# Bridging the gap between agent based models and continuous opinion dynamics

Andrew Nugent  MathSys CDT, University of Warwick  Mathematics Institute,
University of Warwick  Susana N. Gomes  Mathematics Institute, University of
Warwick  Marie-Therese Wolfram  Mathematics Institute, University of Warwick

###### Abstract

There is a rich literature on microscopic models for opinion dynamics; most of
them fall into one of two categories - agent-based models or differential
equation models - with a general understanding that the two are connected in
certain scaling limits. In this paper we show rigorously this is indeed the
case. In particular we show that DEMs can be obtained from ABMs by
simultaneously rescaling time and the distance an agent updates their opinion
after an interaction. This approach provides a pathway to analyse much more
diverse modelling paradigms, for example: the motivation behind several
possible multiplicative noise terms in stochastic differential equation
models; the connection between selection noise and the mollification of the
discontinuous bounded confidence interaction function; and how the method for
selecting interacting pairs can determine the normalisation in the
corresponding differential equation. Our computational experiments confirm our
findings, showing excellent agreement of solutions to the two classes of
models in a variety of settings.

Keywords: Opinion dynamics, agent based model, scaling limit.

Highlights:

  * $\bullet$

Proves the connection between agent-based and differential equation models of
opinion formation.

  * $\bullet$

Explains the relationship between selection noise and mollifying interaction
functions.

  * $\bullet$

Motivates multiplicative diffusion in SDE models using additional noise terms
in an ABM.

  * $\bullet$

Demonstrates the convergence of each model through numerical simulations.

##  1 Introduction

Much of opinion dynamics finds its roots in the models of Hegselmann-Krause
(HK) [1] and Deffuant-Weisbuch (DW) [2]. Both models are based on the
underlying assumption, known as bounded confidence, that people interact only
with those who already share a sufficiently similar opinion. In the DW model
each individual can only interact with one randomly chosen partner at a time,
while in the HK model individuals can interact with everyone in their
interaction radius simultaneously. Computational experiments have shown that
solutions to both models exhibit similar macroscopic behaviours, such as a
transition from consensus to a growing number of opinion clusters as the
bounded confidence radius is reduced, but their exact dynamics can be quite
different.

Since their initial introduction, both models have been studied and adapted
extensively, with their influence and major features visible in a wide range
of modern opinion dynamics models (see review papers [3, 4, 5, 6]).

In this paper we will investigate the two most common approaches in opinion
dynamics: agent-based models (ABMs) and differential equation models (DEMs).
The category of ABMs includes the original DW model, while many DEMs are
generalisations of the ordinary differential equation (ODE) version of the HK
model. In ABMs opinion updates are discrete events in which a pair of randomly
chosen individuals interact and then change their opinion(s). By contrast, in
DEMs opinions change continually as individuals interact constantly with the
entire population, rather than with one individual at a time. This distinction
is more than the difference between discrete and continuous time, but is a
fundamental difference in the nature of interactions and opinion updates. For
example, in ABMs both the order of these random interactions and the distance
by which individuals update their opinions can play a major role in the
dynamics, but these features are essentially absent from DEMs. Despite these
differences, we will show that the models are connected in certain scaling
regimes.

Our analysis begins with the following observation: reducing the update
distance in the DW model leads to a slower convergence to equilibrium [2, 7]
and affects both the location and composition of opinion clusters [7, 8]. We
take this situation to the extreme, considering an ABM in which agents
interact at a very fast rate but with a very small update distance. In this
setting, the effect of many small, random interactions in the ABM mirrors the
continuous opinion updates characteristic of DEMs. Motivated by this, we will
show that under a simultaneous rescaling of time and update distance, the
trajectories of the ABM converge to those of a limiting DEM.

Primarily this clarifies when ABMs and DEMs will give similar results, but
also addresses the question of when the two models are capturing the same
real-world behaviour, highlighting their inherent assumptions. Furthermore, we
will show how changes made in the ABM can translate to those in the
corresponding DEM, drawing parallels between the choices made in each setting.
This link also bridges the gap between two research communities by
demonstrating the essential similarity of two seemingly disparate modelling
approaches. Our effort to establish connections between the many opinion
formation models will help motivate modelling choices and improve our
understanding of these complex processes, we hope this drives others to seek
similar connections.

This paper is organised as follows: firstly in Section 2 we define an ABM and
state our main results about its convergence to an ODE system. We also discuss
several adaptations to the ABM that appear in the relevant literature,
including the introduction of various additional noise terms, and give the
corresponding ODE and stochastic differential equation (SDE) limits. A summary
of the limiting models for all cases examined can be found in Table 1 and
proofs given in the Supplementary Material A. In Section 3 we present
numerical results for each of the ODE and SDE limits described in Section 2.
Finally we conclude with a discussion of the significance and limitations of
our results in Section 4.

##  2 Theory

We begin by introducing several ABMs, based on a generalisation of the DW
model, and discuss their respective limits. Interested readers can find the
rigorous derivation and the mathematical statements of these limits, which are
based on Durrett [9], in the Supplementary Material.

Throughout this paper we will always consider a finite population size $N$,
and do not examine any mean-field limits or density-based models (see for
example [3, 10, 11, 12, 13, 14, 15]). Such models offer one way of comparing
the behaviour of ABMs and DEMs (through the limit of a large population size)
while the aim of this paper is to offer an alternative approach for a fixed
$N<\infty$.

###  2.1 Agent based model

Consider $N$ agents with initial opinions
$x(0)=(x_{1},\dots,x_{N})\in\mathds{R}^{N}$. For generality, we do not
prescribe an interval in which an individual’s opinion should lie, but for the
majority of models we consider opinions will remain within the convex hull of
the initial opinions [16, 17].

Beginning at $t=0$, we proceed in discrete timesteps of size $h>0$. At each
timestep, choose a pair of individuals $i$ and $j$ from the population. This
choice is made uniformly at random with replacement (different selection
mechanisms are considered in Section 2.2). After individuals $i$ and $j$ have
been selected, they go on to interact with probability $p_{ij}(x)$. Note that
$p_{ij}$ does not include the probability of selecting individuals $i$ and
$j$, only the probability of interaction after their selection.

The form of $p_{ij}(x)$ plays a significant role in the consensus dynamics.
The majority of opinion formation models assume that interactions depend on
the opinion distance, that is $p_{ij}(x)=\phi(|x_{i}-x_{j}|)$ for some
interaction function $\phi(d):\mathds{R}^{+}\rightarrow[0,1]$. The bounded
confidence (BC) interaction function with radius $R\in\mathds{R}^{+}$, as used
in the DW model, also falls into this category,

$\phi_{R}(d)=\begin{cases}1&\text{ if }d\leq R\,,\\\ 0&\text{
otherwise}\,.\end{cases}$ (1)

Here the only source of noise is the selection of individuals $i$ and $j$, as
an interaction will occur (or not occur) deterministically based on their
opinion difference. Other possible choices of interaction functions include
exponentially decaying functions and polynomials, see e.g. [15, 18, 16].

An common adaptation to the BC interaction function is the introduction of
selection noise (see [19] and references therein). Hereby, after picking $i$
and $j$, a random number $r$ is drawn from a given distribution $\mathcal{X}$.
Individuals $i$ and $j$ then interact if

$|x_{i}-x_{j}|<R+r.$ (2)

Hence the probability of interaction is

$\displaystyle p_{ij}(x)$
$\displaystyle=1-F_{\mathcal{X}}(|x_{i}-x_{j}|-R)\,,$ (3)

where $F_{\mathcal{X}}$ is the cumulative distribution function of
$\mathcal{X}$. Thus selection noise corresponds to an altered interaction
function. In particular, the introduction of selection noise can provide a
mollification (or smoothing) of the discontinuous BC interaction function.
Several examples are shown in Figure 1 for a variety of normal and uniformly
distributed $\mathcal{X}$’s. Note that for the normal distributions the
interaction function is always strictly positive, while for the uniform
distributions it is zero outside some interval around $R$. This distinction
affects which interactions are possible and has a major impact on the
behaviour of the limiting ODE systm [16].

Figure 1: Examples of interaction functions of the form (3), in which the
discontinuous bounded confidence interaction function (1) (shown in red,
dashed) is mollified with various types of selection noise. All the resulting
interaction functions are Lipschitz continuous, but not necessarily
differentiable everywhere.

The probability of interaction $p_{ij}$ also provides an opportunity to
introduce a social network by setting the probability of interaction equal to
the network adjacency matrix.

In order to establish the connection between ABMs and DEMs, we will require
the following assumptions on $p_{ij}$:

######  Assumption 1.

For all $i,j\in\\{1,\dots,N\\}$, $p_{ij}$ satisfies

  1. a)

$p_{ij}(x)\in[0,1]$ for all $x\in\mathds{R}^{N}$.

  2. b)

$p_{ij}(x)$ is globally Lipschitz continuous in $x$.

Assumption 1a) is required so that $p_{ij}$ can be interpreted as the
probability of $i$ and $j$ interacting once selected. Assumption 1b) is
required so that the limiting system of differential equations is well-defined
on $\mathds{R}^{N}$. In particular this excludes the discontinuous BC
interaction function without mollification by selection noise. Such
assumptions on the continuity of interaction functions are often necessary to
make progress in analysing DEMs [20, 21], making the identification of
realistic mechanisms, such as selection noise, that give rise to smoothed
interaction functions especially important.

If $i$ and $j$ do interact, then individual $i$ updates their opinion
according to,

$\displaystyle x_{i}(t+h)$
$\displaystyle=x_{i}(t)+\mu^{h}\,\big{(}x_{j}(t)-x_{i}(t)\big{)},$ (4)

with all other individuals’ opinions remaining the same in the next timestep.
For a given timestep $h$, let $\mu^{h}=Nh$. If $h\in(0,N^{-1}]$ then $\mu\leq
1$ and individuals’ opinions will remain within the convex hull of the initial
opinions $x(0)$.

######  Remark 2.1.

The setup in which both individuals $i$ and $j$ update their opinions will
give the same limit with $\mu=Nh/2$, under the assumption that
$p_{ij}(x)=p_{ji}(x)$ for all $x\in\mathds{R}^{N}$.

Overall, an interaction between individuals $i$ and $j$ gives the following
possible updates to the opinion of individual $i$,

$x_{i}(t+h)=\begin{cases}x_{i}(t)+\mu^{h}\,\big{(}x_{j}(t)-x_{i}(t)\big{)}&\text{
with probability }p_{ij}(x)\\\ x_{i}(t)&\text{ with probability
}1-p_{ij}(x)\,.\end{cases}$ (5)

This process of selecting a pair of individuals, determining if they interact
and updating the first individual’s opinion, continues until some given time
$T\in[0,\infty)$ is reached.

Main result: Let $x_{0}$ be the opinions at time $t=0$. Then, as $h\rightarrow
0$, trajectories of the ABM obtained using update rule (5) converge in
distribution to the solution $X_{i}=(X_{i}(t)\,:t\in[0,T])$ satisfying,

$\frac{dX_{i}}{dt}=\frac{1}{N}\sum_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})\,,\quad
X(0)=x_{0}.$ (6)

Convergence in distribution means that the distribution of a sequence of
random variables (here the trajectories of the ABM) converges to the
distribution of a limiting random variable. Here the limit is the solution of
the ODE system (6), so it is deterministic, and thus the trajectories of the
ABM also converge in probability. Informally, this means that as $h\rightarrow
0$ the trajectories of the ABM are almost surely arbitrarily close to the
solution of the ODE system. Formal definitions of both types of convergence
are given along with the proof of the main result in the Supplementary
Material.

The ODE (6) is a general model of opinion formation in continuous-time, whose
behaviour is analysed in e.g. [16, 22, 23]. This result provides a link
between the ABM and ODE model, showing that under the assumption of small,
frequent pairwise interactions their behaviours are comparable. In addition,
it shows how the interaction probability $p_{ij}$ in the ABM, which can encode
interaction functions, selection noise and networks, corresponds to opinion
weights in the ODE model. We will continue to explore this connection by
considering various relevant generalisations in the following sections.

######  Remark 2.2.

Here we have defined an ABM that is continuous in opinion space and discrete
in time. It would also be possible to define a continuous-time version of this
ABM, with pairwise interactions occurring randomly at some constant rate
multiplied by $p_{ij}h$. In this continuous-time ABM, interactions (and
therefore opinion updates) are still discrete ‘events’, unlike the continual
opinion updates in DEMs. Hence, changing the ABM from discrete- to continuous-
time would not resolve the fundamental difference between ABMs and DEMs.
However, the re-scaling and convergence results used in this paper could also
be applied to a continuous-time version of the ABM to obtain the same limiting
models.

###  2.2 Normalisation

A common variation of the ODE model (6) is to replace the normalisation by the
population size $N$ with normalisation by either the network degree (e.g. in
[24]) or by the sum of interactions (e.g. in [25, 16, 18]). We show below how
both of these variations can be obtained as the limit of an ABM by changing
the way in which the interacting individuals $i$ and $j$ are selected. Recall
that in (5), individuals $i$ and $j$ are chosen uniformly at random with
replacement.

######  Remark 2.3.

If the choice is made without replacement, the ODE (6) is unchanged if we set
$\mu=(N-1)h$, rather than $\mu=Nh$.

Firstly we consider normalisation by node degree. Define a fixed, weighted
network with adjacency matrix $A\in[0,1]^{N\times N}$, with $A_{ii}=1$ for all
$i=1,\dots,N$. This network will only be used when picking the pair of
individuals $i$ and $j$, and we do not assume that this network appears in the
interaction probabilities $p_{ij}$. For each individual $i$, denote their node
degree by $k_{i}$, this is

$k_{i}=\sum\limits_{\ell=1}^{N}A_{i\ell}\,.$ (7)

At each timestep, select an individual $i$. The second individual $j$ is then
selected with probability

$\pi_{j}=\frac{A_{ij}}{k_{i}}.$ (8)

That is, $j$ is selected from amongst the neighbours of individual $i$, with
probability proportional to the weight of the edge between $i$ and $j$. The
limiting ODE model in this case is given by

$\frac{dX_{i}}{dt}=\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\,p_{ij}(X)\,(X_{j}-X_{i}).$
(9)

Secondly we consider normalisation by interaction probability. In order for
this setup to be well-defined we assume that there exists $c>0$ such that
$p_{ii}(X)>c$ for all $X\in\mathds{R}^{N}$. This can be interpreted as each
individual always maintaining some confidence in their own opinion.

At each timestep, first select an individual $i$. For this individual,
calculate $p_{ij}$ for each $j\in\\{1,\dots,N\\}$. The second individual $j$
is then selected with probability

$\pi_{j}(X)=\frac{p_{ij}(X)}{\sum\limits_{\ell=1}^{N}p_{i\ell}(X)}.$ (10)

That is, $j$ is selected with probability proportional to the likelihood that
they would go on to interact with individual $i$. In this case we do not
consider again the interaction probability $p_{ij}$, instead after individual
$j$ is selected an interaction always then occurs. The limiting ODE model in
this case is given by

$\frac{dX_{i}}{dt}=\Bigg{(}\sum\limits_{\ell=1}^{N}p_{i\ell}(X)\Bigg{)}^{-1}\sum_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i}).$
(11)

This more closely reflects the normalisation by the number of interacting
agents in the original HK model. The key difference is that the ODE system
(11) corresponds to an ABM in which individual $i$ considers the likelihood of
interaction with all individuals before $j$ is chosen, whereas the ODE system
(6) corresponds to an ABM in which individual $i$ considers the likelihood of
interaction only after $j$ is chosen. These variations show how the
interpretation of the method through which $i$ and $j$ are selected in the ABM
can help motivate the choice of normalisation in the ODE. In the case of
opinion dynamics, normalisation by the entire population size may in fact be
the more realistic option.

###  2.3 Additional sources of noise

As described in [19] there are various places that additional noise can be
introduced into the ABM. For many of these additional noise terms, as well as
some not considered in [19], a similar re-scaling as in the previous section
again gives rise to a DEM. The following cases will be considered:

  * $\bullet$

Ambiguity noise: in which the opinion of individual $j$ is not clearly
communicated to individual $i$, meaning that $x_{j}$ is replaced by
$\omega_{j}=x_{j}+\eta^{h}$, where $\eta^{h}$ is some random variable
depending on $h$.

  * $\bullet$

External noise: in which the opinion update of individual $i$ is altered by
adding a random variable $\xi^{h}$. Alternatively, this noise could be added
only when individuals $i$ and $j$ interact, in which case it is referred to as
adaptation noise [19].

  * $\bullet$

Random update distance: in which the fixed value of the update distance $\mu$
is replaced by a random variable $\nu^{h}$.

These changes are made by altering the update rule (5), rather than by
changing the way in which $i$ and $j$ are selected. We return to the original
setup in which $i$ and $j$ are chosen uniformly at random with replacement. In
this section we will introduce assumptions on the distributions of $\eta^{h}$,
$\xi^{h}$ and $\nu^{h}$, and their behaviours as $h\rightarrow 0$, to ensure
convergence of the model with the same drift term as previously. We show that,
under these assumptions, ambiguity noise leads to the same ODE limit (6),
while external noise and random update distance have the potential to give
rise to SDEs.

######  Remark 2.4.

Due to the many possible implementations, we do not consider here any boundary
conditions on either the ABM or SDE system. Such boundary conditions could be
included in the ABM and the changes followed through to determine the
corresponding conditions on the SDE, but such an adaptation is beyond the
scope of this paper. This is discussed further in Section 4.

The following notation will be used throughout this section. Consider a family
of real-valued random variables $\zeta=(\zeta^{h})_{h>0}$, where $\zeta^{h}$
indicates dependence on $h$. We define

$m_{k}(\zeta)=\lim\limits_{h\rightarrow
0}\dfrac{\mathds{E}\big{[}(\zeta^{h})^{k}\big{]}}{h},$ (12)

with $m_{k}(\zeta)$ undefined if a finite limit does not exist.

####  2.3.1 Ambiguity noise

Ambiguity noise represents the fact that an individual may not clearly
communicate their opinion. The noise therefore affects both the probability of
interaction and the location of individual $i$’s new opinion.

We assume here that $p_{ij}(x)=\phi(|x_{j}-x_{i}|)$ for some function
$\phi:\mathds{R}^{+}\rightarrow[0,1]$ that is Lipschitz continuous with
Lipschitz constant $L$. We modify the ABM by replacing the update rule (5)
with

$x_{i}(t+h)=\begin{cases}x_{i}(t)+\mu^{h}\,\big{(}\omega_{j}(t)-x_{i}(t)\big{)}&\text{
with probability }\phi(|\omega_{j}-x_{i}|)\\\ x_{i}(t)&\text{ with probability
}1-\phi(|\omega_{j}-x_{i}|)\,,\end{cases}$ (13)

where $\omega_{j}=x_{j}+\eta^{h}$, with the following assumptions on $\eta$.

######  Assumption 2.

Let $\eta^{h}$ correspond to the ambiguity noise in (13). Assume the family of
real random variables $\eta=(\eta^{h})_{h>0}$ satisfies

  1. a)

$\lim\limits_{h\rightarrow 0}\mathds{E}\big{[}|\eta^{h}|\big{]}=0$.

  2. b)

There exists a constant $C\in(0,\infty)$, independent of $h$, such that
$\mathds{E}\big{[}(\eta^{h})^{2}\big{]}<C$ for all $h>0$.

Broadly speaking, the first assumption ensures that
$\phi(|\omega_{j}-x_{i}|)\rightarrow\phi(|x_{j}-x_{i}|)$ as $h\rightarrow 0$,
while the second is required to ensure convergence of the ABM. Under these
assumptions on $\eta$, this system has the same ODE limit (6) as the ABM
without ambiguity noise. Ambiguity noise affects the likelihood of
interactions, but the first part of Assumption 2 ensures that in the limit
$h\rightarrow 0$ there is no net increase or decrease in this likelihood. As
$\mu$ becomes small, each individual interaction has less significance, with
the average behaviour instead driving the dynamics. Even though ambiguity
noise does affect the ABM [19], this additional stochasticity does not impact
the average behaviour and therefore does not lead to a different limit. This
demonstrates that introducing additional noise and complexity at the ABM level
does not necessarily impact the re-scaled system.

####  2.3.2 External noise

We now adapt the ABM by introducing additive noise to represent random
external influences on an individual’s opinion. We replace the update rule (5)
with

$x_{i}(t+h)=\begin{cases}x_{i}(t)+\mu^{h}\,\big{(}x_{j}(t)-x_{i}(t)\big{)}+\xi^{h}&\text{
with probability }p_{ij}(x)\\\ x_{i}(t)+\xi^{h}&\text{ with probability
}1-p_{ij}(x)\,,\end{cases}$ (14)

where $\xi^{h}\in\mathds{R}$ is a random variable representing all external
influences on individual $i$’s opinion. Motivated by the results applied in
the Supplementary Material, we introduce the following assumptions on $\xi$,

######  Assumption 3.

Let $\xi^{h}$ correspond to the ambiguity noise in (14). Assume the family of
real random variables $\xi=(\xi^{h})_{h>0}$ satisfies

  1. a)

$m_{1}(\xi)\,=0$.

  2. b)

$m_{2}(\xi)$ exists.

  3. c)

If $m_{2}(\xi)>0$, then $m_{3}(\xi)=m_{4}(\xi)=0$.

The first of these assumptions ensures that the noise $\xi$ does not introduce
a drift in opinion.

If $m_{2}(\xi)>0$, meaning the variance of $\xi$ is decreasing $o(h)$, we
obtain an SDE limit with additive noise. An example of this is
$\xi^{h}\sim\mathcal{N}(0,sh)$ for some constant $s>0$, which gives
$m_{2}(\xi)=s$. The final assumption on $\xi$ is required to ensure
convergence in this case. The limiting SDE is given by

$dX_{i}=\frac{1}{N}\sum_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})\,\text{d}t+\bigg{(}\frac{m_{2}(\xi)}{N}\bigg{)}^{\frac{1}{2}}\,\text{d}\beta_{i}\,,$
(15)

where $\beta_{i}$ are independent standard Brownian motions. This shows that
additive noise introduced in the ABM can translate directly to additive noise
in the scaling limit. The noise is additive as $\frac{m_{2}(\xi)}{N}$ is a
constant. Note that for $m_{2}(\xi)=0$ we recover the ODE limit (6).

####  2.3.3 Adaptation noise

Next we consider an adaptation of the ABM in which noise is added only if
individuals $i$ and $j$ interact. This is referred to as adaptation noise in
[19]. We replace the update scheme in (5) with

$x_{i}(t+h)=\begin{cases}x_{i}(t)+\mu^{h}\,\big{(}x_{j}(t)-x_{i}(t)\big{)}+\xi^{h}&\text{
with probability }p_{ij}(x)\\\ x_{i}(t)&\text{ with probability
}1-p_{ij}(x)\,,\end{cases}$ (16)

where $\xi$ is again assumed to satisfy Assumption 3. In this case we obtain
the limiting SDE,

$dX_{i}=\frac{1}{N}\sum_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})\,\text{d}t+\Bigg{(}\frac{m_{2}(\xi)}{N^{2}}\sum_{j=1}^{N}p_{ij}(X)\Bigg{)}^{\frac{1}{2}}\,\text{d}\beta_{i}\,.$
(17)

Here the additional noise term in the ABM has the potential to change the
structure of the limiting DEM, motivating a feasible multiplicative diffusion
function for the SDE.

####  2.3.4 Random update distance

Finally we adapt the ABM by adding a new source of noise: replacing the fixed
value of $\mu$ with a random variable $\nu$. This represents the idea that
opinion updates do not move by a set difference, but that the outcome of an
interaction is also random. At a particular interaction, individual $j$ may be
more or less persuasive and individual $i$ may be more or less receptive or
confident in their existing opinion. Additionally, we do not exclude the
possibility of $\nu^{h}$ taking negative values, only requiring that the mean
returns the fixed value of $\mu=Nh$ used in the standard ABM, meaning
individuals may move apart after an interaction. We replace the update scheme
(5) with

$x_{i}(t+h)=\begin{cases}x_{i}(t)+\nu^{h}\,\big{(}x_{j}(t)-x_{i}(t)\big{)}&\text{
with probability }p_{ij}(x)\\\ x_{i}(t)&\text{ with probability
}1-p_{ij}(x)\,,\end{cases}$ (18)

Unlike in the external noise case, the noise term is also multiplied by
$(x_{j}-x_{i})$, so a consensus state in which all individuals have identical
opinions is still a fixed point.

We make the following assumptions on $\nu$, mirroring those on $\eta$,

######  Assumption 4.

Let $\nu^{h}$ correspond to the ambiguity noise in (18). Assume the family of
real random variables $\nu=(\nu^{h})_{h>0}$ satisfies

  1. a)

$m_{1}(\nu)\,=N$.

  2. b)

$m_{2}(\nu)$ exists.

  3. c)

If $m_{2}(\nu)>0$, then $m_{4}(\nu)=0$.

As before, if $m_{2}(\nu)=0$ we obtain the standard deterministic ODE limit
(6). If $m_{2}(\mu)>0$ we obtain a limiting SDE with a new multiplicative
diffusion function

$dX_{i}=\frac{1}{N}\sum_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})\,\text{d}t+\Bigg{(}\frac{m_{2}(\nu)}{N^{2}}\sum_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})^{2}\Bigg{)}^{\frac{1}{2}}\,\text{d}\beta_{i}\,.$
(19)

As both the drift and diffusion functions of the SDE are zero at a consensus
state, there is no movement away from consensus or change in the position of
consensus once it is established.

###  2.4 Summary table

Table 1 collects the ABMs described in Section 2 and their corresponding
ODE/SDE limits.

Agent-based model | Limiting differential equation
---|---
Standard ABM (2.1) | $\dfrac{dX_{i}}{dt}=\dfrac{1}{N}\sum\limits_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})$
Node degree normalisation (2.2) | $\dfrac{dX_{i}}{dt}=\dfrac{1}{k_{i}}\sum\limits_{j=1}^{N}A_{ij}\,p_{ij}(X)\,(X_{j}-X_{i})$
Interaction probability normalisation (2.2) | $\dfrac{dX_{i}}{dt}=\Bigg{(}\sum\limits_{l=1}^{N}p_{il}(X)\Bigg{)}^{-1}\sum\limits_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})$
Ambiguity noise (2.3.1) | $\dfrac{dX_{i}}{dt}=\dfrac{1}{N}\sum\limits_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})$
External noise (2.3.2) | $dX_{i}=\dfrac{1}{N}\sum\limits_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})\,\text{d}t+\bigg{(}\dfrac{m_{2}(\xi)}{N}\bigg{)}^{\frac{1}{2}}\,\text{d}\beta_{i}$
Adaptation noise (2.3.2) | $dX_{i}=\dfrac{1}{N}\sum\limits_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})\,\text{d}t+\bigg{(}\dfrac{m_{2}(\xi)}{N}\,\sum\limits_{j=1}^{N}p_{ij}(X)\bigg{)}^{\frac{1}{2}}\,\text{d}\beta_{i}$
Random update distance (2.3.4) | $dX_{i}=\dfrac{1}{N}\sum\limits_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})\,\text{d}t+\Bigg{(}\dfrac{m_{2}(\nu)}{N^{2}}\sum\limits_{j=1}^{N}p_{ij}(X)\,(X_{j}-X_{i})^{2}\Bigg{)}^{\frac{1}{2}}\,\text{d}\beta_{i}$
Table 1: Summary of variations on the ABM defined in Section 2.1 and the
corresponding limiting differential equations. $m_{2}(\cdot)$ is defined in
(12).

##  3 Numerical results

In order to demonstrate the convergence results described in Section 2 we
perform numerical simulations of both the ABM and ODE/SDE limits. To allow for
a fair comparison between the different models we use the same setup
throughout.

Let $N=50$. We select a set of initial conditions $x(0)$ uniformly at random
in the interval $[-1,1]$ for each agent, and use these same initial conditions
in all simulations. Let $h=10^{-5}$ and final time $T=20$. We use the BC
interaction function with Gaussian selection noise, given by

$p_{ij}(x)=1-F_{\mathcal{N}(0,1)}\bigg{(}\frac{|x_{i}-x_{j}|-R}{0.01}\bigg{)},$
(20)

where $F_{\mathcal{N}(0,1)}$ is the cumulative distribution function of a
standard normal distribution. Interaction functions of this form can be seen
in the left panel of Figure 1.

In this section we will compare timeseries of ABMs and their corresponding
limits. For two timeseries $X$ and $Y$ we calculate the error timeseries
according to

$\text{Error}(t)=\sum_{i=1}^{N}\big{|}X_{i}(t)-Y_{i}(t)\big{|}.$ (21)

###  3.1 ODE limits

Solutions to the ODE are calculated numerically using a forward Euler scheme
with a timestep of $0.01$. Figure 2 shows a comparison between a single
realisation of the ABM (Section 2.1 with update rule (5)) in solid lines and
the limiting ODE (6) in dashed lines. Colours represent the initial opinions
of the $N=50$ different agents. For this value of $h$ there is clearly an
excellent match between the two timeseries, demonstrating the convergence of
the ABM to the ODE system.

Figure 2: Comparison of the dynamics of the ABM (Section 2.1) and ODE (6) with
$h=10^{-5}$. A single realisation of the ABM is plotted in solid lines, with
the ODE plotted in dashed lines. Colours represent the initial opinions of the
$N=50$ different agents. An inset shows the error, calculated according to
(21). There is an excellent match between the two sets of dynamics.

To quantify this comparison we examine the difference between the ODE model
ABM for various values of $h$. Here the error over the whole timeseries is
calculated as follows: realisations of the ABM are sampled at the timesteps of
the ODE, the Frobenius norm of the difference between these matrices is then
calculated and this value is divided by the length of the simulation to give
the error. For each value of $h$ the ABM is simulated 100 times and the error
calculated for each simulation.

Our results are shown in a violin plot in Figure 3. For each value of $h$ the
violin outline shows the distribution of error values observed over the 100
realisations, with the mean and interquartile range shown within the violin
shape. Figure 3 shows that as $h$ is decreased, both the mean error and spread
in errors decrease. The convergence results we apply in the Supplementary
Material do not provide a bound on the convergence rate between the models,
only guaranteeing weak convergence in the limit $h\rightarrow 0$. However,
these results show, as would be expected, that decreasing the value of $h$
gives a better approximation to the ODE limit.

Figure 3: Violin plot of error between the ABM (Section 2.1) and ODE (6) for
different values of $h$. For each value of $h$ the ABM is run 100 times with
the same initial conditions and the error against the ODE calculated.

####  3.1.1 Normalisation

We also implement the two alternative methods for selecting the interacting
pair $i$ and $j$ discussed in Section 2.2. To consider the limiting ODE (9) we
must first define a network. We generate an Erdős-Rényi random network [26]
and take its adjacency matrix. This adjacency matrix is then adapted by
setting $A_{ii}=1$ for all $i=1,\dots,N$. The ABM is simulated again, and now
at each timestep individual $i$ is selected first, then $j$ is selected
according to (8). The results are shown in Figure 4. Although the dynamics are
more complex than in the case of the fully connected network (as in the
original ABM) there is still an excellent match between the ABM and ODE limit
(9).

Figure 4: Comparison of the dynamics of the ABM on a network (Section 2.1) in
which individual $j$ is chosen according to (8) and the limiting ODE (9) with
$h=10^{-5}$. A single realisation of the ABM is plotted in solid lines, with
the ODE plotted in dashed lines. Colours represent the initial opinions of the
$N=50$ different agents. An inset shows the error, calculated according to
(21). There is again an excellent match between the two sets of dynamics.

Similar results are obtained for the case in which individual $i$ first
calculates the probability of interaction with each individual, then selects
$j$ with probability proportional to the interaction probability (10). This
model does not include a network. These results are shown in Figure 5. Here
the population reaches the clustered state much faster than seen in Figure 2.
As we observe two clusters forming, individuals will have a probability of
interaction that is close to $1$ for approximately half the population, and a
probability of interaction that is close to $0$ for the rest of the
population. Hence the normalisation factor in (11)
($\sum_{\ell=1}^{N}p_{i\ell}$) will be approximately half of that in (6)
($N$), so the dynamics are notably faster. This effect is more pronounced at
the beginning of the dynamics, as the spread of initial opinions reduces the
value of $\sum_{\ell=1}^{N}p_{i\ell}$, thus leading to even faster dynamics.
It is also more pronounced in individuals with extreme opinions, meaning their
movement towards more moderate opinions is even faster. Both the ABM and ODE
limit (11) capture these behaviours.

Figure 5: Comparison of the dynamics of the ABM (Section 2.1) in which
individual $j$ is chosen according to (10) and the limiting ODE (11) with
$h=10^{-5}$. A single realisation of the ABM is plotted in solid lines, with
the ODE plotted in dashed lines. Colours represent the initial opinions of the
$N=50$ different agents. An inset shows the error, calculated according to
(21). There is again an excellent match between the two sets of dynamics.

We do not include numerical results for the variation in which $i$ and $j$ are
chosen without replacement, the variation in which both $i$ and $j$ update
their opinions, or the addition of ambiguity noise, as the limiting models are
exactly the same as the base ODE case.

###  3.2 SDE limits

Realisations of the SDE are calculated using an Euler-Maruyama scheme with a
timestep of $t=0.01$. For each of the comparisons in this section we use the
same initial conditions and simulation setup as before. We run 5000
realisations and calculate the mean and variance between realisations, over
time, for each individual. Due to the multiple sources of stochasticity, this
high number of realisations is chosen to allow sufficient opportunity for the
variances of each model to be representative of their true values.

####  3.2.1 External noise

Firstly we consider the ABM with external noise, given by update rule (14). We
take $\xi^{h}\sim\mathcal{N}(Nh,sh)$ with constant $s=0.05$. Here
$\mathds{E}\big{[}(\nu^{h})^{2}\big{]}=(Nh)^{2}+sh$, meaning $m_{2}(\nu)=s>0$.

Figure 6: Comparison of the ABM with external noise (14) and limiting SDE
system (15) with the setup described at the beginning of Section 3.2. Solid
lines show the mean (top panel) and variance (bottom panel) in the opinion of
each individual across 5000 realisations of ABM, dashed lines show the same
for the corresponding SDE system. Both panels include an inset showing the
error, calculated according to (21).

We observe excellent agreement between both the means and variances of the ABM
and SDE realisations. As in the ODE model, the means reach a clustered state.
For the majority of individuals their mean behaviour is very similar to that
of the ODE model and the variance between realisations is low. This is perhaps
unsurprising given the relatively small size of $m_{2}(\nu)$. However, for a
small number of individuals with initial opinions close to zero (appearing in
blue in Figure 6) their mean behaviour is different and their variance is
significantly higher. This indicates that, with the addition of noise, they
may sometimes enter either of the two major opinion clusters. There are also
some individuals with initial opinions close to one (appearing in yellow in
Figure 6) whose variance is higher, indicating they may sometimes not join the
nearby opinion cluster or make take a long time to do so. Both of these
behaviours can be observed in the selected example timeseries in Figure 8.

These example timeseries also show that, unlike the deterministic case,
individuals’ opinions continue to change after the opinion clusters have
formed. This is due to the fact that interactions are still occurring, meaning
individuals’ opinions are still affected by the external noise, even if their
opinions are very close to those they are interacting with. This can also be
seen in the variances in the lower panel of Figure 6, which continue to
increase as the simulation goes on. If these simulations were allowed to run
indefinitely these variances would continue to increase as the opinion
clusters followed a random walk, driven by the continued external noise.

Figure 8: Example timeseries of the ABM with external noise (14), using the
setup described at the beginning of Section 3.

####  3.2.2 Adaptation noise

We now consider the ABM with adaptation noise, given by the update rule (16).
As previously we take $\xi^{h}\sim\mathds{N}(Nh,sh)$ with constant $s=0.05$.

Figure 9: Comparison of the ABM with adaptation noise (16) and limiting SDE
system (17) with the setup described at the beginning of Section 3.2. Solid
lines show the mean (top panel) and variance (bottom panel) in the opinion of
each individual across 5000 realisations of ABM, dashed lines show the same
for the corresponding SDE system. Both panels include an inset showing the
error, calculated according to (21).

In this case we again observe (in Figure 9) an excellent match between the
behaviours of the ABM and limiting SDE system (17). The behaviour of the means
(across realisations) is again similar to that of the ODE model in Figure 2.
As in the previous case (external noise) we observe one individual with
significantly higher variance than all others, again explained by entering
different opinion clusters, or neither opinion cluster, in different
realisations. In this case the variance of this individual drops slightly
towards the end of the simulations, indicating that they do eventually join
one of the clusters.

A similar behaviour can be observed in the individuals with initial opinion
near one (appearing in yellow) whose variance initially grows, peaks around
$t=6$, then drops again. This indicates that these individuals may take some
time to join an opinion cluster, but always do so eventually. This variation
in the time taken for clusters to form can be seen in the example timeseries
in Figure 11. As the population splits into two non-interacting clusters, and
individuals only experience noise when they interact, this model has
comparatively smaller noise than in the previous case, leading to more
predictable outcomes.

Figure 11: Example timeseries of the ABM with adaptation noise (16), using the
setup described at the beginning of Section 3.

####  3.2.3 Random update distance

Here we give a numerical demonstration of the convergence to the SDE system
(19) shown in Section 2.3.4. Let $\nu^{h}\sim\mathds{N}(Nh,sh)$, with constant
$s=5$. As the noise term $\nu$ is multiplied by the distance between
individuals’ opinions, we choose a larger value of $s$ so that the presence of
noise is still clear.

Figure 12: Comparison of the ABM with random update distance (18) and limiting
SDE system (19) with the setup described at the beginning of Section 3.2.
Solid lines show the mean (top panel) and variance (bottom panel) in the
opinion of each individual across 5000 realisations of ABM, dashed lines show
the same for the corresponding SDE system. Both panels include an inset
showing the error, calculated according to (21).

The top panel of Figure 12 compares the mean opinion of each agent in the ABM
in solid lines and the SDE system in dashed lines. There is clearly a good fit
between the two. Similarly the behaviours of the variances shown in the lower
panel of Figure 12 match well, although the values towards the end of the
timeseries differ slightly for some individuals.

The behaviour of the means and variances mirrors that seen in Figure 6 for the
ABM and SDE system with external noise. However, in this case, there are more
individuals with initial opinions near zero who have a high variance. This may
be caused by the possibility of $\nu$ taking negative values, meaning
interactions can cause individuals’ opinions to move apart, adding to the
uncertainty on which opinion cluster individuals will enter.

Figure 14: Example timeseries of the ABM with random update distance (18),
using the setup described at the beginning of Section 3.

There is a key difference in the example timeseries shown in Figure 14,
compared against those in Figures 8 and 11: towards the end of each
simulation, individuals’ opinions stop changing. This is due to the fact that
the noise term $\nu$ is multiplied by the distance $(x_{j}-x_{i})$. Towards
the end of the simulation, pairs of individuals either have very similar
opinions, meaning $(x_{j}-x_{i})$ is very small, or no longer interact. Hence
the noise arising from $\nu$ also becomes very small and the variance
plateaus. In this case, if the simulations were left to run indefinitely there
would be no further change in individuals’ opinions and no continued increase
in variance, as once a clustered state is reached it is fixed. This also makes
the dynamics less predictable, as there is no possibility that individuals
will eventually, due to continued stochasticity, join a nearby opinion
cluster.

##  4 Discussion

The theoretical results presented in Section 2 provide a framework in which
choices made in the ABM setting can be compared directly to those made in the
ODE/SDE setting. We show that under a simultaneous rescaling of time and
update distance the behaviour of the ABM mirrors that of DEMs. This confirms
the assumptions inherent to DEMs: that pairwise interactions between
individuals produce very small changes in opinion, but occur very frequently.
These assumptions are not always valid in real-world scenarios, but when they
are we have shown that DEMs are a very good approximation of ABMs. Moreover,
we show how including external noise or a random update distance can translate
into stochasticity in the limiting model, motivating the choice of diffusion
in an SDE model. Our numerical results in Section 3 demonstrate the excellent
match between the behaviour of the various ABM and SDE models, showing how
both can capture a variety of complex behaviours. This means that the benefits
of DEMs, such as reduced computational cost and increased analytic
tractability, can be combined with the interpretability of ABMs.

However, the methods we have presented are not without their limitations. As
previously discussed, we cannot take $p_{ij}$ to be the BC interaction
function without selection noise, as this discontinuity prevents the necessary
guarantee of existence and uniqueness of solutions to the corresponding DEM.
This issue occurs whenever we wish to implement the BC interaction function in
continuous time [17] and questions of existence and uniqueness of different
types of solutions to the ODE model have been addressed [20, 22], but further
technical work is required to allow for a comparison against the ABM. In
addition, this approach cannot capture effects such as Pineda noise [27] in
which jumps in opinions cannot be bounded by a function of $h$, although
convergence to a jump-diffusion process could be considered.

Figure 8 shows an example timeseries in which noise causes some individuals’
opinions to leave the initial interval $[-1,1]$. Boundary conditions could be
defined for the ABM and the changes tracked through to identify when a
limiting SDE exists. As there are many different ways that boundary conditions
could be enforced (for example: reflecting or absorbing boundary conditions at
the ends of the opinion interval; restricting the random variables $\eta$,
$\xi$ and $\nu$ to prevent movements outside the convex hull; multiplying
noise terms by a function of $x_{i}$ that is zero at the boundary) we present
instead more general results that could be adapted further if necessary.

This paper confirms the strong connection between agent-based models and
differential equations models. It provides the so-far missing mathematical
underpinning, which will lead to an increased understanding of these two very
popular modeling approaches and an improved communication among communities.

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## Acknowledgements

AN was supported by the Engineering and Physical Sciences Research Council
through the Mathematics of Systems II Centre for Doctoral Training at the
University of Warwick (reference EP/S022244/1). MTW is partly supported by the
Royal Society International Exchange Grant IES/R3/213113.

The authors would also like to thank Peter Steiglechner and Guillaume
Deffuant, whose presentations at ODCD23 provided the initial motivation for
this work.

For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC-BY) license to Any Author Accepted Manuscript version arising
from this submission.

##  Appendix A Supplementary Material

###  A.1 Results from Durrett’s Stochastic Calculus

We begin by formulating the ABM as a Markov process. For each $h>0$ let
$\Pi^{h}(x,\cdot)$ be the transition function on $\mathds{R}^{N}$ and
$Y^{h}_{mh},\,m=0,1,2,\dots$ the corresponding discrete time Markov process.
Define $X^{h}_{t}=Y^{h}_{h\lfloor t/h\rfloor}$, i.e. $X^{h}_{t}$ is constant
on intervals $[mh,(m+1)h]$, as well as

$\displaystyle a^{h}_{ij}(x)$ $\displaystyle=\frac{1}{h}\int_{|y-x|\leq
1}(y_{i}-x_{i})(y_{j}-x_{j})\,\Pi^{h}(x,dy)\,,$ (22) $\displaystyle
b^{h}_{i}(x)$ $\displaystyle=\frac{1}{h}\int_{|y-x|\leq
1}(y_{i}-x_{i})\,\Pi^{h}(x,dy)\,,$ (23)

and

$\Delta^{h}_{\varepsilon}(x)=\frac{1}{h}\Pi^{h}\big{(}x,\mathds{R}^{d}\backslash
B(x,\varepsilon)\big{)}.$ (24)

In addition, define functions $a:\mathds{R}^{d}\rightarrow S_{N}$ and
$b:\mathds{R}^{N}\rightarrow\mathds{R}^{N}$, where $S_{N}$ is the set of non-
negative definite real $N\times N$ matrices.

######  Assumption 5.

The functions $a$ and $b$ are continuous and the martingale problem with
coefficients $(a,b)$ is well-posed.

Assumption 5 is satisfied if $a$ and $b$ are both globally Lipschitz
continuous, see [9]. The stochastic process $X$ being a solution to the
martingale problem for coefficients $a$ and $b$ is equivalent to it being a
weak solution of the corresponding SDE

$dX_{t}=b(X_{t})\,dt+\sqrt{a(X_{t})}\,d\beta_{t}.$ (25)

As we are concerned with weak convergence of the ABM to the SDE, we use this
representation. See Section 5.4, specifically the discussion of Theorem 4.5,
in [9] for full details.

######  Assumption 6.

For each $i,j$ with $1\leq i,j\leq N$, $R<\infty$ and $\varepsilon>0$

  1. a)

$\lim\limits_{h\searrow 0}\sup\limits_{|x|\leq R}|a_{ij}^{h}(x)-a_{ij}(x)|=0$
,

  2. b)

$\lim\limits_{h\searrow 0}\sup\limits_{|x|\leq R}|b_{i}^{h}(x)-b_{i}(x)|=0$ ,

  3. c)

$\lim\limits_{h\searrow 0}\sup\limits_{|x|\leq
R}\Delta^{h}_{\varepsilon}(x)=0$ .

Before stating the main theorem we give definitions of two relevant notions of
convergence for stochastic processes. We will consider processes with
trajectories in the space of càdlàg functions, $D([0,T],\mathds{R}^{N})$ for
any $T<\infty$. Denote by $d$ the metric on this space, as described in [9,
28] (see these for a complete description of this space and its properties).
The following definitions of convergence are adapted from those given in [28].

######  Definition A.1.

For a random element $X_{t}$ and a sequence of random elements
$\\{X^{h}_{t}\\}$, let $P$ and $P^{h}$ denote their respective distributions.
We say $\\{X^{h}_{t}\\}$ converges in distribution to $X_{t}$ if
$P^{h}f\rightarrow Pf$ for each bounded, continuous real-valued function $f$
on $D([0,T],\mathds{R}^{N})$. This is denoted $X^{h}_{t}\Rightarrow X_{t}$.

######  Definition A.2.

For $c\in D([0,T],\mathds{R}^{N})$, we say $\\{X^{h}_{t}\\}$ converges in
probability to $c$ if, as $h\rightarrow 0$,

$\mathds{P}[d(X^{h}_{t},c)<\epsilon]\rightarrow 1.$ (26)

For a given (deterministic) $c\in D([0,T],\mathds{R}^{N})$, convergence in
probability is equivalent to convergence in distribution [28].

We now present the main results from [9] we will use.

Theorem 7.1 statement: If $X_{0}^{h}=x^{h}\rightarrow x$ and Assumptions 5 and
6 hold, we have $X_{t}^{h}\Rightarrow X_{t}$ as $h\rightarrow 0$, where
$X_{t}$ is the solution of the martingale problem with coefficients $a$ and
$b$, with $X_{0}=x$.

####  A.1.1 Useful Lemmas

The first lemma replaces the truncated moments in (22) and (23) with regular
moments and replaces (24) with an easier expression to calculate. Let $1\leq
i,j\leq N$ and $p\in\mathds{N}$, define

$\displaystyle\hat{a}^{h}_{ij}(x)$
$\displaystyle=\frac{1}{h}\int_{\mathds{R}^{N}}(y_{i}-x_{i})(y_{j}-x_{j})\,\Pi^{h}(x,dy)\,,$
(27) $\displaystyle\hat{b}^{h}_{i}(x)$
$\displaystyle=\frac{1}{h}\int_{\mathds{R}^{N}}(y_{i}-x_{i})\,\Pi^{h}(x,dy)\,,$
(28)

and

$\gamma^{h}_{p}(x)=\frac{1}{h}\int_{\mathds{R}^{N}}|y_{i}-x_{i}|^{p}\,\Pi^{h}(x,dy)\,.$
(29)

######  Remark A.1.

As we never apply Theorem 7.1 directly and never calculate $a_{ij}^{h}$ and
$b_{i}^{h}$, only $\hat{a}^{h}_{ij}$ and $\hat{b}^{h}_{i}$, we omit the
’superscript hats’ from now on.

Lemma 8.2: If $p\geq 2$ and for all $0\leq i,j\leq N$, $R<\infty$, we have

  1. a)

$\lim\limits_{h\searrow 0}\sup\limits_{|x|\leq R}|a_{ij}^{h}(x)-a_{ij}(x)|=0$
,

  2. b)

$\lim\limits_{h\searrow 0}\sup\limits_{|x|\leq R}|b_{i}^{h}(x)-b_{i}(x)|=0$ ,

  3. c)

$\lim\limits_{h\searrow 0}\sup\limits_{|x|\leq R}\gamma^{h}_{p}(x)=0$ ,

then Assumption 6 hold.

A second lemma is helpful in the case of deterministic limits.

Lemma 8.5: If for all $0\leq i,j\leq N$, $R<\infty$ we have

  1. a)

$\lim\limits_{h\searrow 0}\sup\limits_{|x|\leq R}|a_{ij}^{h}(x)|=0$ ,

  2. b)

$\lim\limits_{h\searrow 0}\sup\limits_{|x|\leq R}|b_{i}^{h}(x)-b_{i}(x)|=0$ ,

then Assumption 6 hold with $a_{ij}(x)\equiv 0$.

In this case, as the limit $X_{t}$ is deterministic, the process $X^{h}_{t}$
converges to $X_{t}$ both in distribution and in probability.

###  A.2 Convergence results

Consider the ABM defined in Section 2.1. We will use Lemma 8.5 from [9] (given
above) to show that this Markov process converges weakly to the ODE (6). In
the following calculations we assume that all subscript indices $i,j$ satisfy
$1\leq i,j\leq N$.

Let $x$ be the current opinions of the population. As there are $N^{2}$
possible interactions, there are also $N^{2}$ states to which the system could
move. These are given by

$x^{\prime}=x+e_{i}\mu^{h}(x_{j}-x_{i})\,,$ (30)

for $i,j=1,\dots,N$, where $e_{i}$ is the $i^{\text{th}}$ standard basis
vector of $\mathds{R}^{N}$. This state is obtained if $i$ and $j$ are chosen
to interact, which occurs with probability $N^{-2}$, and indeed go on to
interact, which occurs with probability $p_{ij}(x)$. If the same individual is
chosen as both $i$ and $j$, which occurs with probability $N^{-1}$, the system
will not change state. Additionally, if $i\neq j$ are chosen but do not
interact, which occurs with probability $1-p_{ij}(x)$, the system will also
not change state. Hence the transition probability function $\Pi^{h}$ is given
by

$\Pi^{h}(x,y)=\begin{cases}\dfrac{1}{N^{2}}p_{ij}(x)&\text{if
}y=x+e_{i}\mu^{h}(x_{j}-x_{i})\text{ for some $i\neq j$},\\\\[10.00002pt]
\dfrac{1}{N}+\dfrac{1}{N^{2}}\sum\limits_{i\neq
j}\big{(}1-p_{ij}(x)\big{)}&\text{if }y=x,\\\\[11.99998pt]
0&\text{otherwise.}\end{cases}$ (31)

Note that $\Pi^{h}$ depends on $h$ through the update distance $\mu^{h}$,
which determines which states can be reached in one update.

In order to apply Lemma 8.5 we calculate $a^{h}_{ij}$ and $b^{h}_{i}$. These
can be understood as approximations to the coefficients $a_{ii}$ and $b_{i}$
of a limiting SDE model that we wish to determine. If $i\neq j$ then there are
no interactions for which both $(y_{i}-x_{i})$ and $(y_{j}-x_{j})$ are both
nonzero, since the opinion of only one individual is changed at each timestep.
Hence $a^{h}_{ij}\equiv 0$ if $i\neq j$. For $i=1,\dots,N$, we calculate the
following integrals by summing over the possible interactions, noting that
only those interactions involving individual $i$ appear as we multiply by the
change in the $e_{i}$ direction. Recall that we set $\mu=Nh$, and therefore

$\displaystyle b^{h}_{i}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\mu^{h}(x_{j}-x_{i})\,\frac{1}{N^{2}}p_{ij}(x)\,,=\frac{1}{N}\sum_{j=1}^{N}p_{ij}(x)\,(x_{j}-x_{i})\,,$
$\displaystyle a^{h}_{ii}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\big{(}\mu^{h}(x_{j}-x_{i})\big{)}^{2}\,\frac{1}{N^{2}}p_{ij}(x)\,,=h\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)\,(x_{j}-x_{i})^{2}\,\Bigg{)}\,,$

Using these we now check the conditions of Lemma 8.5. Since $b_{i}^{h}=b_{i}$,
as seen in the limiting ODE (6), the second is immediately satisfied. Hence it
remains to check the condition on $a_{ij}$:

$\displaystyle\sup\limits_{|x|\leq R}|a_{ij}^{h}(x)|$
$\displaystyle\leq\sup\limits_{|x|\leq
R}h\,\Bigg{|}\,\sum_{j=1}^{N}p_{ij}(x)\,(x_{j}-x_{i})^{2}\,\Bigg{|}$
$\displaystyle\leq h\,\sum_{j=1}^{N}2R^{2}\,\rightarrow 0\text{ as
}h\rightarrow 0.$

Hence both conditions of Lemma 8.5 are satisfied. This guarantees the weak
convergence of the Markov process (which describes the ABM) to the ODE (6).

####  A.2.1 Variations

We now consider changes to the way in which $i$ and $j$ are selected. If the
selection is made by node degree (8) then the only change is the replacement
of the constant $\frac{1}{N^{2}}$ with $\frac{1}{Nk_{i}}$ in $\Pi^{h}$. The
application of Lemma 8.5 is then essentially the same as in the base case
presented above, with this change in constants following through to give the
normalisation in (9).

The second variation considers normalisation by likelihood of interaction,
given by (10). However, individual $i$ then always interacts with the chosen
individual $j$. This avoids the repetition of the appearance of $p_{ij}$. In
this case $\Pi^{h}$ is given by

$\Pi^{h}(x,y)=\begin{cases}\dfrac{1}{N}p_{ij}(X)\bigg{(}\sum\limits_{l=1}^{N}p_{il}(X)\bigg{)}^{-1}&\text{if
}y=x+e_{i}\mu^{h}(x_{j}-x_{i})\text{ for some $i\neq j$},\\\\[10.00002pt]
\dfrac{1}{N}+\dfrac{1}{N}\sum\limits_{i\neq
j}\Bigg{(}1-p_{ij}(X)\bigg{(}\sum\limits_{l=1}^{N}p_{il}(X)\bigg{)}^{-1}\Bigg{)}&\text{if
}y=x,\\\\[11.99998pt] 0&\text{otherwise.}\end{cases}$ (32)

The change to the case $y=x$ does not affect directly the calculation of
$a^{h}$ and $b^{h}$, so the only change that appears in the calculation is the
replacement of $\frac{1}{N}$ with

$\frac{1}{N}\bigg{(}\sum\limits_{l=1}^{N}p_{il}(X)\bigg{)}^{-1}.$

As in the previous variation this change follows through to affect the
normalisation only. The assumption that there exists $c>0$ such that
$p_{ii}(X)>c$ for all $X\in\mathds{R}^{N}$ is required to ensure that the
normalisation is always well-defined and the limiting coefficients remain
Lipschitz continuous.

It would be possible to consider the case that $i$ and $j$ go on to interact
with probability $p_{ij}$. This would lead to the limiting ODE

$\frac{dX_{i}}{dt}=\Bigg{(}\sum\limits_{l=1}^{N}p_{il}(X)\Bigg{)}^{-1}\sum_{j=1}^{N}\big{(}p_{ij}(X)\big{)}^{2}\,(X_{j}-X_{i}).$
(33)

To the best of the authors’ knowledge this system has not previously been
studied as a model of opinion formation. However, it does appear as a limiting
model when considering a fast adaptive network model in [24].

We next consider a variation in which both individuals $i$ and $j$ update
their opinions. Here the transition function is given by

$\Pi^{h}(x,y)=\begin{cases}\dfrac{1}{N^{2}}\big{(}p_{ij}(x)+p_{ji}(x)\big{)}&\text{if
}y=x+\mu^{h}\,(e_{i}-e_{j})\,(x_{j}-x_{i})\text{ for some
$i<j$},\\\\[10.00002pt] \dfrac{1}{N}+\dfrac{1}{N^{2}}\sum\limits_{i\neq
j}\big{(}1-p_{ij}(x)\big{)}&\text{if }y=x,\\\\[11.99998pt]
0&\text{otherwise.}\end{cases}$ (34)

Note that we specify $i<j$ in the first case to avoid the repetition of
states, but if we select a specific $i$ then interactions with any $j$ are
possible due to the symmetry of the transition function. In this case,

$\displaystyle b^{h}_{i}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\mu^{h}(x_{j}-x_{i})\,\frac{1}{N^{2}}\big{(}p_{ij}(x)+p_{ji}(x)\big{)}\,,=\frac{\mu^{h}}{N^{2}h}\sum_{j=1}^{N}\big{(}p_{ij}(x)+p_{ji}(x)\big{)}\,(x_{j}-x_{i})\,.$

Hence, in order to obtain again the same limiting $b_{i}$ we require that
$p_{ij}(x)=p_{ji}(x)$ for all $x\in\mathds{R}^{N}$. Under this assumption we
have

$\displaystyle b^{h}_{i}(x)$
$\displaystyle=\frac{2\mu^{h}}{N^{2}h}\sum_{j=1}^{N}p_{ij}(x)\,(x_{j}-x_{i})\,,$

and so we require $\mu^{h}=Nh/2$. This also gives,

$\displaystyle a^{h}_{ii}(x)$
$\displaystyle=\frac{h}{2}\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)\,(x_{j}-x_{i})^{2}\,\Bigg{)}\,\text{
and }\,a^{h}_{ij}(x)=-\frac{h}{2}p_{ij}(x)\,(x_{j}-x_{i})^{2}\,,$

so $a^{h}$ has the same limit at in the original ABM. Overall, under the
assumption that $p_{ij}$ are symmetric, the only change required to obtain the
same ODE limit (6) is multiplication by a constant in the value of $\mu^{h}$.

###  A.3 Additional sources of noise

Here we consider adaptations to the ABM described in Section 2.3. The general
construction of the ABM is the same as in Section 2.1, but with changes to the
update rule (5) to include additional noise terms.

The calculations of $a^{h}_{ij}$ and $b^{h}_{i}$ performed below are very
similar to those above. As before, for an individual $i$, we sum over $j$ to
include the effect of each possible interaction. When there is an additional
source of noise we must also, for each $j$, integrate over the possible values
the additional random variable could take. This is equivalent to integrating
over the possible states that can be reached.

####  A.3.1 Ambiguity noise

Consider the ABM with ambiguity noise, defined in Section 2.3.1 with update
rule (13):

$x_{i}(t+h)=\begin{cases}x_{i}(t)+\mu^{h}\,\big{(}\omega_{j}(t)-x_{i}(t)\big{)}&\text{
with probability }\phi(|\omega_{j}-x_{i}|)\\\ x_{i}(t)&\text{ with probability
}1-\phi(|\omega_{j}-x_{i}|)\,.\end{cases}$

We will again use Lemma 8.5 from [9] to show that this Markov process
converges weakly to the ODE (6).

Let $\eta$ be a family of random variables satisfying Assumption 2. Let
$\rho^{h}$ be the probability measure of the random variable $\eta^{h}$ and
$\mathds{E}^{h}[\cdot]$ the expectation with respect to this measure. We
obtain

$\displaystyle b^{h}_{i}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\int_{\mathds{R}}\mu^{h}\,(x_{j}+\eta-
x_{i})\,\frac{1}{N^{2}}\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\,d\rho^{h}(\eta)=\frac{1}{N}\sum_{j=1}^{N}\mathds{E}^{h}\big{[}(x_{j}-x_{i}+\eta)\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}\Bigg{(}(x_{j}-x_{i})\,\mathds{E}^{h}\big{[}\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}+\mathds{E}\big{[}\eta\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}\Bigg{)}\,.$
$\displaystyle a^{h}_{ii}(x)$
$\displaystyle=h\sum_{j=1}^{N}\int_{\mathds{R}}(x_{j}-x_{i}+\eta)^{2}\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\,d\rho^{h}(\eta)$
$\displaystyle=h\sum_{j=1}^{N}\mathds{E}^{h}\big{[}(x_{j}-x_{i}+\eta)^{2}\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}$
$\displaystyle=h\sum_{j=1}^{N}\Bigg{(}(x_{j}-x_{i})^{2}\,\mathds{E}^{h}\big{[}\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}+2(x_{j}-x_{i})\,\mathds{E}\big{[}\eta\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}+\mathds{E}^{h}\big{[}\eta^{2}\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}\Bigg{)}\,.$

We now check that these approximations converge suitably to $b_{i}^{h}$ and
$a_{ii}=0$. Recall that $\phi$ is Lipschitz continuous with Lipschitz constant
$L$ and is bounded above by $1$. Using this Lipschitz condition, together with
triangle inequalities, we obtain,

$\displaystyle\sup\limits_{|x|\leq R}$ $\displaystyle|b_{i}^{h}(x)-b_{i}(x)|$
$\displaystyle\leq\sup\limits_{|x|\leq
R}\frac{1}{N}\sum_{j=1}^{N}\bigg{|}(x_{j}-x_{i})\,\mathds{E}^{h}\big{[}\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}+\mathds{E}^{h}\big{[}\eta\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}-(x_{j}-x_{i})\,\phi\big{(}|x_{j}-x_{i}|\big{)}\bigg{|}$
$\displaystyle\leq\sup\limits_{|x|\leq
R}\frac{1}{N}\sum_{j=1}^{N}2R\,\Big{|}\mathds{E}^{h}\big{[}\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}-\phi\big{(}|x_{j}-x_{i}|\big{)}\Big{|}+\frac{1}{N}\sum_{j=1}^{N}\mathds{E}^{h}\big{[}\big{|}\eta\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{|}\big{]}$
$\displaystyle\leq\sup\limits_{|x|\leq
R}\frac{1}{N}\sum_{j=1}^{N}2R\,\mathds{E}^{h}\Big{[}L\big{|}|x_{j}-x_{i}+\eta|-|x_{j}-x_{i}|\big{|}\Big{]}+\mathds{E}^{h}\big{[}|\eta|\big{]}$
$\displaystyle\leq(2RL+1)\,\mathds{E}^{h}\big{[}|\eta|\big{]}\,.$

As we assumed that $\mathds{E}^{h}\big{[}|\eta|\big{]}\rightarrow 0$ as
$h\rightarrow 0$, this gives the second condition of Lemma 8.5. The
calculation for the bound on $a^{h}_{ii}$ is more straightforward as there is
a factor of $h$, so we need only show that all other terms are bounded.

$\displaystyle\sup\limits_{|x|\leq R}|a_{ii}^{h}(x)|$
$\displaystyle\leq\sup\limits_{|x|\leq
R}h\,\sum_{j=1}^{N}\Bigg{(}\Big{|}(x_{j}-x_{i})^{2}\,\mathds{E}^{h}\big{[}\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}\Big{|}+\Big{|}2(x_{j}-x_{i})\,\mathds{E}^{h}\big{[}\eta\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}\Big{|}$
$\displaystyle\quad\qquad\qquad+\Big{|}\mathds{E}^{h}\big{[}\eta^{2}\,\phi\big{(}|x_{j}-x_{i}+\eta|\big{)}\big{]}\Big{|}\Bigg{)}$
$\displaystyle\leq
h\,\sum_{j=1}^{N}\Bigg{(}2R^{2}+4R\,\mathds{E}^{h}[|\eta|]+\mathds{E}^{h}\big{[}\eta^{2}\big{]}\Big{|}\Bigg{)}\,.$

As $\mathds{E}^{h}\big{[}|\eta|\big{]}\rightarrow 0$ as $h\rightarrow 0$, and
$\mathds{E}^{h}\big{[}\eta^{2}\big{]}\leq C$ we have that the first condition
of Lemma 8.5 is satisfied.

####  A.3.2 Adaptation noise

Consider the ABM with adaptation noise, defined in Section 2.3.2 with update
rule (16):

$x_{i}(t+h)=\begin{cases}x_{i}(t)+\mu^{h}\,\big{(}x_{j}(t)-x_{i}(t)\big{)}+\xi^{h}&\text{
with probability }p_{ij}(x)\\\ x_{i}(t)&\text{ with probability
}1-p_{ij}(x)\,.\end{cases}$

We will now use Lemma 8.2 from [9] to show that this Markov process converges
weakly to the SDE (17). Note that this case is introduced after that of the
ABM with external noise (using update rule (14)) in Section 2.3.2, but will be
considered first here as the calculations for the external noise case are a
simplification of those below.

Let $\rho^{h}$ be the probability measure of the random variable $\xi^{h}$ and
$\mathds{E}^{h}[\cdot]$ the expectation with respect to this measure. Assume
the family of random variables $\xi$ satisfies Assumption 3. We first consider
the case in which noise is added only if individuals $i$ and $j$ interact. In
this case,

$\displaystyle b^{h}_{i}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\int_{\mathds{R}}\big{(}\mu^{h}\,(x_{j}-x_{i})+\xi\big{)}\,\frac{1}{N^{2}}p_{ij}(x)\,d\rho^{h}(\xi)$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})\,+\frac{1}{N^{2}}\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)\Bigg{)}\,\frac{\mathds{E}^{h}[\xi]}{h}\,.$
$\displaystyle a^{h}_{ii}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\int_{\mathds{R}}\big{(}\mu^{h}\,(x_{j}-x_{i})+\xi\big{)}^{2}\,\frac{1}{N^{2}}p_{ij}(x)\,d\rho^{h}(\xi)$
$\displaystyle=h\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})^{2}\,+\frac{2}{N}\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})\Bigg{)}\,\mathds{E}^{h}[\xi]\,+\frac{1}{N^{2}}\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)\Bigg{)}\,\frac{\mathds{E}^{h}[\xi^{2}]}{h}\,.$

As we wish to show that the limiting model is an SDE, rather than an ODE, we
use Lemma 8.2 rather than Lemma 8.5. As before, we check the convergence of
these approximations.

$\displaystyle\sup\limits_{|x|\leq R}|b_{i}^{h}(x)-b_{i}(x)|$
$\displaystyle=\sup\limits_{|x|\leq
R}\Bigg{|}\frac{1}{N^{2}}\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)\Bigg{)}\,\frac{\mathds{E}^{h}[\xi]}{h}\Bigg{|}\,\leq\sup\limits_{|x|\leq
R}\frac{1}{N}\Bigg{|}\frac{\mathds{E}^{h}[\xi]}{h}\Bigg{|}$

As we have assumed $m_{1}(\xi)=0$, the first condition of Lemma 8.2 is
satisfied.

$\displaystyle\sup\limits_{|x|\leq R}|a_{ii}^{h}(x)-a_{ii}(x)|$
$\displaystyle=\sup\limits_{|x|\leq
R}\Bigg{|}h\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})^{2}\,+\frac{2}{N}\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})\Bigg{)}\,\mathds{E}^{h}[\xi]\,$
$\displaystyle\hskip
86.72267pt+\frac{1}{N^{2}}\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)\Bigg{)}\,\frac{\mathds{E}^{h}[\xi^{2}]}{h}-\frac{m_{2}(\xi)}{N^{2}}\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)\Bigg{)}\,\Bigg{|}$
$\displaystyle\leq
h\big{(}4R^{2}\big{)}\,+\frac{4R}{N}\,\mathds{E}^{h}[\xi]\,+\frac{1}{N^{2}}\Bigg{|}\frac{\mathds{E}^{h}[\xi^{2}]}{h}-m_{2}(\xi)\Bigg{|}.$

Clearly the first term in this sum converges to $0$ as $h\rightarrow 0$. As
$m_{1}(\xi)=0$, $\mathds{E}^{h}[\xi]$ also converges to $0$ as $h\rightarrow
0$. In addition, the definition of $m_{2}(\xi)$ ensures that the last term in
the sum also converges to $0$ as $h\rightarrow 0$. Hence the second condition
of Lemma 8.2 is satisfied.

As we are using Lemma 8.2, we also calculate $\gamma^{h}_{p}$ with $p=4$:

$\displaystyle\sup\limits_{|x|\leq R}\gamma^{h}_{4}(x)$
$\displaystyle=\sup\limits_{|x|\leq
R}\frac{1}{h}\sum_{i=1}^{N}\sum_{j=1}^{N}\int_{\mathds{R}}\big{|}\mu^{h}\,(x_{j}-x_{i})+\xi\big{|}^{4}\,\frac{1}{N^{2}}p_{ij}(x)\,d\rho^{h}(\xi)\,,$
$\displaystyle\leq\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}p_{ij}(x)\Bigg{(}16N^{4}h^{3}R^{4}+32N^{3}h^{2}R^{3}\,\mathds{E}^{h}[\xi]+24N^{2}hR^{2}\,\mathds{E}^{h}[\xi^{2}]+8NR\,\mathds{E}^{h}[\xi^{3}]+\frac{1}{h}\mathds{E}^{h}[\xi^{4}]\Bigg{)}$
$\displaystyle\leq
h^{3}\big{(}16N^{4}R^{4}\big{)}+h^{2}\big{(}32N^{3}R^{3}\big{)}\,\mathds{E}^{h}[\xi]+h\big{(}24N^{2}R^{2}\big{)}\,\mathds{E}^{h}[\xi^{2}]+8NR\,\mathds{E}^{h}[\xi^{3}]+\frac{1}{h}\mathds{E}^{h}[\xi^{4}]\,.$

The assumption that $m_{3}(\xi)=m_{4}(\xi)=0$ guarantees that all terms in
this sum converge to $0$ as $h\rightarrow 0$.

####  A.3.3 External noise

Now consider the ABM with external noise, defined in Section 2.3.2 with update
rule (14):

$x_{i}(t+h)=\begin{cases}x_{i}(t)+\mu^{h}\,\big{(}x_{j}(t)-x_{i}(t)\big{)}+\xi^{h}&\text{
with probability }p_{ij}(x)\\\ x_{i}(t)+\xi^{h}&\text{ with probability
}1-p_{ij}(x)\,.\end{cases}$

We will use Lemma 8.2 from [9] to show that this Markov process converges
weakly to the SDE (15). At each timestep, once individuals $i$ and $j$ are
selected, there are two possible updates depending on whether or not an
interaction occurs. This will lead to two integrals in the following
calculation, but in fact gives a simpler final result.

$\displaystyle b^{h}_{i}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\int_{\mathds{R}}\big{(}\mu^{h}\,(x_{j}-x_{i})+\xi\big{)}\,\frac{1}{N^{2}}p_{ij}(x)\,d\rho^{h}(\xi)\,+\frac{1}{h}\sum_{j=1}^{N}\int_{\mathds{R}}(\xi)\,\frac{1}{N^{2}}\big{(}1-p_{ij}(x)\big{)}\,d\rho^{h}(\xi)\,,$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})\,+\frac{1}{N}\,\frac{\mathds{E}^{h}[\xi]}{h}\,.$
$\displaystyle a^{h}_{ii}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\int_{\mathds{R}}\big{(}\mu^{h}\,(x_{j}-x_{i})+\xi\big{)}^{2}\,\frac{1}{N^{2}}p_{ij}(x)\,d\rho^{h}(\xi)\,+\frac{1}{h}\sum_{j=1}^{N}\int_{\mathds{R}}(\xi)^{2}\,\frac{1}{N^{2}}\big{(}1-p_{ij}(x)\big{)}\,d\rho^{h}(\xi)\,,$
$\displaystyle=h\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})^{2}\,+\frac{2}{N}\Bigg{(}\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})\Bigg{)}\,\mathds{E}^{h}[\xi]\,+\frac{1}{N}\frac{\mathds{E}^{h}[\xi^{2}]}{h}\,.$

Comparing these to $b^{h}_{i}(x)$ and $a^{h}_{ii}(x)$ from the previous case,
the only change is the removal of the factor $\sum\limits_{j=1}^{N}p_{ij}(x)$
from the last term in each sum. Due to this similarity we do not show again
the convergence of these approximations and $\gamma^{h}_{p}$, as the
calculations are essentially the same as those above.

####  A.3.4 Random update distance

Finally we consider the ABM with random update distances, defined in Section
2.3.4 with update rule (18):

$x_{i}(t+h)=\begin{cases}x_{i}(t)+\nu^{h}\,\big{(}x_{j}(t)-x_{i}(t)\big{)}&\text{
with probability }p_{ij}(x)\\\ x_{i}(t)&\text{ with probability
}1-p_{ij}(x)\,.\end{cases}$

We will use Lemma 8.2 from [9] to show that this Markov process converges
weakly to the SDE (19).

Let $\rho^{h}$ be the probability measure of the random variable $\nu^{h}$ and
$\mathds{E}^{h}[\cdot]$ the expectation with respect to this measure. Assume
the family of random variables $\nu$ satisfies Assumption 4. Due to the
linearity of the equations in $\mu$, the calculations in this section are much
more straightforward:

$\displaystyle b^{h}_{i}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\int_{\mathds{R}}\nu\,(x_{j}-x_{i})\,\frac{1}{N^{2}}p_{ij}(x)\,d\rho^{h}(\nu)=\frac{\mathds{E}^{h}[\nu]}{h}\frac{1}{N^{2}}\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})\,.$
$\displaystyle a^{h}_{ii}(x)$
$\displaystyle=\frac{1}{h}\sum_{j=1}^{N}\int_{\mathds{R}}\nu^{2}\,(x_{j}-x_{i})^{2}\,\frac{1}{N^{2}}p_{ij}(x)\,d\rho^{h}(\nu)=\frac{\mathds{E}^{h}[\nu^{2}]}{h}\frac{1}{N^{2}}\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})^{2}\,.$

The conditions on $m_{1}(\nu)$ and $m_{2}(\nu)$ clearly ensure the first two
conditions on Lemma 8.2 are satisfied. It only remains to calculate

$\displaystyle\sup\limits_{|x|\leq R}\gamma^{h}_{4}$
$\displaystyle=\sup\limits_{|x|\leq
R}\frac{1}{h}\sum_{i=1}^{N}\sum_{j=1}^{N}\int_{\mathds{R}}\nu^{4}\,(x_{j}-x_{i})^{4}\,\frac{1}{N^{2}}p_{ij}(x)\,d\rho^{h}(\nu)$
$\displaystyle=\frac{\mathds{E}^{h}[\nu^{4}]}{h}\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}p_{ij}(x)(x_{j}-x_{i})^{4}$
$\displaystyle\leq\frac{\mathds{E}^{h}[\nu^{4}]}{h}16R^{4}\,.$

Hence the assumption that $m_{4}(\nu)=0$ gives that the final condition of
Lemma 8.2 is satisfied.