# Unique Nash equilibrium of a nonlinear model of opinion dynamics on networks
with friction-inspired stubbornness

David N. Reynolds<EMAIL_ADDRESS>
School of Mathematics,
Gran Sasso Science Institute
67100 L’Aquila, Italy Francesco Tudisco<EMAIL_ADDRESS>
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh, EH93FD Edinburgh, UK
Gran Sasso Science Institute, 67100 L’Aquila, Italy

###### Abstract

The modeling of opinion dynamics has seen much study in varying academic
disciplines. Understanding the complex ways information can be disseminated is
a complicated problem for mathematicians as well as social scientists. We
present a nonlinear model of opinion dynamics that utilizes an environmental
averaging protocol similar to the DeGroot and Freidkin-Johnsen models. Indeed,
the way opinions evolve is complex and nonlinear effects ought to be
considered when modelling. For this model, the nonlinearity destroys the
translation invariance of the equations, as well as the convexity of the
associated payout functions. The standard theory for well-posedness and
convergence no longer applies and we must utilize the Brouwer topological
degree and nonconvex analysis in order to achieve these results. Numerical
simulations of the model reveal that the nonlinearity behaves similarly to the
well-known Friedkin-Johnsen for so-called “reasonable” opinions, but better
models the way agents that hold “extreme” opinions are more stubborn than
their reasonable counterparts.

#### Keywords

Graphs, networks, continuous-valued opinion dynamics, consensus, compromise,
stubbornness

#### AMS subject classification

91D30, 05C57, 05C50, 34A34, 34D05

##  1 Introduction

Humanity constantly exchanges information amongst itself; from watching the
news and reading articles online, to speaking with acquaintances or simply
overhearing a conversation at a bar. The means by which information is
transmitted are innumerable. From all this input and our inherent
predispositions, we are constantly forming and adjusting our opinions about
any number of topics. With this comes a desire to understand and model the way
opinions change. In recent years there has been an increase in interest in
this exact topic in many academic areas, but in particular, there is much
mathematics being done to rigorously describe and predict how opinions develop
within a group of agents.

Networks represented as undirected graphs
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $N$ nodes $\\{1,\dots,N\\}$ and
undirected edges $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ are one of
the most successful models for opinion dynamics. A wealth of real-world social
and natural interaction scenarios are very well represented via a network
where each agent is represented by a vertex $i\in\mathcal{V}$ and two agents
$i,j\in\mathcal{V}$ exchanging information directly are shown via the edge
$ij\in\mathcal{E}$ that connects them Friedkin & Johnsen (2011); Newman
(2018). Within this framework, many models have been created to describe
certain phenomena like consensus, disagreement, spread of diseases, and even
biological models of flocking dynamics Cucker & Smale (2007a; b); DeGroot
(1974); Friedkin & Johnsen (1990); Hegselmann & Krause (2002); Jia et al.
(2015); Lear et al. (2021). In the context of opinion dynamics, the majority
of the models build on a wealth of empirical evidence which shows that
individuals update their opinions as combinations of their own and others’
opinions, based on the strength of the reciprocal interpersonal ties Altafini
(2012); DeGroot (1974); French Jr (1956); Harary (1959; 1962). This is
represented in the celebrated DeGroot model by means of a linear dynamical
system where the opinion $x_{i}$ of node $i$ evolves as a function of its
neighboring nodes in the graph:
$x_{i}(k+1)=\sum_{j:ij\in\mathcal{E}}x_{j}(k)$. For any aperiodic graph, this
model always reaches consensus, in that all the agents (the nodes) will
eventually share the same opinion. In fact, it is straightforward to show that
in that case a unique limit $\lim_{k\to\infty}x_{i}(k)=x_{i}^{*}$ exists and
$x_{i}^{*}=x_{j}^{*}$ for all $i,j$. The Friedkin-Johnsen model further adds a
linear term to model stubbornness, i.e. the propensity (or resiliency) of each
individual to change their own initial convictions. With this addition,
consensus may not happen, however, due to the linearity (and thus the
translation invariance) of this model, the opinions of the network tend to
become “similar” even when the initial convictions and levels of stubbornness
are very different. In order to better model opinion dynamics it is key to be
able to incorporate nonlinear effects into the model.

In Xia et al. (2020) the authors treat a nonlinear version of the Degroot
model taking into account the psychological phenomena of confirmation bias.
Further, Yu et al. (2011) investigates incorporating a nonlinear force into
the continuous-time Degroot model. In both Xia et al. (2020), and Yu et al.
(2011), the asymptotic state is, in general, consensus. We introduce a
nonlinear variation of the Friedkin-Johnsen model, which better models the
desire of the agents to maintain their own convictions rather than moving
towards a global consensus, i.e. nonlinear stubbornness. The particular
nonlinear term is inspired by the Rayleigh’s self-propulsion and friction
force. Indeed, this forcing term has seen study in the context of flocking and
swarming models of collective dynamics Lear et al. (2021). Understanding the
way this individual forcing term interacts with the consensus operator is also
directly connected to the behaviour of synchronous systems like the Stuart-
Landau oscillators Panteley et al. (2015). Our formulation in terms of a
differential opinion game allows us to prove the existence of a unique Nash
equilibrium to which the system converges along a gradient flow, Li et al.
(2015); Simon (1983). Moreover, it can be applied to previous linear models as
well and allows us to revisit their convergence analysis in terms of Nash
equilibria.

The rest of the paper is structured as follows. First, in Section 1.1 we
introduce our notation and preliminary considerations. Then, the remainder of
this section is devoted to overviewing relevant background, in particular the
Friedkin-Johnsen model, and introducing the nonlinear generalization we
consider in this work. Section 2 is a standard analysis of the continuous
DeGroot model as a warm-up. Section 3 will be dedicated to proving the main
result of the paper, Theorem 1.2. In Section 4 we will apply the technique
developed in the preceding section to the continuous version of the Friedkin-
Johnsen model. Finally, in Section 5 we will present a few numerical examples
to show the different behavior the nonlinear opinion dynamical term
introduces.

###  1.1 Preliminaries

In this paper, each model will represent how the opinions of a group of
agents, on a particular topic, evolve with time. Therefore we will be
considering opinion values $x_{i}(k),x_{i}(t)\in\mathbb{R}_{+}$ where $k$
denotes discrete time steps, $t$ the continuous time evolution, and $i$ the
specific agent. For interpretation as an opinion model we consider small
values close to $0$ representing disagreement about a particular topic, and as
values grow larger the more the agent express agreement, where for large
values $x_{i}\gg 1$, the held opinion would be considered “extreme”.
Similarly, each agent will be allowed a desired or initial opinion value
$u_{i}$, which from now on will be referred to as convictions and will also
reside in the positive reals $u_{i}\in\mathbb{R}_{+}$. The graph $\mathcal{G}$
of the network will be connected and undirected unless otherwise stated, with
node set $\\{1,\dots,N\\}$ representing the group of agents being modeled.
Finally, while the models discussed in the introduction will be based on the
normalized adjacency matrix $A$ as defined in (2), this choice of linear
mapping is not a necessity for the main analysis and in later sections we will
broaden the scope to any symmetric entrywise nonnegative $N\times N$ matrix
$M$. We will specify $M$ in specific application contexts.

###  1.2 DeGroot and Friedkin-Johnsen models

Opinions can be tracked as binary values (agreement or disagreement); as in
the famous Voter Model of Holley and Liggett Holley & Liggett (1975) or
individual words as in the Naming Game seen in Marchetti et al. (2020); Xie et
al. (2011); or they can be continuous values where a larger value could
correspond with agreement and lower with disagreement about a particular
topic, Bizyaeva et al. (2020); Brooks et al. (2022); DeGroot (1974); French Jr
(1956); Friedkin & Johnsen (1990); Hegselmann & Krause (2002); Lear et al.
(2021). The method by which the dynamics themselves occur can be described
through Bayesian Marchetti et al. (2020) or Non-Bayesian processes DeGroot
(1974); Friedkin & Johnsen (1990). Further, there are Bounded Confidence
models that take into account with which other agents interactions can occur
(e.g. only those with similar opinions Deffuant et al. (2000); Hegselmann &
Krause (2002)), and models that incorporate the stubbornness of an agent,
leading to disagreement, or compromises instead of consensus Friedkin &
Johnsen (1990); Lear et al. (2021); Proskurnikov et al. (2017); Xu et al.
(2022). See Proskurnikov & Tempo (2017; 2018) for an overview of the various
mathematical models that have been developed in the field of opinion dynamics.

The early work of French and Harary French Jr (1956); Harary (1959) on
reaching consensus via communication on a graph, set the stage for what is now
known as DeGroot learning DeGroot (1974); a Non-Bayesian process by which a
consensus is achieved via updating beliefs at each time step, based off of
interactions with others in the network. Indeed, the standard model for
achieving consensus is for each agent $i$ in a network to update its current
beliefs $x_{i}(k)$ at time step $k$ via an environmental averaging over the
beliefs of the agents with which it shares a connection:

$x_{i}(k+1)=\frac{1}{\deg(i)}\sum_{j:ij\in\mathcal{E}}x_{j}(k)=\sum_{j=1}^{N}A_{ij}x_{j}(k),$
(1)

where $\deg(i)=|\\{j:ij\in\mathcal{E}\\}|$ is the degree of node $i$, and
$A_{ij}$ are the elements of the row-stochastic “normalized” adjacency matrix

$A_{ij}=\begin{cases}\deg(i)^{-1}&i\sim j\\\ 0&\text{otherwise}\end{cases}\,.$
(2)

Indeed, this environmental averaging protocol is also seen in biological
flocking dynamics Cucker & Smale (2007a; b); Lear et al. (2021); Shvydkoy
(2021). This protocol can be derived from each agent attempting to minimize
the payout function

$p_{i}({\boldsymbol{x}})=\frac{1}{2}\sum_{j=1}^{N}A_{ij}(x_{i}-x_{j})^{2}.$
(3)

It is well known that, so long as the adjacency matrix is not periodic, this
model produces convergence of opinions to a consensus value which is a convex
combination of the initial opinion values.

The Friedkin-Johnsen model Friedkin & Johnsen (1990); Ghaderi & Srikant
(2014); Proskurnikov et al. (2017) uses this mechanic to spread information
between agents, but further includes a stubbornness factor $\lambda_{i}$ as a
means of “anchoring” opinions to a particular fixed value,

$x_{i}(k+1)=\lambda_{i}\sum_{j=1}^{N}A_{ij}x_{j}(k)+(1-\lambda_{i})u_{i},\ \
\lambda_{i}\in[0,1].$ (4)

In this way the opinions $x_{i}$ are torn between the desired consensus value
and its own fixed preferred value $u_{i}$, often taken to be the initial
opinion value $x_{i}(0)$. Each step is taken from a weighted average of these
two values that can similarly be chosen via minimizing the payout function

$p_{i}({\boldsymbol{x}})=\frac{\lambda_{i}}{2}\sum_{j=1}^{N}A_{ij}(x_{i}-x_{j})^{2}+\frac{1-\lambda_{i}}{2}(x_{i}-u_{i})^{2}.$
(5)

If all $\lambda_{i}=1$, then none of the agents are stubborn and we return to
consensus as in (1). Alternatively, if $\lambda_{i}=0$ for all $i$, then that
agent is fully stubborn and its opinion will be exactly $x_{i}=u_{i}$,
regardless of any other opinion values. Finally, for an undirected connected
graph $\mathcal{G}$ with at least one $0<\lambda_{i}<1$, it is well known that
the Perron-Frobenius theory entails this model has a unique equilibrium vector
${\boldsymbol{x}}^{*}$ to which the opinion values will eventually converge.
In general, this equilibrium will not represent consensus, but rather what we
call a compromise. Note that, due to the linearity of the system, the
stability of the fixed point can be immediately seen as the linearization of
the system is represented by a stable diagonal shift of the graph Laplacian.

###  1.3 A nonlinear generalization of the FJ model with friction inspired
stubbornness

Note that we can rewrite the FJ protocol (4) by letting
$y_{i}=\frac{x_{i}}{\lambda_{i}}$ and
$\sigma_{i}=\frac{1-\lambda_{i}}{\lambda_{i}}$ as

$y_{i}(k+1)=\sum_{j=1}^{N}B_{ij}y_{j}(k)+\sigma_{i}u_{i},\ \
\sigma_{i}\in[0,\infty],$ (6)

where $B_{ij}=\lambda_{j}A_{ij}$ is a substochastic matrix. This form leads
nicely to the protocol we introduce here.

The new protocol we introduce is similar to the FJ model, except it introduces
a nonlinearity into the stubbornness term which substantially complicates the
analysis of the model. This protocol has been seen in the Cucker-Smale system
of flocking dynamics, with a Rayleigh friction and self-propulsion forcing
term driving the dynamics Lear et al. (2021); Nguyen & Shvydkoy (2022).
However, while in the flocking case, the Cucker-Smale protocol is a time-
dependent all-to-all coupling, in our settings all couplings are over a fixed
network that allows for more complicated topologies. Interpreting stubbornness
within opinion dynamics as a form of friction yields the following update
protocol with a nonlinear stubbornness forcing term:

$x_{i}(k+1)=\sum_{j=1}^{N}M_{ij}x_{j}(k)+\sigma_{i}(u_{i}-x_{i}^{p}(k))x_{i}(k),$
(7)

where $M$ is any entrywise nonnegative graph-related matrix, $p>0$ is a
parameter responsible for the “strength” of the nonlinearity and we use
$\sigma_{i}\geq 0$ to represent the level of an agent’s desire to believe
$u_{i}^{1/p}$. For $M=B$, the way the update protocol works is similar to the
FJ model, but rather than averaging between the current average opinion and a
fixed value, the agent observes the current average belief of those it is
connected to, $\sum_{j=1}^{N}M_{ij}x_{j}$, as well as how far away their
current belief, $x_{i}(k)$, is from their desired belief $u_{i}^{1/p}$. If the
agent’s current belief is close to its desired belief, or the strength
measured by $\sigma_{i}$ is small, then the update will be closer to the
average belief. However, if the current belief is far from the desired belief,
it will move away from the average, towards the desired belief. In this way,
the model balances the agents’ desire to agree with the average value and its
desire to believe the specific value $u_{i}^{1/p}$ in a nonlinear fashion. In
other words, (7) models agents that are willing to compromise, but the extent
to which is determined by the parameters, and strength of nonlinearity.

Similar to the FJ model, the nonlinear protocol in (7) also has an associated
payout function:

$p_{i}({\boldsymbol{x}})=\frac{1}{2}\sum_{j=1}^{N}M_{ij}(x_{i}-x_{j})^{2}+\sigma_{i}\left(\frac{1}{p+2}x_{i}^{p+2}-\frac{1}{2}u_{i}x_{i}^{2}\right),$
(8)

where the update protocol (7) corresponds with a stationary point of this
payout function. Note that, the lack of convexity in (8) means that we do not
know a-priori that the system has a unique minimizer for (8). Therefore, (7)
is not the only update protocol one can associate with that payout function.
Alternative choices include constant multiples of (7) as well as the following
protocol

$\displaystyle
x_{i}(k+1)=\left(\sum_{j=1}^{N}M_{ij}x_{j}(k)\right)\left(1-\sigma_{i}(u_{i}-x_{i}^{p}(k))\right)^{-1}.$
(9)

Heuristically, (9) would achieve the same goal of balancing between the
average value and the desired value $u_{i}^{1/p}$, but it raises worries if
$(1-\sigma_{i}(u_{i}-x_{i}^{p}(k)))=0.$ For this reason, (7) will be the
nonlinear update protocol, associated with the payout function (8), to which
we will refer.

In this framework, the dynamics are considered to take place at discrete time
steps, and indeed for practical applications and numerical computations this
is an ideal framework. However, if we allow for opinions to evolve
continuously in time, we can re-frame each of these models as a system of
Ordinary Differential Equations (ODEs). To this end, one possibility is to
subtract $x_{i}(k)$ on both sides of the discrete-time models and formally
replace the discrete derivative $x_{i}(k+1)-x_{i}(k)$ with
$\frac{\mathrm{d}}{\mathrm{d}t}x_{i}(t)=\dot{x}_{i}(t)$, where we now use the
symbol $t$ to emphasize that we let $t$ vary continuously. In this way, we
observe that the continuous-time equivalent of the Degroot model (1)
corresponds to the system of ODEs for environmental averaging:

$\dot{x}_{i}(t)=\sum_{j=1}^{N}A_{ij}(x_{j}(t)-x_{i}(t)),$ (10)

with identical payout function (3). Indeed, the continuous analogue has its
own name as the Abelson Model Proskurnikov & Tempo (2017; 2018). Similarly,
for (4) we arrive at:

$\dot{x}_{i}(t)=\lambda_{i}\sum_{j=1}^{N}A_{ij}(x_{j}(t)-x_{i}(t))+(1-\lambda_{i})(u_{i}-x_{i}(t)),$
(11)

obtaining the differential analog of the FJ model with the same corresponding
payout function (5). This model is known as the Taylor Model Proskurnikov &
Tempo (2017; 2018). Last, the continuous version of the nonlinear model
defined in (7) is given by

$\dot{x}_{i}(t)=\sum_{j=1}^{N}M_{ij}(x_{j}(t)-x_{i}(t))+\sigma_{i}(u_{i}-x_{i}(t)^{p})x_{i}(t).$
(12)

We note that in this continuous setting, the payout–protocol correspondence
may change. In fact, while this model corresponds the same payout function
(8), using a different protocol here, as for instance (9), would lead to a
different differential analogue, and would have a different corresponding
payout function. This is one more reason why we prefer (7) over e.g. (9), as
we consider the ‘correct’ continuous analogue of each system to be the one
that seeks stationary points to the same payout function. In this way, each of
these systems can now be viewed as a non-cooperative dynamical opinion game,
where agents continuously update their opinions to minimize the corresponding
payout function.

Throughout the rest of the paper, we will analyze these models in the
framework of differential opinion games, proving that each of them has a Nash
equilibrium to which the system converges. In the case of (10), the
equilibrium will be the consensus of opinions, but the diagonal forcing
included in (11) and (12) makes consensus impossible, leading to the
equilibrium opinion vector being representative of a best-case compromise.

###  1.4 Game theoretic set-up and statement of main results

We will consider each of (10), (11), and (12) as a noncooperative dynamical
opinion game, where the goal of each agent is to minimize their respective
payout functions (3), (5), and (8). In this way we will show that the dynamics
of each of these systems lead to a steady state which is an exponentially
stable Nash equilibrium. For the Abelson and Taylor models these results are
already known, although they are not usually framed in this context, while for
the nonlinear model presented here we must employ different techniques due to
the instability of the diagonal forcing term.

######  Definition 1.1 (Nash Equilibrium).

An $N$-tuple of strategies, ${\boldsymbol{x}}=(x_{1},...,x_{N})$, is a Nash
Equilibria if and only if for each $i$, the payoff function

$\displaystyle
p_{i}({\boldsymbol{x}})=\max_{r_{i}}p_{i}(x_{1},x_{2},...,r_{i},...,x_{N}),$
(13)

i.e., no player can unilaterally increase their payout given the strategy of
each other agent is fixed.

For background on Nash equilibria see Nash (1951). The main result that is to
be shown in this paper is the following.

######  Theorem 1.2.

For any set of parameters $({\bf u},M,\boldsymbol{\sigma})$, such that
$u_{i}\in\mathbb{R}_{+}$, $M_{ij}=M_{ji}\geq 0$, and $\sigma_{i}\geq 0$, there
exists a unique solution ${\boldsymbol{x}}^{*}\in\mathbb{R}^{N}_{+}$ to (19)
which is a locally exponentially stable equilibrium for the system (15). The
map $({\bf u},M,\boldsymbol{\sigma})\mapsto{\boldsymbol{x}}^{*}$ is infinitely
smooth. Moreover, any solution ${\boldsymbol{x}}(t)\in\mathbb{R}^{N}_{+}$
converges to the unique Nash equilibrium.

Before proving this result we present a standard analysis of the convergence
of the Abelson model.

##  2 Continuous Degroot Learning: The Abelson Model

In this section, we will do a quick analysis of model (10), which we recall
below

$\displaystyle\dot{x}_{i}$ $\displaystyle=\sum_{j=1}^{N}A_{ij}(x_{j}-x_{i}),\
\ \ i=1,...,N,$ $\displaystyle x_{i0}$ $\displaystyle=x_{i}(0),\ \ i=1,..,N.$

Let the average opinion value be denoted by
$\bar{x}=\frac{1}{N}\sum_{j=1}^{N}x_{j}$. As $A$ is symmetric, this average
opinion is conserved in time,

$\displaystyle\frac{\mbox{d\,\,}}{\mbox{d}t}\bar{x}=\frac{1}{N}\sum_{i,j=1}^{N}A_{ij}(x_{j}-x_{i})=0$
(14)

Denote $\max_{i}x_{i}(t)=x_{+}(t)$ and $\min_{i}x_{i}(t)=x_{-}(t)$. Then, the
continuous Degroot learning system obeys the maximum and minimum principles.
Indeed, as $x_{+}(t),x_{-}(t)$ are Lipschitz continuous, we can differentiate
almost everywhere to get,

$\displaystyle\frac{\mbox{d\,\,}}{\mbox{d}t}x_{+}(t)$
$\displaystyle=\sum_{j=1}^{N}A_{+j}(x_{j}-x_{+})\leq 0,$
$\displaystyle\frac{\mbox{d\,\,}}{\mbox{d}t}x_{-}(t)$
$\displaystyle=\sum_{j=1}^{N}A_{-j}(x_{j}-x_{-})\geq 0,$

where $A_{+j}$ and $A_{-j}$ are the coefficients associated with the edge
connecting nodes $x_{+}(t)$ (or $x_{-}(t)$) and $x_{j}$.

It is easy to see that any consensus, i.e. any constant vector of opinions
${\boldsymbol{x}}=(x,...,x)$, represents a steady state of the system. As
$\sum_{j}A_{ij}(x_{j}-x_{i})=-\mathbf{1}_{i}^{\top}(D_{A}-A){\boldsymbol{x}}$,
with $\mathbf{1}_{i}$ the $i$-th canonical vector and $D_{A}$ the diagonal
matrix with $i$-th diagonal entry $\sum_{j}A_{ij}$, we see that the first
eigenvalue $\kappa_{1}$ of $D_{A}-A$ is zero, $\kappa_{1}=0$. Further, as $A$
is connected, the second eigenvalue of $D_{A}-A$, the Fiedler number, is
positive, $\kappa_{2}>0$, which means that consensus is the only possible
steady state of the system. As the average is conserved, we now see that the
dynamics must lead to consensus at the average value
${\boldsymbol{x}}=(\bar{x},...,\bar{x})$. Indeed, if the graph is connected,
then we can prove an exponential rate of convergence. From the variational
characterization
$\kappa_{2}=\min_{{\boldsymbol{x}},{\boldsymbol{y}}}\langle(D_{A}-A){\boldsymbol{x}},{\boldsymbol{y}}\rangle/\|{\boldsymbol{x}}\|\|{\boldsymbol{y}}\|$
and the Cauchy-Schwarz inequality, we get

$\langle(D_{A}-A){\boldsymbol{x}},{\boldsymbol{y}}\rangle\geq\kappa_{2}\langle{\boldsymbol{x}},{\boldsymbol{y}}\rangle\,$

for any two vectors ${\boldsymbol{x}},{\boldsymbol{y}}$. Thus,

$\displaystyle\frac{\mbox{d\,\,}}{\mbox{d}t}(x_{+}-x_{-})(t)$
$\displaystyle=\sum_{j=1}^{N}A_{+j}(x_{j}-x_{+})-A_{-j}(x_{j}-x_{-}),$
$\displaystyle=-\langle(D_{A}-A){\boldsymbol{x}},\mathbf{1}_{+}-\mathbf{1}_{-}\rangle\leq-\kappa_{2}(x_{+}-x_{-}),$

yielding $(x_{+}-x_{-})(t)\leq(x_{+}-x_{-})(0)e^{-\kappa_{2}t}$. An important
note for the Abelson model is that it reaches consensus for any entrywise
nonnegative matrix, while in the original Degroot model, periodic matrices
will not necessarily yield consensus, but rather a periodic behavior due to
the synchronous switching of opinion values at each time step. Further, if the
matrix $A$ is not symmetric, then the average opinion value will not be
conserved, and the limiting consensus value will be an emergent convex
combination of the initial values.

In general, consensus is not a realistic result. Introducing stubbornness to
the system leads to the possibility of a different equilibrium where not all
agents agree, as is more often seen in reality. Incorporating stubbornness
leads to a diagonal shift of the graph Laplacian, in the case of the Taylor
model (11), this shift is positive and linear and therefore stable. This
results in the first eigenvalue, $\kappa_{1}$, of $D_{A}-A$ shifting from zero
to being strictly positive, reducing the family of steady states that is
consensus in the Abelson model, to a unique equilibrium in the Taylor model.
The difficulty in analyzing our particular nonlinear forcing is that the
diagonal shift is not stable. Indeed, in general, the Jacobian at the fixed
point will not be diagonally dominant as in the case of the Taylor model.
Therefore, we cannot immediately deduce a unique equilibrium value as we
cannot a priori know where the eigenvalues will shift. In the following
section, we prove first that the dynamics remain bounded away from the trivial
solution at ${\boldsymbol{x}}=0$. Using this fact we can determine that the
Jacobian at any positive equilibrium is not critical. However, due to the lack
of convexity within the payout functions and hence the gradient flow, we
utilize the Brouwer topological degree to achieve uniqueness of the steady
state. The main theorem will then be proved after using the Lojasiewicz
gradient inequality to achieve convergence for all positive initial data.

##  3 Main results: Nonlinear Opinion Dynamics

In this section, we will prove the main result of the paper, Theorem 1.2,
showing existence of a unique Nash Equilibrium for the model (12). Let us
rewrite the system here for reference,

$\displaystyle\dot{x}_{i}$
$\displaystyle=\sum_{j=1}^{N}M_{ij}(x_{j}-x_{i})+\sigma_{i}(u_{i}-x_{i}^{p})x_{i},\
\ \sigma_{i}\in[0,\infty],$ (15) $\displaystyle x_{i0}$
$\displaystyle=x_{i}(0),\ \ i=1,...,N,$ (16)

where $M$ is any symmetric entrywise nonnegative matrix. Then this can be
viewed as a model of opinion dynamics where $x_{i}(t)$ tracks opinion values,
$u_{i}$ represents a fixed conviction value, and each $\sigma_{i}$ is a fixed
measure of each agent’s stubbornness. The first half of the model acts as
averaging amongst the other agents to bring opinions towards a consensus
value. While the second half, derived from the Rayleigh’s friction and self-
propulsion in Lear et al. (2021), attempts to push each opinion toward the
conviction value $u_{i}^{1/p}$. Note that the stubbornness parameters,
$\sigma_{i}$, are allowed to take the values zero or infinity. When
$\sigma_{i}=0$, the agent behaves exactly as an agent in the Abelson model,
converging to the average value of its connections, while for
$\sigma_{i}=\infty$, the agent is perfectly stubborn and remains fixed at
value $u_{i}^{1/p}$. These terms represent agents that are completely flexible
(not stubborn) in the case of $\sigma_{i}=0$, and in the case of
$\sigma_{i}=\infty$, can be seen as a source term like a particular newspaper
or journal. As the dynamics of these cases are already known we need only
consider the case of $0<\sigma_{i}<\infty$ for all $i=1,...,N.$

###  3.1 Boundedness of the dynamics

First, let us prove that all opinion values satisfy a type of Maximum
Principle.

######  Lemma 3.1 (Minimum/Maximum Principle).

For initial conditions $x_{i0}\in\mathbb{R}_{+}$, for each $i=1,...,N$, the
solutions to (15) remain bounded above and below for all time, i.e.,

$\displaystyle\min_{i}\\{\min(x_{i}^{p}(0),u_{i})\\}=c_{-}\leq
x_{i}^{p}(t)\leq c_{+}=\max_{i}\\{\max(x_{i}^{p}(0),u_{i})\\}$ (17)

for all $t\geq 0$.

###### Proof.

Initially (17) is satisfied. Let $x_{-}(t)=\min_{i}x_{i}(t)$, then
differentiating,

$\displaystyle\frac{\mbox{d\,\,}}{\mbox{d}t}x_{-}$
$\displaystyle=\sum_{j=1}^{N}M_{-j}(x_{j}-x_{-})+\sigma_{-}(u_{-}-x_{-}^{p})x_{-}\geq\sigma_{-}(u_{-}-x_{-}^{p})x_{-}\,.$

Suppose $c_{-}=x_{-}^{p}(0)<u_{-}$, then
$\frac{\mbox{d\,\,}}{\mbox{d}t}x_{-}(0)>0$, and $x_{-}^{p}(t)\geq c_{-}$ for
all $t\geq 0$.

Now suppose $c_{-}=u_{-}\leq x_{-}^{p}(0)$. Then, if for some $T\geq 0$,
$x_{-}^{p}(T)=c_{-}$, we have

$\displaystyle\frac{\mbox{d\,\,}}{\mbox{d}t}x_{-}(T)=\sum_{j=1}^{N}M_{-j}(x_{j}(T)-x_{-}(T))\geq
0.$ (18)

Therefore $x_{-}^{p}(t)\geq c_{-}$ for all time. An analogous argument gives
$x_{+}^{p}(t)\leq c_{+}$ as well. ∎

Now, any steady state of the system satisfies

$\sum_{j=1}^{N}M_{ij}(x_{j}-x_{i})+\sigma_{i}(u_{i}-x_{i}^{p})x_{i}=0$ (19)

for each $i=1,...,N$. An immediate estimate on the steady state, from dropping
the sum, gives:

$\min_{i}u_{i}\leq x_{j}^{p}\leq\max_{i}u_{i},$ (20)

which, for identical conviction values, $u_{i}\equiv u$, gives a unique steady
state $x_{i}^{p}=u$, for all $i=1,...,N$, which of course is a consensus
state. Indeed, due to the connectivity of the graph, and along with Lemma 3.1,
this further implies that unless $u_{i}\equiv u$ for all $i$, we have
$x_{i}(t)>c_{-}$ for all $t>0$, and analogously for $x_{i}(t)<c_{+}$.

###  3.2 Uniqueness

Now let us show that there is a unique steady state for the model. Let us
define the map $F:\mathbb{R}^{N}\to\mathbb{R}^{N}$ that assigns to any
${\boldsymbol{x}}\in\mathbb{R}^{N}_{+}$ the vector $F({\boldsymbol{x}})$ with
entries

$\displaystyle
F({\boldsymbol{x}})_{i}=d_{i}x_{i}-\sum_{j=1}^{N}M_{ij}x_{j}+\sigma_{i}(x_{i}^{p}-u_{i})x_{i}\,,$
(21)

with $d_{i}=\deg(i)$. Indeed, when the map $F({\boldsymbol{x}})_{i}=0$ for all
$i=1,...,N$, the original system is at a steady state, i.e. (19) is satisfied.

We show that any steady state of the system is not critical for the Jacobian
of this map.

######  Lemma 3.2.

Let ${\boldsymbol{x}}^{*}$ satisfy equation (19). For any set of parameters
$({\bf u},M,\boldsymbol{\sigma})$, such that $u_{i}\in\mathbb{R}_{+}$, $M$ is
symmetric, entrywise nonnegative, $M_{ij}\geq 0$, and $\sigma_{i}\geq 0$, then
the Jacobian of $F$ at ${\boldsymbol{x}}^{*}$ is not critical.

###### Proof.

We compute the Jacobian Matrix of $F$ as,

$\displaystyle D_{{\boldsymbol{x}}}F({\boldsymbol{x}})=G-M$ (22)

where $G=\mathrm{diag}\\{g_{i}\\}_{i=1}^{N}$ and
$g_{i}=d_{i}+\sigma_{i}((p+1)x_{i}^{p}-u_{i})$. In this way we can see exactly
that the Jacobian is a diagonal shift of the graph Laplacian, with diagonal
shift given by $\sigma_{i}((p+1)x_{i}^{p}-u_{i})$. Indeed if
$\sigma_{i}((p+1)x_{i}^{p}-u_{i})$ were strictly positive for all $i=1,...,N$,
the shift would be stable; however, in general this is not true. Therefore, we
utilize (19) and the fact that $x_{i}>0$ to see that

$\displaystyle
g_{i}=\sigma_{i}px_{i}^{p}+\sum_{j=1}^{N}M_{ij}\frac{x_{j}}{x_{i}}.$ (23)

Note that each $g_{i}>0$ while the off-diagonal entries are given by
$-M_{ij}\leq 0$. Indeed the matrix for which we must compute the determinant
is of the form

$G-M=\begin{bmatrix}g_{1}&-M_{12}&\dots&-M_{1N}\\\ -M_{12}&g_{2}&&\vdots\\\
\vdots&&\ddots&\vdots\\\ -M_{1N}&\dots&\dots&g_{N}\\\
\end{bmatrix}=\begin{bmatrix}c_{11}&c_{12}&\dots&c_{1N}\\\
c_{12}&c_{22}&&\vdots\\\ \vdots&&\ddots&\vdots\\\ c_{1N}&\dots&\dots&c_{NN}\\\
\end{bmatrix}.$

Now, the determinant of $D_{{\boldsymbol{x}}}F({\boldsymbol{x}})=G-M$ is given
by summing over all cycles of the graph (up to the sign of each cycle). Thus,
we can explicitly compute the determinant,

$\displaystyle\det{D_{{\boldsymbol{x}}}F({\boldsymbol{x}})}=\sum_{\delta\in
S_{N}}\mathrm{sgn}(\delta)\prod_{i=1}^{N}c_{i,\delta(i)}$ (24)

where $\delta$ is a cycle of the matrix $G-M$, $S_{n}$ is the permutation
group, and $c_{i,j}$ is the element in the $i$th row and $j$th column of
$D_{{\boldsymbol{x}}}F({\boldsymbol{x}})$. In this particular case, as all the
off-diagonal terms are nonpositive, and the main diagonal is strictly
positive, the only positive summand is the one corresponding to the main
diagonal, while the rest will all be negative. Therefore,

$\displaystyle\det{D_{{\boldsymbol{x}}}F({\boldsymbol{x}})}=\prod_{i=1}^{N}c_{ii}-\sum_{\begin{subarray}{c}\delta\in
S_{N},\\\
\delta\neq\\{1,...,N\\}\end{subarray}}\mathrm{sgn}(\delta)\prod_{k=1}^{N}c_{k,\delta(k)},$
(25)

where $-\mathrm{sgn}(\delta)\prod_{k=1}^{N}c_{k,\delta(k)}\leq 0$ for all
$\delta\neq\\{1,...,N\\}\in S_{N}$.

To see that this is positive let us observe that since $c_{ii}=g_{i}$,

$\displaystyle\prod_{i=1}^{N}c_{ii}$
$\displaystyle=\prod_{i=1}^{N}(\sigma_{i}px_{i}^{p}+\sum_{k=1}^{N}M_{ik}\frac{x_{k}}{x_{i}}),$
(26)
$\displaystyle=p^{N}\prod_{i=1}^{N}\sigma_{i}x_{i}^{p}+p^{N-1}\prod_{l=1}^{N}\prod_{i\neq
j}\sigma_{i}x_{i}^{p}\sum_{j=1}^{N}M_{jl}\frac{x_{l}}{x_{j}}+\dots$ (27)
$\displaystyle\dots+p\sum_{i=1}^{N}\sigma_{i}x_{i}^{p}\prod_{j\neq
i}\sum_{l=1}^{N}M_{jl}\frac{x_{l}}{x_{j}}+\prod_{j=1}^{N}\sum_{l=1}^{N}M_{jl}\frac{x_{l}}{x_{j}},$
(28) $\displaystyle=I_{0}+\dots+I_{N}.$ (29) $\displaystyle I_{N-n}$
$\displaystyle=p^{n}\sum_{i_{1}<i_{2}<...<i_{n}}^{N}\prod_{a\in\\{i_{n}\\}}\sigma_{a}x_{a}^{p}\prod_{l\not\in\\{i_{n}\\}}\sum_{j=1}^{N}M_{jl}\frac{x_{l}}{x_{j}},$
(30)

where $\\{i_{n}\\}$ is the set of $n$ indices that are multiplying the first
terms in the product $p\sigma_{i}x_{i}^{p}$, and the rest are multiplying the
second terms $\sum_{k=1}^{N}M_{ik}\frac{x_{k}}{x_{i}}$. Each summand of
$I_{j}$ is positive, but in particular we see that for any pair $i,j$ the
product $M_{ij}\frac{x_{j}}{x_{i}}\times
M_{ji}\frac{x_{i}}{x_{j}}=M_{ij}M_{ji}.$

Therefore, within the summands of $I_{N}$ we find every cycle of the graph
$M$, all with a positive sign. In fact, we can break $I_{N}$ down into $N$
different parts determined by the length of cycles in the graph.

$\displaystyle I_{N}$ $\displaystyle=\sum_{\delta\in
S_{N}}\prod_{k=1}^{N}M_{k,\delta(k)}+\sum_{j=1}^{N}\sum_{\delta\in
S_{N-1}\backslash j}\left(\sum_{l\neq
j}M_{jl}\frac{x_{l}}{x_{j}}\right)\prod_{k\neq j}M_{k,\delta(k)}+\dots,$ (31)
$\displaystyle:=I_{N,0}+I_{N,1}+\dots,$ (32)

where $S_{N-1}\backslash j$ denotes the permutation group of the indices
$\\{1,...,j-1,j+1,...,N\\}$. Then in general we have for $j=1,...,N-2$

$\displaystyle I_{N,j}$
$\displaystyle=\sum_{k_{1}<...<k_{j}}^{N}\sum_{\delta\in
S_{N-j}\backslash\\{k_{j}\\}}\prod_{k\in\\{k_{j}\\}}\left(\sum_{l\not\in\\{k_{j}\\}}M_{kl}\frac{x_{l}}{x_{k}}\right)\prod_{n\not\in\\{k_{j}\\}}M_{n,\delta(n)},$
(33) $\displaystyle I_{N,N-1}$ $\displaystyle=\sum_{i=1}^{N}M_{ii}\prod_{l\neq
i}M_{li}\frac{x_{i}}{x_{l}},$ (34)

where again $S_{N-j}\backslash\\{k_{j}\\}$ is the permutation group of $N-j$
elements ignoring the $j$ indices within the set $\\{k_{j}\\}$. Thus we have
decomposed $I_{N}$ as,

$\displaystyle I_{N}=\sum_{j=0}^{N-1}I_{N,j}.$ (35)

We do the same for each $I_{N-n}$ so that

$\displaystyle I_{N-n}$ $\displaystyle=\sum_{j=0}^{N-n-1}I_{N-n,j},$ (36)
$\displaystyle I_{N-n,0}$
$\displaystyle=p^{n}\sum_{i_{1}<...<i_{n}}^{N}\prod_{a\in\\{i_{n}\\}}\sigma_{a}x_{a}^{p}\sum_{\delta\in
S_{N-n}\backslash\\{i_{n}\\}}\prod_{k\not\in\\{i_{n}\\}}M_{k,\delta(k)},$ (37)
$\displaystyle I_{N-n,j}$
$\displaystyle=p^{n}\sum_{i_{1}<i_{2}<...<i_{n}}^{N}\prod_{a\in\\{i_{n}\\}}\sigma_{a}x_{a}^{p}\sum_{\begin{subarray}{c}k_{1}<k_{2}<...<k_{j}\not\in\\{i_{n}\\}\\\
\delta\in
S_{N-n-j}\backslash\\{\\{i_{n}\\},\\{k_{j}\\}\\}\end{subarray}}\prod_{l\in\\{k_{j}\\}}\sum_{m\not\in\\{k_{j}\\}}M_{lm}\frac{x_{m}}{x_{l}}\prod_{k\not\in\\{\\{i_{n}\\},\\{k_{j}\\}\\}}M_{k,\delta(k)}.$
(38)

Now, let us break down the negative terms similarly,

$\displaystyle\sum_{\begin{subarray}{c}\delta\in S_{N},\\\
\delta\neq\\{1,...,N\\}\end{subarray}}\mathrm{sgn}(\delta)\prod_{k=1}^{N}c_{k,\delta(k)}=\sum_{j=0}^{N-2}J_{j},$
(39) $\displaystyle J_{j}$ $\displaystyle=\sum_{\begin{subarray}{c}\delta\in
S_{N}\\\
\exists_{j}k:\delta(k)=k\end{subarray}}\prod_{k=1}^{N}c_{k,\delta(k)},$ (40)
$\displaystyle=\sum_{k_{1}<k_{2}<...<k_{j}}\sum_{\delta\in
S_{N-j}\backslash\\{k_{j}\\}}\prod_{k\in\\{k_{j}\\}}m_{k}\prod_{l\not\in\\{k_{j}\\}}M_{l,\delta(l)}.$
(41)

Now, as $m_{k}=p\sigma_{k}x_{k}^{p}+\sum_{j=1}^{N}M_{kj}\frac{x_{j}}{x_{k}}$,
we can further split up each $J_{j}$ into parts determined by how many powers
of $p$ the part has,

$\displaystyle J_{j}$ $\displaystyle=\sum_{k=0}^{j}J_{j,k},$ (42)
$\displaystyle J_{j,k}$
$\displaystyle=p^{k}\sum_{\begin{subarray}{c}k_{1}<...<k_{j}\\\ \delta\in
S_{N-j}\backslash\\{k_{j}\\}\end{subarray}}\prod_{l\not\in\\{k_{j}\\}}M_{l,\delta(l)}\sum_{n_{1}<...<n_{k}}\prod_{n\in\\{n_{k}\\}}\sigma_{n}x_{n}^{p}\prod_{\begin{subarray}{c}m\not\in\\{n_{k}\\}\\\
m\in\\{k_{j}\\}\end{subarray}}\sum_{s\not\in\\{k_{j}\\}}M_{ms}\frac{x_{s}}{x_{m}}.$
(43)

Finally, by combining all parts that have the same powers of $p$, from the
positive $I$ terms, and the negative $J$ terms we have

$\displaystyle I_{N-n,k}=J_{n+k,n},\ \ 0\leq k\leq N-n-2,$ (44)

where all positive $p^{n}$ powers are found in
$I_{N-n}=\sum_{k=0}^{N-n-1}I_{N-n,k}$ and all negative powers within
$\sum_{k=n}^{N-2}J_{k,n}$. In which case,

$\displaystyle I_{N-n}-\sum_{k=n}^{N-2}J_{k,n}=I_{N-n,N-n-1}\geq 0.$ (45)

Thus summing over all $n=0,...N$ we get,

$\displaystyle\det{D_{{\boldsymbol{x}}}F({\boldsymbol{x}})}=\sum_{n=0}^{N}\left(I_{N-n}-\sum_{k=n}^{N-2}J_{k,n}\right)=I_{0}+I_{1}+\sum_{n=0}^{N-2}I_{N-n,N-n-1}>0,$
(46)

Where $I_{0},I_{1}>0$ always, and $\sum_{n=0}^{N-2}I_{N-n,N-n-1}=0$ in the
case of no self loops within the underlying graph, and otherwise is strictly
positive with self loops.

Noting that this computation is identical for any principal minor of the
matrix, we conclude that the determinant of every principal minor is also
strictly positive, and the Jacobian Matrix is in fact an invertible M-Matrix
with strictly positive eigenvalues. ∎

If the payout function (8) were convex, then non-degeneracy of the Jacobian
would immediately grant uniqueness of the steady state. However, without
convexity we utilize the Brouwer topological degree in order to achieve the
result. To achieve the uniqueness we utilize the fact that on a particular
region $\mathcal{W}$, the topological degree of a point ${\boldsymbol{y}}$ is
defined as

$\mathrm{deg}\\{F,\mathcal{W},{\boldsymbol{y}}\\}=\sum_{{\boldsymbol{x}}\in
F^{-1}(\boldsymbol{y})}\mathrm{sgn}(\det{D_{{\boldsymbol{x}}}F({\boldsymbol{x}})}).$
(47)

Now, since we have determined that
$\mathrm{sgn}(\det{D_{{\boldsymbol{x}}}F({\boldsymbol{x}})})=1$, for any
${\boldsymbol{x}}\in F^{-1}(\boldsymbol{0})$, if we can show that the degree
is exactly one, then there must be a unique fixed point within the region
$\mathcal{W}$. For background on the Brouwer topological degree see Cronin
(1995).

######  Lemma 3.3.

For any set of parameters $({\bf u},M,\boldsymbol{\sigma})$, such that
$u_{i}\in\mathbb{R}_{+}$, $M\geq 0$ is symmetric, and $\sigma_{i}\geq 0$,
there exists a unique solution ${\boldsymbol{x}}^{*}$ to (19) which is a
locally exponentially stable equilibrium for the system (15). The map $({\bf
u},M,\boldsymbol{\sigma})\mapsto{\boldsymbol{x}}^{*}$ is infinitely smooth.

###### Proof.

We consider the Brouwer topological degree of $F$ at zero. To define the
degree properly, we restrict $F$ to a wedge region $\mathcal{W}$. Let us
denote

$\displaystyle\langle{\boldsymbol{x}},{\boldsymbol{y}}\rangle=\sum_{i=1}^{N}\sigma_{i}x_{i}y_{i},\
\ \ \ \ \|{\boldsymbol{x}}\|_{p}^{p}=\sum_{i=1}^{N}\sigma_{i}x_{i}^{p}.$ (48)

We define,

$\displaystyle\mathcal{W}=\\{{\boldsymbol{x}}:x_{i}\geq 0,\
\varepsilon\leq\|{\boldsymbol{x}}\|_{\infty},\|{\boldsymbol{x}}\|_{p+1}\leq
R\\},$ (49)

where $R>0$ is large, and $\varepsilon$ small, to be determined momentarily.
We verify that the image of the boundary does not contain the origin,
$0\not\in F(\partial\mathcal{W})$. Indeed, if $x_{i}=0$ for some $i$, then
$F_{i}=-\bar{x}<0.$ Now we compute,

$\displaystyle\sum_{i=1}^{N}F({\boldsymbol{x}})_{i}=-\langle\boldsymbol{u},{\boldsymbol{x}}\rangle+\|{\boldsymbol{x}}\|_{p+1}^{p+1}.$
(50)

Thus, if $\|{\boldsymbol{x}}\|_{p+1}=R$, we have the bound

$\displaystyle\sum_{i=1}^{N}F({\boldsymbol{x}})_{i}\geq\|{\boldsymbol{x}}\|_{p+1}^{p+1}-\|{\boldsymbol{x}}\|_{p+1}\|\boldsymbol{u}\|_{\frac{p+1}{p}}>0,$
(51)

provided $R$ is large enough. Similarly, if
$\|{\boldsymbol{x}}\|_{\infty}=\varepsilon$, then

$\displaystyle\sum_{i=1}^{N}F({\boldsymbol{x}})_{i}\leq-
u_{-}\|{\boldsymbol{x}}\|_{1}+\varepsilon^{p}\|{\boldsymbol{x}}\|_{1}<0,$ (52)

provided $\varepsilon$ is small enough.

Therefore the value $\boldsymbol{0}$ of $F$ is regular, and its degree can be
computed explicitly by

$\displaystyle\mathrm{deg}\\{F,\mathcal{W},\boldsymbol{0}\\}=\sum_{{\boldsymbol{x}}\in
F^{-1}(\boldsymbol{0})}\mathrm{sgn}(\det{D_{{\boldsymbol{x}}}F({\boldsymbol{x}})}).$
(53)

However, we have proved that all Jacobians for ${\boldsymbol{x}}\in
F^{-1}(\boldsymbol{0})$ are strictly positive. Therefore uniqueness can be
shown by proving $\mathrm{deg}\\{F,\mathcal{W},\boldsymbol{0}\\}=1$. This is
certainly true for $\hat{\boldsymbol{u}}=(u,\dots,u)$, since we have a unique
positively oriented solution from (20). Indeed, this is because for identical
conviction values, consensus is achieved at exactly the conviction value,
$u^{1/p}$. Now, fix any such $\hat{\boldsymbol{u}}$ and consider the homotopy
of maps

$\displaystyle F^{(\tau)}:=F_{\tau{\bf u}+(1-\tau)\hat{{\bf u}}}$ (54)

where $F_{{\boldsymbol{y}}}$ denotes the map $F$ in (21) with $u$ replaced by
${\boldsymbol{y}}$. Since $\boldsymbol{0}\not\in
F^{(\tau)}(\partial\mathcal{W})$ for any $\tau$, the Invariance under Homotopy
Principle applies and hence,

$\mathrm{deg}\\{F_{\boldsymbol{u}},\mathcal{W},\boldsymbol{0}\\}=\mathrm{deg}\\{F_{\hat{\boldsymbol{u}}},\mathcal{W},\boldsymbol{0}\\}=1,$
(55)

and the proof of uniqueness is finished.

The smoothness of the steady state, ${\boldsymbol{x}}^{*}$, as a function of
$(\boldsymbol{u},M,\boldsymbol{\sigma})$ follows directly from the non-
degeneracy of the Jacobian and the Implicit Function Theorem. ∎

###  3.3 Convergence

With this lemma in hand we need only prove that for any initial data the
opinion values actually converge to this equilibrium, achieving the Nash
Equilibrium.

Convergence to the equilibrium is proved by first revealing the gradient
structure for the system (15).

$\frac{\mbox{d\,\,}}{\mbox{d}t}{\boldsymbol{x}}=-\nabla\Phi({\boldsymbol{x}}),$
(56)

for

$\displaystyle\Phi({\boldsymbol{x}})=\frac{1}{4}\sum_{i,j=1}^{N}M_{ij}(x_{i}-x_{j})^{2}+\frac{1}{p+2}\sum_{i=1}^{N}\sigma_{i}x_{i}^{p+2}-\frac{1}{2}\sum_{i=1}^{N}\sigma_{i}u_{i}x_{i}^{2}.$
(57)

We note that as $\Phi({\boldsymbol{x}})$ is bounded, both from above and
below, and decreases along the flow, we know that $\Phi({\boldsymbol{x}}(t))$
must converge somewhere.

To prove the convergence of ${\boldsymbol{x}}(t)$ itself we will appeal to the
Lojasiewicz gradient inequality, stated next.

######  Theorem 3.4.

Lojasiewicz (1963) Let $\Phi$ be a real analytic function in a neighborhood of
$U$. Then for any ${\boldsymbol{x}}_{0}\in U$, there are constants $c>0$,
$\delta\in(0,1]$, and $\mu\in[1/2,1)$, such that

$\displaystyle\|\nabla\Phi({\boldsymbol{x}})\|\geq
c|\Phi({\boldsymbol{x}})-\Phi({\boldsymbol{x}}_{0})|^{\mu},\ \ \ \
\forall{\boldsymbol{x}}\in U,\ \ \text{such}\ \text{that}\
\|{\boldsymbol{x}}-{\boldsymbol{x}}_{0}\|\leq\delta.$ (58)

As our $\Phi$ is real analytic for all $x_{i}\in\mathbb{R}_{+}$, the result
applies. Consider a solution ${\boldsymbol{x}}$ in the positive sector,
$\mathbb{R}_{+}^{N}$ to (56). As every solution is bounded there must be an
accumulation point ${\boldsymbol{x}}_{0}$. We now must show that
${\boldsymbol{x}}(t)\to{\boldsymbol{x}}_{0}$, and that
$\nabla\Phi({\boldsymbol{x}}_{0})=0$, establishing that
${\boldsymbol{x}}_{0}={\boldsymbol{x}}^{*}$, the unique steady state. First,
we need a lemma to control the length of the orbit ${\boldsymbol{x}}(t)$ near
the accumulation point ${\boldsymbol{x}}_{0}$.

######  Lemma 3.5.

As long as ${\boldsymbol{x}}(t)\in B_{\delta}({\boldsymbol{x}}_{0})$ for
$t^{\prime}\leq t\leq t^{\prime\prime}$, we have,

$\displaystyle\int_{t^{\prime}}^{t^{\prime\prime}}\|\dot{{\boldsymbol{x}}}(s)\|\,\mbox{d}s\leq\int_{\Phi({\boldsymbol{x}}(t^{\prime\prime}))}^{\Phi({\boldsymbol{x}}(t^{\prime}))}\frac{1}{c|\zeta-\Phi({\boldsymbol{x}}_{0})|^{\mu}}d\zeta.$
(59)

###### Proof.

Let us define the two functions,

$\displaystyle\psi(\zeta)$
$\displaystyle=c|\zeta-\Phi({\boldsymbol{x}}_{0})|^{\mu},$ (60)
$\displaystyle\Psi(x)$
$\displaystyle=\int_{0}^{x}\frac{1}{\psi(\zeta)}d\zeta.$ (61)

Let us compute,

$\displaystyle\frac{\mbox{d\,\,}}{\mbox{d}t}\Psi(\Phi({\boldsymbol{x}}(t)))=\dot{\Psi}(\Phi))\dot{\Phi}=\frac{\dot{\Phi}}{\psi(\Phi)}$
(62)

but as $\dot{\Phi}({\boldsymbol{x}}(t))=-\|\nabla\Phi\|^{2}$, we get,

$\displaystyle-\frac{\mbox{d\,\,}}{\mbox{d}t}\Psi(\Phi({\boldsymbol{x}}(t)))=\frac{\|\nabla\Phi\|^{2}}{c|\Phi({\boldsymbol{x}}(t))-\Phi({\boldsymbol{x}}_{0})|^{\mu}}\geq\|\dot{{\boldsymbol{x}}}\|,$
(63)

by the Lojasiewicz inequality. Integrating over
$(t^{\prime},t^{\prime\prime})$ proves the lemma. ∎

With this lemma in hand, we are ready to prove convergence to the equilibrium.

###### Proof of Theorem 1.2.

As ${\boldsymbol{x}}_{0}$ is an accumulation point, there is a sequence
$t_{n}$ such that ${\boldsymbol{x}}(t_{n})\to{\boldsymbol{x}}_{0}$. Fix an
arbitrary $r<\delta$ and choose a remote time $t_{n}$ far enough so that

$\displaystyle\|{\boldsymbol{x}}(t_{n})-{\boldsymbol{x}}_{0}\|<\frac{r}{2},\ \
\
\int_{\Phi({\boldsymbol{x}}_{0})}^{\Phi({\boldsymbol{x}}(t_{n}))}\psi(\zeta)d\zeta<\frac{r}{2}.$
(64)

Now, to show that the entire trajectory for $t>t_{n}$ remains in
$B_{r}({\boldsymbol{x}}_{0})=\\{{\boldsymbol{y}}:\|{\boldsymbol{x}}_{0}-{\boldsymbol{y}}\|\leq
r\\}$, let $\tilde{t}$ be the first time such that
$\|{\boldsymbol{x}}(t_{n}+\tilde{t})-{\boldsymbol{x}}_{0}\|=r$. Then
${\boldsymbol{x}}(t)$ lies in $B_{r}({\boldsymbol{x}}_{0})$ on
$(t_{n},t_{n}+\tilde{t})$. Then applying Lemma 3.5 we get,

$\displaystyle\|{\boldsymbol{x}}(t_{n}+\tilde{t})-{\boldsymbol{x}}_{0}\|\leq\|{\boldsymbol{x}}(t_{n}+\tilde{t})-{\boldsymbol{x}}(t_{n})\|+\|{\boldsymbol{x}}(t_{n})-{\boldsymbol{x}}_{0}\|<r.$
(65)

Therefore the trajectory must lie in $B_{r}({\boldsymbol{x}}_{0})$ for all
$t>t_{n}$. To conclude that ${\boldsymbol{x}}_{0}={\boldsymbol{x}}^{*}$, we
note that the above argument implies

$\displaystyle\int_{t_{n}}^{\infty}\|\dot{{\boldsymbol{x}}}(s)\|\,\mbox{d}s<\infty.$
(66)

Therefore $\dot{{\boldsymbol{x}}}(s_{n})\to 0$ and thus
$\nabla\Phi({\boldsymbol{x}}(s_{n}))\to 0=\nabla\Phi({\boldsymbol{x}}_{0})$. ∎

##  4 Continuous Friedkin-Johnsen: Taylor Model

The Taylor model (11) is also representative of a gradient flow, and further
its payout function is convex, so the standard convexity theory guarantees
existence, uniqueness and convergence to the Nash Equilibrium. However, the
proof of Theorem 1.2 is powerful as it also applies as an alternate proof for
the Taylor Model (11). Indeed, steady states of (11) satisfy

$\displaystyle\lambda_{i}\sum_{j=1}^{N}A_{ij}(x_{j}-x_{i})+(1-\lambda_{i})(u_{i}-x_{i})=0,$
(67)

and therefore have corresponding $F$-maps,

$\displaystyle
F({\boldsymbol{x}})_{i}=\lambda_{i}x_{i}-\lambda_{i}\sum_{j=1}^{N}A_{ij}x_{j}+(1-\lambda_{i})(x_{i}-u_{i}).$
(68)

The Jacobian Matrix of this map is given by

$\displaystyle D_{{\boldsymbol{x}}}F({\boldsymbol{x}})=I-\Lambda A,$ (69)

where the elements of $\Lambda A$ are given by $\lambda_{i}A_{ij}$. By the
Perron-Frobenius theory, the determinant of this Jacobian must be positive
unless $\lambda_{i}=1$ for all $i$, which of course reduces the model to
$\eqref{eq:EAc}$. The proof then follows exactly as for the nonlinear model,
noting that the gradient structure needed for convergence can be found by the
same change of variable used to transform (4) to (6). If we let
$y_{i}=\frac{x_{i}}{\lambda_{i}}$ and
$\sigma_{i}=\frac{1-\lambda_{i}}{\lambda_{i}}$ then (11) becomes

$\displaystyle\dot{y}_{i}=\sum_{j=1}^{N}B_{ij}(y_{j}-y_{i})+\sigma_{i}u_{i}+(\lambda_{i}-1)y_{i}.$
(70)

Then, the gradient structure can be revealed,

$\displaystyle\frac{\mbox{d\,\,}}{\mbox{d}t}{\boldsymbol{y}}=-\nabla\Phi({\boldsymbol{y}}),$
(71)

for

$\displaystyle\Phi({\boldsymbol{y}})=\frac{1}{4}\sum_{i,j=1}^{N}B_{ij}(y_{i}-y_{j})^{2}+\sum_{k=1}^{N}(\sigma_{k}u_{k}y_{k}+\frac{1}{2}(\lambda_{k}-1)y_{k}^{2}).$
(72)

Therefore, the Taylor model must also converge to a unique Nash Equilibrium.
In the next section, we present some numerical experiments which compare the
nonlinear model with the original Friedkin-Johnsen model.

##  5 Numerical experiments

In this section, we present a few illustrative examples to show the different
behavior of the opinion dynamics modeled by the linear FJ model and he
nonlinear model introduced here, when subject to the same initial convictions
and same stubbornness parameters. When opinion values are close to $1$, the
nonlinear model (12) behaves similarly to the linear version (11), as when
$x_{i}\sim 1$, the linear term $u_{i}-x_{i}$ well-approximates the nonlinear
one $(u_{i}-x_{i}^{p})x_{i}$. However, we see that the nonlinear model becomes
more stubborn for large opinion values than the linear version. Indeed, the
lack of translation invariance leads exactly to this behavior. The behavior
represented by the nonlinear model is that agents which hold an extreme
opinion, even with low stubbornness parameter $\sigma\ll 1$, require much more
external pressure to change their minds, and will anyways continue to hold
extreme opinions.

Figure 1: Adjacency matrix spy plot, and mean and standard deviation of node
dynamics $x_{i}(t)$ in the time window $t\in[0,100]$ for the FJ (73) and the
NFJ (74) models, for different choices of $u_{i}$ and $\sigma_{i}$, as defined
in (75). Top row: Erdős–Rényi random graph with 150 nodes and $p_{e}=0.08$
edge probability; Bottom row: SBM random graph with two blocks of 50 and 100
nodes, respectively, within-block edge probability $p_{in}=0.2$ and between-
block edge probability $p_{out}=0.02$.

This is illustrated in Figure 1 where we present an example of the opinion
dynamics of the two models on a small random graph and a strongly community-
structured random network. The random network is an Erdős–Rényi graph with
$150$ nodes and $p_{e}=0.08$ edge probability. The community-structured
network is a stochastic block model with two clusters of size $50$ and $100$
with within-cluster probability $p_{in}=0.2$ and across-cluster edge
probability $p_{out}=0.02$. For a fair comparison of the choice of the
parameters $\sigma_{i}$ and the initial convictions $u_{i}$, the linear model
considered here is the following modified FJ or Taylor model

$\dot{x}_{i}(t)=\sum_{j=1}^{N}M_{ij}(x_{j}(t)-x_{i}(t))+\sigma_{i}(u_{i}-x_{i}(t))$
(73)

which we compare with the considered nonlinear dynamical system (12) for $p=1$

$\dot{x}_{i}(t)=\sum_{j=1}^{N}M_{ij}(x_{j}(t)-x_{i}(t))+\sigma_{i}(u_{i}-x_{i}(t))x_{i}(t)\,.$
(74)

For both the network configurations, the conviction and the stubbornness
vectors are set to

$u_{i}=\begin{cases}\kappa&1\leq i\leq 50\\\ 1&51\leq i\leq
150\end{cases}\qquad\text{and}\qquad\sigma_{i}=\begin{cases}\delta&1\leq i\leq
50\\\ 1&51\leq i\leq 150\end{cases}$ (75)

and we evaluate the dynamics $x_{i}(t)$ of each node $i$ from $t=0$ to
$t=100$, for $\kappa\in\\{10,100\\}$ and $\delta\in\\{1/2,2\\}$.

As the linear and nonlinear FJ models (11) and (12) attempt to push each
opinion towards the conviction values $u_{i}$ and $u_{i}^{1/p}$, respectively,
the choice of $p=1$ allows us to better appreciate the comparison between the
two models with respect to the same choice of convictions. Although other
values of $p$ would accentuate the system’s nonlinear behavior for values of
$x_{i}$ away from $1$, the choice $p=1$ introduces enough nonlinearity to
significantly change the overall dynamics, and the qualitative difference
between the two models does not change for different values of $p$.

Figure 2: Adjacency matrix spy plot, and mean node dynamics $x_{i}(t)$ in the
time window $t\in[0,100]$ for the FJ (73) and the NFJ (74) models, for
different choices of $u_{i}$ and $\sigma_{i}$, as defined in (75). Top row:
strongly-structured core-periphery network with one single core node $i=1$,
such that $(1,j)\in\mathcal{E}$ for all $j\in\mathcal{V}$, while edges $ij$
with $i,j\neq 1$ exist with small probability $p_{e}=0.01$; Bottom row:
complete (loop-free) graph, with $ij\in\mathcal{E}$ for all
$i,j\in\mathcal{V}$.

Figure 2 illustrates the fact that both the FJ and the NFJ models with
complete all-to-all coupling can be well approximated by a core-periphery
network with a drastically reduced edge-density, where only one single node
$i=1$ is connected to every other node, while edges $ij$ with $i,j\neq 1$
exist with small probability $p_{e}=0.01$. Indeed, the behavior seen in the
first row is identical to that in the second row, while still highlighting the
differences in stubbornness between the FJ and NFJ models. Further, we see
that both FJ and NFJ approach consensus in the final two columns of Figure 2
as the strong mixing of the networks overpowers the small convictions
$\kappa=10$ and small stubbornness $\delta=0.5$. Finally, Figure 3 illustrates
the behavior of FJ and NFJ on two real-world social networks: ‘Jazz’, a
network of Jazz bands, consisting of 198 nodes, being jazz bands, and 2742
edges representing common musicians; ‘CollegeMsg’, a network accounting for
13838 exchanged messages (the edges) among 1899 students (the nodes) in a
north American college. In this case, similarly to what was done in (75), we
defined $u$ and $\sigma$ by randomly assigning $N/2$ entries the value $1$ and
the other $N/2$ the value $\kappa$ and $\delta$, respectively. As expected,
the behavior of the dynamics resembles the one observed on previous synthetic
sparse graphs.

Figure 3: Dynamics on real world networks. Adjacency matrix spy plot, and mean
node dynamics $x_{i}(t)$ in the time window $t\in[0,100]$ for the FJ (73) and
the NFJ (74) models, for different choices of $u_{i}$ and $\sigma_{i}$, as
defined in (75). Top row: ‘Jazz’ network consisting of 198 nodes, jazz bands,
and 2742 edges representing common musicians; Bottom row: ‘CollegeMsg’ network
containing 1899 nodes, students, and 13838 edges representing messages sent
between students.

##  6 Discussion

In this work we introduced a nonlinear update protocol for modelling opinion
dynamics over networks. The nonlinearity was inspired by the Rayleigh friction
and self-propulsion, used in both flocking and oscillatory models like the
Cucker-Smale and Stuart-Landau systems. We prove well-posedness and
convergence of the model to a unique Nash Equilibrium. The techniques involved
directly computing the Jacobian of the steady states, utilizing the Brouwer
topological degree, and analysis of a nonconvex gradient flow. The techniques
used for the nonlinear model carry over directly to the existing literature of
linear models that are also discussed in this article. Further, it expands the
nonlinear theory to more diverse network topologies as well as heterogeneous
stubbornness parameters. Finally, we provide several numerical experiments to
highlight the similarities and differences in outcomes that arise from
utilizing nonlinear effects in the modeling of opinion dynamics. Our model
effectively describes the phenomena: Extreme opinions are less easily swayed
by social pressure.

The model presented here has a strong connection with models for coupled
Stuart-Landau models that will be the topic of a future study. Future
directions within opinion dynamics include generalizing what kind of nonlinear
forcings and consensus mechanisms are permissible to lead to the same sort of
Nash equilibrium.

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