Impact of launch power optimisation in hybrid-amplified links

H. Buglia(1),
E. Sillekens(1),
L. Galdino(2),

R. I. Killey(1), and P. Bayvel(1)

(1) Optical Networks Group, UCL (University College London), UK.

(2) Corning Optical Communications, Ewloe, CH5 3XD, United Kingdom.

Per-channel launch power optimisation in a hybrid-amplified link with optimised pump powers and wavelengths is described. Compared to using the optimum spectrally uniform launch power, an average SNR gain of 0.13 dB is obtained against 0.56 dB for the same system operating with lumped amplifiers only. ©2024 The Author(s)

§ INTRODUCTION

To achieve the maximum throughput in ultra-wideband (UWB) systems using hybrid (Raman + rare-earth doped fibre) amplifiers, optimisation of launch power (LP), pump powers and wavelength allocation is required <cit.>. Such optimisation has become possible thanks to closed-form expressions from the Gaussian noise model in the presence of Raman amplification <cit.>, which estimates the nonlinear interference (NLI) noise and amplified spontaneous emission (ASE) noise in real-time for UWB system operating with hybrid amplifiers.

LP optimisation for UWB has been extensively analysed for lumped-amplified links in <cit.>. It was shown in <cit.> that relevant gains in performance can be achieved by allocating higher power in the S-band and lower in the L-band to counteract the inter-channel stimulated Raman scattering (ISRS) effect. In <cit.> it was also shown that these gains in performance depend on the system configuration including span lengths and optical bandwidths. However, for the case of hybrid-amplified links, very little research has been carried out to date. This is due to the very recent availability of real-time models <cit.> as well complex interactions between ISRS, pump powers and wavelengths, and the signal LP.

Indeed, the first theoretical UWB demonstrations of hybrid-link optimisation were reported in <cit.> for backward pumped Raman systems. In <cit.>, a spectrally uniform LP profile was used to maximise the system throughput, whereas in <cit.>, per-channel LP optimisation was performed. However, in the latter, no quantification of the gains obtained by optimising the LP was reported. This is important as LP optimisation is computationally complex and requires additional shaping components in the deployed systems, increasing the overall system cost and power consumption.

In this work we quantify, for the first time, the gains in total throughout obtained by per-channel LP optimisation in hybrid-amplified links and compare the values with those obtained for lumped systems. A capacity-maximising hybrid amplifier is described, designed using a particle swarm optimisation (PSO) algorithm, where the launch power as well as forward and backward pumps are optimised in terms of power and wavelength to maximise system throughput. The results are compared to an optimum spectrally uniform LP profile. A similar analysis is carried out for a system with the same parameters but designed to operate using lumped amplifiers only.

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Fibre attenuation coefficient and Raman gain spectrum for the fibre used, and optimised forward (green arrows) and backward (blue arrows) pump wavelength allocation.

§ TRANSMISSION SYSTEM SETUP

It was assumed that the modelled system uses a hybrid amplification technology, consisting of two stages: a forward+backward-distributed Raman amplifier followed by an ideal EDFA. Transmission of $N_{\text{ch}}$=131 channels spaced by 100 GHz and centred at 1550 nm was modelled. Each channel was modulated at the symbol rate of 96 GBd, resulting in a total bandwidth of 13 THz (105 nm), spanning 1500 nm to 1605 nm (transmission over the S-, C- and L- bands). Spectral gaps of 10 nm and 5 nm are assumed, respectively, between the S/C and C/L bands. The noise figure of each lumped amplifier placed at the end of each span is 7 dB, 4.5 dB, and 6 dB in the S-, C- and L- bands, respectively. Gaussian constellations, ideal lumped amplification, and ideal transceivers are assumed.

The signal transmission was evaluated over 10 $\times$ 100 km spans (total distance of 1000 km). The  optical fibre  was assumed to have wavelength-dependent attenuation and the Raman gain profile compliant with Fig. <ref>, and an effective area of $80~\mu\text{m}^2$, resulting in a nonlinear coefficient $\gamma = 1.16\text{ W}^{-1} \text{km}^{-1}$.
Dispersion parameters considered are $D = 16.5~\text{ps }\text{nm}^{-1}\text{km}^{-1}$, $S = 0.09~\text{ps }\text{nm}^{-2}\text{km}^{-1}$. For the distributed Raman amplification, forward and backward pumps were placed in the E- and S-bands, as shown in Fig. <ref>, and their wavelengths and powers, and the per-channel system LP were optimised to maximise the system throughput.

§ LAUNCH POWER AND PUMP OPTIMISATION

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3[surf,color=red,opacity=0.4] file Data/ECOC_powerprofile.txt;
Forward + Backward Raman Amplification

Per-channel LP evolution with distance for the optimised launch and pump powers and pump wavelengths.

LP and pump were optimised over a single-span transmission based on a PSO algorithm from the Matlab global optimisation toolbox. The cost function to be maximised was the total system throughput, given by: $C = \sum_{i = 1}^{N_{\text{ch}}} 2\cdot\log_2(1+\SNR_i)$, where $\SNR_i$ is the SNR of the channel $i$, calculated using the model in <cit.>.
Given the large number of degrees of freedom of this problem, the optimisation was divided into two steps. Firstly, pump powers and wavelengths were optimised with a spectrally uniform LP profile. Secondly, the pump powers and wavelengths found previously were kept constant and the channel LP was optimised.

For the first step, 6 forward plus 6 backward pumps were placed in the E- and S-bands (1405 nm-1490 nm).  The PSO algorithm had 13 variables to be optimised (12 pumps + the total LP). The number of particles was chosen to be 50 with a maximum of 50 iterations selected as the stopping criterion. For the algorithm bounds, we allowed the total channel
LP to vary between 10 dBm and 25 dBm, and the power of each pump from
0 mW to 250 mW. The optimisation resulted in an optimum spectrally uniform LP of -1.74 dBm per channel, corresponding to a total LP of 18.75 dBm; 3 forward pumps with non-negligible power, with wavelengths located at 1405 nm, 1410 nm and 1455 nm, and powers of 153.6 mW, 240.1 mW and 31.3 mW, respectively; and 5 backward pumps with non-negligible power, with wavelengths located at 1422 nm, 1428 nm, 1437 nm, 1452 nm, and 1483 nm, and powers of 249.2 mW, 31.7 mW, 250 mW, 80 mW and 225.3 mW, respectively.

For the second step, the pump powers and wavelengths found previously were kept and the per-channel LP was optimised to maximise the system throughout. In this step, the PSO algorithm had $N_{\text{ch}}$ = 131 variables to be optimised, corresponding to the number of WDM channels.
The number of particles was chosen to be $10 \cdot N_{\text{ch}}$ with their values, i.e., the LP for each channel, ranging from -10 dBm to 10 dBm.
A maximum of 75 iterations was selected as the stopping criterion. The optimum per-channel LP evolution of the optimised hybrid amplifier with distance is shown in Fig. <ref>. This optimisation was also carried out for a system operating with lumped amplification only. In that case, the per-channel launch-power bounds ranged from -5 dBm to 15 dBm, and the remaining parameters were kept the same.
Finally, for both cases, the optimum per-channel LP obtained from the PSO algorithm was passed through a second-order Savitzky-Golay filter with a 7-channel window, resulting in a smooth per-channel LP that achieved a total throughput within 0.06 Tb/s of that obtained from the PSO.

Per-channel and spectrally uniform optimum LP profiles for (a) hybrid and (b) lumped amplification.

SNR contributions after 10 spans using per-channel (continuous lines) and spectrally uniform (dashed lines) optimum LP profile for (a) hybrid and (b) lumped amplification.

§ RESULTS AND DISCUSSION

The optimum per-channel and spectrally uniform LP profiles for the optimised hybrid amplifier are shown in Fig <ref> after one span. For the optimum per-channel LP, a total power of 19.14 dBm was obtained with a total per-channel variation of 2.85 dB. For the optimum spectrally uniform LP, the total power was 18.75 dBm. The same analysis was carried out and is shown in Fig. <ref>.b for an optimised system operating with lumped amplifiers only; in that case, these values were 24.44 dBm, with a total per-channel variation of 6.14 dB, and 23.85 dBm respectively.

Fig <ref> shows the system performance after 10 spans for the LP profiles shown in Fig. <ref>.a. For the optimised hybrid amplifier, using the optimum spectrally uniform LP, yields increased values of NLI noise and reduced values of ASE noise in the S-band, resulting mostly from the noise from forward Raman pumps within that band. On the other hand, reduced values of NLI noise and increased values of ASE noise are observed in the C- and L- bands which are most impacted by the noise from the backward Raman pumps. To counteract these noises,  the per-channel optimum LP allocates reduced power levels in the S-band and increased power levels in the C- and L-bands (see Fig. <ref>.a), increasing the system performance.
The results of the same analysis carried out for the optimised lumped amplified system are shown in Fig <ref>.b. As is well known in such cases, the optimum per-channel LP allocates more power in the S-band to counteract the ISRS effect (see Fig. <ref>.b).

In terms of performance for the optimised hybrid amplifier (Fig. <ref>.a), using the per-channel optimum LP profile, yields an average total SNR of 19.78 dB, and total throughputs of 213.14 Tb/s and 141.64 Tb/s after 1 and 10 spans respectively. On the other hand, using the optimum spectrally uniform LP profile yields an average total SNR of 19.65 dB, and a total throughput of 212.48 Tb/s and 140.70 Tb/s, over 1 and 10 spans respectively. The same analysis for the optimised lumped amplifier (Fig. <ref>.b), yields an average total SNR value of 15.07 dB and total throughputs of 179.88 Tb/s and 108.66 Tb/s over 1 and 10 spans for the optimum per-channel LP profile. These values using a spectrally uniform LP profile are 14.50 dB, 175.72 Tb/s and 104.86 Tb/s respectively.

In terms of gains obtained through the use of per-channel LP optimisation, compared to the spectrally uniform case, the hybrid-amplified system yields 0.13 dB in total SNR gain, and 0.66 Tb/s (0.3 %) and 0.94 Tb/s (0.7 %) throughput gains over 1 and 10 spans. For the lumped-amplified system, the gains with per-channel LP optimisation are significantly larger, at 0.56 dB, 4.16 Tb/s (2.4 %) and 3.80 Tb/s (3.6 %) respectively.

§ CONCLUSIONS

The first analysis quantifying the gains of per-channel launch power optimisation in hybrid-amplified links was presented, where optimum spectrally uniform power profile was used for comparison. The modelled system used a hybrid amplifier with forward+backward Raman pumps, where launch power, pump powers and wavelengths were optimised to maximise the total throughput.

For hybrid-amplified systems, in comparison with lumped-amplified systems, it was found that pump power and wavelength optimisation can compensate for almost all the power fluctuations induced by the combination of Raman, ISRS and wavelength-depend fibre parameters, reducing the further gain in SNR achievable with per-channel launch power optimisation. For the optimised hybrid-amplified system only 0.13 dB SNR gain was due to the per-channel power optimisation. In contrast, per-channel power optimisation in the system with lumped amplification was responsible for a larger SNR gain of 0.56 dB.

§ ACKNOWLEDGEMENTS

This work was supported by EPSRC grants EP/R035342/1 TRANSNET
, EP/W015714/1 EWOC
, EPSRC studentship
(EP/T517793/1), and Microsoft Scholarship under the 'Optics for the Cloud' programme.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

booktitleOptical Fiber Communications Conference
titleThroughput Maximisation in Ultra-wideband Hybrid-amplified Links

booktitleOptical Fiber Communications Conference
titleOptimization of channel powers, Raman pumps and EDFAs in the wideband fiber optic transmission systems

journaltitleEuropean Conference on Optical Communication (ECOC)
titleA Modulation-Format Dependent Closed-form Expression for the Gaussian Noise Model in the Presence of Raman Amplification

journaltitleJournal of Lightwave Technology
titleA Closed-Form Expression for the Gaussian Noise Model in the Presence of Raman Amplification

booktitle2022 Optical Fiber Communications Conference and Exhibition (OFC)
titleLink Power Optimization for S+C+L Multi-band WDM Coherent Transmission Systems
Optical fiber amplifiers;Heuristic algorithms;Wavelength division multiplexing;Optical fiber communication;Iterative algorithms;Optimization

journaltitleJournal of Optical Communications and Networking
titleOn the impact of launch power optimization and transceiver noise on the performance of ultra-wideband transmission systems [Invited]
Optimization;Signal to noise ratio;Throughput;Mathematical models;Bandwidth;System performance;Transceivers

booktitle2021 International Conference on Optical Network Design and Modeling (ONDM)
titleChallenges in Extending Optical Fibre Transmission Bandwidth Beyond C+L Band and How to Get There
Analytical models;Computational modeling;Optical fiber networks;Tools;Throughput;Wavelength division multiplexing;Transceivers;Ultra-wideband system;ISRS GN-model;analytical modelling;WDM

journaltitleJournal of Optical Communications and Networking
titlePower control strategies and network performance assessment for C+L+S multiband optical transport
Nonlinear optics;Optical scattering;Wavelength division multiplexing;Power control;Optical fiber networks;Optical noise;Optimization

booktitle2021 International Conference on Optical Network Design and Modeling (ONDM)
titleParameter Optimisation for Ultra-Wideband Optical Networks in the Presence of Stimulated Raman Scattering Effect
Optical fibers;Network topology;Raman scattering;Optical fiber networks;Throughput;Topology;Resource management;parameter optimisation;stimulated Raman scattering;network topology;launch power

Optica Publishing Group

journaltitleOpt. Express
titleOptimization strategy of power control for C$+$L$+$S band transmission using a simulated annealing algorithm
Erbium-doped fiber amplifiers; Mode division multiplexing; Multiplexing; Optical amplifiers; Raman scattering; Space division multiplexing

journaltitleJournal of Optical Communications and Networking
titleModeling and mitigation of fiber nonlinearity in wideband optical signal transmission [Invited]
Wavelength division multiplexing;Optical distortion;Nonlinear optics;Modulation;Optical fiber networks;Wideband

booktitle2022 Optical Fiber Communications Conference and Exhibition (OFC)
titleModeling of Fiber Nonlinearity in Wideband Transmission
Optical fibers;Stimulated emission;Raman scattering;Optical attenuators;Optical fiber communication;Optical scattering;Particle swarm optimization


# Maximising the Influence of Temporary Participants in Opinion Formation

Zhiqiang Zhuang College of Intelligence and Computing, Tianjin University,
China Kewen Wang School of Information and Communication Technology, Griffith
University, Australia Zhe Wang22footnotemark: 2 Junhu Wang22footnotemark: 2
Yinong Yang Department of Mathematics, Liaoning University, China

###### Abstract

DeGroot-style opinion formation presume a continuous interaction among agents
of a social network. Hence, it cannot handle agents external to the social
network that interact only temporarily with the permanent ones. Many real-
world organisations and individuals fall into such a category. For instance, a
company tries to persuade as many as possible to buy its products and, due to
various constraints, can only exert its influence for a limited amount of
time. We propose a variant of the DeGroot model that allows an external agent
to interact with the permanent ones for a preset period of time. We obtain
several insights on maximising an external agent’s influence in opinion
formation by analysing and simulating the variant.

##  1 Introduction

Consider a social network in which people have an opinion about the state of
something in the world, such as the willingness to buy a product, the
effectiveness of a public policy, or the reliability of an economic forecast.
Rather than forming opinions on their own, people tend to learn about the
state of the world via observation and communication with others. Mathematical
models of opinion formation try to formalize these interactions by describing
how people process the other’s opinions and how their opinions evolve as a
result of the interactions [18].

The DeGroot model [11] is a benchmark opinion formation model that has found
usage in many disciplines. The model describes a discrete time opinion
formation process in which agents within a social network have an initial
opinion that they update by repeatedly taking weighted average of their
friends’ opinions. Over the years, some highly influential variants of the
DeGroot model have been proposed to take into account real-world situations
that were neglected in the original model such as the Friedkin and Johnson
model [15] and the bounded confidence model [10, 17]. A recurrent topic in
DeGroot-style opinion formation is the identification of conditions for
reaching a consensus and the quantification of individual influence in forming
the consensus [24].

Implicit in the DeGroot model and its variants is the assumption that the
agents of a social network interact continuously where no agent skips any
interaction at any time. This is a reasonable assumption, given the dynamic
nature of opinion formation and the research focus on its limiting behaviour.
However, it excludes external agents that do not have a permanent presence in
the social network but may have a considerable influence to the permanent
agents. A prominent example is an organisation trying to persuade people to
for instance buy its products or vote for a particular candidate through
advertising. Due to constraints like budget and timing, the organisation can
advertise or exert their influence only for a limited amount of time,
nevertheless, for some people, the organisation’s influence is at least
comparable to that of their friends in the social network.

Traditionally, agents who refused to be influenced by others are termed as
stubborn agents (also called zealots [22] or radicals [25]) [24]. Assuming a
permanent presence, eventually they make every non-stubborn agents submissive
of their opinions. The aforementioned external agents are also stubborn for
having uncompromising opinions, however, a sharp difference with the orthodox
modeling is their temporary nature. To the best of our knowledge we are the
first to consider stubborn agents without a permanent presence. This setting
not only is more realistic, but also opens up new perspectives to investigate
behaviors of such agents. For instance, with a limited time frame to exert
influence, these agents are confronted with the strategic problem of how to
allocate their resource to achieve maximum influence.

Our goal in this paper is twofold. Firstly, we propose a variant of the
DeGroot model that allows an external (stubborn) agent to participate only
temporarily. The variant enables the investigation on how combinations of the
following four factors affect the external agent’s influence. We illustrate
them in the context of a company promoting its products by TV commercials.

Coverage:

the number of agents to which the external agent can exert its influence. This
is the expected number of viewers of the TV commercial each time it is
broadcast.

Duration:

the number of times the external agent can exert its influence. This is the
expected number of times the TV commercial is broadcast.

Intensity:

the amount of influence the external agent can exert to other agents each time
it does so. This reflects the TV commercial’s impact to its viewers each time
they see it.

Timing:

the time points at which the external agent exerts its influence. This
reflects the time points at which the TV commercial is broadcast.

Secondly, we articulate insights on how to allocate the external agents’
resources to the four factors in order to maximise its influence through
mathematical analysis and computer simulation. According to our analysis and
simulations, the timing factor is irrelevant if the coverage factor is at its
maximum (i.e., full coverage); the coverage and the duration factor are
equally important; and it is more effective to allocate resource to scale up
the intensity factor than to scale up the duration factor. We also derive
several other insights that deepen our understanding of opinion formation in
general.

After giving some preliminaries, we present our model that incorporates the
aforementioned factors into the opinion formation process. This is followed by
insights obtained by analytical method and subsequently simulations.

##  2 Preliminaries

In this paper, we write matrices as uppercase letters in boldface such as
$\mathbf{T}$, vectors as lowercase letters in boldface such as $\mathbf{p}$,
and scalars as lowercase letters such as $m$. We denote the set of real
numbers and integers as $\mathbb{R}$ and $\mathbb{Z}$ respectively. The entry
in the $i$-th row and $j$-th column of a matrix $\mathbf{T}$ is denoted as
$t_{ij}$. The _transpose_ of a matrix $\mathbf{T}$ is written as
$\mathbf{T}^{\top}$. A matrix is _non-negative_ if all its entries are non-
negative. A non-negative matrix $\mathbf{T}$ is _stochastic_ if all its rows
sum to 1, that is $\sum_{j}t_{ij}=1$ for all $i$. Vectors are considered as
single column matrices unless otherwise specified. The $i$th component of a
vector $\mathbf{p}$ is denoted as $p_{i}$. The “zero” vector
$(0,\ldots,0)^{\top}$ and the “one” vector $(1,\ldots,1)^{\top}$ are denoted
as $\mathbf{0}$ and $\mathbf{1}$ respectively, and are with dimensions
suitable to the context they appear.

A directed graph is a pair $(V,E)$ where $V$ is the set of nodes and
$E\subseteq V\times V$ the set of _edges_. In opinion formation models, a
directed graph $(V,E)$ is often identified by its adjacency matrix
$\mathbf{T}$ which is a non-negative matrix such that $(i,j)\in E$ iff
$t_{ij}>0$. Paths, cycles and and their lengths in a directed graph are
defined in the standard way. A directed graph or equivalently an adjacency
matrix is _strongly connected_ if there is a path from any node to any other
node and it is _aperiodic_ if the greatest common divisor of the lengths of
its cycles is one.

##  3 Models of Opinion Formation

In this section, we present our opinion formation model. It involves $n$
permanent agents interacting continuously and an external agent interacting
temporarily with the permanent ones. Outside the external agent’s period of
interaction, our model reduces to the DeGroot Model in which the pattern of
interaction is represented as a $n\times n$ stochastic matrix $\mathbf{T}$. An
entry $t_{ij}$ of $\mathbf{T}$ represents the weight agent $i$ places on agent
$j$. The weights $t_{ij}$ for $j=1,2,\ldots,n$ are considered as finite
resources, distributed by $i$ to itself and others. A positive $t_{ij}$
indicates $j$ is able to influence $i$ at each round of interaction; and the
greater $t_{ij}$ is, the stronger the influence. We refer to $\mathbf{T}$ as
the _interaction matrix_ which can be seen as the adjacency matrix that
captures the social network structure. Conventionally, opinions are
represented as real numbers and time is measured in rounds of interactions. We
denote agent $i$’s opinion and the vector of the $n$ agents’ opinions after
the $t$th round of interaction as $p_{i}^{(t)}$ and $\mathbf{p}^{(t)}$
respectively. Moreover, we refer to $p_{i}^{(0)}$ as the _initial opinion_ of
$i$ and $\mathbf{p}^{(0)}$ the _initial opinion vector_. The agents repeatedly
interact by synchronously taking weighted averages of the opinions of agents
who can influence them, that is $\mathbf{p}^{(t)}$ obeys the following
equation for $t\geq 1$

$\mathbf{p}^{(t)}=\mathbf{T}^{t}\mathbf{p}^{(0)}.$ (1)

The weight $t_{ij}$ is therefore the contribution of $j$’s opinion at each
round of interaction to $i$’s opinion at the next round. The weight $t_{ii}$
agent $i$ places on itself represents its openness to other agent’s influence:
$t_{ii}=0$ indicates an open-minded agent who totally relies on the others’
opinions whereas $t_{ii}=1$ indicates a _stubborn_ agent whose opinion remains
unchanged.

An opinion formation (as described by Equation (1)) is _convergent_ if

$\mathbf{p}^{\infty}=\lim_{t\rightarrow\infty}\mathbf{T}^{t}\mathbf{p}^{(0)}$

exists for any $\mathbf{p}^{(0)}$. A convergent opinion formation reaches a
_consensus_ if all components of $\mathbf{p}^{\infty}$ are identical, which
happens when all rows of $\lim_{t\rightarrow\infty}\mathbf{T}^{t}$ are
identical. We refer to $\mathbf{p}^{\infty}$ as the _limiting opinion vector_
and $p_{i}^{\infty}$ the limiting opinion of $i$. It is shown that if the
interaction matrix is strongly connected, then a convergent opinion formation
always reaches a consensus. Moreover, an opinion formation with a strongly
connected interaction matrix $\mathbf{T}$ is convergent iff $\mathbf{T}$ is
aperiodic or equivalently there is a unique left eigenvector $\mathbf{s}$ of
$\mathbf{T}$, corresponding to eigenvalue 1 such that $\sum_{i=1}^{n}s_{i}=1$
and

$p_{i}^{\infty}=\left(\lim_{t\rightarrow\infty}\mathbf{T}^{t}\mathbf{p}\right)_{i}=\mathbf{sp}^{(0)}$
(2)

for $i=1,2,\ldots,n$. The result establishes whether an opinion formation
converges and what it converges to when it does. Also the result implies
$\lim_{t\rightarrow\infty}\mathbf{T}^{t}$ is a matrix with identical rows each
of which is the unique left eigenvector $\mathbf{s}$. As $p_{i}^{\infty}$ are
identical for $i=1,2,\ldots,n$, we refer to all of them as the _limiting
opinion_. See the survey [24] for the other convergence and consensus
conditions and [23] for the technical details on matrix.

According to Equation (2), the limiting opinion is a weighted average of the
initial opinions where agent $i$’s weight is $s_{i}$. These weights are
commonly taken as the measure of an agent’s influence in a DeGroot-style
opinion formation and are sometimes referred to as the agents’ social
influence [11, 18]. We refer to $\mathbf{s}$ as the _social influence vector_.
Note that, $\mathbf{s}$ being a left eigenvector of $\mathbf{T}$ with an
eigenvalue of 1 means $\mathbf{s}\mathbf{T}=\mathbf{s}$ which implies
$s_{i}=\sum_{j=1}^{n}t_{ji}s_{j}$. Thus the social influence of $i$ is a
weighted sum of the social influences of the various agents who can be
influenced by $i$. This is a very natural property of a measure of influence
and entails that an influential person is one who is trusted by other
influential persons. We adopt this measure of influence to quantify the
external agent’s influence in opinion formation.

The novelty of our model lies in the treatment of the external agent that
interacts with the permanent ones for a finite number of rounds. We reserve
the letter $k$ for this finite number which indicates the duration of the
external agent’s influence. To represent the pattern of interaction involving
the external agent, we extend the interaction matrix $\mathbf{T}$ to form a
$(n+1)\times(n+1)$ interaction matrix $\mathbf{A}$ in which the external agent
is identified as the $(n+1)$th agent whose corresponding weights occupy the
$(n+1)$th row and column. We refer to $\mathbf{A}$ as the _extended
interaction matrix_. Since the external agent acts as an organisation or
individual with the solitary goal of persuading the others of its opinion, the
external agent is, in our terminology, a stubborn agent, hence
$a_{(n+1)(n+1)}=1$. For the rest of the entries in the $(n+1)$th column,
$a_{i(n+1)}>0$ means the external agent is able to influence agent $i$. We
reserve the letter $m$ for the number of such positive entries which indicates
the coverage of the external agent’s influence. A simplifying assumption we
make is that the $m>0$ agents place an identical weight of $\lambda$ to the
external agent, that is $a_{i(n+1)}=\lambda$ whenever $a_{i(n+1)}>0$ and
$i\leq n$. The weight $\lambda$ indicates the intensity of the external
agent’s influence. We also assume, without loss of generality, that the $m$
agents occupy the first $m$ rows and columns of $\mathbf{A}$. We reserve the
letter $\Lambda$ for the vector that occupies the first $n$ entries of the
$(n+1)$th column. Lastly, for each entry $a_{ij}$ with $1\leq i,j\leq n$, if
$a_{i(n+1)}=0$, then it inherit the corresponding entry $t_{ij}$ in
$\mathbf{T}$, otherwise it is shrank from $t_{ij}$ by a factor of
$(1-\lambda)$ to make $\mathbf{A}$ stochastic. Putting these together, we have

${\mathbf{A}={\begin{bmatrix}(1-\lambda)t_{11}&\cdots&(1-\lambda)t_{1n}&\lambda\\\
\vdots&\ddots&\vdots&\vdots\\\
(1-\lambda)t_{m1}&\cdots&(1-\lambda)t_{mn}&\lambda\\\
t_{(m+1)1}&\cdots&t_{(m+1)n}&0\\\ \vdots&\ddots&\vdots&\vdots\\\
t_{n1}&\cdots&t_{nn}&0\\\ 0&\cdots&0&1\end{bmatrix}}}$

where $m$ is the number of agents the external agent can influence, $t_{ij}$
are entries of $\mathbf{T}$, and $\lambda$ is the weight placed on the
external agent.

The $n+1$ agents interact exactly as in the DeGroot model only that it is now
governed by both $\mathbf{T}$ and $\mathbf{A}$. The opinion vector
$\mathbf{p}^{(t)}$ obeys Equation (1) when the external agent does not
participate and the following one when it does.

$\mathbf{p}^{(t)}=\left(\mathbf{A}\left(\begin{array}[]{c}\mathbf{p}^{(t-1)}\\\
a\end{array}\right)\right)_{1,\ldots,n}$ (3)

where $a$ is the external agent’s unchanged opinion. For a matrix
$\mathbf{W}$, we denote the matrix formed by the first $n$ rows of
$\mathbf{W}$ as $(\mathbf{W})_{1,\ldots,n}$. For example,
$\big{(}\begin{smallmatrix}a&b\\\ c&d\\\
e&f\end{smallmatrix}\big{)}_{1,2}=\big{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\big{)}$. To illustrate this interaction, suppose the
external agent participates in the third and forth rounds of interaction, then

$\mathbf{p}^{(4)}=\left(\mathbf{A}^{2}\left(\begin{array}[]{c}\mathbf{T}^{2}\mathbf{p}^{(0)}\\\
a\end{array}\right)\right)_{1,\ldots,n}$

and
$\mathbf{p}^{\infty}=\lim_{t\rightarrow\infty}\mathbf{T}^{t}\mathbf{p}^{(4)}$.

Following Equation (2), after expressing the limiting opinion as the weighted
average of the $n+1$ agents’ initial opinions, that is

$p_{i}^{\infty}=w_{1}p_{1}^{(0)}+w_{2}p_{2}^{(0)}+\cdots+w_{n}p_{n}^{(0)}+w_{(n+1)}a$
(4)

we take weight $w_{(n+1)}$ as the external agent’s social influence. Due to
the external agent’s intervention, the social influence vector $\mathbf{s}$ no
longer gives the accurate social influence for the permanent agents. It does
so if the external agent did not participate at all in which case our model
reduces to the DeGroot model.

For the rest of this paper, we assume all interaction matrices are strongly
connected and aperiodic to ensure an opinion formation always reaches a
consensus, for otherwise our measure of influence cannot be defined. Also we
assume, without loss of generality, that the external agent’s unchanged
opinion is $1$ and every permanent agent has an initial opinions of $0$. Note
that agents’ opinions are irrelevant to their influence and we only concern
with the external agent’s influence. By this assumption, it follows from
Equation (4) that the limiting opinion is precisely the external agent’s
social influence.

##  4 Analytical Results

We have argued that an organisation’s influencing effort depends on the
coverage, duration, intensity, and timing of its influence. Our model captures
these factors respectively as the number $m$ of agents the external agent can
influence; the number $k$ of rounds of interactions the external agent
participates; the weight $\lambda$ that is placed on the external agent; and
the time points in which the $k$ rounds of interaction take place. Of those
factors, the first three reflect the amount of resource available for the
influencing effort, the more resource there is, the larger these factors’
values. As organisations usually, if not always, have a finite resource, a
vital question is how and when to allocate it for achieving maximum influence.

Devoted to analytical results, in this section, we express the external
agent’s influence as a function of the dependent factors and obtain influence
maximising insights by analysing the function. For the ease of presentation,
we decompose the multiplication with the matrix $\mathbf{A}$ in Equation (3)
and rewrite it as

$\mathbf{p}^{(t)}=(\mathbf{T}-\lambda(\mathbf{T})_{m})\mathbf{p}^{(t-1)}+\Lambda.$
(5)

Recall that we assume $a=1$ and $\Lambda$ is the n-dimensional vector of which
the first $m$ components are $\lambda$ and the rest are $0$. For a matrix
$\mathbf{T}$, $(\mathbf{T})_{m}$ is the matrix formed by replacing all entries
of $\mathbf{T}$ with zero except those of its first $m$ rows. For example
$\big{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\big{)}_{1}=\big{(}\begin{smallmatrix}a&b\\\
0&0\end{smallmatrix}\big{)}$.

One might have noticed that the timing factor is intrinsically different from
the other three. Apart from being unaffected by the scarcity of resource, more
importantly, the timing factor is not a single factor, but a collection of $k$
factors, one for each of the $k$ rounds of participation. It is impractical
and pointless to consider all variations of the $k$ factors, instead we focus
on three timing options that echo real-world situations: (1) the external
agent only participates after the other agents have reached a consensus; (2)
the external agent participates for the first $k$ rounds of interaction; and
(3) the external agent randomly participates for $k$ rounds of interaction
according to a uniform distribution over a range of time points. And we refer
to them as consensus, start, and uniform respectively. For now we only concern
with the consensus timing option. Later on we will explain how the other two
cause the explosion in the number of variables that makes function analysis
infeasible. In fact, “consensus” is the most important of the three as
simulations show that it gives rise to the largest social influence for the
external agent.

The consensus timing option resembles the real-world situation in which an
organisation always allow sufficient time for its target to thoroughly digest
its influence from the previous intervention before intervening and exerting
its influence again. From a function analysis perspective, it keeps the number
of variables small which results in a succinct expression for the limiting
opinion.

######  Theorem 1.

In an opinion formation, if the external agent can influence $m\leq n$ agents
and it participates in $k\geq 1$ rounds of interaction, each of which is at a
time point the other agents have reached a consensus, then

$p_{i}^{\infty}=\textstyle\sum_{j=0}^{k-1}(1-s\lambda)^{j}s\lambda$

for all $i$, where $s=\sum_{i=1}^{m}s_{i}$ for $\mathbf{s}$ the social
influence vector.

###### Proof.

Let $\mathbf{S}=\lim_{t\rightarrow\infty}\mathbf{T}^{t}$, so each row of
$\mathbf{S}$ is the social influence vector $\mathbf{s}$. We prove by
induction on $k$ that
$\mathbf{p}^{\infty}=\sum_{j=0}^{k-1}(1-s\lambda)^{j}(s\lambda,\ldots,s\lambda)^{\top}$
where $s=\sum_{i=1}^{m}s_{i}$. For the base case of $k=1$. It follows from
Equation (1) and (5) that

$\displaystyle\mathbf{p}^{\infty}$
$\displaystyle=\mathbf{S}((\mathbf{T}-\lambda(\mathbf{T})_{m})\mathbf{0}+\Lambda)$
$\displaystyle=\mathbf{S}(\lambda,\ldots,\lambda,0,\ldots,0)^{\top}$
$\displaystyle=(\textstyle\sum_{i=1}^{m}s_{i}\lambda,\ldots,\textstyle\sum_{i=1}^{m}s_{i}\lambda)^{\top}$
$\displaystyle=(s\lambda,\ldots,s\lambda)^{\top}$

For the induction step, suppose
$\mathbf{p}^{\infty}=\sum_{j=0}^{k-1}(1-s\lambda)^{j}(s\lambda,\ldots,s\lambda)^{\top}$
for $k=l$. We need to show the equality also holds for $k=l+1$. Let
$\mathbf{a}$ be the limiting opinion vector for when the external agent
participates for $l$ rounds of interaction. Due to the induction hypothesis
$\mathbf{a}=\sum_{j=0}^{l-1}(1-s\lambda)^{j}(s\lambda,\ldots,s\lambda)^{\top}$.
Then for $k=l+1$ we have

$\displaystyle\mathbf{p}^{\infty}$
$\displaystyle=\mathbf{S}((\mathbf{T}-\lambda(\mathbf{T})_{m})\mathbf{a}+\Lambda)$
$\displaystyle=\mathbf{S}\mathbf{T}\mathbf{a}-\lambda\mathbf{S}(\mathbf{T})_{m}\mathbf{a}+\mathbf{S}\Lambda$
$\displaystyle=\mathbf{a}-\lambda\mathbf{S}(\mathbf{T}\mathbf{a})_{m}+\mathbf{S}\Lambda$
$\displaystyle=\mathbf{a}-\lambda\mathbf{S}(\mathbf{a})_{m}+(s\lambda,\ldots,s\lambda)^{\top}$
$\displaystyle=\mathbf{a}-\lambda(\textstyle\sum_{i=1}^{m}s_{i})\mathbf{a}+(s\lambda,\ldots,s\lambda)^{\top}$
$\displaystyle=(1-s\lambda)\mathbf{a}+(s\lambda,\ldots,s\lambda)^{\top}$
$\displaystyle=\textstyle\sum_{j=0}^{l}(1-s\lambda)^{j}(s\lambda,\ldots,s\lambda)^{\top}$

This complete the induction step. Thus we have
$p_{i}^{\infty}=\sum_{j=0}^{k-1}(1-s\lambda)^{j}s\lambda$ for all $i$.

∎

Essentially, the limiting opinion is the sum of the first $k$ terms of a
geometric series with start term $s\lambda$ and constant ratio $1-s\lambda$
where the scalar $s$ is the sum of the first $m$ components of the social
influence vector $\mathbf{s}$. Since our model does not specify the identities
of the $m$ agents that can be influenced by the external agent, the same value
of $m$ can lead to different values of $s$. So ultimately it is the value of
$s$ rather than $m$ that matters in determining the limiting opinion.

We can represent the limiting opinion, which is the external agent’s social
influence, as a function $I:X\times Y\times Y\rightarrow\mathbb{R}$ such that

$I(k,\lambda,s)=\textstyle\sum_{j=0}^{k-1}(1-s\lambda)^{j}s\lambda$

where $X=\\{x\in\mathbb{Z}\,|\,x\geq 1\\}$ and
$Y=\\{y\in\mathbb{R}\,|\,0<y<1\\}$. As explained, the variable $m$ should not
appear in the function and $s$ is the combined social influence of the $m$
agents for when the external agent did not participate. Substituting
$I(k,\lambda,s)$ with the formula for the sum of a geometric series, we have

$I(k,\lambda,s)=1-(1-s\lambda)^{k}$ (6)

It is easy to see that $I(k,\lambda,s)$ increases as $s$ gets lager. Since a
few influential agents can have the same combined social influence as that of
many less influential ones, a large value of $m$ does not necessarily give a
large value of $s$. This leads to our first insight through function analysis:

> (A1) if the timing option is consensus, then the external agent should aim
> for the more influential ones to maximise its social influence.

What this analysis also tells us is that the number of agents that can be
influenced by the external agent is not the most accurate measure of
“coverage.” A better choice is the combined social influence of such agents.

With the function $I(k,\lambda,s)$, we note that its three variables have
nothing to do with the structure of the social network, which means the latter
has no impact on the external agent’s social influence. This leads to our
second insight:

> (A2) if the timing option is consensus, then the structure of the social
> network is irrelevant to the external agent’s social influence.

The insight might seem trivial, nevertheless it is of great significance in
practice. Often a social network’s structure is unknown to an external
organisation, so knowing that it is irrelevant brings certainty and assurance
to an organisation’s influencing effort. Hence, we consider this irrelevance
as an advantage of the “consensus” timing option over the others with which
the structure does matter.

With $I(k,\lambda,s)$, we also note that the extra influence accumulated by
participating one more round of interaction is
$I(k+1,\lambda,s)-I(k,\lambda,s)$. Substituting $I(k+1,\lambda,s)$ and
$I(k,\lambda,s)$ with their full expression, we have

$I(k+1,\lambda,s)-I(k,\lambda,s)=(1-s\lambda)^{k}s\lambda.$

Since $0<1-s\lambda<1$, $I(k+1,\lambda,s)-I(k,\lambda,s)$ decreases as $k$
gets larger. This leads to our third insight:

> (A3) if the timing option is consensus, then the external agent’s
> influencing effort become less and less effective as it participates in more
> rounds of interaction.

Next, we will analyse $I(k,\lambda,s)$ to decide which of the three variables
$s$, $k$, and $\lambda$ has a more profound effect on the external agent’s
social influence. As an immediate consequence of Equation (6), the following
lemma shows that $I(k,\lambda,s)$ gains the same increase by scaling up $s$ as
it does by scaling up $\lambda$.

######  Lemma 1.

Let $r\in\mathbb{R}$, $r\geq 1$, $rs<1$ and $r\lambda<1$. Then

$I(k,r\lambda,s)=I(k,\lambda,rs).$

Note that both the weight the other agents place on the external agent and the
combined social influence of the agents that can be influenced by the external
one are percentages, thus the precondition in Lemma 1. The lemma leads to our
four insight:

> (A4) if the timing option is consensus, then it is equally effective to
> scale up the coverage or intensity factor to maximise the external agent’s
> social influence.

Since $\lambda$ and $s$ are equally important in determining the value of
$I(k,\lambda,s)$, it remains to compare either one of them with $k$. The
following lemma shows that $I(k,\lambda,s)$ increases more by scaling up $s$
than it does by scaling up $k$.

######  Lemma 2.

Let $r\in\mathbb{Z}$, $r\geq 2$, $rs<1$ and $r\lambda<1$. Then

$I(k,\lambda,rs)>I(rk,\lambda,s).$

###### Proof.

Since, according to Equation (6), $I(k,\lambda,rs)=1-(1-rs\lambda)^{k}$ and
$I(rk,\lambda,s)=1-(1-s\lambda)^{rk}$, it suffices to show
$(1-rs\lambda)<(1-s\lambda)^{r}$. We will prove by induction on $r$. For the
base case, we have $r=2$. Then $(1-2s\lambda)<(1-s\lambda)^{2}$ follows from
$(1-s\lambda)^{2}=1-2s\lambda+(s\lambda)^{2}$ and $s\lambda>0$.

For the induction step, suppose $(1-rs\lambda)<(1-s\lambda)^{r}$ holds for
$r=l$, we need to show it also holds for $r=l+1$.

$\displaystyle 1-rs\lambda$ $\displaystyle=1-(l+1)s\lambda$
$\displaystyle=(1-ls\lambda)-s\lambda$

and

$\displaystyle(1-s\lambda)^{r}$ $\displaystyle=(1-s\lambda)^{l+1}$
$\displaystyle=(1-s\lambda)^{l}(1-s\lambda)$
$\displaystyle=(1-s\lambda)^{l}-s\lambda(1-s\lambda)^{l}.$

We have by the induction hypothesis that $(1-ls\lambda)<(1-s\lambda)^{l}$.
Also since $(1-s\lambda)^{l}<1$, we have $s\lambda>s\lambda(1-s\lambda)^{l}$.
It follows from $(1-ls\lambda)<(1-s\lambda)^{l}$ and
$s\lambda>s\lambda(1-s\lambda)^{l}$ that
$(1-ls\lambda)-s\lambda<(1-s\lambda)^{l}-s\lambda(1-s\lambda)^{l}$ which
implies $1-rs\lambda<(1-s\lambda)^{r}$. This completes the proof.

∎

Note that it is meaningless to scale up the number of participation rounds by
a non-integer or by the integer 1, thus the precondition of Lemma 2. The lemma
leads to our fifth insight:

> (A5) if the timing option is “consensus,” then it is more effective to scale
> up the coverage or intensity factor than the duration factor to maximise the
> external agent’s social influence.

Finally, it is not uncommon that an organisation can influence everyone in its
target group. For example, this may happen if the group is relatively small
with respect to the organisation’s resource. In the remaining of this section,
we deal with this special but realistic “full coverage” case.

In our model, full coverage means $m=n$ with which Equation (5) reduces to

$\mathbf{p}^{(t)}=(1-\lambda)\mathbf{T}\mathbf{p}^{(t-1)}+\Lambda$ (7)

where $\Lambda$ is the $n$-dimensional vector
$(\lambda,\ldots,\lambda)^{\top}$.

The distinguishing property of the full coverage case is that the timing
factor does not play a part in determining the external agent’s social
influence. The key to appreciating this property is the following easily
verifiable equality.

$(1-\lambda)\mathbf{T}(\mathbf{T}\mathbf{p}^{(t)})+\Lambda=\mathbf{T}((1-\lambda)\mathbf{T}\mathbf{p}^{(t)}+\Lambda)$

By Equation (1) and (7), if the current opinion vector is $\mathbf{p}^{(t)}$
then the LHS of the equality is the opinion vector after two rounds of
interaction where the external agent participates in the second round and the
RHS is the opinion vector after two rounds of interaction where the external
agent participates in the first round. Since this “one-step” change does not
affect the opinion vector after two rounds of interaction neither does it to
the limiting opinion. By realising that we can repeat this “one-step” change
for any number of times to have the $k$ rounds of participation at any $k$
time points without affecting the limiting opinion, the property must hold.
This leads to our final insight obtained through function analysis.

> (A6) If the external agent can influence all agents, then the timing factor
> is irrelevant to its social influence.

##  5 Simulation Results

Function analysis has its limits, in this section, we resort to simulations to
obtain insights tied to the timing options. Like it or not, simulation might
be our last resort for the start and uniform timing options. Suppose the
external agent participates in the first and $r$th rounds where $r$ is $2$ for
“start” and an arbitrary number for “uniform.” Then the limiting opinion and
thus the external agent’s social influence can be derived as follows

$\displaystyle\mathbf{p}^{\infty}$
$\displaystyle=\lim_{t\rightarrow\infty}\mathbf{T}^{t}\mathbf{p}^{(r)}$
$\displaystyle=\lim_{t\rightarrow\infty}\mathbf{T}^{t}((\mathbf{T}-\lambda(\mathbf{T})_{m})\mathbf{p}^{(r-1)}+\Lambda)$
$\displaystyle=\lim_{t\rightarrow\infty}\mathbf{T}^{t}((\mathbf{T}-\lambda(\mathbf{T})_{m})\mathbf{T}^{r-2}\mathbf{p}^{(1)}+\Lambda)$
$\displaystyle=\lim_{t\rightarrow\infty}\mathbf{T}^{t}((\mathbf{T}-\lambda(\mathbf{T})_{m})\mathbf{T}^{r-2}\Lambda+\Lambda)$
$\displaystyle=\lim_{t\rightarrow\infty}\mathbf{T}^{t}(\mathbf{T}^{r-1}\Lambda-\lambda(\mathbf{T}^{r-1})_{m}\Lambda+\Lambda)$
$\displaystyle=2\lim_{t\rightarrow\infty}\mathbf{T}^{t}\Lambda-\lambda\lim_{t\rightarrow\infty}\mathbf{T}^{t}(\mathbf{T}^{r-1})_{m}\Lambda.$

Obviously, the matrix $(\mathbf{T}^{r-1})_{m}$ plays a part in determining
$\mathbf{p}^{\infty}$, meaning that any variation of the social network can
result in a very different social influence of the external agent. Thus, the
function that expresses the social influence must consider all entries of
$\mathbf{T}$ (as well as the $k$ timing factors). This is an overwhelmingly
large number of variables for function analysis to be feasible. For
“concensus” however $(\mathbf{T}^{r-1})_{m}$ is cancelled out in the above
derivation, making it irrelevant for determining the social influence (see the
proof of Theorem 1).

In this paper, all simulations are conducted with 100 agents and with 3000
rounds of interactions. Additionally, 1000 simulations are conducted for each
combination of values for the intensity, duration, coverage and timing factor.
Experimenting with various simulation settings shows that a larger number of
agents and simulations makes no difference to the exhibited patterns, which
are already evident with as little as 10 agents and 10 simulations. Moreover,
the patterns do not rely on any specific factor values, though some values
make them easy to visualise.

Our first set of simulations intends to disentangle the varying effects of the
timing options as the duration value grows. While holding the intensity and
coverage factor constant, for each timing option, we simulate our model for
duration values ranging from 0 to 45 (with an increment value of 1). In Figure
1, we plot the external agent’s average social influence (in 1000 simulations)
induced by each timing option against the duration values.

Figure 1: Comparing the timing options with respect to a growing duration
factor.

Immediately we observe that the plot for “consensus” is virtually always above
those of “start” and “uniform,” with a noticeable gap over the former and a
tiny one over the latter. More specifically, the gap between the plots is
closing towards both ends of the horizontal axis and becomes negligible at the
very ends. The plots suggest “consensus” and “start” respectively gives the
largest and smallest social influence while “uniform” is almost identical to
“consensus.” Closing of the gap, however, cannot be interpreted as suggesting
the less relevance of the timing options with small and large duration values,
rather the pattern is enforced by the bounded nature of social influence
values. Social influence is bounded from below by 0 and from above by 1, thus
if the duration value is sufficiently small or large, then the induced social
influence must be close to 0 or 1 irregardless of the timing option.

Figure 2: Comparing the timing options with respect to a growing coverage
factor

In a similar fashion, while holding the duration and intensity factors
constant, for each timing option, we simulate our model for coverage values
ranging from 0 to 0.9 (with an increment value of 0.1). In Figure 2, the
average social influence induced by each timing option is plotted against the
coverage values. Once again, we observe that “consensus” is the clear winner
against “start” but not so much against “uniform”. Also the gap is closing
towards both ends of the horizontal axis. Likewise, the bounded nature of
social influence plays a part in the closing of the gap. But this time, as the
coverage value grows, the gap closes so drastically that the plots converge at
a social influence (i.e., 0.8) well below 1. This means the growing coverage
value also plays a part. So, unlike the duration value, it is righteous to
interpret the closing of the gap as suggesting the larger the coverage value
the less relevant the timing options. In fact, we have already proved a
limiting case of this pattern in (A6) which concludes the irrelevance of the
timing factor when the coverage value is at its largest. Hence, we obtain the
following insight.

> (S1) Once the coverage value passes certain threshold,111The threshold
> depends on the intensity and duration values. then the larger the coverage
> value, the less relevant the timing options in determining the external
> agent’s social influence.

Figure 3: Comparing the timing options with respect to a growing intensity
factor

Finally, we hold constant the duration and coverage factors and simulate our
model for intensity values ranging from 0 to 0.9 (with an increment value of
0.1). The corresponding plots are given in Figure 3. Yet another time, we
observe the superiority of “consensus” over “start” and “uniform,” but only
slightly over the latter. This unequivocal pattern exhibited in all three sets
of simulations leads to the following insight.

> (S2) Among the three timing options, “consensus” gives rise to the largest
> social influence of the external agent and “start” the least.

Furthermore, the fact that “consensus” and “uniform” are almost
indistinguishable indicates that the key is to spread out the participation
times and avoid having them in a cluster. At last but not least, we observe
that, contrary to the previous simulations, the gap between the plots enlarges
towards the end of the horizontal axis with larger intensity values.
Noticeably, whatever mechanism that drives this enlargement is very effective
as it manages to do so even though the bounded nature of social influence acts
in an opposite direction. This leads to our final insight.

> (S3) The larger the intensity factor, the more relevant of the timing
> options in determining the external agent’s social influence.

##  6 Related Work

Influence maximisation is a recurring topic in studies of opinion formation,
social networks and many more. The term is often referred to as the
algorithmic problem of selecting a predefined number of agents in a social
network to maximise the spread of a binary opinion in an opinion diffusion
process [21]. [12] are the first to pose the algorithmic problem [12]. Later
on, [19] proposed the so-called independent cascade model and gave a greedy
algorithm based on submodular maximisation [19, 20]. These very influential
papers have since then promoted a large amount of follow up works improving
and extending various aspects of the selection techniques [6]. Although the
diffusion process and representation of opinions are different from ours,
these works are, in our terminology, endeavours within the realm of the
coverage factor. Rather than focusing on this single factor, we move on to
consider the intensity, duration and timing factors as well. To the best of
our knowledge, this is the first attempt to understand how the interplay
between the four factors elevate and restrain the overall influencing effort.

Apart from the above works, recent years have seen a surge in the popularity
of various forms of opinion diffusion in artificial intelligence [5, 16, 9, 8,
13, 3, 14, 7, 2]. Some of them also tackled influence maximisation problems
[1, 4]. Unlike ours and the aforementioned ones, they study how the sequence
of opinion update affects the influencing effort. [1] considered the case of
three alternative opinions in an asynchronous mode of opinion update and
investigated the sequence of updates that maximises the spread of an opinion.
[4] continued the work on spread-maximising sequences and provided upper and
lower bounds on the length of such sequences among other results.

##  7 Conclusion

In this paper, we generalised the DeGroot model of opinion formation to allow
a temporary participant. We articulated four factors namely, duration,
intensity, coverage and timing that dominate the temporary participant’s
influencing effort and incorporated them into the opinion formation process.
Through function analysis and simulation, we revealed the degree of importance
and interplay between the factors which lead to crucial insights of influence
maximisation.

In summary, the temporary participant ought to adopt the consensus timing
option and focus its resources on the intensity and coverage factor. The
insights may aid organisations and individuals to make better strategic
choices when facing a limited resource to maximise their influence. Direction
for future work is to investigate the case of multiple external agents.

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