Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations

1.   Introduction

In the last few years, several physicists and mathematicians devoted their attention to opinion dynamics. Different techniques were used, depending on the specific problem. For instance, computer simulations are preferred when agents contact through networks, although they appear also in mean field problems [5,15,32,33,37], Markov processes and other probabilistic tools for finitely many opinions [17,18,21], large systems of ordinary differential equations for active particles, and the associated Boltzmann type equations for the evolution of several observables [12,20,27,29,31,35,38], together with their Fokker-Planck limits. We can cite the books [8,9,19,28], and the surveys [1,24] for details.

Despite these different tools, the microscopic interactions among agents are based on sociological theories like social impact and social pressure theory [3,13,23], where agents modify their opinions trying to fit in some social group, and the persuasive argument theory [11,25], where the new opinion appears after an interchange of arguments among agents.

However, the space of opinions is typically assumed as homogeneous, without considering the advantages of holding some specific opinions. Recently, in [30] we have proposed a model where the opinions are weighted by a function $\lambda(w)$ representing the attractiveness, advantages, or fitness of opinion $w$. If we call $w$, $w_*$ the actual opinions of two interacting agents, the new opinions $w'$, $w_*'$ are obtained from the following microscopic interaction rule

where $0<\gamma<1/2$ is a fixed parameter related to the strength of the interaction. Notice that the new opinion $w'$ is obtained moving $w$ toward $w_*$, thus implementing the tendency to compromise. Moreover the magnitude of the change is proportional to $\lambda(w_*)$ i.e. to the attractiveness of opinion $w_*$.

Let us mention briefly that in [6,7,22] there are two competing opinions, say $\pm L$, and an attitude spectrum $A = \{\pm 1, \pm 2, \ldots, \pm L\}$ which represents the strength of the opinion of an individual or its degree of conviction. Now, the behavior of agents depends on their attitudes, and it is not a characteristic associated with an opinion. A similar mechanism can be found in [2,38], where the microscopic interaction rule is given by

where $P$ is usually equal to zero at the boundary of the space of opinions, representing that extreme opinions are difficult to change. Despite the apparent similarity between (1) and (2), both dynamics are very different, see [30].

In this work we study the long time behavior of a population interacting through rule (1), and we add more heterogeneity on the agents by introducing two parameters $p$ and $q$ modeling the power of persuasion and the stubbornness of each agent, as in [31].

In Section §2 we introduce the model and the mean field first order equation, satisfied by the density of agents in the space of opinions. In Section §3 we focus on the effect of the fitness of opinions in an homogeneous population, and we derive the precise value of the consensus opinion $m_\infty$, namely

where $\Lambda$ is a primitive of $\lambda$, and $f_0$ is the initial distribution of agents on the space of opinions. Let us mention that we need $\lambda>c>0$ for some constant $c$. This hypotheses cannot be relaxed, due to the singular behavior of the population located near the zeros of $\lambda$, and a symmetry breaking phenomena when $\lambda(m_\infty) = 0$. In Section §4 we analyze the formation of consensus in a non-homogeneous population. We include in Section §5 some agent based simulations, showing a good agreement for a finite population of the results predicted by the mean field equation.

2.   Description of the model

2.1.   Opinion, persuasion ability and stubbornness parameters

Let us introduce our model of opinion formation. We consider a population composed of infinitely many agents. The opinion of an agent with respect to some statement is represented by a real number $w\in [-1,1]$ (meaning $-1$ being completely in disagreement with the statement and $1$ in complete agreement).

In addition, we take into account the ability (or difficulty) of an individual to persuade another agent, denoted by $p\in[0,1]$; if $p = 0$ the agent has no persuasion at all, while $p = 1$ corresponds to perfect orators. We also take into account the stubbornness of an agent denoted by $q\in [0,1]$, where $q = 0$ corresponds to a stubborn agent, who will never be affected by other's opinion, and $q = 1$ entails a very volatile agent who always takes into account other agents' opinions.

Each agent is thus characterized by the three parameters $(w,p,q)$. Moreover, we assume that the power of persuasion $p$ and the stubbornness $q$ remain fixed for each of the agents, who may modify their opinion $w$ after binary encounters.

2.2.   Influence functions and microscopic interactions

Our model also takes into account the possibility that the fitness of an individual opinion may affect other agents. We introduce an influence function $\lambda(w)$, with $w\in[-1,1]$, representing the influence exerted by an individual with opinion $w$. Usual examples of influence functions are

● Quadratic : $\lambda(w) = |w|^2;$

● Linear : $\lambda(w) = |w|$;

● Uniform : $\lambda(w) = 1$;

● Co-Linear : $\lambda(w) = 1-|w|$;

● Co-Quadratic : $\lambda(w) = (1-|w|)^2$.

Individuals which held extreme opinions will have more influence under the linear and quadratic functions (extremist influence functions), while the co-linear and co-quadratic functions endow more influence to agents with moderate opinions, (centrist influence functions).

We now describe the up-dating rules of the opinions. Consider two interacting agents with parameters $(w,p,q)$ and $(w_*,p_*, q_*)$ before the encounter. Denote by $(w',p', q'_*)$ and $(w'_*,p'_*,q'_*)$ the new values for the parameters after the interaction, respectively. As we mentioned before, the parameters $p,p_*,q,q_*$ will remain unchanged: $p' = p$, $p_*' = p_*$, $q' = q$ and $q_*' = q_*$. Regarding the up-dating of the opinion, we propose the following rule:

Observe that the term $\gamma \lambda(w^*)p_*(w_*-w)$ reflects the tendency to compromise, which is proportional to the power of persuasion, $p_*$, and the influence exerted by the other agent opinion through $\lambda(w^*)$, as well as her/his own stubbornness, $q$. Here, $\gamma$ is a given real number in $(0,1/2)$ modelling the strength of the tendency to compromise.

2.3.   Macroscopic kinetic model: a first order, mean field, nonlocal transport equation

First of all we establish some notations. Let $K = [-1,1]\times [0,1]\times [0,1]$ be the space of the triple $\varpi = (w,p,q)$. We denote $P(K)$ the convex cone of probability measures on $K$. We endow $P(K)$ with the weak* convergence defined as $f_k\to f$ if $\int \phi\,df_k\to \int\phi\,df$ for any $\phi\in C(K)$. Since $K$ is compact, $P(K)$ is also compact. It is well-known that this topology can be metricized in various ways. We will use the Monge-Kantorovich $W_1$ distance defined for any $\mu,\nu\in P(K)$ by

where the supremum is taken among the 1-Lipschitz functions $\phi:K\to \mathbb{R}$; namely, $|\phi(x)-\phi(y)|\le |x-y|$ for any $x,y\in K$. We refer to [39] for more details on this distance.

Denote as $f^\gamma_t(\varpi)$ the distribution of agents on the triple $\varpi$ at time $t\geq0$ when agents interact following the rule (3). Indeed, $f^\gamma_t$ is a probability measure on $K$, denoted as well in the sequel as $df^\gamma_t$ or $f_t^\gamma(\varpi)d\varpi$. Bear in mind that $f^\gamma_t$ may not necessarily be absolutely continuous with respect to Lebesgue measure. In fact, $f^\gamma_t$ could be a Dirac measure.

In the case of binary interactions and assuming, as usually done, that the joint distibution $f_t^\gamma(\varpi,\varpi_*)d\varpi\,d\varpi_* = f_t^\gamma(\varpi)f_t^\gamma(\varpi_*)d\varpi\,d\varpi_*$, it can be shown (see e.g. [28]) that $f^\gamma_t$ is the unique solution of a Boltzmann type equation that takes into account a mean value of all possible interactions. Namely, for any observable $\phi\in C(K)$,

A fixed point argument yields existence and uniqueness for this equation.

Our next purpose is to describe the time evolution of this density for a given initial condition $f_0 \in P(K)$. To this end, we perform a grazing limit in the Boltzmann type equation above, that is, as the parameter $\gamma$ adjusting the strength of the interactions (3), goes to zero. Indeed we can then approximate

to obtain, after simplification,

with

Notice that $\langle p\lambda\rangle^\gamma$ is the mean value at time $t$ of $p\lambda(w)$, and $m^\gamma(t)$ the mean opinion weighted by the normalized actions of both the power of persuasion and the influence exerted by the agents.

Rescaling time considering $\tau = \gamma t$, so that $\frac{d}{d\tau} = \frac{1}{\gamma}\frac{d}{dt}$, and letting $f^\gamma_{\tau}: = f^\gamma_t$, $m^\gamma(\tau): = m^\gamma(t)$, we obtain the approximation

This procedure can be justified showing that $f^\gamma_{\tau}$ converges as $\gamma\to 0$ to $f_\tau$, the unique solution of the first order transport equation

This is the core of the following Theorem, whose proof (except for minor changes) can be found in [31].

Theorem 2.1. There exists $f\in C([0,+\infty),P(K))$ (where $P(K)$ is endowed with the weak convergence) such that, as $\gamma\to 0$, $f^\gamma_{\tau}: = f^\gamma_t$ with $\tau = \gamma t$ converges to $f_\tau$ in $C([0,T],P(K))$ for any $T>0$. Moreover $f$ is the unique solution in $C([0,+\infty),P(K))$ of

for any $\tau\ge 0$ and any $\phi\in C^1(K)$. Here,

is the mean value at time $\tau$ of $p\lambda(w)$, and

is the mean opinion weighted by the normalized actions of both the power of persuasion and the influence exerted by the agents.

Remark 1. Notice that (6) is the weak formulation of the transport equation

The rest of the paper is devoted to the study of the long-time behaviour of $f_\tau$. The next section deals with the simplest case of an homogeneous population i.e. where there are no parameter $(p,q)$. The case of a hetereogeneous population with stubborn people is presented in section 4.

In both cases we will prove that $f_\tau$ converges as $\tau\to +\infty$ to an explicit measure $f_\infty$ and provide an explicit estimation of the rate of convergence in term of the $W_1$-distance defined in (4). We obtain in particular that the distribution of opinion converges to a Dirac mass located at an explicit limit opinion $w_\infty$. Since $f^\gamma_\tau\to f_\tau$ as $\gamma\to 0$ we deduce that for a fixed $\gamma\ll 1$, the solution $f^\gamma_\tau$ of the Boltzmann equation on the time scale $\tau$ will be very close to $f_\infty$ for $\tau\gg 1$. However for a fixed $\gamma$ the approximation of the Boltzmann equation by the first order equation (6) is valid up to an error term. Indeed (5) shows that the Boltzmann equation will be well-approximated by the first order equation plus an aditional diffusion term $\frac{\gamma}{2} \partial_{ww}(D^\gamma_\tau(\varpi)f^\gamma_\tau)$. We thus expect the solution $f^\gamma_\tau$ of the Boltzmann equation to be very close to a smoothed version of $f_\infty$.

This indeed happens as shown by the numerical experiments at the end of the paper. We also refer to [28] for more details on this issue.

It is worth to emphasize here an important difference with respect to the model studied in [31], which corresponds to taking $\lambda(w) = 1$ as the influence function. Indeed, when $\lambda = 1$ then $m(t) = \frac{1}{\langle p\rangle}\int_K pw\,dg_t$, and the weight $\langle p\rangle$ is constant in time. However, for a non-constant $\lambda$, $\langle p\lambda\rangle$ is a priori non-constant in time. This forced us to develop new arguments in order to tackle this more delicate situation, when studying the asymptotic behaviour of $m(t)$.

From now on we denote $f_t: = f_\tau$ for ease of notation.

Before going further we recall a useful trick concerning transport equations.

2.4.   An useful change of variables

If $f\in P([-1,1])$ is a probablity measure on $[-1,1]$, and $F: \mathbb{R}\to [0,1]$ is its cumulative distribution function, (namely $F(x) = f((-\infty,x])$ - it is a non-decreasing and right-continuous function with left limit), one can consider the generalized inverse of $F$, defined as $F^{-1}:[0,1]\to[-1,1]$

Observe that $F^{-1}$ is also non-decreasing and left-continuous with right limit in $(0,1]$. Furthermore, the following inclusion holds

In addition, for any $x\in [-1,1]$ and any $\rho\in [0,1]$ we have the inequalities

See the note of Embrechts and Hofert [16] for the above (and further) properties of $F^{-1}$. Moreover, it can be proved that

for any $\phi$ integrable.

This change of variables will be a key point in the subsequent arguments. More precisely, consider $f\in C([0,\infty);P([-1,1]))$ satisfying a transport equation of the form

Denote as $F_t$ the cumulative distribution function of $f_t$, and $X_t = F_t^{-1}$ its generalized inverse. With the use of (12) we can rewrite the weak form of equation (13) in a much simpler form. This is the core of the next result:

Proposition 1. Let $v:[0,+\infty)\times [-1,1]\to \mathbb{R}$ be continuous and globally Lipschitz with respect to the second variable. Then, $f \in C([0,+\infty),P([-1,1]))$ is a weak solution of (13) in the sense that for any $\phi\in C^1([-1,1])$ and any $t>0$,

if and only if for any $r\in (0,1]$, $X_t(r)$ is a solution of

Here, $X_0$ is the generalized inverse of $F_0$ (the cumulative distribution function of $f_0$).

The proof can be found essentially in Theorem 3.1 in [2]. See also [31], where it is rewritten under the point of view of the ordinary differential equation for the flux (15).

3.   The effect of attractive opinions in an homogeneous population.

In this section we evaluate the impact of the attractiveness of the opinion independently of any other consideration. Thus, an agent is now completely characterized by its opinion $w\in [-1,1]$, and when interacting with an agent of opinion $w_*\in[-1,1]$ the resulting opinion $w'$ is

This synergy after encounters was indeed introduced in [30]. A simpler form that takes the model with homogeneity, allowed us to find the kinetic equations with the use of the empirical distribution for $N$ agents, $f_N(w,t) = \frac 1N \sum _{i = 1}^N\delta_{w_i(t)}$, and taking limits as $N\to \infty$.

The resultant continuous distribution $f_t\in P([-1,1])$ verifies the mean field first order equation

where

which is consistent with (6) taking $p = q = 1$.

Moreover, in [30] we identified a conserved quantity for the evolution equation (17), whenever $\lambda\in C([-1,1])$. We observed that the function $\int_{-1}^1\Lambda(w)\,df_t(w)$, being $\Lambda$ an antiderivative of $\lambda$, remains constant in time. Indeed, according to (17),

Recall that $\langle\lambda\rangle m_t = \int_{-1}^1 w\lambda(w)\,df_t(w)$, thus the right hand side vanishes and then

Sending $t\to +\infty$ and assuming that consensus occurs in the sense that $f_t\to \delta_{m_\infty}$ for some $m_\infty$, we deduce

Hence, the candidate to be the value of consensus is

In the next result we perform a rigorous proof of the dynamics contraction towards this value.

Theorem 3.1. Assume that $\lambda:[-1,1]\to \mathbb{R}$ is a continuous function such that, for some $\underline{\lambda}>0$,

being $\mathit{\mbox{conv}}(\mathit{\mbox{supp}}(f_0))$ the convex hull of $\mathit{\mbox{supp}}(f_0)$.

Then, there exists $m_\infty: = \lim_{t\to +\infty} m_t \in [-1,1]$ such that

Moreover, the limit opinion $m_\infty$ is given by

where $\Lambda$ is an antiderivative of $\lambda$.

Before the proof, some remarks are in order:

Remark 2. Notice that we just require $\lambda$ to be positive on the convex hull of the support of the initial distribution $f_0$. This is due to the fact that the dynamic is contractive, in the sense that $supp(f_t)\subset supp(f_0)$, $t\ge 0$.

Remark 3. In the case $\lambda\equiv 1$ we have $\Lambda(w) = w$ so that the consensus opinion $m_\infty$ is simply the initial mean opinion, see [34].

The proof of Theorem 3.1 goes as follows:

Proof of Theorem 3.1. Let $X_t$ be the generalized inverse of the cumulative distribution function corresponding to $f_t$. According to Proposition 1 we can rewrite the equation satisfied by $f_t$ as

Moreover,

Since $X_t$ is non-decreasing,

and similarly $m_t \ge X_t(0^+)$. Thus,

Note that $\lambda\ge 0$ implies that $\langle \lambda\rangle\ge 0$. As a result $\partial_tX_t(1)\le 0$, thus $X_t(1)$ is non-increasing. An identical argument shows that $X_t(0^+)$ is non-decreasing. Since $[X_t(0^+),X_t(1)]$ is the convex hull of $\mbox{supp}(f_t)$ this proves that

Indeed, $\lambda(w)\ge \underline{\lambda}>0$ for any $w\in \mbox{supp}(f_t)$. Consequently,

and Gronwall's Lemma gives, for any $r\in (0,1]$ and any $t\ge 0$,

In particular,

which reveals that the length of $\mbox{supp}(f_t)$ goes to 0.

Recalling that $X_t(0^+)\le m_t\le X_t(1)$ for any $t$, $X_t(0^+)$ increases and $X_t(1)$ decreases. Accordingly, $\mbox{supp}(f_t)$ shrinks to a limit point $\lim_{t\to +\infty} m_t: = m_\infty\in[X_t(0^+), X_t(1)]$ for any $t$. In fact, for any $r\in (0,1]$,

Inequality (20) follows now by observing that

It remains to show (21). Notice that $\Lambda$ is $C^1$, $\Lambda$ is positive and increasing on $\mbox{conv}(\mbox{supp}(f_0))$. Thus, $\Lambda$ defines a bijection from the interval $\mbox{conv}(\mbox{supp}(f_0))$ on its image

Furthermore,

and consequently $\int \Lambda \,df_0\in A$. Identity (18) entails that $m_\infty\in \mbox{conv}(\mbox{supp}(f_0))$ and the proof of (21) concludes.

We would like to close this section by emphasizing some relevant considerations of the result above.

3.1.   Positivity of the influence function

First of all, the hypothesis of strict positivity of the influence function on the convex hull of the initial density is absolutely necessary to obtain Theorem 3.1.

Indeed in [30] we already noticed singular behaviour at the roots of $\lambda$ in the simulations. Bear in mind that the value $m_\infty$ specified in (21) is expected to hold in mean. That is the reason why, even starting from identical initial conditions, simulations could give different values for $m_\infty$, due to the random fluctuations from realization to realization.

This fact becomes more evident the larger the order of the zero is. If $\lambda$ has a high order zero at $z$, $\Lambda$ is almost a constant function close to it, and the inverse function $\Lambda$ is defined although it is very sensitive to small changes. If in addition we have that

two phenomena occur. One is the slow formation of consensus due to a frozen dynamics due to the small values that $\lambda$ takes. The second one is a symmetry rupture, and after a long time consensus is reached above or below $z$.

3.2.   Smoothness of the influence function

Let us note that the first order equation is interpreted in a weak sense, so the differentiability of $\lambda$ is not required. However, in the next section we need to impose the condition $\lambda\in W^{2,\infty}([-1,1])$, which seems to be a technical hypothesis. Indeed, we can consider $\lambda(w) = |w|^{1/2}$ and given an uniform initial distribution of agents, the consensus is reached at $m_\infty = 0$, see [30]

3.3.   Comparison with other models

It is worth to mention a related dynamics appearing in [2,38], where agents are influenced by their own opinion when interact,

Typical examples of function $P$ are non-increasing on $|w|$, representing the fact that the extremist people are more likely to remain in their believes. In fact, the roots of the function $P$ represent the reticence of the individual that eventually adopted that opinion, to change it after subsequent encounters. Let us observe that in this model stubborn agents appear dynamically when they approach the zeros of $P$. The qualitative differences in the dynamic between (22) and our model (1) were shown numerically in [30]. Moreover, when consensus occurs in (22), its value is completely different from the value (21) obtained here. In the case of (22), equation (17) reads

where $\langle w\rangle = \int_{-1}^1 w \, df_s(w)$ is the mean opinion at time $s$. If we assume that $P(w)\ge \underline{P}>0$, $w\in [-1,1]$, it follows that an antiderivative $\Pi$ of $1/P$ (and not of $\lambda$ as in our model) is conserved. Indeed

which is equal to $0$. Consequently, if consensus occurs in the sense that $f_t\to \delta_{\tilde m_\infty}$ for some limit opinion $\tilde m_\infty$, then $\tilde m_\infty$ must satisfy

so that $\tilde m_\infty = \Pi^{-1}\Big( \int \Pi \,df_0 \Big)$. We present in section §5 below some numerical experiments to validate the formula for $\tilde m_\infty$.

4.   Asymptotic behaviour of an heterogeneous population given the fitness of opinions.

This section is devoted to study the long-time behaviour of the unique solution to the transport equation (6), arising from an initial distribution $f_0$ of the form

being $\alpha_0\in (0,1]$ the proportion of stubborn agents in the population, and $f_0^0,\,f_0^1\in P(K)$ the initial distributions of agents with parameters $(w,p,q)$ in the stubborn and non-stubborn population, respectively. Observe that since the stubborn agents do not change their opinion in an interaction, $f_t$ will evolve as

4.1.   Heuristic idea of the limit

Before stating our main result, we give an informal deduction of the possible value to be the limiting opinion, presuming that the non-stubborn agents reach a consensus at $m_\infty$.

In view of the transport term, whenever consensus is reached among the non-stubborn agents, it should take place at $m_\infty: = \lim_{t\to\infty} m_t$, whenever this limit exists. To search for the candidate to $m_\infty$, we argue as follows. Accepting that consensus is reached at $m_\infty$, then

where $f_0^1(p,q)dpdq$ is the distribution of the non-stubborn population on the $(p,q)$-parameters. This density is constant in time since $(p,q)$-parameters are unaffected by the dynamics. As a result, we can pass to the limit as $t\to +\infty$ in the definition of $m_t$, namely

On the one hand,

While on the other hand,

Taking now limits in (25) we get

Since $\alpha_0>0$,

Conclusion: If the non-stubborn population reaches consensus, then the consensus opinion $m_\infty$ is specified by

whenever the term

that stands for the mean value of $p\lambda$ within the stubborn population, does not vanish.

This shows that, admitting long-time consensus among the non-stubborn population, its shared opinion $m_\infty$ is the mean opinion value weighted by the normalized $p\lambda$ (the power of conviction times the influence function). Observe that this mean value is taken just within the stubborn population. Thus, if a common limit opinion exists, it is determined by the stubborn agents.

At this stage, it is crucial to underline a relevant fact about the candidate for consensus found above. If the influence function is constant, this model lays on our previous work studied in [31], and hence the value for $m_\infty$ specified in (26) turns out to be

4.2.   Statement and proof of the main result

Our main result shows that non-stubborn agents indeed reach consensus asymptotically at the value $m_\infty$ determined by (26). We also provide an estimate on the rate of convergence towards the consensus, in terms of the $W_1$-distance between $f_t^1$ and its limit $f_0^1(p,q)dpdq\otimes \delta_{m_\infty}$.

Before stating it recall that $f_0^0$ and $f_t^1$ denote the distribution of the stubborn and non-stubborn agents on $(w,p,q)$, and we denote $\alpha_0\in (0,1]$ the proportion of stubborn agents in the population. By $f^1_{t|(p,q)}\in P([-1,1])$ we understand the distribution of opinions within the group of non-stubborn agents having parameters $(p,q)$. Its existence is guaranteed by Jirina's Theorem. There are several classical references on this subject, for example [4,36].

We are now ready to state our main result of this section

Theorem 4.1. Assume $f_0^1\in P(K)$ is supported in $\{q\ge \varepsilon_0\}$ for some $\varepsilon_0>0$ and that the map

is globally Lipschitz for the $W_1$-distance: there exists $L>0$ such that for any $(p,q),(p',q')\in [0,1]\times [\varepsilon_0,1]$,

In addition, assume that $\lambda\in W^{2,\infty}([-1,1])$ verifies that there exists $\underline{\lambda}>0$ such that $\lambda(w)\ge \underline{\lambda}$ for any $w\in [-1,1]$.

Then, for any $t\ge 0$,

where $m_\infty$ is specified in (26) and stands for the mean opinion weighted by the normalized power of persuasion multiplied by the influence function within the group of stubborn agents. The mean value of $p\lambda$ among the stubborn agents, $\langle p\lambda\rangle_0$ is defined in (27). Moreover

The global idea of the proof of Theorem 4.1 follows the lines of the proof in [31], where the case $\lambda\equiv 1$ is treated. Notice however that when $\lambda\equiv 1$ then $\langle p\lambda\rangle = \langle p\rangle$ is constant in time whereas for an arbitrary $\lambda$ it is time-varying quantity. This introduces new difficulties in many of the steps into which the proof is divided.

In the first step we consider $f^1_{t|(p,q)}\in P([-1,1])$, the conditional distribution of opinion among the agents with parameter $(p,q)$. This conditional distribution turns out to be the unique solution to the following transport equation. Furthermore, it is $(p,q)$-Lipschitz with respect to the Wasserstein distance. These facts are summarized below and the proof can be easily adapted from the case $\lambda$ constant [31].

Step 4.1. For any $(p,q)\in {\text{supp}}(f_0(p,q)dpdq)$, $f^1_{t|(p,q)}$ is the unique solution to

in $C([0,+\infty),P([-1,1]))$.

Moreover, the function $(p,q)\to f^1_{t|(p,q)}$ is Lipschitz with respect to the Wasserstein distance $W_1$. Namely, for any $(p,q),(p',q')\in [0,1]\times [\varepsilon_0,1]$,

Furthermore, it fulfils

In the next item, we take advantage of the tendency to compromise modeled by the interaction rules (3) to prove that non-stubborn agents with given $(p,q)$ parameters tend to synchronize their opinions. Conditioning to values $(p,q)$ we declare

and

the mean value of $\lambda$ and the mean opinion among the agents with parameter $(p,q)\in [0,1]\times [\varepsilon_0,1]$, respectively.

Step 4.2. For any $(p,q)\in [0,1]\times [\varepsilon_0,1]$ there holds

being $\langle p\lambda\rangle_0$ defined in (27).

Proof. Let $X_t$ be the generalized inverse of the cumulative distribution function corresponding to $f_{t|(p,q)}$. According to Proposition 1 we can rewrite (30) as

In particular

The nonnegativity of $\lambda$ entails that

hence

Gronwall's Lemma implies now that

which sending $r\to 0^+$ gives

This shows that the length of the support of $f_{t|(p,q)}$ goes to 0.

Invoking (12), we express $m(t,p,q)$ in terms of the generalized inverse,

Since $X_t$ is non-decreasing,

and similarly $m(t,p,q) \ge X_t(0^+)$. Thus,

Therefore, for any $r\in (0,1]$,

The proof now concludes by noticing that

As a result of the previous step, it is desirable to study the asymptotic behavior of the function $m(t,.)$ as $t\to +\infty$.

Step 4.3. For any $t\ge 0$ and any $(p,q)\in [0,1]\times [\varepsilon_0,1]$ the function $m(t,p,q)$ declared in (33) satisfies

where $m_\infty$ and $\langle p\lambda\rangle_0$ are given in (26) and (27) respectively, and

Proof. Using equation (30), the evolution in time of the conditioned mean $\langle\lambda\rangle_{(p,q)}$ defined in (32) behaves as

and

Thus,

Summing up,

Estimate (34) allows to bound the intergral in the r.h.s. as follows. Let $\phi(w) = (m_t-w) \lambda'(w) (w-m(t,p,q))$ so that

According to the definition of the $W_1$-distance we obtain

It is easily seen that $Lip(\phi)\le 4(\|\lambda'\|_\infty + \|\lambda''\|_\infty)$. Accordingly to (34), we deduce

Thus,

with

given that the assumption $\lambda(w)\ge \underline{\lambda}>0$ for any $w\in [-1,1]$ entails that $\langle \lambda\rangle_{(p,q)}\ge \underline{\lambda}>0$ for any $(p,q)$.

We now focus on the second term in the right hand side of (40). First,

where the integral can be written using (31) as

Recalling in addition that $\langle pw\lambda\rangle_0 = \langle p\lambda\rangle_0 m_\infty$ with $m_\infty$ given in (26), we obtain

On the other hand,

Merging the former identities together,

According to (34) we have for any $(p,q)$ that

As a result,

with

where we used that $m(t,p,q)\in [-1,1]$ for any $(t,p,q)$. We deduce (38) taking $R(t,p,q) = \tilde{R}(t,p,q) + \hat{R}(t,p,q)$.

To understand intuitively the infinite system of equations (38) it is useful to consider the elementary situation where only a finite number of values for $(p,q)$, $q\neq 0$, are present in the population, namely $(p_1,q_1),...,(p_N,q_N)$. Then, $f^1_t$ takes the simpler form

where $g_t^i: = f_{t|(p_i,q_i)}$ is the distribution of opinion in the $(p_i,q_i)$-population and $\alpha_i\in (0,1]$ is the proportion of $(p_i,q_i)$ agents in the non-stubborn population. Letting $m_t^i: = m(t,p_i,q_i)$ we can rewrite the system (38) as

where $R_t^i: = R(t,p_i,q_i)$, $A = \alpha_0\langle pw\lambda\rangle_0$, $B = \alpha_0\langle p\lambda\rangle_0$, $c_j = (1-\alpha_0)\alpha_j p_j.$ The right hand side of (41) is composed of three terms. The first one $A-Bm_t^i$ drives $m_t^i$ towards $A/B = m_\infty$, which is the desired asymptotic state. The sum includes a coupling between $m_t^i$ and all the $m_t^j$, $j = 1,\ldots, N$, whose effect contributes to synchronize them as in, e.g. the Cucker-Smale model [14]. The error term decreases exponentially fast to 0, thus it lacks relevance in the asymptotic behaviour. We thus expect that $\lim_{t\to +\infty}m_t^i = m_\infty$ for any $i = 1,..,N$. We will prove in the sequel that this intuition is indeed correct in general.

The regularity of $m(t,p,q)$ with respect to $(p,q)$ plays an important role in the convergence of $m(t,p,q)$ to $m_\infty$ as $t\to +\infty$.

Step 4.4. For any $t\ge 0$, the function $(p,q)\in supp\,(f_0^1(p,q)dpdq)\to m(t,p,q)$ is Lipschitz.

Proof. According to Step 4.1, $(p,q)\to f^1_{t|(p,q)}$ is Lipschitz with respect to the Wasserstein distance $W_1$: for any $(p,q),(p',q')\in [0,1]\times [\varepsilon_0,1]$,

The functions $\langle\lambda\rangle_{(p,q)}$ and $\langle w\lambda\rangle_{(p,q)}$ are also Lipschitz in $(p,q)$ for a given $t$, whenever we assume that $\lambda$ is Lipschitz. Indeed for a given $t\ge 0$ and any $(p,q),(p',q')$,

and in the same way

Consequently, for a given $t\ge 0$, the functions $\langle\lambda\rangle_{(p,q)}$ and $\langle w\lambda\rangle_{(p,q)}$ are continuous in $(p,q)$. Since $\langle\lambda\rangle_{(p,q)}\ge \underline{\lambda}>0$ for any $(p,q)$, it follows that $m(t,p,q) = \langle w\lambda\rangle_{(p,q)}/\langle \lambda\rangle_{(p,q)}$ is continuous in $(p,q)$ for any $t\ge 0$.

Step 4.5. For any $(p,q)\in {\text{supp}}\,(f_0^1(p,q)dpdq)$ and any $t\ge 0$ it holds that

where

Proof. Relation (38) implies that for any $q\in [\varepsilon_0,1]$ and $t\geq 0$,

Recall that $m(t,.)$ is continuous by the previous Step and $\text{supp}(f_0^1(p,q)dpdq)$ is compact. Take some $(p^*,q^*)$ such that

In particular, we can write (42) at $(p^*,q^*)$ as

The choice of $q^*$ assures that

The cancelation of these two terms and $q^*\ge \varepsilon_0$ gives

Let $h(t;(p,q)) = |m(t,p,q)-m_\infty|^2$. Notice that $t\mapsto h(t;(p,q))$ is a $C^1$ function, since $m$ is $C^1$ in time. Moreover, from (42) it follows that $| \partial_th(t;(p,q))|\le C$. Since $h(t,.)$ is continuous for any $t$ by the previous Step, we obtain that $h$ is continuous in $(t,p,q)$. According to the Measurable Selection Theorem (see e.g. §18.19 in [10]) we can choose $(p^*,q^*)$ to be a measurable function of $t$.

The Envelope Theorem (see Theorem 4.2 below) ensures that the function $V(t)$ defined by

is absolutely continuous with derivative

Furthermore, in view of (43) and (39)

where

Dividing by $2\sqrt{V}$ we obtain

which integrated gives

This shows that

for any $(p,q)\in \mbox{supp}(f_0^1(p,q)dpdq)$, as desired.

In the course of the proof of the previous step we used the following envelope Theorem, due to Milgrom and Segal in [26]:

Theorem 4.2. Consider the function $V(t): = \max_{x\in X} h(x,t)$, $t\in [0,1]$ being $X$ a set. Suppose that $h$ is absolutely continuous with respect to $t$, for any $x$. Moreover, admit that there exists $b\in L^1([0,1])$ such that $| \partial_th(x,t)|\le b(t)$ for any $x\in X$ and almost any $t\in [0,1]$. Then $V$ is absolutely continuous.

Assuming further that $h$ is differentiable in $t$, for any $x\in X$, and that for any $t\in [0,1]$ the set $X(t): = \mathit{\text{argmax}}h(.,t)$ is non-empty. Then, for any selection of $x^*(t)\in X(t)$ we have

We are now in position to finish the proof of Theorem 4.1.

Step 4.6. There holds

for any $t\ge 0$.

Proof. Step 4.5 ensures that for any $t\ge 0$ and any $(p,q)\in \text{supp }f_0^1(p,q)dpdq$,

while according to (34), we have that

In fact, we claim that

Let $\psi:K\to \mathbb{R}$ be an arbitrary 1-Lipschitz function. Then,

The inner integral is bounded above by $W_1( f_{t|(p,q)}, \delta_{m_\infty})$ since $\psi(.,p,q)$ is 1-Lipschitz, which implies that

The claim follows by taking supremum among the functions $\psi$ 1-Lipschitz.

5.   Numerical experiments

In this section we perform some agent based simulations of the results studied along this work. We assess qualitatively the effect of the influence function

on the dynamics. We first consider a homogeneous population and ratify that, indeed, the conclusions of Theorem 3.1 are true. Then, to illustrate the conclusion of Theorem 4.1, we add to this scenario several stubborn agents positioned at two specific values for $\omega$. We finally analyze the evolution of a completely heterogeneous population with different values of $q$.

In all of the simulations presented here, we consider a population of $N = 1000$ agents. The opinion of the non-stubborn agents are initially uniformly distributed in $[-1,1]$. At every time slot, each one of the $N$ agents interacts with a randomly selected agent and then, updates its opinion following the interaction rule (3) with $\gamma = 0.01$.

5.1.   Numerical experiments depicting the opinion attractiveness in a homogeneous population

Since opinions are initially distributed uniformly in $[-1,1]$, i.e. $f_0 = \frac12 1_{[-1,1]}$, the theoretical value $m_\infty$ of the consensus(given by (21) in Theorem 3.1), satisfies

where $\Lambda$ is an antiderivative of $\lambda(w) = (w-0.5)^2+\varepsilon$, $\varepsilon = 0.01$. A numerical resolution of this nonlinear equation gives

In Figure 1 (left) we show the time evolution of the opinions of 10 agents of the population (blue curves). Consensus clearly occurs at the value $m_\infty$, indicated by the horizontal red dashed line.

This is completely in contrast to consider that the influence is exerted, instead, by one's own opinion agent, see [2,38]. Indeed, under the interaction rules (22), we prove in subsection §3.3 that if consensus is reached then, the consensus opinion $\tilde m_\infty$ must satisfy

where $\Pi$ is an antiderivative of $1/P$. Taking as $f_0$ the uniform distribution on $[-1,1]$ and $P(w) = \lambda(w) = (w-0.5)^2+\varepsilon$, we numerically obtain

The result of the agent-based simulation shown in the right figure of Figure 1 confirms that consensus takes place at $\tilde m_\infty$ (indicated by the red dashed horizontal line).

We can also observe from the simulation that the consensus is attained much faster for the interaction rule (16) than for (22). Recall that the interaction rule (22) takes into account the attractiveness of the opinion of the agent one is interacting with. Intuitively, this is due to the fact that agents with opinion $w$ such that $P(w)\approx 0$ are almost stubborn agents: they change opinion very slowly. This is clear from the figures in Figure 2, where we plot for the two interaction rules (16) (left) and (22) (right) the logarithm of the length of the convex hull of $\text{supp}\,f_t$, the support of the distribution $f_t$ of opinion at time $t$. Namely, we depict

In the rest of this section we examine the impact of stubborn agents combined with the influence function.

We consider a population of $N = 1000$ agents with a proportion $\alpha_0$ of stubborn agents. To balance the effect between stubbornness and the attractiveness of the other's opinions, every agent has $p = 1$. As before the influence function is $\lambda(w) = (w-0.5)^2+\varepsilon$, $\varepsilon = 0.01$, and $\gamma = 0.01$. Assume that half of the stubborn agents have opinion $w = 1/4$ and the other half opinion $w = 3/4$, so that

Notice that, in particular

and

The theoretical limit opinion is then

To evaluate the impact driven just by the proportion $\alpha_0$ of stubborn agents on the dynamics, we suppose that all of the non-stubborn agents have $q = 1$. We show in Figure 3 the time evolution of the opinion of 10 agents belonging to the non-stubborn population (blue curves). We consider a proportion of stubborn agents $\alpha_0 = 2\%$ (left) and $\alpha_0 = 60\%$ (right). The theoretical limit opinion $m_\infty$, is depicted on red dashed line.

We observe a perfect compliance between the agent-based simulations and the theoretical prediction. Furthermore, in the simulations the consensus is clearly achieved in two steps. First non-stubborn agents quickly reach a consensus, and then all together move slowly towards the final limit opinion $m_\infty$.

This fact is specially well observed on the left figure, where a smaller proportion of stubborn population is considered ($\alpha_0 = 2\%$). At first the impact of the stubborn agents is almost negligible, so that the opinions of the non-stubborn population evolve first at a value close to the predicted consensus opinion in absence of stubborn agents, namely here $-0.355$. In other words, the influence function $\lambda$ drives the dynamics at early stages, and then is wiped out by the stubborn agents. Comparing both figures, we also notice that a high proportion of stubborn agents accelerates the convergence towards the consensus, in accordance with estimation (29).

Finally, we wish to appreciate the qualitative impact of the parameter $q$ on the dynamics. We now consider that the values of $q$ for the non-stubborn population are distributed uniformly in $(0.2;1)$. The rest of the parameters are kept as before with $\alpha_0 = 0.6$. In Table 4 we show several snapshots at different times of the distribution of $(w,q)$ in the non-stubborn population ($w$ in the horizontal axis, $q$ in the vertical axis). We can clearly appreciate that agents with a high $q$, i.e. the most volatile agents, are indeed changing opinion quicker than the rest.

Acknowledgments

The authors appreciated the fine comments of both referees which helped to enhance the previous version of the manuscript.