# A Two-Functional-Network Framework of Opinion Dynamics

Wentao Zhang, Zhiqiang Zuo, and Yijing Wang This work was supported by the
National Natural Science Foundation of China No. 61933014, No. 61773281, No.
61673292.The authors are with the Tianjin Key Laboratory of Process
Measurement and Control, School of Electrical and Information Engineering,
Tianjin University, Tianjin, 300072, P. R. China. (e-mail: {wtzhangee, zqzuo,
yjwang}@tju.edu.cn).

###### Abstract

A common trait involving the opinion dynamics in social networks is an anchor
on interacting network to characterize the opinion formation process among
participating social actors, such as information flow, cooperative and
antagonistic influence, etc. Nevertheless, interacting networks are generally
public for social groups, as well as other individuals who may be interested
in. This blocks a more precise interpretation of the opinion formation process
since social actors always have complex feeling, motivation and behavior, even
beliefs that are personally private. In this paper, we formulate a general
configuration on describing how individual’s opinion evolves in a distinct
fashion. It consists of two functional networks: interacting network and
appraisal network. Interacting network inherits the operational properties as
DeGroot iterative opinion pooling scheme while appraisal network, forming a
belief system, quantifies certain cognitive orientation to interested
individuals’ beliefs, over which the adhered attitudes may have the potential
to be antagonistic. We explicitly show that cooperative appraisal network
always leads to consensus in opinions. Antagonistic appraisal network,
however, causes opinion cluster. It is verified that antagonistic appraisal
network affords to guarantee consensus by imposing some extra restrictions.
They hence bridge a gap between the consensus and the clusters in opinion
dynamics. We further attain a gauge on the appraisal network by means of the
random convex optimization approach. Moreover, we extend our results to the
case of mutually interdependent issues.

###### Index Terms:

Social networks, appraisal network, cooperative/antagonistic interaction,
random convex optimization.

##  I Introduction

The study of opinion dynamics in social networks has a long history. Compared
with many natural or man-made systems (networks), social actors (agents) in
social networks rarely display a common interest. In contrast, they usually
aggregate into several small groups where the agents in the same group achieve
a unanimous behavior while the opinions of the whole network comprise several
clusters. Opinion dynamics in social networks are universal topics and have
captured massive interests from different disciplines, such as sociology,
social anthropology, economics, ideological political science, physics,
biology and control theory, even in the field of military [1, 2].

A simple yet instructive mathematical model is fundamental for the study of
opinion dynamics in social networks. As a backbone for opinion dynamics,
DeGroot’s iterative opinion pooling configuration shows that social actors can
share a common viewpoint if a convex combination mechanism is performed (cf.
[3]). In many practical situations, agents often interact with those who are
like-minded, and agree on more deviant viewpoints with discretion. For
Hegselmann-Krause model, each agent only communicates with those whose
opinions are constrained into its confidence interval [4, 5]. Actually, this
model is implicitly based on the principle of biased assimilation or
homophily. It is always the case that some individuals in social networks have
their own prejudices, no matter what kind of opinion formation mechanism is
applied. An attempt towards this direction is the Friedkin-Johnsen (FJ) model
where some of the individuals (stubbornness) are affected by external signal
[6]. Unlike Hegselmann-Krause model, FJ model achieves opinion clusters even
in the linear opinion formation process. Conventionally, FJ model mainly
focuses on the issue free opinion dynamics or the scalar opinion dynamics.
Refs. [7, 8] extended the FJ model to the vector-based opinion dynamics where
the opinions tightly relate to several issues, see also [9]. As hinted by
belief systems (cf. [10]), topic-specific opinion dynamics are usually
entwined whenever agent’s opinion is involved into several interdependent
issues. Parsegov _et al_. [11] introduced a row stochastic matrix (MiDS
matrix) which clarifies what the attitude of agent towards the issue sequence
is. Similarly, the property of belief system dynamics subject to logical
constraints was elaborated in [12], and a case study was provided to show how
the fluctuations of population’s attitudes evolve. The gossip-based version of
the FJ model was proposed in [13]. More recently, an approach building on the
DeGroot model and the FJ model was given to investigate the evolutions with
respect to the self-appraisal, interpersonal influences and social power over
issue sequence for the star, the irreducible and the reducible communication
topologies [14].

Nowadays, people have realized that postulate of “cognitive algebra” on
heterogeneous information (cf. [15]) for opinion formation process among
individuals may not attribute all complex behaviors in social networks due to
the negative interpersonal influences that often emerge in many community of
dissensus (cf. [16, 17]). Typical instances include but are not limited to
multiple-party political systems, biological systems, international alliances,
bimodal coalitions, rival business cartels, duopolistic markets, as well as
boomerang effect in dyad systems. Based on multi-agent system theory, a lot of
research interests have been devoted to the opinion polarization and the
stabilization problems. By signed graph theory, Altafini [18] proved that
structurally balanced graph is the necessary and sufficient condition for
bipartite consensus. Proskurnikov _et al_. [19] further reported the necessary
and sufficient criterion to stabilize Altafini’s model. For general
communication graph, [20] discussed the interval bipartite consensus problem.
The second-order/high-order multi-agent systems in the presence of
antagonistic information were also discussed in [21, 22], as well as finite-
time consensus [23]. More related work about the signed graph theory refers to
[24, 25]. Cooperative control with antagonistic reciprocity was discussed in
[26, 27, 28] using the node-based mechanism.

Unlike complex systems or multi-agent systems, agents (individuals) in social
networks usually display diversity behaviors, such as attitude, feeling and
belief, etc. In business negotiation or alliance, for instance, each member
tends to collect more “sides information” with respect to the remaining
members before organizing a meeting, by which they aim at achieving the profit
maximization. They possess introspective ability to access what and how they
should do. Notice that the above research is based on a common hypothesis that
merely the public network is used to quantify opinion formation process.
However, a downside of public networks111As hinted by [1] and [11], we treat
the interaction topologies (communication graphs) in the aforementioned
literature by the public networks. in the aforementioned literature to
describe the antagonistic interactions among participating individuals is that
individuals in social networks can completely get access to the opinion, the
attitude and the belief of the other individuals towards the interested
individuals. But this is not always the case since the individuals in social
networks feature complex behaviors, even complex thoughts [29], naturally
leading to the complex opinion dynamics that the natural or the man-made
complex networks cannot show. Moreover, as discussed in [30, 11], public
networks generally characterize the social influence structure, and henceforth
they are assumed to be thoroughly known. For this reason, potential hostile
information will become transparent whenever only the public network is
utilized to model the opinion formation process. Obviously, this is not always
true for social networks where it is rather arduous to have access to the
individual’s opinion in prior.

In this paper, we propose a new framework with functional networks, that is,
the appraisal network and the interacting network, to describe the opinion
formation process. The opinion evolution in social network is firstly governed
by an appraisal network characterizing how each individual assigns its
attitude or influence towards other individuals. Afterwards, each individual
updates its opinion through an interacting network as the conventional DeGroot
model (we call it the interacting network or the public network). To the best
of the authors’ knowledge, there are no results available to model the
evolution of opinion dynamics using two functionally independent networks.
More importantly, we will show that the proposed formulation indeed provides
some intriguing phenomena that cannot be preserved by the existing setups. We
summarize the main features of this paper as follows:

(a)

Cooperative appraisal networks lead to consensus in opinion whereas the final
aggregated value may not be contained in a convex hull spanned by the initial
opinions of the participating individuals. This potentially implies that the
way of the decision among individuals in social networks formulated in this
paper is not necessarily constrained into the convex hull as the usual models.
In fact, non-convex interactions in social networks are common, and it is
natural to model the evolution of the opinions by taking this factor into
account [29];

(b)

Antagonistic appraisal networks result in clusters in opinions. In particular,
we show that consensus in opinion dynamics appears provided that the
considered antagonistic appraisal networks enjoy certain requirements. It is
illustrated that most of the existing results merely guarantee the clusters in
opinions subject to hostile interactions (cf. [24, 25]), or stability of the
agents [19];

(c)

Random convex optimization (cf. [31, 32]) is formulated to provide a feasible
estimation on the appraisal network, with the purpose of finite-horizon sense.
Also, a bound on the number of “required observations”, which enables us to
get rid of a-prior specified level of probabilistic violation, is explicitly
given. Therefore we can make a justification of the postulate on self-
preservation of the appraisal network for each individual, as opposed to the
hypothesis on interacting network in the literature;

(d)

The proposed setup could be further extended to the multiple mutually
entangled issues that are quantified by an issue-dependence matrix, upon which
we deduce the criteria associated with the convergence (resp. stability) of
the agents. More interestingly, we point out that the introduction of the
issue-dependence matrix enables us to steer the leader’s opinion, which has
been verified to be fixed all the time in the context of multi-agent systems
community.

The layout of this paper is outlined as follows: Section II describes some
basic preliminaries and the problem formulation as well as the dynamic model.
Section III presents the results in the context of the cooperative and the
antagonistic appraisal networks. Section IV reports the results for the
interdependent issues. Section V gives numerical examples to support the
derived theoretical results as well as some discussions. Finally, a conclusion
is drawn in Section VI.

##  II Preliminaries and Problem Formulation

###  II-A Basic Notations

The real set, the nonnegative real set and the nonnegative integer set are
denoted by $\mathbb{R}$, $\mathbb{R}_{+}$ and $\mathbb{Z}$. Symbol $``\prime"$
denotes the transpose regarding a vector or a matrix. Symbol $|\cdot|$
represents the modulus or the cardinality, and $|\cdot|_{2}$ the $2$-norm.
$|Q|$ stands for the spectral norm for matrix $Q$. $\lambda(Q)$ and $Q^{-1}$
denote the eigenvalue and the inverse of an invertible matrix $Q$. $Q\succ 0$
indicates that matrix $Q$ is symmetric, and positive definite. We denote
$\\{1,...,N\\}$ and $(1,...,1)^{\prime}$ by $\mathbb{I}_{N}$ and
$\textbf{1}_{N}$, where $N$ is the number of agents in social networks. $I$
and $\mathcal{O}$ are, respectively, the identity matrix and the zero matrix.
The diagonal matrix is represented by ${\rm diag}(\cdot)$. The sign function
is abbreviated by ${\rm sgn}(\cdot)$. For any complex number $\lambda$,
$\lambda\triangleq{\rm Re}(\lambda)+\mathbbm{i}{\rm
Im}(\lambda)=|\lambda|(\cos(\arg(\lambda))+\mathbbm{i}\sin(\arg(\lambda)))$
where $\mathbbm{i}^{2}=-1$ and $\arg(\lambda)$ represents the value of
argument principle.

In addition, we denote $(\Omega,\mathcal{F},\mathcal{P})$ the probability
space, upon which $\Omega$ represents the sample space, $\mathcal{F}$ the
Borel $\sigma$-algebra, and $\mathcal{P}$ the probability measure.

An interacting graph $\mathcal{G}$ is commonly represented by a triple
$\\{\mathbb{V},\mathbb{E},\mathbb{A}\\}$ where $\mathbb{V}$ is the node set,
$\mathbb{E}$ is the edge set and $\mathbb{A}=(a_{ij})_{N\times N}$ is the
adjacent matrix with $a_{ij}>0$ provided that $(j,i)\in\mathbb{E}$ and
$a_{ij}=0$ otherwise. No self-loop is allowed throughout the paper, i. e.,
$a_{ii}=0$. The associated Laplacian matrix $\mathcal{L}=(l_{ij})_{N\times N}$
is given by $l_{ii}=\sum^{N}_{j=1,j\neq i}a_{ij}$ and $l_{ij,j\neq
i}=-a_{ij,j\neq i}$222In the context of the opinion dynamics, $p_{ij}$
represents the attitude of agent $i$ towards agent $j$, while $l_{ij}$ in the
framework of the multi-agent systems means that agent $j$ is a neighbor of
agent $i$. Therefore, if no confusion arises, we treat both $p_{ij}$ and
$l_{ij}$ as the influence of agent $j$ imposed on agent $i$ with the purpose
of the notion consistency.. A digraph is strongly connected if for any two
distinct nodes, they are connected by a path. A root of $\mathcal{G}$ is a
special node, from which we can arrive at any other nodes. A graph has a
directed spanning tree if and only if it contains at least a root.

###  II-B Dynamic Model for Social Networks

Consider a group of interacting individuals (agents or social actors), whose
opinions evolve according to

$\displaystyle\xi_{i}(k+1)=$
$\displaystyle~{}\xi_{i}(k)+\varrho_{i}\sum_{j\in\mathcal{N}_{i}}a_{ij}(z_{j}(k)-z_{i}(k))$
(1a) $\displaystyle z_{i}(k)=$
$\displaystyle~{}\sum^{N}_{j=1}\delta_{ij}\xi_{j}(k),~{}i\in\mathbb{I}_{N},~{}k\in\mathbb{Z}$
(1b)

where $\xi_{i}(k)\in\mathbb{R}$ stands for the opinion of the $i$th agent at
instant $k$, $\mathcal{N}_{i}$ is the neighboring set of agent $i$. Similar to
the FJ model, we call parameter $\varrho_{i}$ ($\varrho_{i}\neq 0$) the
susceptibility factor with respect to the $i$th social actor. $z_{i}(k)$ in
$(\ref{20191eq1b})$ represents the appraisal or the self-appraisal
($z_{i}(k)=\delta_{i}\xi_{i}(k)$) of agent $i$. The constant $\delta_{ij}$
specifies the weighted influence assigned by individual $i$ towards individual
$j$, and satisfies $0<\sum^{N}_{j=1}|\delta_{ij}|\leq 1$. The compact form of
$(\ref{20191eq1})$ is expressed by

$\displaystyle\xi(k+1)=$
$\displaystyle~{}(I-\Lambda\mathcal{L}\mathcal{D})\xi(k),~{}k\in\mathbb{Z}$
(2)

where $\xi(k)=(\xi_{1}(k),...,\xi_{N}(k))^{\prime}$, $\Lambda={\rm
diag}(\varrho_{1},...,\varrho_{N})$ and $\mathcal{D}=(\delta_{ij})_{N\times
N}$.

It is easy to see that system $(\ref{20191eq2})$ involves two networks, and we
call them the interacting or the public network (quantified by matrix
$\mathcal{L}$) and the appraisal network (quantified by matrix $\mathcal{D}$),
respectively. We emphasize that the susceptibility factor $\varrho_{i}$ is
crucial for the convergence of system $(\ref{20191eq2})$ since the appraisal
network and the interacting network are entwined, leading to a significant
difference in contrast with the DeGroot model. It is noted that a system might
collapse due to the hostility. And as will be shown later, a system subject to
antagonistic information fails to converge, regardless of the connection
property of the underlying interacting graph333For Ref. [18], the proposed
consensus algorithm always assures the convergence of the reciprocal agents as
long as the communication topology attains a directed spanning tree, that is,
bipartite consensus (interval bipartite consensus) for structurally balanced
graph (cf.[18, 20]) and stability for graph that contains the in-isolated
structurally balanced subgraphs, or is structurally unbalanced (cf. [19])..

###### Remark 1

One can easily see that system $(\ref{20191eq2})$ boils down to the classical
DeGroot model (cf. [3]) or the multi-agent systems (cf. [33]) when we fix
$\mathcal{D}$ to be an identity matrix and $\varrho_{i}$ some positive
constant $\varrho$ fulfilling
$0<\varrho<\max_{i\in\mathbb{I}_{N}}\\{l_{ii}\\}$ (in such a scenario we
always treat $\varrho$ as the step size). It should be pointed out that the
two networks described above have nothing to do with the multilayer networks
in the context of complex networks [34] where all layer networks functionally
inherent with the interacting network $\mathcal{L}$. $\blacklozenge$

We are now in a position to give some interpretations about the reason why we
introduce the appraisal network: (1) Unlike the natural and the man-made
complex networks where the interacting agents are creatures and smart
machines, the individuals in social networks are people. In one word, the
subject in social networks possesses emotion, belief as well as attitude
towards a specific object. Therefore, it is a common sense that people try to
search and collect as much information as possible before they make a
decision, such as organizing a conference, exchanging ideas with colleagues,
etc. That is to say, people in social networks always evaluate and self-
reflect their opinions, beliefs, and behaviors before interacting with others;
(2) As mentioned before, the hostile interaction is a key element in the study
of opinion dynamics in social networks. It is notable that the interacting
networks are generally assumed to be completely known (cf. [30, 11]).
Therefore, the private opinions of the participating agents may be leaked if
merely the interacting networks are applied to model the opinion evolution in
social networks, especially, involving antagonistic interactions.

Before moving on, we give some useful definitions regarding system
$(\ref{20191eq2})$.

###### Definition 1

System $(\ref{20191eq2})$ achieves:

  * (i)

the _consensus in opinions_ , if

$\displaystyle\lim_{k\rightarrow\infty}\xi_{i}(k)=\varphi,~{}~{}i\in\mathbb{I}_{N}$

where $\varphi\in\mathbb{R}$ is a constant.

  * (ii)

the _convergence in opinions_ , if

$\displaystyle\lim_{k\rightarrow\infty}\xi_{i}(k)=\varphi_{i},~{}~{}i\in\mathbb{I}_{N}$

where $\varphi_{i}\in\mathbb{R}$ is a constant.

  * (iii)

the _stability in opinions_ , if

$\displaystyle\lim_{k\rightarrow\infty}\xi_{i}(k)=0,~{}~{}i\in\mathbb{I}_{N}$

The mechanism behind Definition 1 is that a cooperative appraisal network
gives rise to the opinion aggregation while an antagonistic appraisal network
leads to the clusters in opinion in general. Moreover, the consensus of the
reciprocal agents is a special type of the convergence where all interacting
individuals share a common viewpoint eventually.

Figure 1: A holistic paradigm of the opinion evolution for system
$(\ref{20191eq2})$ where the appraisal network (characterized by
$\mathcal{D}$) is cooperative, i.e., $\delta_{ij}\geq 0$.

##  III Main Results

In the following discussions, we will show that the opinion formation process
with setup $(\ref{20191eq1})$ is more general comparing with the DeGroot
model. More specific, system $(\ref{20191eq1})$ permits non-convex
interactions of the individuals, and it provides an intriguing viewpoint on
how the individual’s introspective process influences the opinion evolution of
social networks.

###  III-A Cooperative Appraisal Network

In this subsection, we are concerned with the situation where the appraisal
network is cooperative, i.e., each influence weight $\delta_{ij}$ is
nonnegative. A simple illustration for this case is drawn in Fig. 1. Hence,
$\sum^{N}_{j=1}\delta_{ij}=1$. In such a case,
$\mathcal{D}=(\delta_{ij})_{N\times N}$ is a nonnegative stochastic matrix.
Before proceeding on, we introduce a simple proposition which bridges the gap
between the stochastic matrix and the Laplacian matrix in connection with the
communication topology in a unified manner.

###### Proposition 1

([33]) Given any nonnegative stochastic matrix
$\mathcal{D}=(\delta_{ij})_{N\times N}$, there always exists a Laplacian
matrix $L=(l_{ij})_{N\times N}$ such that

$\displaystyle\mathcal{D}=I-\epsilon L$ (3)

where $\epsilon>0$ is the step size.

It should be emphasized that the Laplacian matrix $L$ depicted in
$(\ref{20191eq37})$ generally has little connection with the interacting
network $\mathcal{L}$ reported in $(\ref{20191eq2})$. In the sequel, we give
the main results on cooperative appraisal network.

###### Theorem 1

Suppose that the appraisal network is cooperative, i.e., $\delta_{ij}\geq 0$.
Then _consensus_ in system $(\ref{20191eq2})$ is achieved if and only if
matrix $I-\Lambda\mathcal{L}\mathcal{D}$ has a simple $1$ eigenvalue and the
remaining eigenvalues are preserved in the unit disk.

###### Proof:

Suppose that the communication graph associated with $\mathcal{L}$ has a
directed spanning tree. By Sylvester rank inequality, one has

$\displaystyle{\rm rank}(\mathcal{L})+{\rm rank}(\mathcal{D})-N\leq$
$\displaystyle~{}{\rm rank}(\mathcal{L}\mathcal{D})$ $\displaystyle\leq$
$\displaystyle~{}\min\bigg{\\{}{\rm rank}(\mathcal{L}),{\rm
rank}(\mathcal{D})\bigg{\\}}$

In fact, the zero eigenvalue is simple if the graph induced by $\mathcal{L}$
has a directed spanning tree. Then, zero is an eigenvalue of matrix
$\Lambda\mathcal{L}\mathcal{D}$. Additionally, vector one is the eigenvector
of the zero eigenvalue. Hence, system $(\ref{20191eq2})$ can achieve the
consensus.

We proceed with the condition for consensus. Actually, matrix
$I-\Lambda\mathcal{L}\mathcal{D}$ has a simple $1$ eigenvalue and the
remaining eigenvalues are preserved in the unit disk if and only if
$\Lambda\mathcal{L}\mathcal{D}$ contains a simple zero eigenvalue and the
remaining eigenvalues share positive real parts (here we can redefine the
coupling matrix by $\Lambda=\epsilon_{1}\Lambda^{{\dagger}}$ where
$\epsilon_{1}$ is a small step size, by doing so we could access to the
continuous version of $(\ref{20191eq2})$ by
$\dot{\xi}=-\Lambda^{{\dagger}}\mathcal{L}\mathcal{D}\xi$). It further assumes
that the left and the right eigenvectors with respect to the zero eigenvalue
are, respectively, $\varsigma\in\mathbb{R}^{N}$ and $\iota\in\mathbb{R}^{N}$
with the constraint $\varsigma^{\prime}\iota=1$. Note that
$\iota=\textbf{1}_{N}$ for such a scenario since $\mathcal{D}\iota=\iota$. We
define the disagreement error by
$\theta(k)=(I-\iota\varsigma^{\prime})\xi(k)$, then one has

$\displaystyle\theta(k+1)=(I-\Lambda\mathcal{L}\mathcal{D})\theta(k)$ (4)

We can see that the eigenvalues of $I-\Lambda\mathcal{L}\mathcal{D}$ are
entirely constrained in the unit disk over the space
$\mathbb{R}^{N}\backslash\\{\varphi\\}$ where
$\varphi=\iota\varsigma^{\prime}\xi(0)=\alpha\iota$. Thus the error system
$(\ref{20191eq11})$ is exponentially stable, which ensures the consensus of
system $(\ref{20191eq2})$. This ends the proof by Definition 1. ∎

###### Remark 2

Different from the DeGroot model[3] and multi-agent systems[33], the final
shared common opinion of the interacting individuals governed by
$(\ref{20191eq2})$ may not be restricted in a convex hull spanned by the
initial opinions of the individuals. This is because that the appraisal
network specifying the interaction rule in this paper is no long to be convex,
to a large extent. We emphasize that non-convex interactions among
participating individuals in social networks are rather common, which have
been clearly pointed out by Wang _et al_. in [29]. This intriguing issue will
be verified via an example later. $\blacklozenge$

Next, we consider a special case where the appraisal network is the same as
the interacting network, i.e., $L=\mathcal{L}$. In order to achieve the
consensus, it suffices to fix $\varrho_{i}=\varrho$ for all $i$. One hence
arrives at

$\displaystyle\xi(k+1)=$
$\displaystyle~{}(I-\varrho\mathcal{L}+\epsilon\varrho\mathcal{L}^{2})\xi(k)$
(5)

where $\epsilon>0$ is a constant.

###### Corollary 1

Suppose that the graph induced by $\mathcal{L}$ has a directed spanning tree.
Then system $(\ref{20191eq6})$ achieves the _consensus in opinions_ if and
only if

$\displaystyle 0<\varrho<\min_{{\rm
Re}(\lambda^{\star}_{i})>0}\bigg{\\{}\frac{2{\rm
Re}(\lambda^{\star}_{i})}{{\rm Re}^{2}(\lambda^{\star}_{i})+{\rm
Im}^{2}(\lambda^{\star}_{i})}\bigg{\\}}$ (6)

where $\lambda^{\star}_{i}$ stands for the $i$th eigenvalue of
$\mathcal{L}-\epsilon\mathcal{L}^{2}$, and constant $\epsilon$ satisfies

$\left\\{\begin{aligned} &\epsilon\in\mathbb{R},~{}~{}|{\rm
Re}(\lambda_{i})|=|{\rm Im}(\lambda_{i})|\\\ &\epsilon<\min_{\lambda_{i}\neq
0}\frac{{\rm Re}(\lambda_{i})}{{\rm Re}^{2}(\lambda_{i})-{\rm
Im}^{2}(\lambda_{i})},~{}~{}|{\rm Re}(\lambda_{i})|>|{\rm Im}(\lambda_{i})|\\\
&\epsilon>\max_{\lambda_{i}\neq 0}-\frac{{\rm Re}(\lambda_{i})}{{\rm
Im}^{2}(\lambda_{i})-{\rm Re}^{2}(\lambda_{i})},~{}~{}|{\rm
Re}(\lambda_{i})|<|{\rm Im}(\lambda_{i})|\end{aligned}\right.$ (7)

where $\lambda_{i}$ is the $i$th eigenvalue of matrix $\mathcal{L}$. Moreover,
if $\epsilon$ is further required to be positive, then we have

$\displaystyle\epsilon\in\bigg{(}0,~{}~{}\min_{\lambda_{i}\neq 0,~{}|{\rm
Re}(\lambda_{i})|>|{\rm Im}(\lambda_{i})|}\frac{{\rm Re}(\lambda_{i})}{{\rm
Re}^{2}(\lambda_{i})-{\rm Im}^{2}(\lambda_{i})}\bigg{)}$

Moreover, the final aggregated value is restricted in a convex hull spanned by
the initial opinions of the roots.

###### Proof:

Let us first study an auxiliary system of $(\ref{20191eq6})$ by

$\displaystyle\dot{\xi}(t)=$ $\displaystyle~{}-W\xi(t),~{}t\in\mathbb{R}_{+}$
(8)

where $W=\mathcal{L}-\epsilon\mathcal{L}^{2}$. It is known that system
$(\ref{20191eq7})$ guarantees the consensus if and only if matrix $W$ contains
a simple zero eigenvalue and the remaining eigenvalues share positive real
parts. When the induced graph by Laplacian matrix $\mathcal{L}$ has a directed
spanning tree, one has that the zero eigenvalue of matrix $W$ is simple. We
continuous to show that the nonzero eigenvalues of $W$ have positive real
parts. One can easily find that the nonzero eigenvalues of $W$ are of the form

$\displaystyle\lambda^{\star}_{i}=$ $\displaystyle~{}{\rm
Re}(\lambda_{i})-\epsilon({\rm Re}^{2}(\lambda_{i})-{\rm
Im}^{2}(\lambda_{i}))$ (9) $\displaystyle+\mathbbm{i}({\rm
Im}(\lambda_{i})-2\epsilon{\rm Re}(\lambda_{i}){\rm Im}(\lambda_{i}))$

By $(\ref{20191eq8})$, the nonzero eigenvalues in $W$ share positive real
parts if and only if $(\ref{ch7_20191eq10_1})$ is desirable.

It is noticeable that the relationship between the eigenvalues of system
matrix in $(\ref{20191eq7})$ and those in $(\ref{20191eq6})$ can be formulated
by

$\displaystyle\lambda^{\ast}_{i}=$
$\displaystyle~{}1-\varrho\lambda^{\star}_{i},~{}i\in\mathbb{I}_{N}$

where $\lambda^{\ast}_{i}$ represents the $i$th eigenvalue of
$I-\varrho\mathcal{L}+\epsilon\varrho\mathcal{L}^{2}$. As a result, one can
check that $I-\varrho\mathcal{L}+\epsilon\varrho\mathcal{L}^{2}$ has a simple
$1$ eigenvalue and the remaining eigenvalues are restricted in the unit disk
if and only if $(\ref{20191eq10})$ is desirable, by which
$|\lambda^{\ast}_{i}|<1$ is always guaranteed as long as
$\lambda^{\star}_{i}\neq 0$.

The second part is trivial, and hence is omitted. ∎

An extension of $\epsilon=\varrho$ would be interesting since the developed
method in Corollary 1 does not work in such a case. Actually, $\varrho$ is
involved in both $(\ref{20191eq6})$ and $(\ref{20191eq7})$, leading to the
failure of computing the allowable range for $\varrho$. Apart from these
concerns, the argument induced within a general interacting network is far
from obvious in contrast to the case of bidirectional interacting network, as
suggested by Corollary 1.

###### Theorem 2

Suppose that the graph induced by $\mathcal{L}$ has a directed spanning tree.
Then system $(\ref{20191eq6})$ achieves the _consensus in opinions_ if and
only if the step size $\varrho$ is bounded with the following constraints,

  * (i)

If ${\rm Im}(\lambda_{i})=0$ for $\lambda_{i}\neq 0$,

$\displaystyle\varrho\in\min_{\lambda_{i}}\bigg{(}0,\frac{1}{\lambda_{i}}\bigg{)}$

where $\lambda_{i}>0$ is the eigenvalue of $\mathcal{L}$.

  * (ii)

If ${\rm Im}(\lambda_{i})\neq 0$ with $\lambda_{i}\neq 0$,

$\left\\{\begin{aligned} &\min_{\varrho>0,\lambda_{i}\neq
0}f_{i}(\varrho,\lambda_{i},\arg(\lambda_{i}))>0\\\
&\varrho\not\in\bigg{\\{}\varrho_{i,1},\varrho_{i,2}\bigg{\\}}\bigcup\bigg{\\{}\varrho_{i,3},\varrho_{i,4}\bigg{\\}}\end{aligned}\right.$

where

$\displaystyle f_{i}(\varrho,\lambda_{i},\arg(\lambda_{i}))$ $\displaystyle=$
$\displaystyle~{}-\varrho^{3}|\lambda_{i}|^{3}+\varrho^{2}|\lambda_{i}|^{2}\cos^{2}(\arg(\lambda_{i}))$
$\displaystyle-2\varrho|\lambda_{i}|\sin(2\arg(\lambda_{i}))-\varrho|\lambda_{i}|+2\cos(\arg(\lambda_{i}))$

and $\varrho_{i,1}$, $\varrho_{i,2}$, $\varrho_{i,3}$, $\varrho_{i,4}$ are
depicted in $(\ref{20191eq107})$

$\left\\{\begin{aligned}
\varrho_{i,1}=&~{}\frac{\cos(\arg(\lambda_{i}))+\sqrt{\cos^{2}(\arg(\lambda_{i}))(8\cos(\theta)-7)+4(1-\cos(\theta))}}{2|\lambda_{i}|\cos(2\arg(\lambda_{i}))}\\\
\varrho_{i,2}=&~{}\frac{\cos(\arg(\lambda_{i}))-\sqrt{\cos^{2}(\arg(\lambda_{i}))(8\cos(\theta)-7)+4(1-\cos(\theta))}}{2|\lambda_{i}|\cos(2\arg(\lambda_{i}))}\\\
\varrho_{i,3}=&~{}\frac{\sin(\arg(\lambda_{i}))+\sqrt{\cos^{2}(\arg(\lambda_{i}))(8\cos(\theta)-7)+4(1-\cos(\theta))}}{2|\lambda_{i}|\sin(2\arg(\lambda_{i}))}\\\
\varrho_{i,4}=&~{}\frac{\sin(\arg(\lambda_{i}))-\sqrt{\cos^{2}(\arg(\lambda_{i}))(8\cos(\theta)-7)+4(1-\cos(\theta))}}{2|\lambda_{i}|\sin(2\arg(\lambda_{i}))}\end{aligned}\right.$
(10)

where $\theta\in[0,2\pi)$.

###### Proof:

The proof of Theorem 2 is self-contained, and is reported in Appendix for the
sake of concinnity. ∎

From Theorem 2, the requirement on $\varrho$ coincides with the statement in
[33] if the underlying network is bidirectional. For the general interpersonal
network, the condition on $\varrho$ is far from trivial as the previous case
since it tightly links to the amplitude and the argument principal value of
nonzero eigenvalues corresponding to $\mathcal{L}$. In fact, there is an
alternative to guarantee the consensus in opinion for cooperative antagonistic
network, even though it falls short of elegance as Theorem 2.

###### Corollary 2

Suppose that the graph induced by $\mathcal{L}$ attains a directed spanning
tree. Then system $(\ref{20191eq6})$ achieves the _consensus in opinions_ if
and only if

$\displaystyle\varrho\in\bigcap_{\varrho>0,\lambda_{i}\neq
0}\bigg{\\{}a_{i}\varrho^{3}+b_{i}\varrho^{2}+c_{i}\varrho+d_{i}<0\bigg{\\}}$
(11)

where

$\displaystyle a_{i}=$ $\displaystyle~{}4{\rm Re}^{2}(\lambda_{i})+({\rm
Re}^{2}(\lambda_{i})-{\rm Im}^{2}(\lambda_{i}))^{2}$ $\displaystyle b_{i}=$
$\displaystyle~{}2{\rm Re}(\lambda_{i})({\rm Re}^{2}(\lambda_{i})-{\rm
Im}^{2}(\lambda_{i}))-4{\rm Re}(\lambda_{i}){\rm Im}(\lambda_{i})$
$\displaystyle c_{i}=$ $\displaystyle~{}3{\rm Re}^{2}(\lambda_{i})-{\rm
Im}^{2}(\lambda_{i})$ $\displaystyle d_{i}=$ $\displaystyle~{}-2{\rm
Re}(\lambda_{i})$

In addition, consensus in system $(\ref{20191eq6})$ is achieved for the
undirected graph if and only if

$\displaystyle\varrho\in\min_{\lambda_{i}}\bigg{(}0,\frac{1}{\lambda_{i}}\bigg{)}$

where $\lambda_{i}>0$ is the eigenvalue of $\mathcal{L}$.

###### Proof:

System $(\ref{20191eq6})$ can be rewritten as

$\displaystyle\xi(k+1)=$
$\displaystyle~{}(I-\varrho\mathcal{L}+\varrho^{2}\mathcal{L}^{2})\xi(k)$ (12)

The eigenvalue of matrix $I-\varrho\mathcal{L}+\varrho^{2}\mathcal{L}^{2}$ is
of the form

$\displaystyle\lambda^{\ast}_{i}=$
$\displaystyle~{}1-\varrho\lambda_{i}+\varrho^{2}\lambda^{2}_{i}$
$\displaystyle=$ $\displaystyle~{}1-{\rm Re}(\lambda_{i})\varrho+\bigg{(}{\rm
Re}^{2}(\lambda_{i})-{\rm Im}^{2}(\lambda_{i})\bigg{)}\varrho^{2}$
$\displaystyle+\bigg{(}2{\rm Re}(\lambda_{i}){\rm
Im}(\lambda_{i})\varrho^{2}-{\rm Im}(\lambda_{i})\varrho\bigg{)}\mathbbm{i}$

Therefore, $|\lambda^{\ast}_{i}|<1$ with ${\rm Re}(\lambda_{i})>0$ is equal to

$\displaystyle 1\geq$ $\displaystyle~{}\bigg{(}1-{\rm
Re}(\lambda_{i})\varrho+\bigg{(}{\rm Re}^{2}(\lambda_{i})-{\rm
Im}^{2}(\lambda_{i})\bigg{)}\varrho^{2}\bigg{)}^{2}$
$\displaystyle+\bigg{(}2{\rm Re}(\lambda_{i}){\rm
Im}(\lambda_{i})\varrho^{2}-{\rm Im}(\lambda_{i})\varrho\bigg{)}^{2}$

By tedious computation, it yields the requirement in $(\ref{20191eq63})$. And
the second statement is consistent with that in Theorem 2, and is hence
omitted. ∎

Figure 2: A holistic paradigm of the opinion evolution for system
$(\ref{20191eq2})$ where the appraisal network (characterized by
$\mathcal{D}$) is antagonistic, that is, the red arcs mean $\delta_{ij}<0$
while the black ones indicate that $\delta_{ij}>0$.

###  III-B Antagonistic Appraisal Network

A remarkable feature of the social networks is that social actors rarely share
unanimous opinions. It has been recognized that biased assimilation principle
or homophily principle (cf. [35]) is vital to explain the opinion clusters.
Another reason for the opinion dynamics giving rise to clusters is antagonism.
The main goal of this subsection is to investigate the case where there exists
antagonistic interactions among social actors.

By following the basic route as that in subsection III-A, the public network
characterized by $\mathcal{L}$ is the same as before. Meanwhile, the appraisal
network characterized by $\mathcal{D}$ involves hostile interactions. A simple
illustration for system $(\ref{20191eq2})$ is depicted by Fig. 2.

Suppose that agent $i$ possesses an opposite attitude to the opinion of agent
$j$, then ${\rm sgn}(\delta_{ij})=-1$. In addition, since $|\delta_{ij}|$
quantifies the degree of the influence among the total social influence that
the $i$th social actor has, we require $\sum^{N}_{j=1}|\delta_{ij}|=1$ for all
$i$. Before moving on, a condiment is needed.

###### Definition 2

Two graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ share the same topology if
there exists an edge from the $i$th node to the $j$th node in
$\mathcal{G}_{1}$, then the edge $(i,j)$ also belongs to $\mathcal{G}_{2}$,
and vice versa.

Actually, two graphs satisfying Definition 2 differ only in the weighted
values of edges. Therefore, we can define a new stochastic matrix
$\mathcal{D}^{\star}\triangleq(|\delta_{ij}|)_{N\times N}$. With the help of
Definition 2, the graphs induced by $\mathcal{D}$ and $\mathcal{D}^{\star}$
have a topology structure in common.

###### Theorem 3

Suppose that the appraisal network is antagonistic. Then system
$(\ref{20191eq2})$ _converges in opinions_ if and only if matrix
$I-\Lambda\mathcal{L}\mathcal{D}$ has a simple $1$ eigenvalue and the
remaining eigenvalues are preserved in the unit disk.

###### Proof:

The proof of Theorem 3 is similar to that in Theorem 1. Generally speaking,
the right eigenvector $\iota$ associated with the zero eigenvalue of
$\Lambda\mathcal{L}\mathcal{D}$ may not be equal to vector one when the
appraisal network contains hostile interactions. Therefore, it follows that
$\varphi=\iota\varsigma^{\prime}\xi(0)$. This implies the convergence in
opinions of the social actors. ∎

According to Theorem 3, the clusters in opinions occur due to the presence of
antagonistic interactions in the appraisal network. In view of Theorems 1 and
3, it can be concluded that cooperative appraisal networks lead to the
consensus in opinions, while hostile appraisal networks attain the clusters in
opinions. Here we emphasize that it is rather arduous to give some specific
indices on the weighted influence matrix $\Lambda$ due to the potential
complexity of the appraisal network. Another interesting welfare behind
Theorems 1 and 3 is that the agents in system $(\ref{20191eq2})$ will not
converge to zero for almost all initial values. This is because that
$I-\Lambda\mathcal{L}\mathcal{D}$ contains at least an eigenvalue one.
Fortunately, this intriguing feature could be addressed by virtue of the issue
dependence matrix that is about to be elucidated later.

Figure 3: (a) Opinion evolution for the antagonistic appraisal network where
the initial opinions $\xi(0)=(25,75,85)$ are partly borrowed from [11,
Equation (15)]; (b) Opinion evolution for the antagonistic appraisal network
with the same parameters as (a) except for replacing $\Lambda$ by $0.5I$.

###  III-C Connection between Cooperative and Antagonistic Appraisal Networks

Subsections III-A and III-B discuss the opinion evolution on cooperative and
antagonistic appraisal networks, respectively. Here we are interested in a
question: what is the bridge between the cooperative and the antagonistic
appraisal networks? To answer this question, we first look at a simple
example.

###### Example 1

Consider a network with three social actors whose interacting and appraisal
networks are of the forms

$\displaystyle\mathcal{L}=\begin{bmatrix}2&-1&-1\\\ -1&2&-1\\\
-1&-1&2\end{bmatrix},~{}\mathcal{D}=\begin{bmatrix}0.5&-0.5&0\\\ 0&0.5&-0.5\\\
-0.5&0&0.5\end{bmatrix}$

We choose $\Lambda={\rm diag}(-0.05,0.5,0.5)$, and then compute the
eigenvalues of $I-\Lambda\mathcal{L}\mathcal{D}$

$\displaystyle\lambda(I-\Lambda\mathcal{L}\mathcal{D})\in\bigg{\\{}0,0.9448,1.9052\bigg{\\}}$

Opinion evolutions of the social actors are plotted in Fig. 3(a). It reveals
an interesting phenomenon that antagonistic appraisal networks could achieve
consensus in opinions. More importantly, we can see that the final aggregated
opinion is not restricted in the convex hull spanned by the initial opinions.
Next, we select another group of susceptibility factors by $0.5I$. In this
case, the consensus in opinions could still be preserved while it is contained
in the convex hull spanned by the initial opinions (see Fig. 3(b) for more
details). Therefore, we could treat the susceptibility factor $\varrho_{i}$ as
a design parameter quantifying how does the opinion formation process work.
$\blacklozenge$

Note that consensus in opinions could be preserved for cooperative appraisal
network. It implicitly manifests that the opinions either achieve consensus or
diverge whenever the appraisal network is cooperative. However, opinions
either achieve consensus or clusters if the underlying appraisal network is
antagonistic, except for the divergence in opinions. We summarize them as
below.

###### Proposition 2

Suppose that the appraisal network is antagonistic. Then system
$(\ref{20191eq2})$ achieves _consensus in opinions_ if and only if the
following properties hold:

  * (i)

System $(\ref{20191eq2})$ converges;

  * (ii)

$\mathcal{D}\textbf{1}_{N}=\mathcal{O}_{N\times 1}$ or
$\mathcal{D}\textbf{1}_{N}=-\textbf{1}_{N}$.

###### Proof:

(Necessity) According to Definition 1, the consensus in $(\ref{20191eq2})$
indicates that
$\lim_{k\rightarrow\infty}\xi_{i}(k)=\lim_{k\rightarrow\infty}\xi_{j}(k)$ for
arbitrary $i,j\in\mathbb{I}_{N}$. Hence, $\textbf{1}_{N}$ is the right
eigenvector associated with the eigenvalue one of matrix
$I-\Lambda\mathcal{L}\mathcal{D}$. It further arrives at

$\displaystyle\mathcal{O}_{N\times 1}=$
$\displaystyle\Lambda\mathcal{L}\mathcal{D}\textbf{1}_{N}\Rightarrow\mathcal{O}_{N\times
1}=\mathcal{L}\mathcal{D}\textbf{1}_{N}$

The above formula is true since matrix $\Lambda$ is nonsingular. The directed
spanning tree preserved in the interacting network implies that merely one of
the following equalities is satisfied,

$\displaystyle\mathcal{D}\textbf{1}_{N}=$ $\displaystyle~{}\textbf{1}_{N}$
(13a) $\displaystyle\mathcal{D}\textbf{1}_{N}=$
$\displaystyle~{}\mathcal{O}_{N\times 1}$ (13b)
$\displaystyle\mathcal{D}\textbf{1}_{N}=$ $\displaystyle~{}-\textbf{1}_{N}$
(13c)

Next we will show that only $(\ref{20191eq48b})$ or $(\ref{20191eq48c})$ hold
for the antagonistic appraisal network. Suppose that $(\ref{20191eq48a})$
holds, i.e.,

$\displaystyle\sum^{N}_{j=1}\delta_{ij}=1,~{}\forall~{}i\in\mathbb{I}_{N}$
(14)

Moreover, we also require

$\displaystyle\sum^{N}_{j=1}|\delta_{ij}|=1,~{}\forall~{}i\in\mathbb{I}_{N}$
(15)

One can see that $(\ref{20191eq49})$ and $(\ref{20191eq50})$ suggest

$\displaystyle|\delta_{ij}|=\delta_{ij},~{}\forall~{}i,j\in\mathbb{I}_{N}$

which implies that the appraisal network is cooperative. It is a
contradiction. Therefore, we always have
$\mathcal{D}\textbf{1}_{N}=\mathcal{O}_{N\times 1}$ or
$\mathcal{D}\textbf{1}_{N}=-\textbf{1}_{N}$. Obviously,
$\mathcal{D}\textbf{1}_{N}=-\textbf{1}_{N}$ means that all entries of
$\mathcal{D}$ are non-positive, which indicates that the appraisal network is
antagonistic.

(Sufficiency) The convergence of system $(\ref{20191eq2})$ means that
$I-\Lambda\mathcal{L}\mathcal{D}$ has a simple eigenvalue one (note that
system $(\ref{20191eq2})$ cannot guarantee the stability since matrix
$\Lambda\mathcal{L}\mathcal{D}$ always has at least a zero eigenvalue). In
addition, $\mathcal{D}\textbf{1}_{N}=\mathcal{O}_{N1}$ or
$\mathcal{D}\textbf{1}_{N}=-\textbf{1}_{N}$ suggests that the vector
$\textbf{1}_{N}$ is an eigenvector associated with the eigenvalue one of
matrix $I-\Lambda\mathcal{L}\mathcal{D}$. Therefore, system $(\ref{20191eq2})$
preserves consensus in opinions. ∎

With the help of Proposition 2 and Theorem 3, we could conclude that
cooperative appraisal network achieves consensus in opinions, while
antagonistic appraisal network shows both consensus and clusters in opinions.
However, antagonistic appraisal network that can achieve consensus in opinions
has certain special requirements for its structure.

###  III-D Estimation for Appraisal Network

Briefly speaking, it is a thorny problem to determine the structure of the
appraisal network. Fortunately, a large number of efforts have been poured on
identifying the dynamic structure and topologies of the networks with the help
of the experiment data. The emerging fields include sociology, signal
processing and statistics (see, e.g., [1, 36, 37]). As hinted before,
interacting network is always known to all; while appraisal network is
generally private, and thus is of great significance to be estimated. The
reason lies that it is the first step to understand the mechanism behind the
emerging and the evolution of the opinions in social networks.

To cope with the estimation issue related to appraisal network, a technical
lemma is needed.

###### Lemma 1

([38]) For any matrix $Q=(q_{1},...,q_{n})$, its vectorization, denoted by
${\rm vec}(Q)$, is

$\displaystyle{\rm vec}(Q)=(q^{\prime}_{1},...,q^{\prime}_{n})^{\prime}$

For matrices $Q$, $W$ and $R$, the vectorization with respect to their product
is given by

$\displaystyle{\rm vec}(QWR)=$ $\displaystyle(R^{\prime}\otimes Q){\rm
vec}(W)$

By virtue of Lemma 1, two properties can be obtained.

###### Corollary 3

For any variable $\xi\in\mathbb{R}^{N}$ and matrix
$\Lambda\mathcal{L}\mathcal{D}$, the following facts are true

$\left\\{\begin{aligned} &{\rm vec}(\xi)=~{}\xi\\\ &{\rm
vec}(\Lambda\mathcal{L}\mathcal{D}\xi)=~{}(\xi^{\prime}\otimes\Lambda\mathcal{L}){\rm
vec}(\mathcal{D})\end{aligned}\right.$

###### Proof:

The proof is trivial by Lemma 1. ∎

With the above preparations, we are about to formulate the estimation problem
on appraisal network. Apart from $\Lambda$ and $\mathcal{L}$, we postulate
that one has access to $m$ observed opinions of length $1$ during opinion
formation process under $(\ref{20191eq1})$, i.e., $m$ sequences of opinions
$(\xi_{t}(k-1),\xi_{t}(k))$ with $t\in\mathbb{I}_{m}$ and
$k\in\mathbb{Z}_{+}$, in a manner of independent and identically distribution
(i.i.d.). More specifically, we associate a uniform observation of $m$ by

$\displaystyle\Omega_{m}=~{}\bigg{\\{}(\xi_{t}(k-1),\xi_{t}(k)),1\leq t\leq
m,\forall~{}k\in\mathbb{Z}_{+}\bigg{\\}}$ (16)

Upon collecting the $m$ (randomly with uniform) opinion sets and using
Corollary 3, we formulate the following optimization problem:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\min_{\zeta,~{}Q}~{}\gamma$
(17) $\displaystyle{\rm
s.t.}~{}~{}~{}~{}~{}~{}f(\zeta,m)=\frac{1}{m}\sum^{m}_{t=1}g(t)\leq\gamma$
$\displaystyle g(t)=X^{\prime}_{t}(k)QX_{t}(k)$ $\displaystyle
X_{t}(k)=\xi_{t}(k)-\xi_{t}(k-1)+\bigg{(}\xi^{\prime}_{t}(k-1)\otimes\Lambda\mathcal{L}\bigg{)}\zeta$
$\displaystyle\gamma\geq 0,~{}Q\succ 0$

###### Remark 3

For optimization problem $(\ref{20191eq31})$, we emphasize: (i) The
utilization of Corollary 3 is helpful since the coupling between the
interacting network and the appraisal network is entangled; (ii) In general,
it is unrealistic to require parameter $m$ to be infinite because system
matrix in $(\ref{20191eq2})$ is not stable.

Algorithm 1 depicted below gives a lower bound on the “observations” needed
for a desirable performance.

Algorithm 1 Lower Bound for the Observations in $\Omega_{m}$

1: Given a sequence of observations of length $m$, and a prior level of
performance index $0<\gamma_{0}<\infty$;

2: Set $m=m_{0}$ with $m_{0}\geq 1$;

3: while $\gamma>\gamma_{0}$; do

4: Print: Gauge of appraisal network fails;

5: Set $m\leftarrow 1+m$;

6: Optimize $(\ref{20191eq31})$;

7: end while

8: return $m$;

Although Algorithm 1 may provide an estimation on appraisal network, we still
cannot give specific value of sample length $m$ that is essential for solving
$(\ref{20191eq31})$. Obviously, too much observations bring computation
complexity, while less observation may yield inaccuracy estimation.
Furthermore, even Algorithm 1 may mitigate the downside on $(\ref{20191eq31})$
to a certain degree, a lot of computing resources are needed apart from the
time consumption since Algorithm 1 executes in a way of try-once-discard. All
the discussions above naturally raise an interesting question: What is the
confidence can we possess with the aid of a finite sets on opinions of
empirical observations, on which the appraisal network can work for the whole
opinion space? To this end, we resort to the tool from random convex
optimization (also called by chance-constrained optimization); see Ref. [31,
32] for more details.

Now, we are going to formulate the considered problem. For fixed probability
space $(\Omega,\mathcal{F},\mathcal{P})$, compact set
$\mathbb{X}\subseteq\mathbb{R}^{N\times N}$ is convex and contains the origin
as its interior. A measurable function
$f:\mathbb{X}\times\Omega\mapsto\mathbb{R}$ is convex on its first argument
for any fixed second argument, and bounded for the second argument whenever
the first is fixed. Thereby, for random finite opinion set $\Omega_{m}$ in
$(\ref{20191eqnew1})$ and any level parameter $\varepsilon\in(0,1)$, given a
confidence level $\beta\in(0,1)$ and some symmetric positive definite matrix
$Q$, the probability of violation (cf. [31, Definition 1]) is

$\displaystyle
V(\zeta,m)=\mathcal{P}\\{m\in\mathbb{Z}_{+}:f(\zeta,m)>\gamma^{\star}\\}$

where $\gamma^{\star}$ is the optimal solution of the optimization problem
$(\ref{20191eq31})$.

We then focus on the following optimization problem:

$\displaystyle\min_{\zeta}{}{}{}{}{}{}$ $\displaystyle c^{\prime}\zeta$ (18)
$\displaystyle{\rm s.t.}{}{}{}{}{}{}$
$\displaystyle\mathcal{P}\\{(\zeta,m):V(\zeta,m)>\varepsilon\\}\leq\beta$
$\displaystyle\zeta\in\mathbb{X}$

where $c$ is a certain “cost” vector (also known as the objective direction).

For optimization problem $(\ref{20191eqnew2})$, we get the probability with
respect to $f(\zeta,m)\leq\gamma^{\star}$ at least $1-\beta$ using the samples
with length $m$. A smaller violation level $\varepsilon$ is a more desirable
estimation on the appraisal network. As a result, the number of samples
increases. The next theorem gives a confirmative answer on how many opinion
observations are needed to obtain a satisfactory performance.

###### Theorem 4

Consider optimization problem $(\ref{20191eqnew2})$ with $m\geq N^{2}$ where
$\Omega_{m}\subseteq\mathbb{X}$, in the sense of i.i.d. Then, for
$\forall\varepsilon\in[0,1]$, it follows that

$\displaystyle\mathcal{P}\\{(\zeta,m):V(\zeta,m)>\varepsilon\\}\leq\beta(\varepsilon,m)$

where

$\displaystyle\beta(\varepsilon,m)=\sum^{m}_{\ell=0}\begin{pmatrix}N^{2}\\\
\ell\end{pmatrix}\varepsilon^{\ell}(1-\varepsilon)^{m-\ell}$

Moreover, the low bound on $m$ is

$\displaystyle
m(\varepsilon,\beta)=\min\bigg{\\{}m\in\mathbb{Z}_{+}\bigg{|}\sum^{m}_{\ell=0}\begin{pmatrix}N^{2}\\\
\ell\end{pmatrix}\varepsilon^{\ell}(1-\varepsilon)^{m-\ell}\leq\beta\bigg{\\}}$

###### Proof:

By [32, Theorem 3.3] and [39, Theorem 1], the proof follows directly. ∎

###### Remark 4

Theorem 4 explicitly provides the number of required samples to obtain a
“good” estimation for the true appraisal network. In [11], the authors
concentrated on the estimation of multi-issue dependence matrix. We do some
further work in this paper. That is, on the one hand, we can give a desirable
estimation on the appraisal network. On the other hand, a bound on the samples
is provided which has no been addressed in the previous work.

##  IV Topic Specific Opinion

Unlike man-made systems, the actors in social networks are generally affected
by a couple of topics that are interdependent. For example, the leader of the
corporate should account for many factors if he/she intends to operate a
policy, such as the cost, the external and the internal environment, the
potential market as well as the potential customers, etc. As a matter of fact,
the issue dependence problem has been well addressed for a long time, in
particular the disciplines such as social anthropology, sociology and
political science and psychology where they share a common ground that certain
objects are coupled by interdependent cognitive orientations.

The first step on how agents’ interpersonal influences form a belief system
was proposed by the FJ model (cf. [6])

$\displaystyle\xi_{i}(k+1)=$
$\displaystyle\lambda_{ii}C\sum^{N}_{j=1}p_{ij}\xi_{j}(k)+(1-\lambda_{ii})\mu_{i}$
(19)

where $\xi_{i}(k)\in\mathbb{R}^{n}$, $\lambda_{ii}$ and $\mu_{i}$ are,
respectively, the susceptibility and the initial opinion of the $i$th agent.
$P=(p_{ij})_{N\times N}$ with $p_{ij}\geq 0$ is a stochastic matrix. Matrix
$C=(c_{ij})_{n\times n}$ stands for the introspective transformation called
the multi-issues dependence structure (MiDS) (cf. [11]), and satisfies
$\sum^{n}_{j=1}|c_{ij}|=1$.

For topic specific issues, the opinion evolving equation $(\ref{20191eq1a})$
becomes

$\displaystyle\xi_{i}(k+1)=$
$\displaystyle~{}C\xi_{i}(k)+\varrho_{i}C\sum_{j\in\mathcal{N}_{i}}a_{ij}(z_{j}(k)-z_{i}(k))$
(20)

where the constant matrix $C\in\mathbb{R}^{n\times n}$ describes the MiDS,
while the remaining variables are the same as those in $(\ref{20191eq1a})$.
Combining $(\ref{20191eq1b})$ and $(\ref{20191eq53})$ gives

$\displaystyle\xi(k+1)=$
$\displaystyle~{}(I-\Lambda\mathcal{L}\mathcal{D})\otimes C\xi(k)$ (21)

where $\otimes$ denotes the Kronecker product.

Here we discuss the similarities and the differences between
$(\ref{20191eq52})$ and $(\ref{20191eq54})$.
Similarity:
They both model how the interdependent issues affect the opinion’s evolution.
Differences:
$(\ref{20191eq52})$ is concerned with the opinion evolution in the context of
the cooperative interacting networks while $(\ref{20191eq54})$ studies the
opinion evolution with antagonistic interactions characterized by an appraisal
network. We emphasize that the interacting network in $(\ref{20191eq54})$
merely quantifies whether or not there exists an information flow between a
pair of agents. Moreover, we can further extend $(\ref{20191eq54})$ to the
case where some agents may never forget their initial opinions. However, this
is beyond the scope of this paper, and is thus omitted.

Note that $(\ref{20191eq2})$ usually ensures the convergence in opinions.
However, agents in $(\ref{20191eq54})$ may converge to zero due to the
appearance of matrix $C$.

###### Theorem 5

Suppose that matrix $I-\Lambda\mathcal{L}\mathcal{D}$ has a simple eigenvalue
one. The agents in $(\ref{20191eq54})$ achieve the _stability_ if and only if
$|\lambda_{\max}(C)|<1$.

###### Proof:

It is more vulnerable to check that the system is stable if and only if the
eigenvalues in matrix $(I-\Lambda\mathcal{L}\mathcal{D})\otimes C$ are
constrained in the unit disk. Note that the eigenvalues of matrix
$(I-\Lambda\mathcal{L}\mathcal{D})\otimes C$ are
$\lambda(I-\Lambda\mathcal{L}\mathcal{D})\lambda(C)$ according to the matrix
theory. Hence, $(\ref{20191eq54})$ is stable if and only if
$|\lambda_{\max}(I-\Lambda\mathcal{L}\mathcal{D})\lambda_{\max}(C)|<1$. This
is equivalent to $|\lambda_{\max}(C)|<1$. The proof hence follows. ∎

Theorem 5 suggests that we can achieve the stability of the agents by just
imposing the restriction on matrix $C$ provided that the issue-free cases are
convergent. Note that the number of issues are drastically less than that of
the participating individuals in general. An interesting welfare from Theorem
5 in contrast to Theorems 1 and 3 is the stability of the interacting agents,
which is another motivation to introduce the issue-interdependence matrix $C$
for setup $(\ref{20191eq1})$. Indeed, only the convergence of the opinions is
generally assured for the issue-free scenario (see Theorems 1 and 3 for more
information).

As discussed before, the social networks rarely achieve unanimous behavior (we
treat the stability of the agents by a special case of the consensus).
Therefore, it is of great importance to further study the convergence
condition on $(\ref{20191eq54})$ in the presence of matrix $C$.

###### Theorem 6

Suppose that matrix $I-\Lambda\mathcal{L}\mathcal{D}$ has a simple eigenvalue
one. The agents in $(\ref{20191eq54})$ are _convergent_ if and only if
$|\lambda^{\star}_{\max}(I-\Lambda\mathcal{L}\mathcal{D})\lambda_{\max}(C)|<1$
where $\lambda^{\star}_{\max}(I-\Lambda\mathcal{L}\mathcal{D})$ stands for the
eigenvalue of $I-\Lambda\mathcal{L}\mathcal{D}$ with the second largest
magnitude compared with eigenvalue $1$.

###### Proof:

Here we prove the theorem by following the idea from $(\ref{20191eq11})$ and
$(\ref{20191eq54})$. The error system is revised as

$\displaystyle\theta(k+1)=(I-\Lambda\mathcal{L}\mathcal{D})\otimes C\theta(k)$
(22)

where
$\theta(k)\in\bigg{(}\mathbb{R}^{N}\backslash\\{\varphi\\}\bigg{)}\otimes\mathbb{R}^{n}$.
As $I-\Lambda\mathcal{L}\mathcal{D}$ has a simple eigenvalue one over the
space
$\theta(k)\in\bigg{(}\mathbb{R}^{N}\backslash\\{\varphi\\}\bigg{)}\otimes\mathbb{R}^{n}$,
the eigenvalues in matrix $I-\Lambda\mathcal{L}\mathcal{D}$ are completely
contained in the unit disk. Therefore, we can see that the error system
$(\ref{20191eq55})$ is stable if and only if
$|\lambda^{\star}_{\max}(I-\Lambda\mathcal{L}\mathcal{D})\lambda_{\max}(C)|<1$.
The proof hence follows. ∎

Based on the conclusion in Theorem 6, it does not require
$|\lambda_{\max}(C)|<1$ as Theorem 5. Hence, there is a matrix $C$ with
$|\lambda_{\max}(C)|>1$ that we could still guarantee the convergence of the
participating agents as long as
$|\lambda^{\star}_{\max}(I-\Lambda\mathcal{L}\mathcal{D})\lambda_{\max}(C)|<1$
is desirable. The following corollary gives some specific requirements on the
matrix for guaranteeing the convergence of the agents.

###### Corollary 4

Suppose that matrix $I-\Lambda\mathcal{L}\mathcal{D}$ has a simple eigenvalue
one. The agents in $(\ref{20191eq54})$ are _convergent_ only if
$\lim_{k\rightarrow\infty}C^{k}$ exists.

###### Proof:

One can see that

$\displaystyle\xi(k)=$
$\displaystyle~{}(I-\Lambda\mathcal{L}\mathcal{D})^{k}\otimes C^{k}\xi(0)$
(23)

Therefore, $(\ref{20191eq56})$ converges only if
$\lim_{k\rightarrow\infty}C^{k}$ exists. This ends the proof. ∎

Due to the antagonistic information, matrix $I-\Lambda\mathcal{L}\mathcal{D}$
may not be a nonnegative stochastic matrix in general. Consequently, the
developed methods in the framework of the multi-agent systems are no longer
applicable. By virtue of the approach developed in Subsection III-D, it is
possible to estimate matrix $C$ if $\Lambda$ and $\mathcal{D}$ are available.

Figure 4: Interacting graph of four social actors associated with
$(\ref{20191eq44})$.
Figure 5: Opinion evolution of four social actors according to the DeGroot
model where the initial opinions are randomly selected from $[-10,10]$.
Figure 6: (a)-(b) Opinion evolution with the issues independence under the
cooperative appraisal network; (c)-(d) Opinion evolution with the issues
dependence under the cooperative appraisal network.
Figure 7: (a)-(b) Opinion evolution with the issues independence under the
antagonistic appraisal network; (c)-(d) Opinion evolution with the issues
dependence under the antagonistic appraisal network.
Figure 8: (a)-(b) Stability of the agents with the cooperative appraisal
network depicted in Section V-A; (c)-(d) Stability of the agents with the
antagonistic appraisal network depicted in Section V-B.

##  V Numerical Example

Consider a social network with four individuals. The interacting matrix ($P$
corresponds to the DeGroot model) is

$\displaystyle P=\begin{bmatrix}0.22&0.12&0.36&0.3\\\
0.147&0.215&0.344&0.294\\\ 0&0&1&0\\\ 0.09&0.178&0.446&0.286\end{bmatrix}$
(24)

It should be pointed out that the elements of matrix $P$ in
$(\ref{20191eq40})$ come from real experiment (see [6]). More specific, each
entry of $P$ denotes the inter-personal influence by individual $i$ to
individual $j$. By Proposition 1, the Laplacian matrix is

$\displaystyle\mathcal{L}=\begin{bmatrix}0.78&-0.12&-0.36&-0.3\\\
-0.147&0.785&-0.344&-0.294\\\ 0&0&0&0\\\
-0.09&-0.178&-0.446&0.714\end{bmatrix}$ (25)

with $\epsilon=1$, and the associated interacting graph is presented in Fig.
4. The opinion evolution with respect to four social actors are drawn in Fig.
5.

###  V-A Cooperative Appraisal Network

In the sequel, we examine the opinion evolution of the interacting agents with
the cooperative appraisal network

$\displaystyle\mathcal{D}_{1}=\begin{bmatrix}0.2&0.2&0.3&0.3\\\
0.1&0.5&0&0.4\\\ 0.1&0.4&0&0.5\\\ 0.4&0.3&0.2&0.1\end{bmatrix}$ (26)

The opinion evolution of the individuals for the issue independence
($C_{1}=I$) is plotted in Fig. 6(a)-(b) with the susceptibility factor matrix
$\Lambda_{1}={\rm diag}(-1,1,1,-1)$, where the initial opinions are borrowed
from [11, Equation (15)],

$\displaystyle\xi(0)\in\bigg{\\{}\overbrace{25,25}^{\xi_{1}(0)},\overbrace{25,15}^{\xi_{2}(0)},\overbrace{75,-50}^{\xi_{3}(0)},\overbrace{85,5}^{\xi_{4}(0)}\bigg{\\}}$

From Fig. 6(a)-(b), the opinions of agents aggregate to the opinion of the
leader’s, i.e., the $3$rd agent.

In what follows, we discuss the opinion evolution of the interacting agents
with the issue interdependence influence. Matrix $C_{1}$ has the form as that
in [11, Section VII],

$\displaystyle C_{1}=\begin{bmatrix}0.9&0.1\\\ 0.1&0.9\end{bmatrix}$

With other parameters unchanged, the opinion evolution of agents is depicted
in Fig. 6(c)-(d). Analogous to the issues independence case, the opinions
achieve the consensus. However, some interesting phenomena arise: (1) Unlike
the issue free case, the leader’s opinion varies over time. Traditionally, the
leader’s opinion remains the same even if the JF model (including the issue
interdependence version [11]) is applied; (2) By introducing the issue
dependence matrix $C_{1}$, the final aggregated opinion may be steered to an
opposite direction of the leader’s initial opinion. In one word, cooperative
appraisal network leads to consensus in opinions, which coincides with the
conclusion drawn in Theorem 1.

###  V-B Antagonistic Appraisal Network

This subsection focuses on the opinion evolution with antagonistic appraisal
network, which is of the form

$\displaystyle\mathcal{D}_{2}=\begin{bmatrix}0.2&-0.2&-0.3&-0.3\\\
0.1&0.5&0&0.4\\\ -0.1&0.4&0&0.5\\\ 0.4&0.3&-0.2&0.1\end{bmatrix}$

Similar to Subsection V-A, we first consider the issue free case. The agents’
opinion evolutions are shown in Fig. 7(a)-(b) where the parameters are the
same except for $\mathcal{D}_{2}$ and $\Lambda_{2}={\rm diag}(-1.5,2,1,-0.5)$.
Using the method in [11], the agents’ final opinions share the same direction
with the leader’s, see [11, Fig. 5(a)]. However, the second agent’s opinion
has an opposite sign with the leader’s, even if their initial opinions have
the same direction (see Fig. 7(a)-(b)).

We proceed with the issue interdependence case with

$\displaystyle C_{2}=\begin{bmatrix}0.6&0.4\\\ 0.3&0.7\end{bmatrix}$ (27)

Using the same parameters as before, the opinions of the agents are depicted
in Fig. 7(c)-(d). It is clear that the leader’s opinion changes along with the
time-evolution, as opposed to the issue free case. In addition, we can see
that although some agents have the same direction on the initial opinion of
the leader’s at the beginning, the agents’ final opinions appear an opposite
direction with the leader’s. Furthermore, the opinions tend to clusters in
such a case, which is in accordance with Theorem 3. One can check that $C_{2}$
in $(\ref{20191eq59})$) does not satisfy the conditions in Theorem 5. Indeed,
we compute the eigenvalues of matrix $C_{2}$, i.e.,
$\lambda(C_{2})\in\\{1,0.3\\}$. They, however, meet the requirement in Theorem
6.

###  V-C Stability of Interacting Agents

Now we are dedicated to studying the stability of agents with cooperative and
antagonistic appraisal networks. For the cooperative appraisal network, we
perform the simulation using the parameters in Subsection V-A by replacing
matrix $C_{1}$ with $C^{\star}_{1}$

$\displaystyle C^{\star}_{1}=0.85\begin{bmatrix}0.9&0.1\\\
0.1&0.9\end{bmatrix}$

It can be verified that the requirement in Theorem 5 is fulfilled. The
opinions of the agents are depicted in Fig. 8(a)-(b). For the case of
antagonistic appraisal network, we use the same parameters as Subsection V-B
by replacing matrix $C_{2}$ with $C^{\star}_{2}$

$\displaystyle C^{\star}_{2}=0.95\begin{bmatrix}0.6&0.4\\\
0.3&0.7\end{bmatrix}$

The opinion evolution of the agents can be found in Fig. 8(c)-(d).

###  V-D Further Discussion

With the framework of $(\ref{20191eq1})$, we can achieve both consensus and
clusters in opinions, and the consensus in opinions is not necessarily
restricted into a convex hull as the classical DeGroot model (see Fig. 3(a)).
Moreover, formulation $(\ref{20191eq1})$ extends the protocol of [27] in
several aspects: (i) it could be utilized to describe more general behaviors
in social networks; (ii) it further manifests the importance on the weighted
gain matrix $\Lambda$. In other words, system $(\ref{20191eq2})$ may diverge
without the help of $\Lambda$.

In [11, Equation (15)], Parsegov _et al_. endowed both the initial opinions
and the final opinions of the social actors with some specific meanings: the
positive (resp. the negative) opinions correspond to the vegetarian (resp. the
all-meat) diets by introducing an interdependent issue matrix. According to
[11, Examples $3$ and $4$], all social actors are the vegetarian which is
coincide with their initial opinions, $\xi^{1}(0)=(25,25,75,85)$ (cf. [11,
Equation (15)]), i.e., the initial opinions on the first issue. However, we
can see from Fig. 6, some agents become the all-meat diets eventually, even if
they are the vegetarian at the beginning. This also indicates that some agents
have the opposite attitudes compared with the leader’s ever though they all
have the same direction of attitudes at first. Additionally, from Figs.
6(c)-(d) and 7(c)-(d), the leader’s final opinion may be affected by the
evolutions of other agents, which is new from the perspective of the classical
DeGroot model. In a nutshell, the proposed setup in this paper brings some
interesting phenomena comparing with the existing literature.

##  VI Conclusion

This paper has studied the opinion dynamics in social networks by introducing
an appraisal network to quantify the cooperative or antagonistic information.
We have shown that the cooperative appraisal network achieves the consensus in
opinions while the antagonistic appraisal network leads to the opinion
clusters. The tool of random convex optimization is used to estimate the
appraisal network with a confident level of robustness, along with the lower
bound on the amounts of sampled observations. Moreover, the proposed setup has
been extended to the case of multiple issues interdependence. Some discussions
have also been given to compare with the existing literature.

[Proof of Theorem 2] To prove Theorem 2, the Hermite-Biehler Theorem (cf.
[40]) and the Bilinear Transformation Theorem (cf. [41]) are needed. To begin
with, we first introduce a lemma.

###### Lemma 2

([41]) Given two polynomials $\mathbb{S}(z)$ of degree $d$ and $\mathbb{Q}(z)$
with

$\displaystyle\mathbb{Q}(z)=~{}(z-1)^{d}\mathbb{S}\bigg{(}\frac{z+1}{z-1}\bigg{)}$

Then the Schur stability of $\mathbb{S}(z)$ implies the Hurwitz stability on
$\mathbb{Q}(z)$, and vice versa.

For complex polynomial $\mathbb{Q}(z)$, replacing $z$ with $\mathbbm{i}w$
yields

$\displaystyle\mathbb{Q}(\mathbbm{i}w)=~{}S(w)+\mathbbm{i}Q(w)$

where both $S(w)$ and $Q(w)$ are real polynomials. A relationship among the
roots of $S(w)$ and $Q(w)$ is depicted as follows.

###### Definition 3

([40]) For any real polynomials $S(w)$ and $Q(w)$, they are interlaced if

  * (i)

The roots of $S(w)$ (denoted by $\\{S_{1},...,s_{\ell_{S}}\\}$) and $Q(w)$
(denoted by $\\{Q_{1},...,Q_{\ell_{Q}}\\}$) satisfy

$\left\\{\begin{aligned} &S_{1}<S_{2}<\cdots<S_{\ell_{S}}\\\
&Q_{1}<Q_{2}<\cdots<Q_{\ell_{Q}}\end{aligned}\right.$

  * (ii)

$|\ell_{S}-\ell_{Q}|\leq 1$, and one of the following facts holds

$\left\\{\begin{aligned}
&S_{1}<Q_{1}<S_{2}<Q_{2}\cdots<Q_{\ell_{Q}}<S_{\ell_{S}},~{}\ell_{S}=\ell_{Q}+1\\\
&Q_{1}<S_{1}<Q_{2}<S_{2}\cdots<S_{\ell_{S}}<Q_{\ell_{Q}},~{}\ell_{Q}=\ell_{S}+1\\\
&Q_{1}<S_{1}<Q_{2}<S_{2}\cdots<Q_{\ell_{Q}}<S_{\ell_{S}},~{}\ell_{Q}=\ell_{S}\\\
&S_{1}<Q_{1}<S_{2}<Q_{2}\cdots<S_{\ell_{S}}<Q_{\ell_{Q}},~{}\ell_{S}=\ell_{Q}\\\
\end{aligned}\right.$

The next lemma develops a judgement on the Hurwitz stability associated with
$\mathbb{Q}(z)$ that is closely linked to the roots of polynomials $S(w)$ and
$Q(w)$ and the interlaced property defined in Definition 3.

###### Lemma 3

([40]) Polynomial $\mathbb{Q}(z)$ is Hurwitz stable if and only if

  * (i)

Polynomials $S(w)$ and $Q(w)$ are interlaced;

  * (ii)

Polynomials $S(w)$, $\frac{\partial Q(w)}{\partial w}$, $\frac{\partial
S(w)}{\partial w}$ and $Q(w)$ at the origin fulfill

$\displaystyle S(0)\frac{\partial Q(0)}{\partial w}-\frac{\partial
S(0)}{\partial w}Q(0)>0$

With the above preparations, we are about to give the proof of Theorem 2.

###### Proof:

It is easy to see that the updating dynamics of the individuals is determined
by $(\ref{20191eq64})$. Accordingly, the characteristic equation of
$(\ref{20191eq64})$ can be written as

$\displaystyle{\rm
det}\bigg{(}zI-(I-\varrho\mathcal{L}+\varrho^{2}\mathcal{L}^{2})\bigg{)}$
$\displaystyle=$
$\displaystyle~{}(z-1)\prod^{N}_{i=2}(z-1+\varrho\lambda_{i}-\varrho^{2}\lambda^{2}_{i})$

Before proceeding further, it is indispensable to consider two cases where
${\rm Im}(\lambda_{i})$ is equal to zero or not. And we start with the
scenario that ${\rm Im}(\lambda_{i})=0$.

As argued before, consensus in opinions indicates that $|z|<1$ if $z\neq 1$.
In such a setting, it is enough to demonstrate that

$\displaystyle\varrho^{2}\lambda^{2}_{i}-\varrho\lambda_{i}+1<1$ (28a)
$\displaystyle\varrho^{2}\lambda^{2}_{i}-\varrho\lambda_{i}+1>-1$ (28b)

with the constraints $\varrho>0$ and $\lambda_{i}\neq 0$. By computation, one
derives that the feasible region on $\varrho$ for inequality
$(\ref{20191eq79a})$ is $(0,\frac{1}{\lambda_{i}})$, while the feasible region
on $\varrho$ associated with inequality $(\ref{20191eq79b})$ is $[0,\infty)$.
Thus, the feasible region for $\varrho$ is

$\displaystyle\varrho\in\bigg{(}0,\frac{1}{\lambda_{i}}\bigg{)}$

In the sequel, we are dedicated to the case of ${\rm Im}(\lambda_{i})\neq 0$.
Let $\mathbb{S}_{i}(z)$ be of the form

$\displaystyle\mathbb{S}_{i}(z)=~{}z-1+\varrho\lambda_{i}-\varrho^{2}\lambda^{2}_{i},~{}i=2,...,N$
(29)

where $\lambda_{i}$ stands for the $i$th eigenvalue of $\mathcal{L}$. Applying
the bilinear transformation

$\displaystyle\mathbb{Q}_{i}(z)=~{}(z-1)\mathbb{S}_{i}\bigg{(}\frac{z+1}{z-1}\bigg{)}$
(30)

One gets

$\displaystyle\mathbb{Q}_{i}(z)=$
$\displaystyle~{}2+\varrho\lambda_{i}(1-\varrho\lambda_{i})z+\varrho\lambda_{i}(\varrho\lambda_{i}-1)$
$\displaystyle=$
$\displaystyle~{}2+\varrho|\lambda_{i}|\bigg{(}\cos(\arg(\lambda_{i}))+\mathbbm{i}\sin(\arg(\lambda_{i}))\bigg{)}z$
$\displaystyle-\varrho|\lambda_{i}|\bigg{(}\cos(\arg(\lambda_{i}))+\mathbbm{i}\sin(\arg(\lambda_{i}))\bigg{)}$
$\displaystyle-\varrho^{2}|\lambda_{i}|^{2}\bigg{(}\cos^{2}(\arg(\lambda_{i}))-\sin^{2}(\arg(\lambda_{i}))$
$\displaystyle+2\mathbbm{i}\sin(\arg(\lambda_{i}))\cos(\arg(\lambda_{i}))\bigg{)}z$
$\displaystyle-\varrho^{2}|\lambda_{i}|^{2}\bigg{(}\cos^{2}(\arg(\lambda_{i}))-\sin^{2}(\arg(\lambda_{i}))$
$\displaystyle+2\mathbbm{i}\sin(\arg(\lambda_{i}))\cos(\arg(\lambda_{i}))\bigg{)}$

Substituting $z=\mathbbm{i}w$ into $\mathbb{Q}_{i}(z)$ results in

$\displaystyle\mathbb{Q}_{i}(\mathbbm{i}w)=$
$\displaystyle~{}~{}2+\varrho|\lambda_{i}|\bigg{(}\mathbbm{i}\cos(\arg(\lambda_{i}))-\sin(\arg(\lambda_{i}))\bigg{)}w$
(31)
$\displaystyle-\varrho|\lambda_{i}|\bigg{(}\cos(\arg(\lambda_{i}))+\mathbbm{i}\sin(\arg(\lambda_{i}))\bigg{)}$
$\displaystyle-\varrho^{2}|\lambda_{i}|^{2}\bigg{(}\mathbbm{i}\cos^{2}(\arg(\lambda_{i}))-\mathbbm{i}\sin^{2}(\arg(\lambda_{i}))$
$\displaystyle-2\sin(\arg(\lambda_{i}))\cos(\arg(\lambda_{i}))\bigg{)}w$
$\displaystyle-\varrho^{2}|\lambda_{i}|^{2}\bigg{(}\cos^{2}(\arg(\lambda_{i}))-\sin^{2}(\arg(\lambda_{i}))$
$\displaystyle+2\mathbbm{i}\sin(\arg(\lambda_{i}))\cos(\arg(\lambda_{i}))\bigg{)}$
$\displaystyle\triangleq$ $\displaystyle~{}{\rm
Re}(\mathbb{Q}_{i}(w))+\mathbbm{i}{\rm Im}(\mathbb{Q}_{i}(w))$

With the constraint ${\rm Re}(\mathbb{Q}_{i}(w))=0$, it yields

$\displaystyle w_{1}$ $\displaystyle=$
$\displaystyle~{}\frac{2+\varrho^{2}|\lambda_{i}|^{2}(2\cos^{2}(\arg(\lambda_{i}))-1)-\varrho|\lambda_{i}|\cos(\arg(\lambda_{i}))}{\varrho|\lambda_{i}|\sin(\arg(\lambda_{i}))-2\varrho^{2}|\lambda_{i}|^{2}\sin(\arg(\lambda_{i}))\cos(\arg(\lambda_{i}))}$
$\displaystyle\triangleq$ $\displaystyle~{}\frac{2+\mathbbm{x}}{\mathbbm{y}}$

Analogously, the requirement of ${\rm Im}(\mathbb{Q}_{i}(w))=0$ immediately
leads to

$\displaystyle w_{2}$ $\displaystyle=$
$\displaystyle~{}\frac{2\varrho^{2}|\lambda_{i}|^{2}\sin(\arg(\lambda_{i}))\cos(\arg(\lambda_{i}))-\varrho|\lambda_{i}|\sin(\arg(\lambda_{i}))}{\varrho^{2}|\lambda_{i}|^{2}(2\cos^{2}(\arg(\lambda_{i}))-1)-\varrho|\lambda_{i}|\cos(\arg(\lambda_{i}))}$

In accordance with $w_{1}$, $w_{2}$ hence is of the form

$\displaystyle w_{2}=~{}\frac{-\mathbbm{y}}{\mathbbm{x}}$

We now determine the condition guaranteeing

$\displaystyle w_{1}\frac{1}{w_{2}}\neq 1$

For ease of discussion, we resort to the opposite statement, i.e.,

$\displaystyle 1=$ $\displaystyle~{}w_{1}\frac{1}{w_{2}}$ (32)
$\displaystyle=$
$\displaystyle~{}\frac{2\mathbbm{x}+\mathbbm{x}^{2}}{-\mathbbm{y}^{2}}$

We can see that $(\ref{20191eq84})$ is equivalent to

$\displaystyle 1=~{}(\mathbbm{x}+1)^{2}+\mathbbm{y}^{2}$

By the polar coordination, for $\theta\in[0,2\pi)$, it attains

$\displaystyle\mathbbm{x}=$ $\displaystyle~{}\cos(\theta)-1$ (33a)
$\displaystyle\mathbbm{y}=$ $\displaystyle~{}\sin(\theta)$ (33b)

For $(\ref{20191eq85a})$, it is more prone to access that

$\displaystyle\mathbbm{x}-\cos(\theta)+1=$
$\displaystyle~{}\varrho^{2}|\lambda_{i}|^{2}\cos(2\arg(\lambda_{i}))$
$\displaystyle-\varrho|\lambda_{i}|\cos(\arg(\lambda_{i}))-\cos(\theta)+1$
$\displaystyle\triangleq$ $\displaystyle~{}f_{\theta}(\varrho)$

For $\forall~{}\theta\in[0,2\pi)$, denote

$\displaystyle\Delta_{\mathbbm{x}}\triangleq$
$\displaystyle~{}|\lambda_{i}|^{2}+4|\lambda_{i}|^{2}\cos(2\arg(\lambda_{i}))(\cos(\theta)-1)$
$\displaystyle=$
$\displaystyle~{}|\lambda_{i}|^{2}\bigg{(}\cos^{2}(\arg(\lambda_{i}))(8\cos(\theta)-7)+4(1-\cos(\theta))\bigg{)}$

In the sequel, two cases involving the real roots of $f_{\theta}(\varrho)=0$
are formulated.

Case 1) $\theta\in[-\arccos(\frac{7}{8}),\arccos(\frac{7}{8})]$,
$\Delta_{\mathbbm{x}}\geq 0$ follows directly.

Case 2)
$\theta\in(-\frac{\pi}{2},-\arccos(\frac{7}{8}))\bigcup(\arccos(\frac{7}{8}),\frac{\pi}{2})$.
In such a circumstance, one has $\Delta_{\mathbbm{x}}\geq 0$ if and only if
$\arg(\lambda_{i})$ is preserved in the set

$\displaystyle\bigg{[}\arccos(\sqrt{\frac{4(1-\cos(\theta))}{7-8\cos(\theta)}},\frac{\pi}{2})\bigg{)}$
$\displaystyle\bigcup$
$\displaystyle\bigg{(}-\frac{\pi}{2},-\arccos(\sqrt{\frac{4(1-\cos(\theta))}{7-8\cos(\theta)}})\bigg{]}$

The two real roots of $f_{\theta}(\varrho)=0$ are given by

$\left\\{\begin{aligned}
\varrho_{i,1}=&~{}\frac{|\lambda_{i}|\cos(\arg(\lambda_{i}))+\sqrt{\Delta_{\mathbbm{x}}}}{2|\lambda_{i}|^{2}\cos(2\arg(\lambda_{i}))}\\\
\varrho_{i,2}=&~{}\frac{|\lambda_{i}|\cos(\arg(\lambda_{i}))-\sqrt{\Delta_{\mathbbm{x}}}}{2|\lambda_{i}|^{2}\cos(2\arg(\lambda_{i}))}\\\
\end{aligned}\right.$

Moreover, we calculate

$\displaystyle|\lambda_{i}|^{2}\cos^{2}(\arg(\lambda_{i}))-\Delta_{\mathbbm{x}}=4|\lambda_{i}|^{2}(1-\cos(\theta))\cos(2\arg(\lambda_{i}))$

Therefore, the sign of the calculation value is entirely contingent on
$\cos(2\arg(\lambda_{i}))$, which in turn indicates that both $\varrho_{1}$
and $\varrho_{2}$ are positive.

For $(\ref{20191eq85b})$, we find

$\displaystyle\sin(\theta)-\mathbbm{y}=$
$\displaystyle~{}2\varrho^{2}|\lambda_{i}|^{2}\sin(2\arg(\lambda_{i}))-\varrho|\lambda_{i}|\sin(\arg(\lambda_{i})$
$\displaystyle+\sin(\theta)$ $\displaystyle\triangleq$
$\displaystyle~{}g_{\theta}(\varrho),~{}\theta\in[0,2\pi)$

For $\forall~{}\theta\in[0,2\pi)$, denote

$\displaystyle\Delta_{\mathbbm{y}}\triangleq$
$\displaystyle~{}|\lambda_{i}|^{2}\bigg{(}\sin^{2}(\arg(\lambda_{i}))-4\sin(2\arg(\lambda_{i}))\sin(\theta)\bigg{)}$

Evidently, for $\theta\in[0,2\pi)$, $\Delta_{\mathbbm{y}}\geq 0$ implies

$\displaystyle\sin^{2}(\arg(\lambda_{i}))\geq
8\sin(\arg(\lambda_{i}))\cos(\arg(\lambda_{i}))\sin(\theta)$ (34)

In a similar manner, two scenarios should be argued.

Case i) $\arg(\lambda_{i})\in(0,\frac{\pi}{2})$. In this case,
$(\ref{20191eq91})$ is desirable provided that

$\displaystyle\arg(\lambda_{i})\in\bigg{[}\max\\{0,\arctan(8\sin(\theta))\\},\frac{\pi}{2}\bigg{)}\backslash\\{0\\},~{}\theta\in[0,2\pi)$

Case ii) $\arg(\lambda_{i})\in(-\frac{\pi}{2},0)$. It is trivial that

$\displaystyle 0\leq\Delta_{\mathbbm{y}},~{}\forall~{}\theta\in[0,\pi]$

And for $\theta\in[-\pi,0]$, $(\ref{20191eq91})$ could be further expressed by

$\displaystyle\sin^{2}(\arg(\lambda_{i}))\geq
8|\sin(\arg(\lambda_{i}))|\cos(\arg(\lambda_{i}))|\sin(\theta)|$

which is true as long as

$\displaystyle\arg(\lambda_{i})\in\bigg{[}-\frac{\pi}{2},\min\\{0,-\arctan(8\sin(\theta))\\}\bigg{)}\backslash\\{0\\}$

With the foregoing arguments, $g_{\theta}(\varrho)=0$ has two real roots,
which can be described as

$\left\\{\begin{aligned}
\varrho_{i,3}=&~{}\frac{|\lambda_{i}|\sin(\arg(\lambda_{i}))+\sqrt{\Delta_{\mathbbm{y}}}}{2|\lambda_{i}|^{2}\sin(2\arg(\lambda_{i}))}\\\
\varrho_{i,4}=&~{}\frac{|\lambda_{i}|\sin(\arg(\lambda_{i}))-\sqrt{\Delta_{\mathbbm{y}}}}{2|\lambda_{i}|^{2}\sin(2\arg(\lambda_{i}))}\\\
\end{aligned}\right.$

Therefore, $w_{1}$ and $w_{2}$ are interlaced (cf. Definition 3) with the
constraints on $\varrho$, i.e.,

$\displaystyle\varrho\not\in\\{\varrho_{i,1},\varrho_{i,2}\\}\bigcup\\{\varrho_{i,3},\varrho_{i,4}\\}$

By virtue of the specifications on ${\rm Re}(\mathbb{Q}_{i}(w))$ and ${\rm
Im}(\mathbb{Q}_{i}(w))$ (cf. $(\ref{20191eq78})$), one gets

$\left\\{\begin{aligned} \frac{\partial{\rm Re}(\mathbb{Q}_{i}(0))}{\partial
w}=&~{}\varrho^{2}|\lambda_{i}|^{2}\sin(2\arg(\lambda_{i}))-\varrho|\lambda_{i}|\sin(\arg(\lambda_{i}))\\\
{\rm
Re}(\mathbb{Q}_{i}(0))=&~{}2+\varrho^{2}|\lambda_{i}|^{2}(2\cos^{2}(\arg(\lambda_{i}))-1)\\\
&-\varrho|\lambda_{i}|\cos(\arg(\lambda_{i}))\\\ \frac{\partial{\rm
Im}(\mathbb{Q}_{i}(0))}{\partial
w}=&~{}\varrho^{2}|\lambda_{i}|^{2}(1-2\cos^{2}(\arg(\lambda_{i})))\\\
&+\varrho|\lambda_{i}|\cos(\arg(\lambda_{i}))\\\ {\rm
Im}(\mathbb{Q}_{i}(0))=&~{}\varrho^{2}|\lambda_{i}|^{2}\sin(2\arg(\lambda_{i}))-\varrho|\lambda_{i}|\sin(\arg(\lambda_{i}))\end{aligned}\right.$

We perform the calculation

$\displaystyle{\rm Re}(\mathbb{Q}_{i}(0))\frac{\partial{\rm
Im}(\mathbb{Q}_{i}(0))}{\partial w}-\frac{\partial{\rm
Re}(\mathbb{Q}_{i}(0))}{\partial w}{\rm Re}(\mathbb{Q}_{i}(0))$
$\displaystyle=$
$\displaystyle~{}\varrho|\lambda_{i}|\bigg{(}-\varrho^{3}|\lambda_{i}|^{3}+\varrho^{2}|\lambda_{i}|^{2}\cos^{2}(\arg(\lambda_{i}))$
$\displaystyle-2\varrho|\lambda_{i}|\sin(2\arg(\lambda_{i}))-\varrho|\lambda_{i}|+2\cos(\arg(\lambda_{i}))\bigg{)}$
$\displaystyle\triangleq$
$\displaystyle~{}\varrho|\lambda_{i}|f_{i}(\varrho,\lambda_{i},\arg(\lambda_{i})),~{}i=2,...,N$

Obviously, ${\rm Re}(\mathbb{Q}_{i}(0))\frac{\partial{\rm
Im}(\mathbb{Q}_{i}(0))}{\partial w}-\frac{\partial{\rm
Re}(\mathbb{Q}_{i}(0))}{\partial w}{\rm Re}(\mathbb{Q}_{i}(0))$ is positive if
and only if $f_{i}(\varrho,\lambda_{i},\arg(\lambda_{i}))>0$.

Consequently, by Lemma 3, we can see that $\mathbb{Q}_{i}(z)$ in
$(\ref{20191eq76})$ is Hurwitz stable. With the aid of Lemma 2, one could
state that $\mathbb{S}_{i}(z)$ in $(\ref{20191eq75})$ is Schur stable. This
ends the proof. ∎

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