remarkRemark hypothesisHypothesis claimClaim C. Kann and M. Feng ex_supplement

# A Repulsive Bounded-Confidence Model of Opinion Dynamics in Polarized
Communities††thanks: Submitted to the editors December 8, 2022. This work was
partially funded by the James S. McDonnell Foundation Postdoctoral Fellowship

Claudia Kann Department of Humanities and Social Sciences, California
Institute of Technology<EMAIL_ADDRESS>Michelle Feng Computing +
Mathematical Sciences Department, California Institute of Technology ().
<EMAIL_ADDRESS>
###### Abstract

Collective opinions affect civic participation, governance, and societal
norms. Due to the influence of opinion dynamics, many models of their
formation and evolution have been developed. A commonly used approach for the
study of opinion dynamics is bounded-confidence models. In these models,
individuals are influenced by the opinions of others in their network. They
generally assume that individuals will formulate their opinions to resemble
those of their peers. In this paper, inspired by the dynamics of partisan
politics, we introduce a bounded-confidence model in which individuals may be
repelled by the opinions of their peers rather than only attracted to them. We
prove convergence properties of our model and perform simulations to study the
behavior of our model on various types of random networks. In particular, we
observe that including opinion repulsion leads to a higher degree of opinion
fragmentation than in standard bounded-confidence models.

###### keywords:

opinion dynamics, bounded confidence, mathematical political science,
congressional voting

91D30, 91F10

##  1 Introduction

Opinions dictate how individuals interact with society. They influence who we
are friends with, how we vote, and what we consume. At the individual and
collective level, opinions shape our lives and our social interactions.
Understanding how opinions are formed and their dynamics provides a framework
for studying changes in our society. The role of opinions in politics and
governance is a prominent part of public discourse in the U.S. Inspired by
discussions of political polarization and partisan politics, this paper
presents a mathematical approach to modelling polarized opinion dynamics where
individuals feel both attractive and repulsive forces.

The influence of public opinion on politics have been studied by philosophers,
sociologists, and social theorists[15, 6, 30]. Contemporary approaches to
studying opinions frequently seek to quantify them. In this paper, we focus on
the dynamics of opinions. We are interested in studying how opinions in a
society shift as a result of relationships between individuals. Various models
for studying individual opinions exist[14, 9, 17, 21]. We will focus on
bounded-confidence models. Bounded-confidence models are a class of models
that suppose individuals change their opinions based on their relationships,
when their opinions are already close to those of their peers. That is, if
someone’s opinion is very far away from my own, even if I have a relationship
with them, I will not base my opinions on theirs. Many bounded-confidence
models have been developed and studied. They include examinations of consensus
formation[11, 13], polarization[16, 29], and a large variety of model
extensions for application to real-world opinions [20, 18, 8, 1].

We consider polarization, and the notion that individuals may form their
opinions by being contrarian. If I have an adversarial relationship with
someone, I may specifically choose to hold an opinion that is different from
their’s. Similar to other bounded-confidence models, we maintain the idea that
individuals are mostly influenced by others whose opinions are already
somewhat close to our own. We are most interested in understanding how
collective opinions in this model behave. What types of relationships and
community structures lead to strong polarization within a society? How might
we extend those observations to real-world applications and data?

The paper is organized as follows. We introduce the motivation for our model
in Section 2 and define our model in Section 3. We present analytical results
in Section 4, and perform numerical simulations on synthetic networks (Section
5). Conclusions follow in Section 6.

##  2 Background and motivation

In this section, we introduce the motivation for our proposed model of opinion
dynamics. In Section 2.1, we discuss political science research which
motivates our modelling choices, and in Section 2.2 we introduce the
Hegselmann–Krause model for opinion dynamics, which we use as a starting point
in the formulation of our model.

###  2.1 Political Science motivation

In political science it is common to think of ideologies as points in space,
as being on the left or the right, liberal or conservative. This spatial view
of politicians and individuals drives much of the work that is done on voting
behavior, both at the individual and legislative levels, as well as the models
of strategic behavior within Congress. The original conception of this model
is often attributed to Downs and his median voter theory [12]. This work was
followed by further theoretical work on legislative organization [4, 25, 26,
19], electoral competition [2], and the courts [22] to name a few.

The most common method of obtaining ideological spacial estimates for members
of congress is NOMINATE [23]. It uses the observed voting choices and an item
response model (IRT) to recover spatial distances. This work has been expanded
to include bridges over time to estimate changes in the distribution og
congressional representatives across congresses [24]. More recently, such
bridging techniques and new data sources have been used in order to get
consistent measurements for politicians in different chambers as well as
candidates who do not win their election [7, 10, 3, 27].

In this article we present a bounded confidence model in which there are both
attractive and repulsive links between members. This is motivated by the idea
of varying salience of issues among members of congress. While representatives
may have ideological positions that can be uncovered through voting behavior,
there is reason to believe that politicians are drawn to fellow
representatives with similar priorities. Therefore, working with other members
of congress causes their ideologies to converge. In contrast, they make a
point of distancing themselves from representatives who’s salient issues run
in opposition to them, regardless of other similarities. This would cause them
to attempt to distinguish themselves. From an electoral perspective, this
distinguishing is important and has not yet, to the our knowledge, been
accounted for in spatial models.

###  2.2 Bounded-Confidence models

The model we propose is a variant of the Hegselmann–Krause (HK) model[17]. The
HK model considers the opinions of a group of interacting agents who influence
each other. In the HK model, agents are modelled in a network, with
connections between them. Agents who are connected to each other will affect
each others’ opinions, but only if their opinions are sufficiently close. That
is, even if two agents are connected, if their opinions are far apart, they
will not take each other into consideration as they form new opinions.

The precise mathematical statement of HK is as follows. Suppose $G=(V,E)$ is a
network, with associated adjacency matrix $A$. Then at each time step $t$, we
denote the opinions of nodes $i\in V$ with the opinion vector $\mathbf{x}(t)$.
We associate to the model a confidence bound $c$. Opinions are updated
according to the following rule:

(1) $x_{i}(t+1)=\frac{\sum_{j\in
V}A_{ij}x_{j}(t)\mathbf{1}_{|x_{j}(t)-x_{i}(t)|<c}}{\sum_{j\in
V}A_{ij}\mathbf{1}_{|x_{j}(t)-x_{i}(t)|<c}}$

That is, at timestep $t+1$, we examine all neighbors of $i$ which are within
the confidence bound, and then average their opinions. Note that we can
reformulate this as

(2) $x_{i}(t+1)=x_{i}(t)+\frac{\sum_{j\in
V}A_{ij}(x_{j}(t)-x_{i}(t))\mathbf{1}_{|x_{j}(t)-x_{i}(t)|<c}}{\sum_{j\in
V}A_{ij}\mathbf{1}_{|x_{j}(t)-x_{i}(t)|<c}}$

While the model formulation given above is for 1-dimensional opinions, by
expanding the notation, the same averaging scheme can be used for higher-
dimensional opinions as well.

Previous studies of the HK model have found that it converges in polynomial
time[5]. In [17], Hegselmann and Krause also investigated the steady states of
the model. Specifically, the HK model suggests that as the confidence bound
increases, there is a transition between three types of steady states. For low
confidence bounds, the steady state has many possible opinions and no
particularly dominant opinions (we refer to this as fragmentation). As the
confidence bound increases, steady states begin to exhibit only a small number
of dominant opinions (polarization). For confidence bounds beyond a certain
threshold, we observe only a single dominant opinion (consensus). These three
different steady states can be seen in Fig. 1. In later sections, we will
discuss how the steady states of our model compare.

Figure 1: In this figure a Erdős–Renyi Random Graph is created with connection
probability of 25%, the evolution three confidence intervals (0.05, 0.2, 0.6)
are shown in order to demonstrate the three steady states of the model

##  3 Model statement

The key mechanism that drives our model is the inclusion of repulsive edges,
so that individual nodes can push each other way. Suppose $G=(V,E)$ is a
network with associated adjacency matrix $A$. For any pair of nodes $i,j$, if
$A_{ij}=1$, there is an attractive edge between them. If $A_{ij}=-1$, there is
a repulsive edge. Otherwise, $A_{ij}=0$ and there is no edge between the
nodes. As in the original model, we assume that for all $i$, $A_{ii}=1$. This
assumption ensures that a node’s existing position is included in the
calculation for the next step. We let $\mathbf{x}$ be the vector of opinions,
with $x_{i}(t)$ representing the opinion of node $i$ at time $t$.

We define the variable $M$ as follows:

(3) $M_{ij}(t)=\left\\{\begin{array}[]{c c}x_{j}(t)-x_{i}(t)&A_{ij}\geq 0\\\
\text{sign}(x_{j}(t)-x_{i}(t))|c-|x_{j}(t)-x_{i}(t)||&A_{ij}=-1,|x_{j}(t)-x_{i}(t)|>0\\\
\text{sign}(j-i)c&A_{ij}=-1,x_{j}(t)=x_{i}(t)\end{array}\right.$

Intuitively, $M_{ij}$ represents a signed distance which node $i$ will
potentially travel because of node $j$. The effect of $M$ is that repulsive
forces grow weaker as nodes move farther away from each other. Note that the
third row of $M_{ij}$ covers the case where two nodes have the same opinion
and repulse each other. In this case, the node with the higher index is pushed
towards a higher opinion, while the node with the lower index is pushed
towards a lower opinion. In simulation, this situation is unlikely, as it is
rare that two nodes which are repulsed share the precise same value. We
updated opinions using the following rule:

(4) $x_{i}(t+1)=x_{i}(t)+\frac{\sum_{j\in
V}A_{ij}M_{ij}(t)\mathbf{1}_{|x_{j}(t)-x_{i}(t)|<c}}{\sum_{j\in
V}|A_{ij}|\mathbf{1}_{|x_{j}(t)-x_{i}(t)|<c}}$

Note that if there are no repulsive edges, Eq. 4 reduces precisely to the HK
model as stated in Eq. 2.

The incorporation of Eq. 3 into the model is to aid convergence. To see why,
suppose that we instead naively replaced $M_{ij}(t)$ in Eq. 4 with
$x_{j}(t)-x_{i}(t)$. We can quickly see from the following three node example
in Fig. 2 that opinions might ping pong forever, with attractions pulling
opinions together which then rocket away from each other. We further discuss
the steady states of this model in Section 4 and Section 5.

Figure 2: Example with three nodes, where the top and bottom node repulse each
other, while the central node attracts both of the others. Without the
incorporation of distance scaling, the central node pulls the two outer nodes
towards it until they are within confidence of each other, at which point they
push each other away until they are no longer within confidence of each other.
However, because they are still within confidence of the central node, they
are pulled back in, and the cycle repeats so that the model never converges.
Using the value of $M_{ij}(t)$ represented in Eq. 3.

When there exist repulsive edges, our model gives rise to several forms of
behaviors that differ from the standard HK model. First, the initial range of
opinions does not necessarily bound the final set of opinions. In our model,
if there exist enough repulsive edges, it is possible for the final opinions
to span a much wider range than the initial opinions (as shown in Fig. 3). We
prove a bound on final opinions in Theorem 4.18.

Figure 3: This image shows a simulation where final opinion width was wider
than initial opinion width. The edges are created as described in Equation Eq.
7 where $p_{1}=40\%$, $p_{2}=80\%$ and a confidence bound of 1.6

Second, with repulsive edges, connected nodes within confidence of each other
may not converge to a single opinion. In HK, we can consider the receptivity
subgraph, or the subgraph of $G$ where edges are pruned if they connect nodes
outside confidence bound of each other. In HK, the connected components of the
receptivity subgraph will converge to single opinions. In our model, because
tensions between attractive and repulsive edges exist, it is possible for
nodes to converge to an opinion which is different from its neighbors at
stopping time. For example, the same three-node example in Fig. 2 converges to
a state where all three nodes are still connected and within confidence of
each other, but do not have the same opinion.

##  4 Analytical results

In this section we give some simple proofs about steady states of our model.
We note that while the model converges in most cases (we suspect all, based on
the analytical results in Section 5), it is not necessarily true that the
final opinions are bounded by the initial opinions. The inclusion of negative
edges means that repulsive forces between nodes can push the final opinions
well outside the bounds of the initial opinions, though this process can only
continue so far before nodes are no longer within confidence of each other. To
that end, we propose a simple bound on final opinions based on the number of
negative edges in Theorem 4.18.

We first prove the bound in the simplest case (that of 2 nodes).

######  Proposition 4.1.

Let $G=(V,E)$ be a network with 2 nodes , so that $V=\\{0,1\\}$. Then the
dynamical process described by Eq. 4 converges, and at time of convergence
$T$,

$\left|x_{0}(T)-x_{1}(T)\right|\leq\max\left\\{c,\left|x_{0}(0)-x_{1}(0)\right|\right\\}$

######  Proof 4.2.

Suppose that there are no edges, or $E=\emptyset$. Then the model converges at
time $T=1$, and

$\left|x_{0}(T)-x_{1}(T)\right|=\left|x_{0}(0)-x_{1}(0)\right|$

Now suppose that $E=\\{(x_{0},x_{1})\\}$, and $|x_{0}(0)-x_{1}(0)|\geq c$.
Then the update rule will result in no changes, the model converges at $T=1$

$\left|x_{0}(T)-x_{1}(T)\right|=\left|x_{0}(0)-x_{1}(0)\right|$

Now suppose that $|x_{0}(0)-x_{1}(0)|<c$. If $A_{01}=-1$, the two nodes repel
each other. Then

$\displaystyle x_{1}(1)$
$\displaystyle=x_{1}(0)+\frac{(x_{1}(0)-x_{1}(0))+c-(x_{1}(0)-x_{0}(1))}{2}$
$\displaystyle x_{0}(1)$
$\displaystyle=x_{0}(0)+\frac{(x_{0}(0)-x_{0}(0))-(c-(x_{1}(0)-x_{0}(0))}{2}$
$\displaystyle x_{1}(1)-x_{0}(1)$
$\displaystyle=x_{1}(0)-x_{0}(0)+\frac{2(c-(x_{1}(0)-x_{0}(0)))}{2}$
$\displaystyle=c$

and $x_{1}(1)-x_{0}(1)>=c$, so that after this time, these two nodes will no
longer affect each other, and cannot push each other further, so the model has
converged, and $\max_{i,j}\left|x_{0}(T)-x_{1}(T)\right|\leq c$.

If $A_{01}=1$, the two nodes attract each other, and the model is equivalent
to standard Hegselmann–Krause, so that we have convergence to a single point
and

$|x_{0}(T)-x_{1}(T)|=0$

This covers all possible cases, and the proposition is proven.

The main point to note from this two-node proof is that the repulsive forces
between any two nodes will contribute to attempting to push them apart to a
distance of precisely $c$. Also note that any node cannot move more than $c$
in any direction over the course of one timestep, because $|M_{ij}(t)|\leq c$.

We now prove several lemmas that we will use to prove Theorem 4.18.

######  Lemma 4.3.

Suppose $i\in V$ a node. Define the following sets:

$\displaystyle V_{i}^{+}(t)$ $\displaystyle=\left\\{j\in V:A_{ij}=1\text{ and
}|x_{j}(t)-x_{i}(t)|<c\right\\}$ $\displaystyle U_{i}(t)$
$\displaystyle=\left\\{j\in V:A_{ij}=-1\text{ and
}\left[(0<x_{j}(t)-x_{i}(t)<c)\text{ or }\left(x_{j}(t)=x_{i}(t)\text{ and
}j>i\right)\right]\right\\}$ $\displaystyle L_{i}(t)$
$\displaystyle=\left\\{j\in V:A_{ij}=-1\text{ and
}\left[(0<x_{i}(t)-x_{j}(t)<c)\text{ or }\left(x_{j}(t)=x_{i}(t)\text{ and
}i>j\right)\right]\right\\}$

Then

(5) $x_{i}(t+1)=\frac{\sum_{j\in V_{i}^{+}(t)}x_{j}(t)+\sum_{j\in
U_{i}(t)}(x_{j}(t)-c)+\sum_{j\in
L_{i}}(x_{j}(t)+c)}{|V_{i}^{+}(t)|+|U_{i}(t)+|L_{i}(t)|}$

######  Proof 4.4.

From Eq. 4, we rearrange

$\displaystyle x_{i}(t+1)$ $\displaystyle=x_{i}(t)$
$\displaystyle+\frac{\sum_{j\in
V}A_{ij}M_{ij}(t)\mathbf{1}_{|x_{j}(t)-x_{i}(t)|<c}}{\sum_{j\in
V}|A_{ij}|\mathbf{1}_{|x_{j}(t)-x_{i}(t)|<c}}$ $\displaystyle=x_{i}(t)$
$\displaystyle+\frac{\sum_{j\in V_{i}^{+}(t)\cup U_{i}(t)\cup
L_{i}(t)}A_{ij}M_{ij}(t)}{\sum_{j\in V_{i}^{+}(t)\cup U_{i}(t)\cup
L_{i}(t)}|A_{ij}|}$ $\displaystyle=x_{i}(t)$ $\displaystyle+\frac{\sum_{j\in
V_{i}^{+}(t)}(x_{j}(t)-x_{i}(t))}{|V_{i}^{+}(t)|+|U_{i}(t)|+|L_{i}(t)|}$
$\displaystyle+\frac{\sum_{j\in
U_{i}(t)}(-1)(1)(c-(x_{j}(t)-x_{i}(t)))}{|V_{i}^{+}(t)|+|U_{i}(t)|+|L_{i}(t)|}$
$\displaystyle+\frac{\sum_{j\in
L_{i}(t)}(-1)(-1)(c-(x_{i}(t)-x_{j}(t)))}{|V_{i}^{+}(t)|+|U_{i}(t)|+|L_{i}(t)|}$
$\displaystyle=\frac{\sum_{j\in V_{i}^{+}(t)}x_{j}(t)+\sum_{j\in
U_{i}(t)}(x_{j}(t)-c)+\sum_{j\in
L_{i}}(x_{j}(t)+c)}{|V_{i}^{+}(t)|+|U_{i}(t)+|L_{i}(t)|}$

Intuitively, this lemma tells us that the update rule moves $x_{i}(t)$ to
$x_{i}(t+1)$ by taking an average of several opinions. The set $V_{i}^{+}(t)$
contains nodes $i$ is attracted to at time $t$. The set $U_{i}(t)$ contains
nodes which repulse $i$ at time $t$, and which will push $i$’s opinion lower.
The set $L_{i}(t)$ contains nodes which repulse $i$ at time $t$, and which
will push $i$’s opinion higher. Equation 5 tells us that we can take the
average of $x_{j}(t)$ for $j\in V_{i}^{+}(t)$, $x_{j}(t)-c$ for $j\in
U_{i}(t)$, and $x_{j}(t)+c$ for $j\in L_{i}(t)$ to determine $x_{i}(t+1)$.

######  Lemma 4.5.

Let $i\in V$ at time $t$, and let $W(t)\subset V$ be a set of nodes such that
$W(t)$ is completely contained in $V_{i}^{+}(t)\cup U_{i}(t)\cup L_{i}(t)$.
Define

$\overline{W}(t)=\frac{\sum_{j\in W(t)}x_{j}(t)}{|W(t)|}$

to be the average of $x_{j}(t)$ for all $j\in W(t)$. Then we can rewrite Eq. 5
as

$x_{i}(t+1)=\frac{\sum_{j\in\left(V_{i}^{+}(t)\cup U_{i}(t)\cup
L_{i}(t)\right)\setminus W(t)}x_{j}(t)+\sum_{j\in
W(t)}\overline{W}(t)+\left(|L_{i}(t)|-|U_{i}(t)|\right)c}{|V_{i}^{+}(t)|+|U_{i}(t)+|L_{i}(t)|}$

######  Proof 4.6.

We rearrange Eq. 5 as follows:

$\displaystyle x_{i}(t+1)$ $\displaystyle=\frac{\sum_{j\in
V_{i}^{+}(t)}x_{j}(t)+\sum_{j\in U_{i}(t)}(x_{j}(t)-c)+\sum_{j\in
L_{i}}(x_{j}(t)+c)}{|V_{i}^{+}(t)|+|U_{i}(t)+|L_{i}(t)|}$
$\displaystyle=\frac{\sum_{j\in\left(V_{i}^{+}(t)\cup U_{i}(t)\cup
L_{i}(t)\right)}x_{j}(t)+\left(|L_{i}(t)|-|U_{i}(t)|\right)c}{|V_{i}^{+}(t)|+|U_{i}(t)+|L_{i}(t)|}$
$\displaystyle=\frac{\sum_{j\in\left(V_{i}^{+}(t)\cup U_{i}(t)\cup
L_{i}(t)\right)\setminus W(t)}x_{j}(t)+\sum_{j\in
W(t)}\overline{W}(t)+\left(|L_{i}(t)|-|U_{i}(t)|\right)c}{|V_{i}^{+}(t)|+|U_{i}(t)+|L_{i}(t)|}$

This lemma allows us to replace a group of opinion values of individual nodes
with the average of opinion values across the group, in certain situations.

######  Lemma 4.7.

Let $G=(V,E)$ be a network with $n$ nodes and $m$ edges with confidence bound
$c$. Suppose that every edge in $G$ is repulsive. At time $t$, suppose
$x_{i}(t)>x_{j}(t)$ for all other nodes $j$, so that $i$ is the node with the
highest opinion value at time $t$. Then $x_{i}(t+1)>x_{j}(t+1)$ for all $j$.

######  Proof 4.8.

Note that $V_{k}^{+}(t)=\\{x_{k}\\}$ for all $k,t$, since every edge in $G$ is
repulsive. For convenience we define the following sets:

$\displaystyle U_{ij}(t)$ $\displaystyle=U_{j}(t)\bigcap U_{i}(t)$
$\displaystyle U_{ij}^{\prime}(t)$ $\displaystyle=U_{i}(t)\setminus U_{ij}(t)$
$\displaystyle L_{ij}(t)$ $\displaystyle=L_{i}(t)\bigcap L_{j}(t)$
$\displaystyle L_{ji}^{\prime}(t)$ $\displaystyle=L_{j}(t)\setminus L_{ij}(t)$
$\displaystyle W_{ij}(t)$ $\displaystyle=L_{i}(t)\bigcap U_{j}(t)$

Unpacking this notation, $U_{ij}(t)$ consists of all nodes that repel both $i$
and $j$ downward, while $L_{ij}(t)$ consists of all nodes that repel both $i$
and $j$ upward. $U_{ij}^{\prime}(t)$ consists of nodes which repel $i$
downward, but not $j$ (note that if $x_{i}(t)<x_{j}(t)$, this is automatically
empty), while $L_{ji}^{\prime}(t)$ consists of nodes which repel $j$ upward,
but not $i$ (again, if $x_{i}(t)<x_{j}(t)$, this is empty). Finally,
$W_{ij}(t)$ consists of nodes which repel $i$ upward and $j$ downward (empty
if $x_{j}(t)>x_{i}(t)$).

Now, suppose $x_{i}(t)>x_{j}(t)$ for all $j\in V$. Then we can write

$\displaystyle V_{i}^{+}(t)$ $\displaystyle=\\{i\\}$ $\displaystyle U_{i}(t)$
$\displaystyle=\emptyset$ $\displaystyle L_{i}(t)$
$\displaystyle=L_{ij}(t)\bigcup W_{ij}(t)\bigcup\\{j\\}$

and

$\displaystyle V_{j}^{+}(t)$ $\displaystyle=\\{j\\}$ $\displaystyle U_{i}(t)$
$\displaystyle=\\{i\\}\bigcup W_{ij}(t)$ $\displaystyle L_{i}(t)$
$\displaystyle=L_{ij}(t)\bigcup L_{ji}^{\prime}(t)$

Then we observe the following from the knowledge that nodes only effect each
other if they are within confidence of each other.

$\displaystyle x_{j}(t)+c$ $\displaystyle>x_{i}(t)$
$\displaystyle\overline{L_{ij}}(t)+c$ $\displaystyle>x_{i}(t)$
$\displaystyle\overline{W_{ij}}(t)+c$ $\displaystyle>x_{i}(t)$ $\displaystyle
x_{i}(t)$ $\displaystyle>\overline{L_{ji}^{\prime}}+c$

Then applying Eq. 5 and Lemma 4.5:

$\displaystyle x_{i}(t+1)$
$\displaystyle=\frac{x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|W_{ij}(t)|(\overline{W_{ij}}(t)+c)}{2+|L_{ij}(t)|+|W_{ij}(t)|}$
(6)
$\displaystyle>\frac{x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|W_{ij}(t)|(\overline{W_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(\overline{L_{ji}^{\prime}}(t)+c)}{2+|L_{ij}(t)|+|W_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle>\frac{(x_{i}(t)-c)+x_{j}(t)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|W_{ij}(t)|(\overline{W_{ij}}(t)-c)+|L_{ji}^{\prime}(t)|(\overline{L_{ji}^{\prime}}(t)+c)}{2+|L_{ij}(t)|+|W_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle=x_{j}(t+1)$

where the inequality in Eq. 6 follows because $x_{i}(t+1)$ is a weighted
average, and $\overline{L_{ji}^{\prime}}(t)$ is less than all of the other
values being averaged in the previous line. The next inequality follows
straightforwardly by replacing each value in the average with a smaller or
equal value.

So if $x_{i}(t)$ has the highest value opinion at time $t$, it will always
have the highest value opinion.

######  Corollary 4.9.

Let $G=(V,E)$ be a network with $n$ nodes and $m$ edges with confidence bound
$c$. Suppose that every edge in $G$ is repulsive. At time $t$, let
$M=\\{i:x_{i}(t)\geq x_{j}(t)\forall j\in V\\}$. Then
$x_{\max_{M}i}(t+1)>x_{j}(t+1)\forall j\in V$.

######  Proof 4.10.

From the definitions of $U_{i}(t),L_{i}(t)$, we can observe that the member of
$M$ with highest index will have the largest corresponding set $L_{i}(t)$ and
the smallest corresponding $U_{i}(t)$, so that at time $t+1$, that member of
$M$ will have the highest-valued opinion of all members of $M$. By the same
logic as in the proof of Lemma 4.7, that opinion will also be the highest-
valued opinion overall.

######  Corollary 4.11.

Let $G=(V,E)$ be the complete network with $n$ nodes with confidence bound
$c$. Suppose that every edge in $G$ is repulsive. At time $t$, let
$M=\\{i:x_{i}(t)\leq x_{j}(t)\forall j\in V\\}$. Then
$x_{\min_{M}i}(t+1)<x_{j}(t+1)\forall j\in V$.

######  Proof 4.12.

Proves that $x_{i}(t)<x_{j}(t)$ for all $j\in V$, then $x_{i}(t+1)<x_{j}(t+1)$
for all $j\in V$, by segmenting $U_{i}(t),L_{i}(t),U_{j}(t),L_{j}(t)$ into the
appropriate subsets and reversing inequalities as needed as in Lemma 4.7. Then
the same logic as in Corollary 4.9 proves the statement.

######  Lemma 4.13.

Let $G=(V,E)$ be a network with $n$ nodes and $m$ edges with confidence bound
$c$. Suppose that every edge in $G$ is repulsive. At time $t$, suppose
$x_{i}(t)>x_{j}(t)$ for all other nodes $j\in V$, so that $i$ is the node with
the highest-valued opinion at time $t$. Suppose that there is some node $j$
such that $x_{i}(t)-x_{j}(t)<c$, and that $j$ has the highest-valued opinion
of all such nodes. Then

$\frac{2c}{2+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}\leq
x_{i}(t+1)-x_{j}(t+1)\leq\frac{(|L_{ji}^{\prime}(t)|+2)c}{2+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$

######  Proof 4.14.

By assumption, since $j$ has the highest-valued opinion of all nodes within
confidence of $i$, $W_{ij}(t)$ is empty. To prove the lower bound,

$\displaystyle x_{i}(t+1)$
$\displaystyle=\frac{x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)}{2+|L_{ij}(t)|}$
$\displaystyle\geq\frac{x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(\overline{L_{ji}^{\prime}}(t)+c)}{2+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle x_{j}(t+1)$
$\displaystyle=\frac{(x_{i}(t)-c)+x_{j}(t)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(\overline{L_{ji}^{\prime}}(t)+c)}{2+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle x_{i}(t+1)-x_{j}(t+1)$
$\displaystyle\geq\frac{2c}{2+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$

To prove the upper bound,

$\displaystyle x_{i}(t+1)$
$\displaystyle=\frac{x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+}{2+|L_{ij}(t)|}$
$\displaystyle\leq\frac{x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(x_{j}(t)+c)}{2+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle x_{j}(t+1)$
$\displaystyle=\frac{(x_{i}(t)-c)+x_{j}(t)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(\overline{L_{ji}^{\prime}}(t)+c)}{2+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle x_{i}(t+1)-x_{j}(t+1)$
$\displaystyle\leq\frac{c+c+|L_{ji}^{\prime}(t)|(x_{j}(t)-\overline{L_{ji}^{\prime}}(t))}{2+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle\leq\frac{(|L_{ji}^{\prime}(t)|+2)c}{2+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$

Notice that if $L_{ji}^{\prime}(t)$ is empty, both inequalities become
equalities, so that

$x_{i}(t+1)-x_{j}(t+1)=\frac{2c}{2+|L_{ij}(t)|}$

Notice also that if both $L_{ij}(t)$ and $L_{ji}^{\prime}(t)$ are empty, that
the distance between $x_{i}(t+1)-x_{j}(t+1)$ is precisely $c$.

######  Corollary 4.15.

Let $G=(V,E)$ be a network with $n$ nodes and $m$ edges with confidence bound
$c$. Suppose that every edge in $G$ is repulsive. At time $t$, suppose
$x_{i}(t)<x_{j}(t)$ for all other nodes $j\in V$, so that $i$ is the node with
the lowest-valued opinion at time $t$. Suppose that there is some node $j$
such that $x_{j}(t)-x_{i}(t)<c$, and that $j$ has the lowest-valued opinion of
all such nodes. Then

$\frac{2c}{2+|U_{ij}(t)|+|U_{ij}^{\prime}(t)|}\leq
x_{i}(t+1)-x_{j}(t+1)\leq\frac{(|U_{ij}^{\prime}(t)|+2)c}{2+|U_{ij}(t)|+|U_{ij}^{\prime}(t)|}$

Lemma 4.13 and Corollary 4.15 are interesting because they give us precise
conditions under which the nodes with the most extreme opinions will no longer
be within confidence of any other nodes. Specifically, in order for the node
with the highest-value opinion to lose connection with all other nodes, it
must be true that the only node it is still influenced by is the node with the
second-highest-value opinion, and that neither of the two nodes is influenced
by any other nodes. Otherwise, they will remain within confidence of each
other, even as the node with highest-value opinion remains the most extreme
node and continues to have its opinion pushed upward.

We conclude with one more lemma about the bound on the width of the gap
between consecutive nodes.

######  Lemma 4.16.

Let $G=(V,E)$ be the complete network with $n$ nodes and confidence bound $c$.
Suppose that every edge in $G$ is repulsive. At time $t$, suppose that $i$ and
$j$ are nodes such that $(i,j)\in E$, $x_{i}(t)>x_{j}(t)$, and
$x_{i}(t)-x_{j}(t)<c$, and there exist no nodes $k$ connected to $i$ or $j$
such that $x_{i}(t)>x_{k}(t)>x_{j}(t)$. Then

$|x_{i}(t+1)-x_{j}(t+1)|\leq c$

######  Proof 4.17.

By the assumption that no nodes have values between $x_{i}(t)$ and $x_{j}(t)$,
we have that $W_{ij}(t)=W_{ji}(t)=\emptyset$. Then to prove one direction of
the bound,

$\displaystyle x_{i}(t+1)$
$\displaystyle=\frac{|U_{ij}^{\prime}(t)|(\overline{U_{ij}^{\prime}}(t)-c)+|U_{ij}(t)|(\overline{U_{ij}}(t)-c)+x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|}$
$\displaystyle\leq\frac{|U_{ij}^{\prime}(t)|(\overline{U_{ij}^{\prime}}(t)-c)+|U_{ij}(t)|(\overline{U_{ij}}(t)-c)+x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(x_{j}(t)+c)}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle x_{j}(t+1)$
$\displaystyle=\frac{|U_{ij}(t)|(\overline{U_{ij}}(t)-c)+(x_{i}(t)-c)+x_{j}(t)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(\overline{L_{ji}^{\prime}}(t)+c)}{2+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle\geq\frac{|U_{ij}^{\prime}(t)|(x_{i}(t)-c)+|U_{ij}(t)|(\overline{U_{ij}}(t)-c)+(x_{i}(t)-c)+x_{j}(t)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(\overline{L_{ji}^{\prime}}(t)+c)}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$

Combining both equations,

$\displaystyle x_{i}(t+1)-x_{j}(t+1)$
$\displaystyle\leq\frac{|U_{ij}^{\prime}(t)|\left(\overline{U_{ij}^{\prime}}(t)-x_{i}(t)\right)+c+c+|L_{ji}^{\prime}(t)|\left(x_{j}(t)-\overline{L_{ji}^{\prime}}(t)\right)}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle\leq\frac{\left(2+|U_{ij}^{\prime}(t)|+|L_{ji}^{\prime}(t)|\right)c}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle\leq c$

To prove the other direction,

$\displaystyle x_{j}(t+1)$
$\displaystyle=\frac{|U_{ij}(t)|(\overline{U_{ij}}(t)-c)+(x_{i}(t)-c)+x_{j}(t)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(\overline{L_{ji}^{\prime}}(t)+c)}{2+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle\leq\frac{|U_{ij}^{\prime}(t)|(\overline{L_{ij}}(t)+c)+|U_{ij}(t)|(\overline{U_{ij}}(t)-c)+(x_{i}(t)-c)+x_{j}(t)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(\overline{L_{ji}^{\prime}}(t)+c)}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle x_{i}(t+1)$
$\displaystyle=\frac{|U_{ij}^{\prime}(t)|(\overline{U_{ij}^{\prime}}(t)-c)+|U_{ij}(t)|(\overline{U_{ij}}(t)-c)+x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|}$
$\displaystyle\geq\frac{|U_{ij}^{\prime}(t)|(\overline{U_{ij}^{\prime}}(t)-c)+|U_{ij}(t)|(\overline{U_{ij}}(t)-c)+x_{i}(t)+(x_{j}(t)+c)+|L_{ij}(t)|(\overline{L_{ij}}(t)+c)+|L_{ji}^{\prime}(t)|(\overline{U_{ij}}(t)-c)}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$

Combining both inequalities yields

$x_{j}(t+1)-x_{i}(t+1)\leq\frac{|U_{ij}^{\prime}(t)|\left(\overline{L_{ij}}(t)-\overline{U_{ij}^{\prime}}(t)+2c\right)+c+c+|L_{ji}^{\prime}(t)|\left(\overline{L_{ji}^{\prime}}(t)-\overline{U_{ij}}(t)+2c\right)}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$

Note, however, that

$\displaystyle 2c$ $\displaystyle=(x_{i}(t)+c)-(x_{i}(t)-c)$
$\displaystyle>\overline{U_{ij}^{\prime}}(t)-\overline{L_{ij}}(t)$
$\displaystyle>(x_{j}(t)+c)-x_{j}(t)=c$

and similarly $c<\overline{U_{ij}}(t)-\overline{L_{ji}^{\prime}}(t)<2c$ so
that we have

$\displaystyle x_{j}(t+1)-x_{i}(t+1)$
$\displaystyle\leq\frac{|U_{ij}^{\prime}(t)|\left(\overline{L_{ij}}(t)-\overline{U_{ij}^{\prime}}(t)+2c\right)+c+c+|L_{ji}^{\prime}(t)|\left(\overline{L_{ji}^{\prime}}(t)-\overline{U_{ij}}(t)+2c\right)}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle\leq\frac{\left(2+|U_{ij}^{\prime}(t)|+|L_{ji}^{\prime}(t)|\right)c}{2+|U_{ij}^{\prime}(t)|+|U_{ij}(t)|+|L_{ij}(t)|+|L_{ji}^{\prime}(t)|}$
$\displaystyle\leq c$

and the proof is finished.

######  Theorem 4.18.

Let $G=(V,E)$ be a network with $n$ nodes and confidence bound $c$. Suppose
that $G$ is the complete graph, and that every edge $(i,j)\in E$ is repulsive
(that is $A_{ij}=-1$). Suppose also that we have initial opinions $x_{i}(0)$
such that $|x_{i}(0)-x_{j}(0)|<c$. Then the model converges, and
$\max_{i,j}\left|x_{i}(T)-x_{j}(T)\right|=(n-1)c$.

######  Proof 4.19.

The intuition for this theorem is as follows: for any repulsive edge $(i,j)$,
nodes $i$ and $j$ will repel each other until

$|x_{i}(t)-x_{j}(t)|>=c$

at some future timestep $t$. If every edge is repulsive, we must have distance
at least $c$ between every pair of nodes connected by an edge in order for the
model to converge. Intuitively, the nodes will always continue to push each
other outward until they reach a distance of $c$, and no further, so that the
final convergent state of the model will occur when there are gaps of at least
$c$ between all of the $m$ edges in the original graph. However, from Lemma
4.13, the gaps will have precisely width $c$, so that the bound holds.

From Corollary 4.9 and Corollary 4.11, at time 1, there must be a highest and
lowest-value opinion node. By Lemma 4.7, for $t>1$, these nodes will always be
the highest and lowest-value opinion nodes. Call these nodes
$i_{max},i_{min}$.

Because $G$ is the complete graph, and all edges are repulsive, we can observe
that $i_{max}$ and $i_{min}$ will have their opinions pushed outward, since
initially every node is within confidence of every node. Additionally, from
Lemma 4.5, we can observe that $i_{max}$ will be pushed in the direction of
$\overline{\\{j\neq i\\}}(0)+c$, so that the nodes with opinions much lower
valued than the average will start to drop out of confidence of $i_{max}$.
Further, from Lemma 4.13, $i_{max}$ will remain within confidence of at least
one node as long as it is within confidence of at least 2 nodes in the
previous timestep. Combining these lemmas, we can see that eventually at time
$t^{\prime}$, $i_{max}$ will be within confidence of exactly one other node.

Let $i_{max}^{\prime}$ be the singular node for which
$x_{i_{max}}(t^{\prime})-x_{i_{max}^{\prime}}(t^{\prime})<c$. Then we can
follow the same proof procedure as in Lemma 4.7 to prove that
$x_{i_{max}^{\prime}}(t^{\prime}+1)>x_{j}(t^{\prime}+1)$ for all $j\in V$
other than $j=i_{max}$, and that none of the remaining nodes can be pushed
into confidence of $i_{m}ax$. We do not include the procedure here because of
its similarity to Lemma 4.7, but the key observation that drives the proof is
that there is only a single node $i_{max}$ exerting downward pressure on
$i_{max}^{\prime}$ (if a very high number of nodes were exerting downward
pressure on $i_{max}^{\prime}$, it would be possible for $i_{max}^{\prime}$ to
lose its position as the node with second-highest-value opinion). This allows
us to rewrite the $x_{i_{max}^{\prime}}$ as an average of values which
preserve the order of $i_{max},i_{max}^{\prime},$ and the remaining nodes.
Similarly, we can show that there is some time after which the node with the
second-lowest-value opinion will always remain the node with the second-
lowest-value opinion.

We continue in this manner, proceeding from the nodes with the highest and
lowest-value opinions inwards until we show that after some time, the nodes’
opinions must remain in a fixed order.

From this point on, we observe that from Lemma 4.16, the gap between any two
consecutive nodes is bounded by $c$. Because of our initial conditions on
$x_{i}(t)$, it is impossible for any gap between consecutive nodes to be
larger at any point. If any two nodes have a gap smaller than $c$, we will not
have converged, as the repulsion between the two nodes will push them apart in
the next time step. All nodes will push each other apart until the gap between
any two consecutive nodes is precisely $c$, at which point the model has
converged. Because there are $n$ nodes, this tells us

$\max\\{|x_{i}(T)-x_{j}(T)|\\}=(n-1)c$

The proof for Theorem 4.18 relies on all edges being repulsive, thereby
preserving the ordering of the nodes. This property does not necessarily hold
when there are both attractive and repulsive edges. However, we suspect based
on numerics that the following theorem is also true:

######  Theorem 4.20.

Suppose $G=(V,E)$ is a network with $n$ nodes, $m$ edges, and confidence bound
$c$. Let $m_{r}$ be the number of repulsive edges in the network. Then the
model converges and

$\max_{i,j\in V}\\{x_{i}(T)-x_{j}(T)\\}\leq\max\\{\max_{i,j\in
V}\\{x_{i}(0)-x_{j}(0)\\},mc\\}$

######  Proof 4.21 (Intuition).

The worst case for this model assumes that all repulsive nodes end up at least
$c$ apart from each other, so if all nodes start out within confidence of each
other, the worst case is one in which all nodes with repulsive edges are
chained together in consecutive order along a line of $m$ edges, in which case
the width of their opinions cannot exceed $mc$, since the bounds in Lemma 4.16
should apply and prevent any individual gap from growing wider. The only way a
gap could grow wider is if there are attractive nodes pulling the repulsed
nodes further apart, in which case those attractive nodes either have
repulsive forces between them, and have already been considered in the line,
or must have started farther apart to begin with, in which case we look at
$\max_{i,j\in V}\\{x_{i}(0)-x_{j}(0)\\}$.

Because we cannot rely on nodes remaining in fixed order in this case, we
cannot use the same technique as in Theorem 4.18 to prove convergence and a
bound. However, in practice, we observe that the range of final opinions
increases with the number of repulsive edges, and that in practice the bound
of $mc$ is not very tight (this is to be expected, as, for example, in the
case of the complete graph in Theorem 4.18, the bound is considerably
smaller). To see numerics showing that the range of final opinions scales with
number of repulsive edges and $c$, see Fig. 5 and associated discussion.

##  5 Numerical results on synthetic networks

In this section we present analysis of numerical simulations on a variety of
random networks, chosen for their usage in modelling social structures[28].

###  5.1 Erdős–Renyi

We begin with an adaptation of Erdős–Renyi (ER) networks as a simple random
network model. To achieve a random network with both positive and negative
edges, we generate two ER networks, $G_{1}=G(n,p_{1})$ and $G_{2}=G(n,p_{2})$,
with associated adjacency matrices $A_{1}$ and $A_{2}$. The total network $G$,
is then the network derived from the adjacency matrix $A_{1}-A_{2}$. A visual
of this method can be seen in Fig. 4. In the subsequent network the
probability of each type of edge between any set of nodes can be written as:

(7) $P((i,j)\in
E)=\left\\{\begin{array}[]{cc}0&(1-p_{1})(1-p_{2})+p_{1}p_{2}\\\
1&p_{1}(1-p_{2})\\\ -1&(1-p_{1})p_{2}\end{array}\right.$

(a) G1: positive node network with $p_{1}=0.6$

(b) G2: negative node network with $p_{2}=0.2$

(c) G3: final network with $p_{1}=0.6$ and $p_{2}=0.2$.

Figure 4: An example of the generation of the ER network with attractive and
repulsive edges

To create the simulation results, 100 trials were run with all combinations of
the following parameters:

$\displaystyle p_{1}$ $\displaystyle\in(0.2,0.4,0.6,0.8,1)$ $\displaystyle
p_{2}$ $\displaystyle\in(0.0,0.2,0.4,0.6,0.8)$ $\displaystyle c$
$\displaystyle\in(0.05,0.4,0.8,1.2,1.6)$

For each trial, a random set of initial opinions is generated and the model is
applied for 10000 iterations. In Fig. 5, one trial is shown for each set of
parameters when $p_{1}$ is set to 0.4. This trial was chosen randomly and all
other trials qualitatively look the same.

The final range of opinions gets wider with both $p_{2}$ and $c$ once
repulsive edges are included. These results are in line with expectations. As
$p_{2}$ increases, so does the number of negative connections, resulting in
more repulsive forces between nodes, pushing opinions apart. As $c$ increases,
nodes have more neighbors. For $p_{2}<<p_{1}$, the attractive forces overpower
the repulsive ones, so that higher $c$ leads to more consensus, as in standard
HK models. For $p_{2}>>p_{1}$, the opposite is true – repulsive forces
overpower attractive ones, and nodes push each other further apart for higher
$c$, resulting in a wider spread of opinions.

Figure 5: For all plots, $p_{1}=.4$. The horizontal axis represents $p_{2}$,
while the vertical axis represents $c$. Note that as $p_{2}$ increases, the
range of final opinions gets wider. For low values of $p_{2}$, as $c$
increases, the range of final opinions becomes narrower (closer to consensus).
By contrast, for high values of $p_{2}$, as $c$ increases, the range of final
opinions becomes wider.

In particular, we observe that the proportion of $\frac{p_{2}}{p_{1}}$ seems
to be the driving factor in the range of final opinions. To draw clearer
conclusions, we look at opinion spread, which we define as the following
quantity:

(8) $\frac{\max_{i,j}|x_{i}(T)-x_{j}(T)|}{\max_{i,j}|x_{i}(0)-x_{j}(0)|}$

Figure 6: Heat map of opinion spread as a function of the probability of a
connection in $G_{1}$ and $G_{2}$ iterated over confidence bound $c$. Data
drawn from the mean of 100 trials for each set of parameters with parameters
$p_{1}\in\\{0.2,0.4,0.6,0.8,1\\}$, $p_{2}\in\\{0,0.2,0.4,0.6,0.8,1\\}$, and
$c\in\\{0.05,0.4,0.8,1.2,1.6\\}$.

In Fig. 6, we plot average opinion spread across trials as a function of the
proportion $\frac{p_{2}}{p_{1}}$ and confidence bound $c$ in a heat map. We
observe similar trends as in Fig. 5, with higher proportions
$\frac{p_{2}}{p_{1}}$ leading to higher values of opinion spread, and the
influence of $c$ on opinion spread depending on $\frac{p_{2}}{p_{1}}$. In the
following examples, we will similarly see that opinion spread is largely
controlled by the negative edges in the network, but that the addition of more
structure to the network will influence opinion formation in interesting ways.

###  5.2 Stochastic Block Models

Next, we adapt a Stochastic Block Model (SBM) in order to incorporate both
positive and negative edges. In these networks, each node is assigned to a
group $k\in K$. The probabilities of connections when $i_{k}=j_{k}$ is
different than when $i_{k}\not=j_{k}$. This enforces structure within the
network. Similarly to in Section 5.1 we generate this network through two sub
networks. In this case, the process begins with two SBM networks,
$G_{1}=G(n,p_{1},\rho)$ and $G_{2}=G(n,p_{2},\rho)$, with associated adjacency
matrices $A_{1}$ and $A_{2}$. The variable $p$ is the probability of having a
connection with another node in the same cluster while $p\rho$ is the
probability of having a connection with a node in a different cluster. If G is
network represented by the adjacency matrix given by $A_{1}-A_{2}$ we have the
generated network edge probabilities:

(12) $\displaystyle P((i,j)\in E)_{i_{k}=i_{j}}$
$\displaystyle=\left\\{\begin{array}[]{cc}0&(1-p_{1})(1-p_{2})+p_{1}p_{2}\\\
1&p_{1}(1-p_{2})\\\ -1&(1-p_{1})p_{2}\end{array}\right.$ (16) $\displaystyle
P((i,j)\in E)_{i_{k}\not=i_{j}}$
$\displaystyle=\left\\{\begin{array}[]{cc}0&(1-p_{1}\rho)(1-p_{2}\rho)+\rho^{2}p_{1}p_{2}\\\
1&p_{1}\rho(1-p_{2}\rho)\\\ -1&(1-p_{1}\rho)p_{2}\rho\end{array}\right.$

A sample of the generative process can be seen in Fig. 7, where the blue edges
represent positive edges and the black negative. Each color of nodes
represents a group $k\in K$.

(a) G1: positive node network with $p_{1}=0.85$ and $\rho=0.05$

(b) G2: negative node network with $p_{2}=0.3$ and $\rho=0.05$

(c) G3: final network $p_{1}=0.85$, $p_{2}=0.3$ and $\rho=0.05$

Figure 7: An example of the generation of the SBM network with attractive and
repulsive edges

Again, in order to create simulation results, 100 trials are run for all
combinations of the parameters

$\displaystyle\rho$ $\displaystyle\in(0,0.2,0.4,0.6,0.8,1)$ $\displaystyle
p_{1}$ $\displaystyle\in(0.2,0.4,0.6,0.8,1)$ $\displaystyle p_{2}$
$\displaystyle\in(0.0,0.2,0.4,0.6,0.8)$ $\displaystyle c$
$\displaystyle\in(0.05,0.4,0.8,1.2,1.6)$

For each trial, a random set of initial opinions is generated and the model is
applied. The full histories of an example run when $p_{1}=0.8$ and $p_{2}=0.2$
can be seen in Fig. 8. It can be seen that each group fully converges on
itself, while the confidence interval determines how dispersed the groups are
from each other. As $\rho$ increases, the full set begins to converge. As is
the case with the ER models, for certain combinations the spread at steady-
state is larger than that at the start. From these values we can see that when
$p_{1}$ and $p_{2}$ are locked, higher confidence bounds result in a larger
terminal spread, as does lower percentages of cross-group edges ($\rho$).

Figure 8: Example of the paths taken when the graph is distributed according
to the Stochastic Block Model scheme laid out in 16. In this case $p_{1}=0.8$
and $p_{2}=0.2$ are both locked the confidence bound varies between the rows
and the percent of cross group links varies over columns. Each color
represents a different group. There are five groups in these examples.

As with the ER model, with the SBM we first look at steady-state opinion
spread. The results can be seen in Fig. 9. Note, that when $\rho=1$ the SBM
model is equivalent to the ER model for the same parameters. We therefore have
that the final row is identical to Fig. 6. We note that as the value of $\rho$
shrinks, the final spread increases. Otherwise, the trends found for the ER
model are consistent.

Figure 9: Heat map of the opinion spread of Stochastic Block Model trials. The
rows represent $\rho$, the percent of positive(negative) connection values
that are out (in) group, while the columns are $c$ the confidence bound.

In the case of SBMs, since each vertex $i\in V$ is assigned to a group in $k$,
we are also interested in clustering in addition to opinion spread. We
introduce a measures of how close vertices are to in-group vertices versus
out-group vertices. In order to calculate this, which we call proportional
spread, first we find the average in and out-group distances as:

(17) $\mathcal{I}_{T}=\frac{1}{|k|}\sum_{k_{i}\in
k}\frac{1}{|k_{i}|^{2}}\sum_{j\in k_{i}\subset V}\sum_{\ell\in k_{i}\in\subset
V}|x_{j}(T)-x_{\ell}(T)|$ (18) $\mathcal{O}_{T}=\frac{1}{|k|}\sum_{k_{i}\in
k}\frac{1}{|k_{i}|}\sum_{j\in k_{i}\subset
V}\frac{1}{|k|-|k_{i}|}\sum_{\ell\in\subset V/k_{i}}|x_{j}(T)-x_{\ell}(T)|$

Since the end-spread of the samples differ, in order to appropriately compare
them we look at the ratio, that is

(19) $\mathcal{PS}_{T}=\frac{\mathcal{I}_{T}}{\mathcal{O}_{T}}$

Smaller values of proportional spread imply same-group nodes are significantly
closer to each other than different-group nodes. Thus, small values imply
increased separation by group. In Fig. 10 the spread as well as
$\mathcal{O}_{0}$ and $\mathcal{I}_{0}$ can be seen. It is clear that the
values range most of the space, in addition
$\mathcal{O}_{0}\approx\mathcal{I}_{0}$. Thus, uniformly, at the beginning of
the trials we have $\mathcal{PS}_{0}=1$. The plots of proportional spread at
steady-state over trials can be seen in Fig. 11. We observe that clustering is
most clear with higher values of $p_{1}$ and lower values of $\rho$. To a
lesser extent, clustering also increases with lower values of $p_{2}$. This
implies that the higher rates of in-group connection and fewer cross group
connections lead to increased polarization.

Figure 10: Histograms of metrics at the the start of each trial, useful for
comparing the final results Figure 11: Proportional spread for Stochastic
Block Model trials. The rows represent $p_{2}$, the base probability of an
out-group negative connection, while the columns are $p_{1}$ the base in-group
positive probability. The x-axis is $\rho$ the percent of cross group edges
and the color represents the confidence bound. A value of 1 means that nodes
are equally as close to in and out-group nodes. Smaller values mean groups are
increasingly clustered.

##  6 Future Work and Conclusions

Previous models of opinion dynamics have focused on the attractive nature of
network connections. These result in a complete convergence in the steady-
state for each receptivity subgraph. When the term polarization is used, it
still implies a shrinking of the overall opinion space. The model introduced
in this paper, in contrast, acknowledges that there are circumstances in which
individuals seek to differentiate themselves from those in their network. This
behavior leads to the possibility of an overall opinion space expansion.

In this paper, the basic bounds of this expansion were proven analytically in
certain cases, with intuition provided for the general case. In addition, the
steady-state behavior for random networks were analyzed numerically. These
simulation results offered insight into the effects of various structural
parameters on the model. It was clear that when there was structure in the
network links (for instance in the Stochastic Block Model) group-based
clustering emerged. This clustering was despite the random initial conditions
provided. In addition, the size of the confidence bounds and the density of
repulsive edges are both pivotal in the opinion spread.

The model introduced in this paper lends itself to complex opinion dynamics,
where politicians or individuals want to differentiate themselves due to
factors orthogonal to their expressed opinions. In future work we hope to
explore how this model can help us understand political behavior. Initial
ideas include using informative initial conditions for congressional networks.
Alternatively, this model can be used to look at ideological opinions of
individuals who are influenced by pop culture associating ideological beliefs
with other factors. This would introduce a variable connection to an ideal
point, which then attracts or repulses the individual. The addition of
repulsive forces make bounded-confidence models increasingly relevant for
empirical and theoretical studies of human behavior.

## Acknowledgments

One of the authors (M. Feng) on this project is funded by the James S.
McDonnell Foundation Postdoctoral Fellowship. In addition, we would like to
thank Danny Ebanks, R. Michael Alvarez, Jonathan Katz, and Mason A. Porter for
helpful comments and insights.

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