# Social clustering reinforces external influence on the majority opinion
model

Niels Van Santen  Jan Ryckebusch  Luis E. C. Rocha

###### Abstract

Public opinion is subject to peer interaction via social networks and external
pressures from the media, advertising, and other actors. In this paper, we
study the interaction between external and peer influence on the stochastic
opinion dynamics of a majority vote model. We introduce a model where agents
update their opinions based on the combined influence from their local
neighbourhood (peers) and from an external actor in the transition rates. In
the first model, the external influence is only felt by agents non-aligned
with the external actor (”push strategy”). In the second model, agents are
affected by external influence, independently of their opinions (”nudging
strategy”). In both cases, the external influence increases the possible
macroscopic outcomes. These outcomes are determined by the chosen strategy. We
also find that the social network structure affects the opinion dynamics, with
high social clustering positively reinforcing the external influence while
degree heterogeneity weakens it. These findings are relevant to businesses and
policy making, helping to understand how groups of individuals collectively
react to external actors.

###### keywords:

opinion dynamics , complex networks , complex contagion , external influence ,
peer influence

††journal: Physica A: Statistical Mechanics and its Applications

[inst1]organization=Department of Data-Analysis, Ghent
University,addressline=Henri Dunantlaan 1, city=Ghent, postcode=9000,
country=Belgium

[inst2]organization=Department of Physics and Astronomy, Ghent
University,addressline=Proeftuinstraat 86, city=Ghent, postcode=9000,
country=Belgium

[inst3]organization=Department of Economics, Ghent
University,addressline=Sint-Pietersplein 5, city=Ghent, postcode=9000,
country=Belgium

The interaction between peer pressure and an external field leads to more
complex behaviour than reinforcing the external opinion.

The intensity and weight of an external field push the population into a
greater variation of macroscopic states.

Macroscopic outcomes depend on the alignment of agents to the external field.

Social clustering reinforces the influence of an external field.

Degree heterogeneity and hubs hinder the effect of an external field.

##  1 Introduction

Statistical physics offers helpful tools to study social systems involving
collective and emergent phenomena [1, 2, 3, 4, 5]. In particular, physical
models have been used to explore and understand the collective behaviour of
opinions in society [6, 7]. Supporting the idea that human behaviour is not
wholly unpredictable [8]. Given the universality in order-disorder
transitions, it is appropriate to use physics-based models to investigate
large-scale regularities as collective effects. Models composed of adaptive
agents have been effective at describing complex social behaviour [9, 10, 11].
The rules and constraints of those models, for example, threshold imitation or
homophily [12], should be informed by psychology and social science. Opinion
dynamics is a multidisciplinary field built on early sociological observations
that people tend to conform to each other’s behaviours and opinions. This led
to questions about collective behaviour and inspired early models in sociology
and economics, such as the Schelling model [13] and Axelrod’s cultural
dissemination model [14]. Behaviour and opinions can only spread when there is
interaction, implying the importance of the structure of the underlying
contact networks [15].

In public opinion formation, next to interpersonal social interaction, also
mass media and other external actors play a role [16, 17, 18, 19, 20]; thus
opinion dynamics is subject to the interaction of personal predispositions,
external media influence and social interaction [21]. We are interested in
exploring the interaction between the internal peer pressure towards
conforming, and an external influence. In the last decades, many models for
opinion dynamics have been created for discrete and continuous opinion spaces
or a combination of both [22]. Voting problems have served as early
applications. Examples include the well-known Voter Model [23, 24] and its
many variations [25]. We focus here on binary opinion models, particularly
complex contagions [26], where opinions require reinforcement to become more
easily adapted. This idea of reinforcement has been implemented in different
ways. For example, by interacting with more than one neighbour [27, 28, 29,
30, 31], via the use of memory [32], by introducing different levels of
confidence in the opinion [33], or by repeated interactions [34] and
algorithmic personalisation [35, 36]. We will follow the first approach and
have individuals take their entire social neighbourhood into account when
updating their opinions.

Several investigations have been conducted to include an external influence,
such as the presence of media, in opinion dynamics models. In the binary
opinion case, for the Voter Model and some of its extensions [37, 38, 39], the
Majority Rule model [40] and the Sznajd Model [41]. Often, at every time step,
there is a probability $r$ with which an agent interacts with its environment,
feeling only internal pressure, and a probability $1-r$ they only feel the
external media influence in that time step. The model we introduce in this
paper makes a different assumption that agents feel both internal and external
pressure at every time step. This allows us to understand the effect of
opposing and reinforcing interactions simultaneously; an agent does not feel
external influence independently from their neighbourhood pressure. This paper
also addresses the impact of external influence applied to complex contagion.
It proposes asymmetric scenarios where the external influence is felt
depending on the (opinion) alignment of the agent with the media. The
concurrent influence on complex contagions and the comparison between
homogeneous and alignment-based influence have been subject to little
investigation in the literature [7].

This paper introduces an opinion dynamics model containing the following
properties: (i) complex contagion spread, (ii) concurrent interaction between
internal and external pressure, (iii) social network structure, and (iv)
stochasticity. The latter is chosen to resemble real-world noise and
incomplete information during decision-making. To capture the above
properties, we propose a model where, at each time step, a randomly chosen
agent updates their opinion with a certain transition rate that results from
the interaction of internal and external influence. Internal influence
contributes to the transition rate according to the fraction of nearest
neighbours disagreeing with the chosen agent. The external influence adds or
subtracts from that transition rate based on the disagreement or agreement
between the agent and the media, respectively. The media influence will have a
strength fixed in time, and the relative weight for internal and external
pressures is set by a parameter $\alpha$. We test two versions of this model:
an asymmetric version where the media only influences agents that disagree
with it and a symmetric version where every agent, independently of its own
opinion, feels the external media influence such that the media adjusts
transition rates in both directions. Both versions enrich the space of
possible consensus regimes over the absence of external influence, with the
asymmetric version showing the most variation concerning possible final
consensus states.

##  2 Model

###  2.1 Networks

A network comprises a set of $N$ nodes $i$ connected by $M$ links $(i,j)$,
where we exclude self-links $(i,i)$ and multiple links between the same node
pair. We will restrict ourselves to undirected networks; a link $(i,j)$
implies a symmetric link $(j,i)$. The topology is characterised by a symmetric
matrix $\boldsymbol{A}$, the adjacency matrix, with entries $a_{ij}=1$ if a
link exists between node $i$ and $j$, and zero otherwise. Each node $i$ has a
degree $k_{i}=\sum_{j}a_{ij}$ representing the number of other nodes $j$ to
which $i$ is connected. All networks used in this work are connected, i.e.
$k_{i}>0,\;\forall i$, and have $N=1000$ nodes with $\langle k\rangle\approx
10$. The average shortest path length is given by $\langle
l\rangle=\frac{1}{N(N-1)}\sum_{i\neq j}l_{ij}$, where $l_{ij}$ is the
shortest-path between nodes $i$ and $j$, and the number of local bridges $\mu$
are the links whose endpoints have no common neighbours. The average
clustering coefficient is $\langle
cc\rangle=\frac{1}{N}\sum_{i}\frac{2M_{i}}{k_{i}(k_{i}-1)}$, where $M_{i}$ is
the number of links between common neighbours or $i$, and the betweenness
centrality of node $k$ is $B_{k}=\sum_{ij}\frac{\sigma(i,k,j)}{\sigma(i,j)}$,
where $\sigma(i,j)$ is the number of shortest-paths between $i$ and $j$, and
$\sigma(i,k,j)$ is the number of shortest-paths between $i$ and $j$ passing
through node $k$.

We will use three theoretical network models to study the opinion dynamics and
investigate the sensitivity of our results on the different structures
observed in real-world social networks. We are particularly interested in
studying short path length, the combination of short path length and high
clustering, and the presence of high-degree nodes (i.e. hubs). The first model
is the Erdős-Rényi (ER) model, where nodes are randomly connected with
probability $p$. This model exhibits short path lengths $\langle d\rangle$ and
low (local) clustering $\langle cc\rangle$. The second model is the Watts-
Strogatz (WS) model, which combines short path lengths with high local
clustering. It is constructed from a regular lattice by rewiring edges with a
probability $q$. According to this probability, the WS model leads to
structures intermediate between the regular lattice ($q=0$) with high
clustering and long path lengths and the ER model ($q=1$) with low clustering
and short path lengths. Finally, the Barabási-Albert (BA) preferential
attachment growth model, where nodes are added to the network and connected
preferentially to existing nodes with higher degrees, is used to obtain a
structure with heterogeneous degree distribution. Table 1 shows the mean
values of relevant network measures characterising those networks.

| ER | WS | BA
---|---|---|---
$\langle k\rangle$ | 9.926 $\pm$ 0.077 | 10 $\pm$ 0 | 9.95 $\pm$ 0
$\langle cc\rangle$ | 0.010 $\pm$ 0.001 | 0.089 $\pm$ 0.003 | 0.040 $\pm$ 0.003
$\langle d\rangle$ | 3.267 $\pm$ 0.009 | 3.382 $\pm$ 0.004 | 2.977 $\pm$ 0.012
$\langle\mu\rangle/M$ | 0.909 $\pm$ 0.009 | 0.558 $\pm$ 0.007 | 0.685 $\pm$ 0.014
$\langle B_{k}\rangle$ | 0.002 $\pm$ 0.001 | 0.002 $\pm$ 0.001 | 0.002 $\pm$ 0.007
Table 1: The mean and the standard deviation ($\pm$) of relevant network
measures over ten realisations of the ER, WS, and BA network models with
$N=1000$ nodes.

###  2.2 Opinion Dynamics

We devise a model with $N$ agents (nodes) $i=0,1,\ldots,N-1$. At time step t,
each agent has a binary opinion $o_{i}(t)\in\mathcal{O}=\\{A,B\\}$. We assume
that the opinion of agent $i$ can change as a result of two competing
mechanisms: (i) pressure from its social contacts $\nu(i)$, i.e. the ratio of
neighbours $j$ of $i$ having opinion $o_{j}(t)\neq o_{i}(t)$ (Eq. 1) and (ii)
an external fixed source promoting opinion $A$ and affecting all agents
equally $\mathcal{E}^{A(B)}\in[0,1]\subset\mathbb{R}$, i.e. an external field
representing, for example, mass media, institutions, or nudging.

$\mathcal{I}_{i}(t)=\frac{\sum_{<ij>}^{o_{j}(t)\neq
o_{i}(t)}}{k_{i}}o_{j}(t).$ (1)

Agents with opinion $o_{i}(t)=A(B)$ have a transition probability towards
opinion $B(A)$ at time $t$ given by $W_{i}^{A(B)\rightarrow B(A)}(t)$. The
relative importance of the external field concerning social pressure is
regulated via the parameter $0\leq\alpha\leq 1$, reflecting the trust agents
have in the external opinion. Larger $\alpha$ means that agents give more
weight to the external field than to pressure from social contacts. We
consider two stochastic versions of the model, differing in how agents are
subject to the pressure of the external field.

In the first model (hereafter called asymmetric model), the external field
only influences agents with an opinion opposite to the field, i.e. agents
agreeing with the external field (defined to be opinion $A$) are only
influenced by their social contacts (there is no reinforcement). Equations 2
give the transition rates for agents with opinions $A$ and $B$. In this case,
the external field can promote the transition from $B\rightarrow A$ but not
vice-versa. This corresponds to a scenario where the external field acts
towards one of the opinions, e.g. an employee has to decide on adopting a
method or software and is subject to both “outside” influence (employer
suggests using software $A$) and “internal” peer pressure (colleagues
recommend $A$ or $B$). An employee who uses the default option $A$, as set by
the company, does not feel pressure from the employer but only from colleagues
using $B$. Meanwhile, when using $B$, the employee feels the influence of
colleagues and the employer. In the limiting cases, the model reduces to
complex contagion for $\alpha=0$ whereas for $\alpha=1$, transitions from
$B\rightarrow A$ only depend on the external field. We could also model the
case of outside pressure affecting only those with the same opinion as the
external field, but that would lead to confirmation bias and, thus, simple
reinforcement.

$\begin{split}W_{i}^{A\rightarrow B}(t)&=\mathcal{I}_{i}^{B}(t)\\\
W_{i}^{B\rightarrow
A}(t)&=(1-\alpha)\mathcal{I}_{i}^{A}(t)+\alpha\mathcal{E}^{A}.\end{split}$ (2)

In the second model (hereafter called symmetric model), all agents may be
influenced by the external field (Eqs. 3). This version will not only
facilitate that agents with opposing opinions switch to the external opinion
(i.e. through nudging opinion $A$ by controlling $\alpha$) but will also
hinder agents who agree with the external field from changing their opinion
(i.e. reinforcing opinion $A$ to counter-balance social pressure).

$\begin{split}W_{i}^{A\rightarrow
B}(t)&=\text{max}[0,(1-\alpha)\mathcal{I}^{B}(t)-\alpha\mathcal{E}^{A}]\\\
W_{i}^{B\rightarrow
A}(t)&=(1-\alpha)\mathcal{I}_{i}^{A}(t)+\alpha\mathcal{E}^{A}\end{split}$ (3)

These two versions of the model are simulated computationally using agent-
based modelling. All simulations start with an initial configuration where
each agent $i$ has an opinion $A$ or $B$ chosen uniformly. At each time step
$t$, an agent $i$ is chosen uniformly. The fraction of disagreeing social
contacts (Eq. 1) is calculated and used to determine the opinion flip rate
$W_{i}(t)$, based on social pressure, the external field, and $\alpha$. Agent
$i$ changes its opinion with probability $W_{i}(t)$ and keeps it with
probability $1-W_{i}(t)$. This procedure is repeated for $T$ time steps (Fig.
1), chosen long enough for the system to reach the stationary state
($T_{ER}\approx 40\times 10^{4}$; $T_{WS}\approx 80\times 10^{4}$;
$T_{BA}\approx 60\times 10^{4}$). Because time-dependent measures only depend
on other time-dependent measures at the same time step, the explicit time
dependence will be omitted for the sake of brevity.

Figure 1: An example of the transition rates with external influence
$\mathcal{E}^{A}=0.7$, where $\mathcal{I}_{i}^{B}=0.6$
($\mathcal{I}_{i}^{A}=0.4$) and $\alpha=0.5$.

We use different measures to quantify the consensus formation in the opinion
dynamics model across scales [2]. The opinions $o_{i}=A$ and $o_{i}=B$ are
modeled by spin variables $\sigma_{i}=+1$ and $\sigma_{i}=-1$. A
magnetization-like measure is used to quantify the level of global consensus
$m=\frac{1}{N}\sum_{i}\sigma_{i}$, which is the fraction of the total
population holding an opinion ($A$ or $B$). The values lie in the interval
$[-1,1]$, where the extremes $m=1$ and $m=-1$ resemble the total population
holding opinion $A$ and $B$ (full consensus), respectively.

The local consensus is measured using the so-called density of interfaces (Eq.
4), where $N_{p}$ is the total number of nearest neighbour pairs (i.e. ṡocial
contacts $(i,j)$) and $N_{a}$ the number of such pairs with neighbours having
different opinions (i.e. $o_{i}\neq o_{j}$ for $(i,j)$). The case $n_{a}=1/2$
corresponds to the fully mixed state, and $n_{a}=0$ indicates complete order.

$n_{a}=\frac{N_{a}}{N_{p}},$ (4)

The two measures are redundant in the case of full consensus but not in a
mixed state. In the latter, $n_{a}$ can be used to gain additional information
about how the opinions are locally ordered, i.e. to measure opinion clusters.
These measures provide no information about how the population relates to the
external opinion. An energy-like measure is used to measure the combined
social stress in the population due to local disagreement and disagreement
with the opinion of the external field (Eq. 5). This quantity is not conserved
and thus will not be referred to as energy but social stress $S_{E}$.

$S_{E}=-(1-\alpha)\sum_{<ij>}\sigma_{i}\sigma_{j}-\alpha\mathcal{E}\sum_{i}\sigma_{i}.$
(5)

Social stress increases when two contacts disagree, and agents disagree with
the external field opinion. An algorithm based on breadth-first search [42] is
used to count distinct opinion clusters in the network. An opinion cluster
($\Omega_{o}$) is thus here defined as a group of agents with the same opinion
$o_{i}$ for which there exists a path between any two agents in the group
without having to go through an agent of the opposite opinion (Fig. 2).

Figure 2: An example network highlighting the different opinion clusters. Six
distinct opinion clusters are identified, two for opinion “red” and four for
opinion “green”.

For a given network model and set of parameters, averages are computed using
100 points (10 realizations for each of the ten starting configurations).

##  3 Results

###  3.1 Consensus Formation

We start by studying consensus formation and how the external influence
affects the stationary population opinion. Without loss of generality, we
assign opinion $A$ to the external field. We study the consensus formation by
analysing how the strength of the external field $\mathcal{E}$ affects the
prevalence of opinion $P_{A}$ for a fixed $\alpha$, which can be interpreted
as the amount of trust placed in the external actor relative to one’s peers.
At $t=0$, both opinions are distributed uniformly at random over the
population such that the prevalence of both opinions is approximately equal,
$P_{A}(0)\approx P_{B}(0)\approx 0.5$. Figure 3 shows the temporal evolution
of $P_{A}(t)$ for different levels of the external field in both versions of
the model.

Figure 3: The prevalence $P_{A}(t)$ of opinion $A$ on the ER, WS, and BA
network models (See Methods) for the (a) asymmetric model with
$\mathcal{E}^{A}=0$, (b) asymmetric model with $\mathcal{E}^{A}=0.5$, (c)
asymmetric model with $\mathcal{E}^{A}=1$ (d) symmetric model with
$\mathcal{E}^{A}=0$, (e) symmetric model with $\mathcal{E}^{A}=0.5$, and (f)
symmetric model with $\mathcal{E}^{A}=1$. The parameter $\alpha=0.5$ is fixed.
The shaded area represents the standard error.

The dynamics lead to a stationary state where the fraction of agents following
the external opinion $A$ plateaus below the maximum value, except for the WS
network, where the dynamics need longer times to reach stationarity. The
plateau values are higher for the symmetric than for the asymmetric version.
For $\mathcal{E}^{A}=0.5$ in the asymmetric model and $\mathcal{E}^{A}=0$ in
the symmetric model, the prevalence of the opinions remains balanced, except
by the increasing prevalence observed for opinion $A$ in the WS network,
likely due to reinforcement. The prevalence of opinion $A$ on the WS networks
are relatively larger, whereas the prevalence is relatively lower for the BA
networks. Generally, increasing $\mathcal{E}^{A}$ leads to increasing $P_{A}$,
but there is a difference between both models for low $\mathcal{E}^{A}$. The
asymmetric version allows for parameter combinations where the opposing
opinion becomes most prevalent on average, whereas this is not possible with
the symmetric version. These differences are more significant at the
stationary state for various combinations of parameters (Figs. 4-5).

Figure 4: Results of the asymmetric model for various parameter combinations.
(a) global consensus $m$ and ER model, (b) local consensus $n_{a}$ and ER
model, (c) social stress $S_{E}$ and ER model, (d) global consensus $m$ and WS
model, (e) local consensus $n_{a}$ and WS model, (f) social stress $S_{E}$ and
WS model, (g) global consensus $m$ and BA model, (h) local consensus $n_{a}$
and BA model, (i) social stress $S_{E}$ and BA model, for different values of
the model parameters $\alpha$ and $\mathcal{E}^{A}$.

For the asymmetric model, the stationary state results show two regions of
high (average) global consensus and order for all network structures. Agents
move more towards global consensus when they are subjected to low or high
$\mathcal{E}^{A}$ external field, with much trust in the external source
($\alpha>0.5$), than when they are subject to moderate levels of external
pressure ($\mathcal{E}^{A}\approx 0.5$). The asymmetric model allows each
opinion to become the majority opinion in the population, depending on
$\alpha$-$\mathcal{E}^{A}$ combinations. The effect of strong consensus
formation in the case of high trust in the external field but low field
intensity is due to this model’s asymmetry in transition rates (Eq. 2). In
this region with large $\alpha$, the asymmetric model is subject to two
different influences: $W_{i}^{A\rightarrow B}=\mathcal{I}_{i}^{B}$ and
$W_{i}^{B\rightarrow A}=\mathcal{E}^{A}$ (eq. 2). Thus, the region of high
$\alpha$ and low $\mathcal{E}^{A}$ is driven by local peer pressure alone,
while the region with large $\alpha$ and $\mathcal{E}^{A}$ is driven by an
interaction of peer and external pressure (Fig. 4). When the agents are only
subject to social pressure ($\alpha=0$), the population ends up in a mixed
state. This is due to the averaging over multiple stochastic realisations
where various levels of consensus are reached for both opinions. Another
region with, on average, low global order is visible for values of $\alpha\neq
0$ and medium $\mathcal{E}^{A}$. The higher local disorder and social stress
in this area compared to the $\alpha\approx 0$ area could indicate that this
average global disorder results from actual disordered states instead of
averaging over more or less ordered states in both opinion directions. This
area appears as a disordered transition region between going from one majority
opinion to the other by increasing or decreasing the intensity of the external
field.

Figure 5: Results of various parameter combinations for the symmetric model.
(a) global consensus $m$ and ER model, (b) local consensus $n_{a}$ and ER
model, (c) social stress $S_{E}$ and ER model, (d) global consensus $m$ and WS
model, (e) local consensus $n_{a}$ and WS model, (f) social stress $S_{E}$ and
WS model, (g) global consensus $m$ and BA model, (h) local consensus $n_{a}$
and BA model, (i) social stress $S_{E}$ and BA model, for different values of
the model parameters $\alpha$ and $\mathcal{E}^{A}$.

In the symmetric model, the resistance to moving away from the opinion of the
external field significantly reduces the possible final states, which are
qualitatively similar for every network model (Fig. 5). All measures point to
high levels of consensus and order in the large $\alpha$ and large
$\mathcal{E}^{A}$ upper half of the parameter space. The average fully mixed
state at $\alpha=0$ is still visible and appears for $\mathcal{E}^{A}=0$,
regardless of $\alpha$. For a fixed $\alpha>0$ and $\mathcal{E}^{A}>0$, there
is generally a monotonous increase in consensus visible with increasing
external field and trust, respectively. Thus, higher external pressure results
in higher levels of consensus as well as a reduction in the variance around
the stationary state (Fig. 3 (d - f)). Global consensus is now only possible
towards the external field opinion, in contrast with the asymmetric model,
where both forces can define consensus on average. For $\alpha>0$ and
$\mathcal{E}>0$, the global consensus and local ordering levels are
significantly higher than in the asymmetric model, as reflected in the low
levels of social stress attained. This restraining effect of the external
field is present in all network structures; the difference can be attributed
to how much the structure facilitates the spread of opinion. For the WS
networks, high levels of consensus are reached for a larger region of the
parameter space. In contrast, BA networks show signs of stronger resistance
towards opinion spread, requiring a stronger external field. In the WS model,
agents reinforce each other due to the redundancy in the social structure
(i.e. high clustering), leading to values of $\mathcal{I}_{i}$ larger than
$0.5$. Therefore, even a small influence of the external field is amplified
due to social reinforcement. On the other hand, the hubs (high-degree nodes)
in the BA network act as a type of bottleneck. Hubs can affect a large part of
the network, composed mainly by low degree nodes, keeping $\mathcal{I}_{i}$
close to 0.5. If a hub is chosen to update its opinion, the chance that all
its neighbours agree on a specific opinion is relatively low because they are
not exposed to reinforcement, leading to $\mathcal{I}_{i}\sim 0.5$ for hubs.
The poor clustering of the BA network structure hinders the creation of a
majority opinion and resists changing towards the external force.

###  3.2 Opinion Clusters

Figures 6-9 show the evolution of the number $|\Omega|$ and average size
$\langle\Omega\rangle$ of the opinion clusters for different levels of the
external field $\mathcal{E}^{A}$. In the asymmetric version (Figs. 6-7), the
majority of opinions $B$ and $A$ at respectively low and high
$\mathcal{E}^{A}$ is reflected in the number and size of their respective
clusters. When an opinion reaches a majority, its opinion clusters grow and
decrease in number. In contrast, for an opinion that declines in its number of
followers, the clusters become more numerous but smaller. As an opinion loses
ground and agents acting as bridges in their cluster switch to the opposite
opinion, the clusters fracture, and agents of the losing opinion become more
isolated as the opinion keeps losing ground. The opposite is visible when an
opinion grows in popularity; new followers combine nearby clusters into larger
ones.

Figure 6: The temporal evolution of the number of opinion clusters $|\Omega|$
with (a-c) opinion $A$ and (d-f) opinion $B$, for the asymmetric model.
Different values of the external field $\mathcal{E}^{A}$ are combined with
fixed $\alpha=0.5$ (equal weight to peers and external influence). Shaded
areas represent the standard error. These results are averaged over 25
realisations of a given network/parameter combination. Figure 7: The temporal
evolution of the average size of opinion clusters with opinion $A$,
$\langle\Omega_{A}\rangle$ (a-c), and opinion B, $\langle\Omega_{B}\rangle$
(d-f), for the asymmetric model. Different values of the external field
$\mathcal{E}^{A}$ are combined with fixed $\alpha=0.5$ (equal weight to peers
and external influence). Shaded areas represent the standard error. These
results are averaged over 25 realisations of a given network/parameter
combination. Figure 8: The temporal evolution of the number of opinion
clusters $|\Omega|$ with (a-c) opinion $A$ and (d-f) opinion $B$, for the
symmetric model. Different values of the external field $\mathcal{E}^{A}$ are
combined with fixed $\alpha=0.5$ (equal weight to peers and external
influence). Shaded areas represent the standard error. These results are
averaged over 25 realisations of a given network/parameter combination. Figure
9: The temporal evolution of the average size of opinion clusters
$\langle\Omega\rangle$ with (a-c) opinion $A$ and (d-f) opinion $B$, for the
symmetric model. Different values of the external field $\mathcal{E}^{A}$ are
combined with fixed $\alpha=0.5$ (equal weight to peers and external
influence). Shaded areas represent the standard error. These results are
averaged over 25 realisations of a given network/parameter combination.

This process is also observed in the symmetric version (Figs. 8-9), where low
$|\Omega_{A}|$ for every $\mathcal{E}^{A}$ indicates that indeed the media
opinion $A$ does not, on average, become the minority opinion in the symmetric
version. Fragmentation of the minority opinion clusters and conglomeration of
the majority opinion clusters also occur in the symmetric version. Still, this
effect seems more extreme than the asymmetric version. The symmetric version
shows higher $|\Omega|$ and lower $\langle\Omega\rangle$ for a minority
cluster and visa versa for a majority cluster. The values for $|\Omega|$ and
$\langle\Omega\rangle$ approach those at $\mathcal{E}^{A}=1$ for lower values
of $\mathcal{E}^{A}$ in the symmetric version, showing that here a strong
impact of the external opinion is already present for median values of
$\mathcal{E}^{A}$.

For both model versions, $|\Omega|$ and $\langle\Omega\rangle$ display
differences for the different network structures. These differences are most
pronounced in the regimes that lead to an $|m|>0$, i.e. where, on average, we
have a majority and minority opinion. We notice that the BA and ER topologies
have opinion communities that evolve similarly concerning their average size
but not when it comes to their numbers. When there is fragmentation of a
minority opinion, this is more extreme in BA than in any other network
topology. The asymmetrical degree distribution and hub structure of the BA
network could lead to the many low-degree nodes being isolated in their single
or small opinion clusters. For WS, cluster size and number evolution differ
from ER and BA. When fragmentation occurs, $|\Omega|$ first peaks, after which
it declines to settle at a lower value. This effect can be explained by the
fact that much higher values of $m$ are reached for WS. So, the initial rise
is fragmentation, as seen for BA and ER. Still, the following decline is
because WS ends up with fewer agents that adhere to the minority opinion and,
thus, fewer opinion communities. This is also seen by looking at the values of
$\langle\Omega_{A}\rangle$ for WS, which are significantly more extreme than
other network topologies.

##  4 Discussion

The asymmetric model results in three regimes that can be mapped in the
$\alpha-\mathcal{E}^{A}$ parameter space (Fig. 10(a)). In the first regime B,
the population consensus moves towards the opinion opposite to the external
field. This regime is for high $\alpha$-low $\mathcal{E}^{A}$ combinations.
The second regime A/B, showing average low consensus and order with high
social stress, is at intermediate values of $\mathcal{E}^{A}$ or low values of
$\alpha$. A third regime A of average high global and local consensus is
formed in favour of the external field opinion. It is located in the parameter
space’s high $\alpha$-high $\mathcal{E}^{A}$ part. This picture is recurrent
across all studied network models with differences in the stationary state
values and the sizes of and transition regions between the different regimes
(Figs. 3-4). The average stationary consensus in the A/B regime is
qualitatively similar to the situation where agents ignore the external
influence, i.e. when $\alpha=0$. For a fixed $\alpha>0$, increasing the
strength of the media opinion results in breaking up majority opinion clusters
(regime B) into smaller ones, increasing $n_{a}$ or the number of disagreeing
nearest neighbour pairs. This leads to fracturing of those clusters,
increasing social stress at first (regime A/B), decreasing again when enough
agents take the media opinion. Large clusters of opinion are formed (regime A)
(Figs. 3-7). The existence of regime B for the asymmetric version can be
understood from eq. 2 where $W_{i}^{B\rightarrow A}$ is small for high
$\alpha$ and small $\mathcal{E}^{A}$. In contrast, $W_{i}^{A\rightarrow B}$ is
unaffected, meaning that the asymmetry results in the internal pressure away
from the field being felt the strongest in that region.

(a)

(b)

Figure 10: Qualitative diagram of the macroscopic population opinion of the
(a) asymmetric and (b) symmetric models at the stationary state.

The symmetric model exhibits two regimes in the $\alpha-\mathcal{E}^{A}$
parameter space (Fig. 10(b)). A disordered regime A/B presents itself at low
$\alpha$ and low $\mathcal{E}^{A}$ and for $\alpha=0$ or $\mathcal{E}^{A}=0$.
An ordered regime A with high consensus in favour of the external opinion
emerges for higher values of $\alpha$ and $\mathcal{E}$. Fixing a positive
$\alpha$ and increasing $\mathcal{E}^{A}$ leads to a shift from a state where
(on average) consensus towards both opinions is possible (with a significant
variation on the ordering – regime A/B), to a state where holding a different
opinion to the external field is unlikely, with the emergence of consensus and
order (regime A). This pattern is observed for all the network structures,
with the main difference being that for WS networks the regime A is larger.
The WS case shows the most different patterns in both directions. This results
from the larger social clustering and fewer bridges concerning the ER and BA
networks. These topological characteristics facilitate the spread of complex
contagions, which can amplify the effect of the external opinion. The
difference in the structure also leads to the higher prevalence of the
majority opinions and the more extreme fragmentation of the minority opinion
clusters. The exit probability for the voter model on degree heterogeneous
networks (BA model) strongly favours the opinion held by large degree vertices
[43], implying that high degree nodes (i.e. hubs) play an important role in
simple (i.e. without reinforcement) contagion. In complex contagion, a hub
requires several other agents to change its opinion and enforce it to its
neighbours. This may explain why the BA structure is more resistant to
external influence than the random structure (ER model). Similarly, the
difference in degree distribution explains the higher opinion fragmentation in
the BA structure compared to the ER structure.

The results suggest that an external field can affect consensus formation. In
the absence of external influence ($\alpha=0$), consensus (global order) ($m$)
and local order ($n_{a}$) can form, and both opinions are equally likely to
reach a majority. The amount depends on the underlying network structure, but
on average, no preference toward any specific opinion is observed, resulting
in the absence of consensus. Introducing an external field pushing a
particular opinion creates a more diverse set of possibilities where different
regimes appear depending on $\mathcal{E}^{A}$ and the weight $\alpha$. The
external opinion tends to disrupt the equality between both opinions, going
from being equally likely to achieve a majority, towards a situation where a
favoured opinion is more likely than the other one to gain a majority. Without
this external field, the difference between the average numbers and sizes of
the resulting opinion clusters of both opinions is small, implying that both
opinions can achieve a majority. In general, the symmetric model reliably
results in lower levels of social stress because it restricts opinion change
away from the media opinion. In contrast, the asymmetric model allows for more
fluctuations.

##  5 Conclusions

We studied the concurrent impact of media influence (as an external field) and
social interactions (peer pressure) on the opinion dynamics of a stochastic
majority model. We introduced one model version in which the external field
only influences those agents who disagree with the field, and another version
in which all agents are subject to the external field. The external field led
the population to different macroscopic regimes in which different opinions
will likely become the majority. In the first model, the population could
converge to one of the opinions or to a mixed state where both opinions
coexist. In the mixed case, the population behaves qualitatively in the same
way in the presence or absence of external influence. In the second model,
either the external opinion dominated, or the population had mixed opinions.
These macroscopic patterns depended on the intensity of the external influence
and on how much weight of the external field compared to the weight of social
contacts. If the external influence is not strong enough, the population can
move towards the opposite opinion to the one being enforced. Furthermore, the
external field is reinforced by social clustering, which helps align the
population towards the external opinion, even in the case of low-intensity
external influence. On the other hand, high-degree nodes help to prevent the
population from moving towards the external field. By controlling whether
agents feel the external field equally, we showed that agents being exposed to
the external influence, independently of their opinions, led the population to
states where one of the opinions (the one not being enforced) was never
dominant. On the other hand, if only agents aligned with the external field
were affected by this field, the space of possible macroscopic states was
equal for both opinions. This can be relevant to business advertising, health,
and other governmental intervention campaigns.

The impacts of the concurrent influence of peers and an external actor have
been investigated in relatively simple social structures. More complex network
topologies could reveal other underlying structures affecting the opinion
dynamics in real populations. An example could be using stochastic block
models to represent community structure or network models combining homophily
with preferential attachment [44]. Additionally, agents could have
heterogeneous weights representing differences in their susceptibility to
external actors. A more in-depth study of the connection between complex
opinion spread and topological characteristics should include using measures
adapted to complex contagions [45]. The WS network is the sole topology where
full consensus was reached. This result, of complete order, is understood as a
finite-size effect for the classical voter model on WS networks [46]. Despite
limitations, our study shows that introducing simultaneous external and peer
influence in opinion dynamics can lead to a more diverse behaviour than a
monotonous increase in consensus for increasing external influence. The
emergence of different macroscopic population opinions can help our
understanding of how groups with varying trust in the media and subject to
certain levels of media presence, react.

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# Hecke–Farey coordinates for Bring’s surface

Doha Kattan1 1Department of Mathematics, Faculty of Science and Arts, King
Abdulaziz University, Rabigh, Saudi Arabia  and  David Singerman2 2School of
Mathematical Sciences, University of Southampton, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The modular group $\Gamma$ (which is the Hecke group $H_{3}$) can be used to
study triangular maps, Here we use the Hecke group $H_{4}$ to study the
regular map that underlies Bring’s surface of genus 4. Our main result is the
determination of the 20-sided hyperbnolic polygon that gives Bring’s surface.

##  1\. Introduction

The Farey map $\mathbb{F}$, defined in section 2, is the universal triangular
map, in that every triangular map on an orientable surface is a quotient of
$\mathbb{F}$ by a subgroup of the modular group, $\Gamma$ and every regular
triangular map is a quotient of $\mathbb{F}$ by a normal subgroup of $\Gamma.$
This gives rise to many important Riemann surfaces such as the Klein surface.
In [4] we used the Farey map to study the Klein surface. In particular we
showed how the Farey structure could be used to construct the famous 14-sided
fundamental polygon that represents the Klein surface. (Other Riemann surfaces
coming out of triangular maps are described in [11]). In this paper we
describe an important Riemann surface that comes from a regular 4-gonal map,
namely Bring’s surface. We now use the universal 4-gonal map and the modular
group is replaced by the Hecke group $H_{4}$. Our main result is to use Hecke-
Farey coordinates to describe the 20-sided polygon that represents Bring’s
surface and describe the side pairings needed to construct the surface. This
polygon was also given by [9] and discussed by [1] but here we provide
detailed proofs that this polygon and side pairings does indeed give Bring’s
surface.

##  2\. Bring’s surface

We start with the following result which comes from Conder and Dobcsanyi’s
classifications of regular maps of small genus [2].

######  Theorem 2.1.

There is a unique regular map of type $\\{5,4\\}$ of genus 4 with 120
automorphisms. The automorphism group of this map here is the symmetric group
$S_{5}$.

Underlying this map is a unique Riemann surface of genus 4 with automorphism
group $S_{5}$. This surface is called Bring’s Surface. (Type $\\{5,4\\}$ means
that all faces are quadrilaterrals and all vertices have valency 5.)

We let $\Gamma(2,5,4)$ denote the triangle group with periods 2, 5, 4. This
triangle group acts as a group of conformal automorphisms of the upper-half
complex plane $\mathbb{H}$ and it has a presentation

$\Gamma(2,5,4)=\langle X,Y,Z\,|\,X^{2}=Y^{5}=Z^{4}=XYZ=1\rangle.$

There is an epimorphism $\theta:\Gamma(2,5,4)\mapsto S_{5}$ given by $X\mapsto
x=(1,5),Y\mapsto y=(5,4,3,2,1)$, and $Z\mapsto z=(2,3,4,5).$

The kernel $K$ of this epimorphism is a surface group of index 120 in $\Gamma$
and so by the Riemann-Hurwitz formula $\mathbb{H}/K$ is a Riemann surface of
genus 4 with $S_{5}$ as automorphism group and so by Theorem 2.1, this surface
is Bring’s Surface.

As $K$ is a normal subgroup of the triangle group $\Gamma(2,5,4)$, it follows
from [5] that underlying Bring’s surface there is a regular map of type
$\\{5,4\\}$ which we consider in more detail later.

All compact Riemann surfaces arise from compact algebraic curves. Bring’s
surface is the underlying Riemann surface of Bring’s Curve which is the
complete intersection of of the three hyper-surfaces

(2.1)
$\sum_{i=1}^{5}x_{i}=0\;\;\;\;\sum_{i=1}^{5}x_{i}^{2}=0\;\;\;\;\sum_{i=1}^{5}x_{i}^{3}=0.$

Another way of considering Bring’s Surface is in terms of the Hecke group
$H_{4}$.

Consider a group with the two generators $T:z\mapsto z+\lambda$ and
$S:z\mapsto\frac{-1}{z}$.

Hecke showed that this group is discrete, i.e., a Fuchsian group, if and only
if

(i) $\lambda=\lambda_{q}=2\cos\frac{\pi}{q}$ or

(ii) $\lambda\geq 2$.

We are interested in case (i).

We then find that

$R(z)=TS(z)=\frac{\lambda_{q}z-1}{z}.$

This transformation is represented by the matrix

$\begin{pmatrix}\lambda_{q}&-1\\\ 1&0\\\ \end{pmatrix}.$

The eigenvalues of this matrix are $e^{\pm\pi i/q}$ and so $R$ has period $q$.
It is known that as a group $H_{q}\cong C_{2}*C_{q}$.

In this paper we are interested in the case $q=4$ and then
$\lambda_{4}=\sqrt{2}$. The group $H_{4}$ has two generators

$T:z\mapsto z+\sqrt{2}\;\;\;\;and\;\;\;\;S:z\mapsto\frac{-1}{z}.$

Writing the Möbius transformations as matrices we find that the
transformations of $H_{4}$ are of two types
(i)

$\begin{pmatrix}a&b\sqrt{2}\\\ c\sqrt{2}&d\\\
\end{pmatrix}\;a,b,c,d\in\mathbb{Z}\;,\;ad-2bc=1$

and
(ii)

$\begin{pmatrix}a\sqrt{2}&b\\\ c&d\sqrt{2}\\\
\end{pmatrix}\;a,b,c,d\in\mathbb{Z}\;,\;2ad-bc=1.$

The matrices of type (i) form a subgroup of index 2 in $H_{4}$ and are called
even matrices. We denote this subgroup by $H_{4}^{e}$. The matrices of type
(ii) are called odd matrices and they form a coset.

##  3\. The congruence subgroups of the Hecke groups

Let $I$ be an ideal of $\mathbb{Z}[\lambda_{q}]$.
We define

$PSL(2,\mathbb{Z}[\lambda_{q}],I)=\left\\{\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in PSL(2,\mathbb{Z}[\lambda_{q}])\;|\;a-1,b,c,d-1\;\in
I\right\\}.$

Now for any ideal $I$ of $\mathbb{Z}[\lambda_{q}]$ we define the
$\mathit{principal\;congruence\;subgroup}$ of the Hecke group $H_{q}$ as,

$H_{q}(I)=PSL(2,\mathbb{Z}[\lambda_{q}],I)\cap H_{q},$

that is, the subgroup of $H_{q}$ consisting of elements in
$PSL(2,\mathbb{Z}[\lambda_{q}])$. For our interest we take the special case
when $I=(n)$ where $2\leq n\in\mathbb{Z}^{+}$, i.e. $(n)$ is a principal ideal
of $\mathbb{Z}[\sqrt{m}]$ where $m=2,3$ for $q=4,6$. The even principal
congruence subgroup is now

(3.1) $H_{q}^{e}(n)=\left\\{\left(\begin{matrix}a&b\sqrt{m}\\\ c\sqrt{m}&d\\\
\end{matrix}\right)\in H_{q}^{e}\;|\;a\equiv d\equiv\pm
1\;\text{mod}\;n,\;b\equiv c\equiv 0\;\text{mod}\;n\right\\}.$

As $a\sqrt{m}$ and $d\sqrt{m}$ are not congruent to $\pm 1$ mod $n$ we can not
have odd elements in $H_{q}(n).$

For $q=$4 or 6 and for $n>2$, the index of $H_{q}(n)$ in $H_{q}$ was found by
Parson [8, Theorem 2.3].

######  Theorem 3.1.

(3.2)
$\mu_{q}(n)=|H_{q}:(H_{q})(n)|=\begin{cases}n^{3}\prod_{p|n}(1-\frac{1}{p^{2}})&if\;(n,m)=1\\\
n^{3}(1-\frac{1}{m})\prod_{p|n,p\neq
m}(1-\frac{1}{p^{2}})&if\;(n,m)=m.\end{cases}$

Now we are interested in the quotient spaces $\mathbb{H}/H_{4}(n)$.
The fundamental region $F_{\lambda_{4}}$ for $H_{4}$ is the region of the
upper-half plane bounded by the unit circle and the lines
$Re(z)=\pm{\sqrt{2}/2}.$

Figure 1. The fundamental region $F_{\lambda_{4}}$ for $H_{4}$

This fundamental domain is not compact and in fact the quotient space
$\mathbb{H}/H_{4}$ has one puncture corresponding to the point at infinity in
the fundamental domain. Thus $\mathbb{H}/H_{4}(5)$ is a punctured surface.
However its compactification $\overline{\mathbb{H}/H_{4}(5)}$ (found by adding
the punctures) is a compact Riemann surface. Now by Theorem 3.1, $H_{4}(5)$ is
a normal subgroup of index 120 in $H_{4}$ and so
$\overline{\mathbb{H}/H_{4}(5)}$ is a compact Riemann surface with 120
automorphisms.

We now show its genus is equal to 4. $H_{4}$ is the $\Gamma(2,\infty,4)$
triangle group so that by [5] $\mathbb{H}/H_{4}(5)$ has the structure of a
topological map. In fact, as $H_{4}(5)$ is a normal subgroup of $H_{4}$ (of
index 120 by Theorem 3.1) it is a regular map with 120/2 =60 edges, 120/4=30
vertices and 120/5=24 faces. Thus if $g$ is the genus then $2-2g=30-60+24=-6,$
giving $g=4.$

As $\mathbb{H}/H_{4}(5)$ has 120 automorphisms then by Theorem 3.1 we deduce

######  Theorem 3.2.

$\overline{\mathbb{H}/H_{4}(5)}$ is Bring’s surface.

##  4\. Universal $q$-gonal maps

In [10, 7] it was shown that the Farey map $\hat{\mathscr{M}}_{3}$ is the
universal triangular map. The Farey map is a map on the upper-half plane whose
vertices are the extended rationals $\mathbb{Q}\cup\\{\infty\\}$ and where two
rationals $\frac{a}{c}$ and $\frac{b}{d}$ are joined by an edge if and only if
$ad-bc=\pm 1.$ We sometimes refer to the rational $\frac{a}{c}$ as a Farey
coordinate. The automorphism group of $\hat{\mathscr{M}}_{3}$ is the classical
modular group $\Gamma.$ Another way of constructing $\mathbb{F}$ is as the map
whose vertex set is the extended rationals (the images of $\infty$ under
$\Gamma$) and whose edge set is the images of the imaginary axis under
$\Gamma.$

In [7] the authors constructed the universal $q$-gonal maps
$\hat{\mathscr{M}}_{q}.$ The vertices of this map are the images of $\infty$
under $H_{q}$ and the edges are the images of the imaginary axis under
$H_{q}.$ In [7] these universal maps were drawn for $q=4,5,6,7$. The principal
face $F_{q}$ has as its vertices, the images of $\infty$ under the $q$ powers
of $R.$
In this paper we are interested in Bring’s map and so we are particularly
interested in the case $q=4.$ The universal 4-gonal map is drawn in Figure 3.

The vertices of ${\hat{\mathscr{M}}_{4}}$ are the images of $\infty$ under
$H_{4}$ and these are all real numbers of the form $\frac{a\sqrt{2}}{c}$ or
$\frac{b}{d\sqrt{2}}$ where $a,b,c,d\in\mathbb{Z}$ and $(a,c)=(b,d)=1.$ We
call $\frac{a}{c\sqrt{2}}$ and $\frac{b\sqrt{2}}{d}$ as Hecke-Farey
coordinates. An edge joins $\frac{a\sqrt{2}}{c}$ and $\frac{b\sqrt{2}}{d}$ if
and only if $ad-2bc=\pm 1.$ The principal face $F_{4}$ is the $4$-gon whose
vertices are $\infty$, $\frac{0}{1},\frac{1}{\sqrt{2}},\frac{\sqrt{2}}{1}.$

Figure 2. Principal face when $q=4$ Figure 3. The universal 4-gonal
tessellation $\hat{\mathscr{M}}_{4}$

##  5\. Farey fractions modulo $n$

Farey fractions were first introduced in [12] for the case $q=3$. A Farey
fraction modulo $n$ is a pair $(a,c)$ where $(a,c,n)=1$ and where we identify
$\frac{a}{c}$ with $\frac{-a}{-c}.$ We think of these fractions as being
points on $\mathbb{H}/\Gamma(n)$. Two Farey fractions mod $n$, $(a,c)$ and
$(b,d)$ are adjacent when $ad-bc\equiv\pm 1\;\text{mod}\;n$.

######  Example 5.1.

$n=5.$ The Farey fractions mod 5 are

$\frac{1}{0},\frac{2}{0},\frac{0}{1},\frac{1}{1},\frac{2}{1},\frac{3}{1},\frac{4}{1},\frac{0}{2},\frac{1}{2},\frac{2}{2},\frac{3}{2},\frac{4}{2}.$

We join $\frac{a}{c}$ and $\frac{b}{d}$ by a edge if and only if $ad-
bc\equiv\pm 1mod\ 5$

There are 12 vertices and by drawing the diagram we see that we get a regular
icosahedron.

Figure 4. Drawing of $\mathscr{M}_{3}(5)$ with Farey coordinates [4]

##  6\. Hecke-Farey fractions for the cube

The main aim of this paper is to find Hecke-Farey fractions for Bring’s curve,
$\mathscr{M}_{4}(5).$ In general, for a positive integer $n$ the map
$\mathcal{M}_{4}(n)$ has vertices of the form $\frac{a}{\sqrt{2}c}$ or
$\frac{b\sqrt{2}}{d}$. where $a,b,c,d\in\mathbb{Z}_{n}.$ These numbers are
called Hecke-Farey coordinates.These two vertices are joined by an edge If and
only if $ad-2bc\equiv\pm 1$ mod$\ n.$

Before we consider $\mathscr{M}_{4}(5)$ we look at the simpler case
$\mathscr{M}_{4}(3)$ which gives the cube.

We are now considering the triangle group $\Gamma(2,3,4)$. This has a
presentation $\langle{X,Y,Z|X^{2}=Y^{3}=Z^{4}=XYZ=1}\rangle.$ It is well-known
that $\Gamma(2,3,4)$ is isomorphic to the symmetric group $S_{4},$ the
rotation group of the cube. In fact,
$X\mapsto(1,2),Y\mapsto(1,3,4),Z\mapsto(1,2,4,3)$ gives this isomorphism. Now
from (3.2), $|H_{4}:H_{4}(3)|=24$ so that $\mathscr{M}_{4}(3)$ has 24 darts,
12 edges, 8 vertices, and 6 faces. $\mathscr{M}_{4}(3)$ is a regular map, it
must be the cube. We now see that there are exactly 8 Hecke-Farey fractions
modulo 3.

These are

$\frac{1}{0\sqrt{2}},\frac{0}{1\sqrt{2}},\frac{1}{1\sqrt{2}},\frac{1}{2\sqrt{2}},\frac{0\sqrt{2}}{1},\frac{2\sqrt{2}}{1},\frac{1\sqrt{2}}{0},\frac{1\sqrt{2}}{1}.$

(Note for example that
$\frac{\sqrt{2}}{2}=\frac{-\sqrt{2}}{-2}=\frac{2\sqrt{2}}{1}$ so that this is
the complete list of Hecke–Farey fractions modulo 3.)

By drawing a diagram we notice that the corresponding Farey map is a cube.

Figure 5. $\mathscr{M}_{4}(3)$; Cube embedded in a sphere

##  7\. The Bring curve $\mathscr{M}_{4}(5)$

We want want to give each vertex of $\mathscr{M}_{4}(5)$ a Hecke-Farey
coordinate. These will have the form $\frac{a}{c\sqrt{2}}$ or
$\frac{b\sqrt{2}}{d}$ where $a,b,c,d$ lie in the set $\\{0,1,2,3,4\\}$. Now
$\frac{a}{c\sqrt{2}}=\frac{x}{y\sqrt{2}}$ if and only if $a=\pm x$ and $c=\pm
y$, etc. so we can easily work out all the 24 Hecke- Farey fractions
representing the points of $\mathscr{M}_{4}(5)$. These are presented in Table
1.

Table 1. Table of Correspondence for $\mathscr{M}_{4}(5)$ $A_{1}:\frac{1}{0\sqrt{2}}$ | $B_{1}:\frac{2}{0\sqrt{2}}$ | $C_{1}:\frac{0}{1\sqrt{2}}$ | $D_{1}:\frac{1}{1\sqrt{2}}$
---|---|---|---
$E_{1}:\frac{2}{1\sqrt{2}}$ | $F_{1}:\frac{3}{1\sqrt{2}}$ | $G_{1}:\frac{4}{1\sqrt{2}}$ | $H_{1}:\frac{0}{2\sqrt{2}}$
$I_{1}:\frac{1}{2\sqrt{2}}$ | $J_{1}:\frac{3}{2\sqrt{2}}$ | $K_{1}:\frac{2}{2\sqrt{2}}$ | $L_{1}:\frac{4}{2\sqrt{2}}$
$A_{2}:\frac{0\sqrt{2}}{1}$ | $B_{2}:\frac{0\sqrt{2}}{2}$ | $C_{2}:\frac{1\sqrt{2}}{0}$ | $D_{2}:\frac{1\sqrt{2}}{1}$
$E_{2}:\frac{1\sqrt{2}}{2}$ | $F_{2}:\frac{1\sqrt{2}}{3}$ | $G_{2}:\frac{1\sqrt{2}}{4}$ | $H_{2}:\frac{2\sqrt{2}}{0}$
$I_{2}:\frac{2\sqrt{2}}{1}$ | $J_{2}:\frac{2\sqrt{2}}{2}$ | $K_{2}:\frac{2\sqrt{2}}{3}$ | $L_{2}:\frac{2\sqrt{2}}{4}$

##  8\. Poles

In a Farey map (the case $q=3$) a pole is a vertex with a Farey coordinate of
the form $\frac{a}{0}$, where $a\not=0.$ For example, in the icosahedron
$\mathscr{M}_{3}(5)$ the poles are $\frac{1}{0}$ and $\frac{2}{0}.$ which from
Figure 4 are the north and south poles of the icosahedron. Also, in [4] Farey
fractions were used to construct the fundamental region for the Klein quartic.
We found that $\frac{1}{0}$ was at the centre whilst $\frac{2}{0}$ and
$\frac{3}{0}$ lie on the boundary. This is a typical situation. The pole
$\frac{1}{0}$ always lies at the centre and the other poles lie on the
boundary. In a rough sense these other poles lie at "infinity". The poles of
$\mathscr{M}_{4}(5)$ are

$A_{1}=\frac{1}{0\sqrt{2}},\,B_{1}=\frac{2}{0\sqrt{2}},\,C_{2}=\frac{\sqrt{2}}{0},\,H_{2}=\frac{2\sqrt{2}}{0}.$

Figure 6.

##  9\. Hecke–Farey coordinates for Bring’s surface

There are two important papers where Bring’s surface is drawn as a 20-sided
polygon with side identifications. The first of these is by G. Riera and R. E.
Rodriguez [9] from 1992. The second is by H. W. Braden and T. P. Northover
[1], who just used the Riera–Rodriguez diagram. This diagram is slightly
clearer so we use this. This is given in Figure 6, but without the Hecke–Farey
coordinates $G_{1},J_{2}$ etc. In the diagram we also use their notation for
the poles, So $P1=H2,\;P2=B2,\;P3=C2$. Otherwise we use the original notation.

Riera and Rodriguez did not give a proof that that their polygon represents
Bring’s surface. They write that the consider a Cayley diagram for $S_{5}$ and
embed it into the surface but no details are given, nor did they justify the
side pairings. Our aim is to give Hecke–Farey coordinates for the vertices of
the 20-sided polygon. This will tell us how to draw the polygon and also how
to find the side-pairings. We start with a simple observation.

######  Lemma 9.1.

The map $z\mapsto z+\sqrt{2}$ of the Hecke-Farey coordinates induces a
rotation of $\mathscr{M}_{4}(5)$ through $2\pi/5$ about the centre
$\frac{1}{0}.$

###### Proof.

As we are working modulo 5 in in $\mathscr{M}_{4}(5)$ the transformation has
order 5 and fixes the centre $\frac{1}{0}.$ (Here we are identifying the
Hecke-Farey coordinate with its corresponding vertex.) ∎

For example, consider the pole $H_{2}=\frac{2\sqrt{2}}{0}$, which we regard as
the top vertex of our 20-sided polygon (See Figure 6). As
$H_{2}+\sqrt{2}=H_{2}$ we find that there are 5 vertices of our polygon,
labelled $H_{2}.$ The same applies to the pole $C_{2}$. There are 5 poles with
label $C_{2}.$ However, as we now see, there are 10 poles with label $B_{1}$.

######  Definition 9.2.

A Farey circuit is a collection of points $v_{0},v_{1},\cdots v_{n}$ each with
a Hecke-Farey coordinate such that $v_{i}$ is adjacent to $v_{i+1}$ and
$v_{0}=v_{n}.$

A Farey circuit on $\mathscr{M}_{4}(5)$ is given by
$H_{2},E_{1},F_{2},B_{1},J_{2},K_{1},C_{2},L_{1},E_{2},B_{1},\\\
F_{2},D_{1},H_{2}$ (12 vertices). For example $H_{2}=\frac{2\sqrt{2}}{0}$ is
adjacent to $E_{1}=\frac{2}{\sqrt{2}}$ and $E_{1}$ is adjacent to
$F_{2}=\frac{\sqrt{2}}{3}$, etc. We continue this Farey sequence just by
adding multiples of $\sqrt{2}.$ For example, if we add $\sqrt{2}$ to our
circuit we get $H_{2}+\sqrt{2}=H_{2}$, $E_{1}+\sqrt{2}=G_{1}$,
$F_{2}+\sqrt{2}=E_{2}$ etc., and we get another 12 vertices. We get 12 more
vertices each by adding $2\sqrt{2},\;3\sqrt{2}$ and $4\sqrt{2}$. In this way
we find Hecke-Farey coordinates for all of the 60 boundary vertices as
illustrated in Figure 6. Now in the first 12 vertices we found that there was
one pole labelled $H_{2}=P_{1}$, one pole labelled $C_{2}=P_{3}$ and two poles
labelled $B_{2}=P_{2}$. Thus in total there are 5 poles of the form $P_{1}$, 5
poles labelled $P_{3}$ and 10 poles labelled $P_{2}$. Thus in total there are
20 poles on the boundary and we regard these as the vertices of a 20-sided
hyperbolic polygon. We use the diagram in [1], where the edges are labelled
$1,2,\cdots,20.$

In the diagram of Bring’s surface with its Hecke-Farey coordinates, each at
the 20 edges has one vertex of valency 5 in its interior. For example, edge 1
has $K2$ as the vertex of valency 5 in its interior. In Table 2 we just list
the edges together with these vertices.

$1\leftrightarrow K_{2}$ | $2\leftrightarrow B_{2}$
---|---
$3\leftrightarrow L_{1}$ | $4\leftrightarrow I_{1}$
$5\leftrightarrow B_{2}$ | $6\leftrightarrow J_{2}$
$7\leftrightarrow J_{1}$ | $8\leftrightarrow H_{1}$
$9\leftrightarrow J_{2}$ | $10\leftrightarrow F_{2}$
$11\leftrightarrow K_{1}$ | $12\leftrightarrow L_{1}$
$13\leftrightarrow F_{2}$ | $14\leftrightarrow E_{2}$
$15\leftrightarrow I_{1}$ | $16\leftrightarrow J_{1}$
$17\leftrightarrow E_{2}$ | $18\leftrightarrow K_{2}$
$19\leftrightarrow H_{1}$ | $20\leftrightarrow K_{1}$
Table 2.

We now just pair the sides of the polygon which have the same Hecke-Farey
coordinate. For example both side 1 and side 18 have the vertex $K_{2}$ so we
pair them.

We give all the side pairings in Table 3.

$2\leftrightarrow 5$ | $6\leftrightarrow 9$
---|---
$10\leftrightarrow 13$ | $14\leftrightarrow 17$
$18\leftrightarrow 1$ | $3\leftrightarrow 12$
$7\leftrightarrow 16$ | $11\leftrightarrow 20$
$15\leftrightarrow 4$ | $19\leftrightarrow 8$
Table 3.

We notice that if $k\equiv 2\,\;\text{mod}\;4$ then we pair $k$ with $k+3$,
while if $k\equiv 3\;\text{mod}\;4$ then we pair $k$ with $k+9$. This is
similar, but not exactly the same as in [9] or [1] where they give their
answers without proof.

##  10\. Equivalence classes of vertices

.

With these side identifications we can easily find the equivalence classes of
vertices.
Label the vertices $a_{1},a_{2},\cdots a_{20}$ with edge 1 having vertices
$a_{1},a_{2}$ (going anti-clockwise), side 2 having vertices $a_{2},a_{3}$
etc.

As 2 is paired to 5, we get $a_{2}=a_{6}$, $a_{3}=a_{5}$.

As 6 ia paired to 9, we get $a_{6}=a_{10}$, $a_{7}=a_{9}.$

As 7 is paired with 16, we get $a_{7}=a_{17},\;a_{8}=a_{16}$.

As 10 is paired to 13, we have $a_{11}=a_{13}$, $a_{10}=a_{14}.$

As 11 is paired to 20 is paired with 13, we get $a_{11}=a_{13}$,
$a_{10}=a_{14}.$

As 14 is pared with 17 we gat $a_{15}=a_{17}$, $a_{14}=a_{18}.$

As 4 is paired to 15 $a_{4}=a_{16}$, $a_{3}=a_{15}.$

As 18 is paired to 1, we get $a_{1}=a_{19}$, $a_{18}=a_{2}.$

As 8 is paired with 19 , we get $a_{8}=a_{19}$, $a_{8}=a_{20}.$

Thus we find that $a_{k}$ where $k$ is odd represent one vertex in
$\mathscr{M}_{4}(5)$, $a_{2},\;a_{6},\;a_{10},\;a_{14},\;a_{18}$ represent
another vertex and $a_{4},\;a_{8},\;a_{12}$ and $a_{16}$ represent a third
vertex. Thus $\mathscr{M}_{3}(5)$, Bring’s surface has 3 vertices, 10 edges
and 1 face. We then get $2-2g=3-10+1=-6$ dgiving $g=4$ which it should be.

.

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