}-1}e^{{\langle}\beta,f(\alpha,x-z){\rangle}-{\langle}\alpha,f(\beta,z-x){\rangle}}Y^{f}(e_{\beta},z)Y^{f}(e_{\alpha},x)$
(5.34) $\displaystyle=\ $
$\displaystyle\varepsilon(\alpha,\beta)Y^{f}(e_{\alpha+\beta},z)x^{-1}\delta\left(\frac{z}{x}\right)$

for $a,b\in{\mathfrak{h}},\ \alpha,\beta\in L$.

###### Proof.

For the first assertion, the uniqueness is clear, so we only need to show the
existence. With Propositions 5.4 and 5.7, by Theorems 2.25 and 3.6 we have a
quantum vertex algebra $\mathfrak{D}_{Y_{M}^{f}}^{\rho}(V_{L})$ with $V_{L}$
as its underlying space. From Theorem 2.25, for $h\in{\mathfrak{h}},\
\alpha\in L$, we have

$\displaystyle Y^{f}(h,x)$ $\displaystyle=$ $\displaystyle
Y^{\sharp}(\rho(h),x)=Y^{\sharp}(h\otimes{\bf 1}+{\bf 1}\otimes
h,x)=Y(h,x)+Y_{M}^{f}(h,x)$ $\displaystyle=$ $\displaystyle
Y(h,x)+\Phi(f^{\prime}(h,x)),$ $\displaystyle Y^{f}(e_{\alpha},x)$
$\displaystyle=$ $\displaystyle
Y^{\sharp}(\rho(e_{\alpha}),x)=Y^{\sharp}(e_{\alpha}\otimes
e^{\alpha},x)=Y(e_{\alpha},x)Y_{M}^{f}(e^{\alpha},x)$ $\displaystyle=$
$\displaystyle Y(e_{\alpha},x)\exp(\Phi(f(\alpha,x))).$

Thus we have a quantum vertex algebra structure on $V_{L}$ as desired.

For the second assertion, since $V_{L}$ is an irreducible $V_{L}$-module, it
suffices to prove that any $V_{L}^{f}$-submodule $U$ of $V_{L}^{f}$ is also a
$V_{L}$-submodule of $V_{L}$. For $h\in{\mathfrak{h}}$, since
$f^{\prime}(h,x)\in{\mathfrak{h}}\otimes x{\mathbb{C}}[[x]]$, we have

(5.35) $\displaystyle Y^{f}(h,x)^{-}=Y(h,x)^{-}=\sum_{n\geq 0}h(n)x^{-n-1},$
(5.36) $\displaystyle Y^{f}(h,x)^{+}=Y(h,x)^{+}+\Phi(f^{\prime}(h,x)).$

From (5.35) we conclude $h(n)U\subset U$ for $h\in{\mathfrak{h}},\ n\geq 0$.
Furthermore, we get $\Phi(f^{\prime}(h,x))U\subset U[[x]]$ for
$h\in{\mathfrak{h}}$. Then from (5.36) we deduce $Y(h,x)^{+}U\subset U[[x]]$.
This proves that $U$ is an $\widehat{{\mathfrak{h}}}$-submodule of $V_{L}$.
Furthermore, for $\alpha\in L$, as
$Y(e_{\alpha},x)=Y^{f}(e_{\alpha},x)\exp(-\Phi(f(\alpha,x)))$ we obtain
$Y(e_{\alpha},x)U\subset U[[x,x^{-1}]]$. Then it follows that $U$ is also a
$V_{L}$-submodule of $V_{L}$. Therefore, $V_{L}^{f}$ is an irreducible
$V_{L}^{f}$-module, which implies that $V_{L}^{f}$ is a simple quantum vertex
algebra and it is non-degenerate by [Li5].

For the furthermore assertion, note that for $u,v\in{\mathfrak{h}},\
g(x)\in{\mathbb{C}}((x))$, we have

(5.37) $\displaystyle\Phi(u\otimes g(x))v=\sum_{n\geq
0}(-1)^{n}\frac{1}{n!}g^{(n)}(x)u_{n}v=-{\langle}u\otimes
g^{\prime}(x),v{\rangle}.$

Combining this with (5.25) and Theorem 3.6, we get that

(5.38) $\displaystyle{S}(x)(b\otimes a)=b\otimes a+{\bf 1}\otimes{\bf
1}\otimes({\langle}f^{\prime\prime}(b,x),a{\rangle}-{\langle}f^{\prime\prime}(a,-x),b{\rangle}),$
(5.39) $\displaystyle S(x)(b\otimes e_{\alpha})=b\otimes e_{\alpha}-{{\bf
1}}\otimes
e_{\alpha}\otimes({\langle}f^{\prime}(\alpha,-x),b{\rangle}+{\langle}f^{\prime}(b,x),\alpha{\rangle}),$
(5.40) $\displaystyle S(x)(e_{\alpha}\otimes b)=e_{\alpha}\otimes
b+e_{\alpha}\otimes{{\bf
1}}\otimes({\langle}f^{\prime}(\alpha,x),b{\rangle}+{\langle}f^{\prime}(b,-x),\alpha{\rangle}),$
(5.41) $\displaystyle{S}(x)(e_{\beta}\otimes e_{\alpha})=(e_{\beta}\otimes
e_{\alpha})\otimes
e^{{\langle}\beta,f(\alpha,-x){\rangle}-{\langle}\alpha,f(\beta,x){\rangle}},$

where $a,b\in{\mathfrak{h}}$ and $\alpha,\beta\in L$ and $S(x)$ is the quantum
Yang-Baxter operator of $V_{L}^{f}$. Since
$f({\mathfrak{h}},x)\subset{\mathfrak{h}}\otimes x{\mathbb{C}}[[x]]$, we have
that

$\displaystyle\Phi(f(a,x))e_{\beta}\in{\mathbb{C}}e_{\beta}\otimes
x{\mathbb{C}}[[x]]\quad\textnormal{{and}}\quad\Phi(f(a,x))b\in{\mathbb{C}}{{\bf
1}}\otimes x{\mathbb{C}}[[x]].$

It follows that

(5.42)
$\displaystyle\textnormal{{Sing}}_{x}\Phi(f^{\prime}(a,x))b=0=\textnormal{{Sing}}_{x}\Phi(f^{\prime}(a,x))\beta=\textnormal{{Sing}}_{x}\exp\left(\Phi(f(a,x))\right)b$

and

$\displaystyle\exp\left(\Phi(f(a,x))\right)e_{\beta}\in
e_{\beta}+{\mathbb{C}}e_{\beta}\otimes x{\mathbb{C}}[[x]].$

Then we get that

$\displaystyle\textnormal{{Sing}}_{x}Y^{f}(a,x)b=\textnormal{{Sing}}_{x}Y(a,x)b,\quad\textnormal{{Sing}}_{x}Y^{f}(a,x)e_{\beta}=\textnormal{{Sing}}_{x}Y(a,x)e_{\beta},$
$\displaystyle\textnormal{{Res}}_{x}x^{-1}Y^{f}(\alpha,x)e_{\alpha}=\partial
e_{\alpha}+{\langle}f^{\prime}(\alpha,0),\alpha{\rangle}e_{\alpha},$
$\displaystyle\textnormal{{Sing}}_{x}x^{{\langle}\alpha,\beta{\rangle}}Y^{f}(e_{\alpha},x)e_{\beta}=\varepsilon(\alpha,\beta)e_{\alpha+\beta}.$

Combining these with Lemma 2.6, Definition 2.7 and relations (5.38), (5.39),
(5.41), we complete the proof. ∎

Combining Proposition 5.2 and Theorem 5.9, we immediately get that

######  Corollary 5.10.

Let $f:{\mathfrak{h}}\to{\mathfrak{h}}\otimes x{\mathbb{C}}[[x]]$ be a linear
map such that

$\displaystyle{\langle}f(a,x),b{\rangle}={\langle}f(b,-x),a{\rangle},\quad\textnormal{{for
}}a,b\in{\mathfrak{h}}.$

Then the identity map on ${\mathfrak{h}}\oplus{\mathbb{C}}_{\varepsilon}[L]$
determines a nonlocal vertex algebra isomorphism from $V_{L}$ to $V_{L}^{f}$.

Recall that ${\mathfrak{h}}={\mathbb{C}}\otimes_{\mathbb{Z}}L\subset V_{L}$.
Let $G$ be an automorphism group of $V_{L}$ with a linear character $\chi$
such that $G$ preserves ${\mathfrak{h}}$. It follows that $G$ preserves the
bilinear form on ${\mathfrak{h}}$ and the standard conformal vector $\omega$
of $V_{L}$. Thus $gL(0)=L(0)g$ on $V_{L}$ for $g\in G$. Then $V_{L}$ is a
$(G,\chi)$-module vertex algebra with $R(g)=\chi(g)^{L(0)}g$ for $g\in G$.
Furthermore, we have:

######  Proposition 5.11.

Let $G$ be an automorphism group of $V_{L}$ such that
$G({\mathfrak{h}})={\mathfrak{h}}$ and let $\chi$ be a linear character of
$G$. Assume $\eta(\cdot,x):{\mathfrak{h}}\rightarrow{\mathfrak{h}}\otimes
x{\mathbb{C}}[[x]]$ is a linear map satisfying the condition

(5.43) $\displaystyle(\mu\otimes 1)\eta(h,x)=\eta(\mu(h),\chi(\mu)x)\ \ \
\mbox{ for }\mu\in G,\ h\in{\mathfrak{h}}.$

Then $V_{L}^{\eta}$ is a $(G,\chi)$-module quantum vertex algebra with $R$
defined by $R(g)=\chi(g)^{L(0)}g$ for $g\in G$.

###### Proof.

Recall that for $v\in V_{L},\ f(t)\in{\mathbb{C}}((t))$,

$\Phi(v\otimes f(t))(z)=\text{Res}_{x_{1}}Y(v,x_{1})f(z-x_{1}).$

For $\mu\in G$, set $R_{\mu}=\chi(\mu)^{L(0)}\mu$. For $h\in{\mathfrak{h}}$,
we have

$\displaystyle R_{\mu}\Phi(\eta(h,t))(z)=\ $
$\displaystyle\text{Res}_{x_{1}}R_{\mu}Y(\eta(h,z-x_{1}),x_{1})$
$\displaystyle=\ $ $\displaystyle\text{Res}_{x_{1}}\chi(\mu)Y((\mu\otimes
1)\eta(h,z-x_{1}),\chi(\mu)x_{1})R_{\mu}\mu$ $\displaystyle=\ $
$\displaystyle\text{Res}_{x_{1}}\chi(\mu)Y(\eta(\mu(h),\chi(\mu)(z-x_{1})),\chi(\mu)x_{1})R_{\mu}$
$\displaystyle=\ $
$\displaystyle\text{Res}_{z_{1}}Y(\eta(\mu(h),\chi(\mu)z-z_{1}),z_{1})R_{\mu}$
(5.44) $\displaystyle=\ $
$\displaystyle\Phi(\eta(\mu(h),t))(\chi(\mu)z)R_{\mu}.$

Then for $\alpha\in L$ we get

$\displaystyle R_{\mu}Y^{\eta}(e_{\alpha},z)R_{\mu}^{-1}=\ $ $\displaystyle
R_{\mu}Y(e_{\alpha},z)\exp(\Phi(\eta(\alpha,t))(z))R_{\mu}^{-1}$
$\displaystyle=\ $ $\displaystyle
Y(R_{\mu}e_{\alpha},\chi(\mu)z)\exp\left(R_{\mu}\Phi(\eta(\alpha,t))(z)R_{\mu}^{-1}\right)$
$\displaystyle=\ $
$\displaystyle\chi(\mu)^{\frac{1}{2}{\langle}\alpha,\alpha{\rangle}}Y(\mu
e_{\alpha},\chi(\mu)z)\exp\left(\Phi(\eta(\mu(\alpha),t)(\chi(\mu)z)\right)$
$\displaystyle=\ $
$\displaystyle\lambda_{\mu,\alpha}\chi(\mu)^{\frac{1}{2}{\langle}\mu(\alpha),\mu(\alpha){\rangle}}Y(e_{\mu(\alpha)},\chi(\mu)z)\exp\left(\Phi(\eta(\mu(\alpha),t))(\chi(\mu)z)\right)$
$\displaystyle=\ $
$\displaystyle\lambda_{\mu,\alpha}\chi(\mu)^{\frac{1}{2}{\langle}\mu(\alpha),\mu(\alpha){\rangle}}Y^{\eta}(e_{\mu(\alpha)},\chi(\mu)z)$
$\displaystyle=\ $ $\displaystyle Y^{\eta}(R_{\mu}e_{\alpha},\chi(\mu)z),$

where $\mu(e_{\alpha})=\lambda_{\mu,\alpha}e_{\mu(\alpha)}$ with
$\lambda_{\mu,\alpha}\in{\mathbb{C}}$. Note that (5.43) implies

$(\mu\otimes 1)\eta^{\prime}(h,x)=\chi(\mu)\eta^{\prime}(\mu(h),\chi(\mu)x).$

Following the argument in (5.2) we obtain

$\displaystyle
R_{\mu}\Phi(\eta^{\prime}(h,t))(z)R_{\mu}^{-1}=\chi(\mu)\Phi(\eta^{\prime}(\mu(h),t))(\chi(\mu)z)=\Phi(\eta^{\prime}(R_{\mu}(h),t))(\chi(\mu)z).$

Using this we get

$R_{\mu}Y^{\eta}(h,x)R_{\mu}^{-1}=R_{\mu}\left(Y(h,x)+\Phi(\eta^{\prime}(h,t))(x)\right)R_{\mu}^{-1}=Y^{\eta}(R_{\mu}h,\chi(\mu)x).$

Note that $\\{e_{\alpha}\ |\ \alpha\in L\\}\cup{\mathfrak{h}}$ is a generating
subset of $V_{L}^{\eta}$ as a nonlocal vertex algebra. By using weak
associativity (or by Lemma 2.11) it is straightforward to show that

$R_{\mu}Y^{\eta}(v,x)R_{\mu}^{-1}=Y^{\eta}(R_{\mu}v,\chi(\mu)x)\ \ \ \mbox{
for all }\mu\in G,\ v\in V_{L}.$

Therefore, $V_{L}^{\eta}$ is a $(G,\chi)$-module quantum vertex algebra as
desired. ∎

######  Remark 5.12.

We here show the existence of a linear map $\eta(\cdot,x)$ satisfying the
condition (5.43) in Proposition 5.11 with $G$ a finite group. Let
$g(\cdot,x):\ {\mathfrak{h}}\rightarrow{\mathfrak{h}}\otimes
x{\mathbb{C}}[[x]]$ be any linear map. Define a linear map
$\widetilde{g}(\cdot,x):\ {\mathfrak{h}}\rightarrow{\mathfrak{h}}\otimes
x{\mathbb{C}}[[x]]$ by

(5.45) $\displaystyle\widetilde{g}(\alpha,x)=\sum_{\sigma\in G}(\sigma\otimes
1)g(\sigma^{-1}\alpha,\chi(\sigma^{-1})x)\ \ \ \mbox{ for
}\alpha\in{\mathfrak{h}}.$

Then it is straightforward to show

(5.46) $\displaystyle(\sigma\otimes
1)\widetilde{g}(\alpha,x)=\widetilde{g}(\sigma\alpha,\chi(\sigma)x)\quad\textnormal{{
for }}\sigma\in G,\ \alpha\in{\mathfrak{h}}.$

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