Recent advances in opinion modeling:
control and social influence
Abstract
We survey some recent developments on the mathematical modeling of opinion dynamics. After an introduction on opinion modeling through interacting multiagent systems described by partial differential equations of kinetic type, we focus our attention on two major advancements: optimal control of opinion formation and influence of additional social aspects, like conviction and number of connections in social networks, which modify the agents’ role in the opinion exchange process.
1 Preliminaries
We introduce some of the essential literature on the opinion formation, by emphasizing the role of the kinetic description. New problems recently treated in the scientific community are outlined. Then, the mathematical description of the core ideas of kinetic models for opinion formation are presented in details.
1.1 Introduction
The statistical physics approach to social phenomena is currently attracting much interest, as indicated by the huge and rapidly increasing number of papers and monographies based on it [16, 37, 86, 89]. In this rapidly increasing field of research, because of its pervasiveness in everyday life, the process of opinion formation is nowadays one of the most studied application of mathematics to social dynamics [18, 25, 28, 58, 71, 99].
Along this survey, we focus on some recent advances in opinion formation modeling, which aims in coupling the process of opinion exchange with other aspects, which are closely related to the process itself, and takes into account the dependence on new variables which are usually neglected, mainly in reason of the mathematical difficulties that the introduction of further dimensions add to the models.
These new aspects are deeply connected and range from opinion leadership and opinion control, to the role of conviction and the interplay between complex networks and the spreading of opinions. In fact, leaders are recognized to be important since they can exercise control over public opinion. It is a concept that goes back to Lazarsfeld et al. [75]. In the course of their study of the presidential elections in the USA in 1940, it was found interpersonal communication to be much more influential than direct media effects. In [75] a theory of a twostep flow of communication was formulated, where socalled opinion leaders who are active media users select, interpret, modify, facilitate and finally transmit information from the media to less active parts of the population. It is clear that various principal questions arise, mainly linked to this twostep flow of communication. The first one is related to the ability to effectively exercise a control on opinion and to the impact of modern communication systems, like social networks, to the dynamics of opinions. The second is related to the fact that the less active part of the population is in general adapting to leaders opinion only partially. Indeed, conviction plays a major role in this process, by acting as a measurable resistance to the change of opinion.
These enhancements will be modeled using the toolbox of classical kinetic theory [89]. Within this choice, one will be able to present an almost uniform picture of opinion dynamics, starting from few simple rules. The kinetic model of reference was introduced by one of the authors in 2006 [99], and was subsequently generalized in many ways (see [86, 89] for recent surveys). The building block of kinetic models are pairwise interactions. In classical opinion formation, interactions among agents are usually described in terms of few relevant concepts, represented respectively by compromise and selfthinking. Once fixed in binary interactions, the microscopic rules are responsible of the formation of coherent structures.
The remarkably simple compromise process describes mathematically the way in which pairs of agents reach a fair compromise after exchanging opinions. The rule of compromise has been intensively studied [18, 19, 20, 51, 71, 105]. The second one is the selfthinking process, which allows individual agents to change their opinions in an unpredictable way. It is usually mathematically described in terms of some random variable [18, 99]. The resulting kinetic models are sufficiently general to take into account a large variety of human behaviors, and to reproduce in many cases explicit steady profiles, from which one can easily elaborate information on the opinion behavior [11, 27, 28, 29, 30]. For the sake of completeness, let us mention that many other models with analogous properties have been introduced and studied so far [17, 23, 25, 43, 63, 65, 64, 74, 91, 94, 97, 100, 104].
Kinetic models have been also the basis for suitable generalizations, in which the presence of leaders and their effect of the opinion dynamics has been taken into account [4, 33, 41, 42, 45, 57, 58]. Also, the possibility to establish an effective control on opinion, both through an external media or through the leaders’ action, has attracted the interest of the research community [6, 7, 21]. The methods here are strongly connected to analogous studies in crowd dynamics and flocking phenomena [5, 8, 26, 36, 60].
Further, the effect of conviction in the formation of opinion started to be studied. While in general conviction is assumed to appear as a static parameter in the opinion dynamics [47, 48, 84, 105], in [31] conviction has been assumed to follow a proper evolution in the society on the basis of interactions with an external background. Recently, a similar approach has been used to model the effect of competence and the socalled equality bias phenomena [91]. This point of view was previously applied to the study of the formation of knowledge in [90], as a starting point to investigate its role in wealth distribution. Indeed, many of the aforementioned models share a common point of view with the statistical approach to distribution of wealth [38, 44, 86, 90].
More recently, in reason of their increasing relevance in modern societies, the statistical mechanics of opinion formation started to be applied to extract information from complex social networks [1, 2, 12, 25, 15, 68, 69, 92, 103]. In these models the number of connections of the agents play a major role in characterizing the dynamic [9, 10, 50, 55, 67]. In [9] the model links the graph evolution modeled by a discrete connection distribution dynamic with the spreading of opinion along the network.
Before starting our survey, it is essential to outline the peculiar aspects od the microscopic details of the binary interactions which express the microscopic change of opinion. Indeed, these interactions differ in many aspects from the usual binary interactions considered in classical kinetic theory of rarefied gases. The first difference is that opinion is usually identified with a continuum variable which can take values in a bounded interval. Second, the postinteraction opinions are not a linear transformation of the preinteraction ones. Indeed, it is realistic to assume that people with a neutral opinion is more willing to change it, while the opposite phenomenon happens with people which have extremal opinions.
Once the details of the pairwise interactions are fixed, the explicit form of the bilinear kinetic equation of Boltzmann type follows [89]. One of the main consequences of the kinetic description is that it constitute a powerful starting point to obtain, in view of standard asymptotic techniques, continuous meanfield models with a reduced complexity, which maintain most of the physical properties of the underlying Boltzmann equation. The main idea, is to consider important only interactions which are grazing, namely interactions in which the opinion variable does not change in a sensible way, while at the same time the frequency of the interactions is assumed to increase. This asymptotic limit (hereafter called quasiinvariant opinion limit) leads to partial differential equations of FokkerPlanck type for the distribution of opinion among individuals, that in many cases allow for an analytic study.
The rest of the survey is organized as follows. In the remaining part of Section 1 we describe the basic kinetic model for opinion formation. It represents the building block for the binary opinion dynamic which is used in the subsequent Sections. Next in Section 2 we deal with control problems for opinion dynamics. First by considering an external action which forces the agents towards a desired state and subsequently by introducing a leaders’ population which acts accordingly to a prescribed optimal strategy. Here we start from the optimal control problem for the corresponding microscopic dynamic and approximate it through a finite time horizon or model predictive control technique. This permits to embed the feedback control directly into the limiting kinetic equations. Section 3 is then devoted to the modeling through multivariate distribution functions where the agents’ opinion is coupled with additional variables. Specifically we consider the case where conviction is also an evolving quantity playing a role in the dynamic and the case where agents interact over an evolving social network accordingly to their number of connections. Some final remarks are contained in the last Section and details on numerical methods are given in a separate Appendix.
1.2 Kinetic modelling
On the basis of statistical mechanics, to construct a model for opinion formation the fundamental assumption is that agents are indistinguishable [89]. An agent’s state at any instant of time is completely characterized by his opinion , where and denote two (extreme) opposite opinions.
The unknown is the density (or distribution function) , where and the time , whose time evolution is described, as shown later, by a kinetic equation of Boltzmann type.
The precise meaning of the density is the following. Given the population to study, if the opinions are defined on a subdomain , the integral
represents the number of individuals with opinion included in at time . It is assumed that the density function is normalized to , that is
As always happens when dealing with a kinetic problem in which the variable belongs to a bounded domain, this choice introduces supplementary mathematical difficulties in the correct definition of binary interactions. In fact, it is essential to consider only interactions that do not produce opinions outside the allowed interval, which corresponds to imposing that the extreme opinions cannot be crossed. This crucial limitation emphasizes the difference between the present social interactions, where not all outcomes are permitted, and the classical interactions between molecules, or, more generally, the wealth trades (cf. [89], Chapter 5), where the only limitation for trades was to insure that the postcollision wealths had to be nonnegative.
In order to build a realistic model, this severe limitation has to be coupled with a reasonable physical interpretation of the process of opinion forming. In other words, the impossibility of crossing the boundaries has to be a byproduct of good modeling of binary interactions.
From a microscopic viewpoint, the binary interactions in [99] were described by the rules
(1.1)  
In (1.1), the pair , with , denotes the opinions of two arbitrary individuals before the interaction, and their opinions after exchanging information between each other and with the exterior. The coefficient is a given constant, while and are random variables with the same distribution, with zero mean and variance , taking values on a set . The constant and the variance measure respectively the compromise propensity and the degree of spreading of opinion due to diffusion, which describes possible changes of opinion due to personal access to information (selfthinking). Finally, the functions and take into account the local relevance of compromise and diffusion for given opinions.
Let us describe in detail the interaction on the righthand side of (1.1). The first part is related to the compromise propensity of the agents, and the last contains the diffusion effects due to individual deviations from the average behavior. The presence of both the functions and is linked to the hypothesis that openness to change of opinion is linked to the opinion itself, and decreases as one gets closer to extremal opinions. This corresponds to the natural idea that extreme opinions are more difficult to change. Various realizations of these functions can be found in [89, 99]. In all cases, however, we assume that both and are nonincreasing with respect to , and in addition , . Typical examples are given by and .
In the absence of the diffusion contribution (), (1.1) implies
(1.2)  
Thus, unless the function is assumed constant, , the mean opinion is not conserved and it can increase or decrease depending on the opinions before the interaction. If is assumed constant, the conservation law is reminiscent of analogous conservations which take place in kinetic theory. In such a situation, thanks to the upper bound on the coefficient , equations (1.1) correspond to a granulargaslike interaction [89] where the stationary state is a Dirac delta centered on the average opinion. This behavior is a consequence of the fact that, in a single interaction, the compromise propensity implies that the difference of opinion is diminishing, with . Thus, all agents in the society will end up with exactly the same opinion.
We remark, moreover, that, in the absence of diffusion, the lateral bounds are not violated, since
(1.3)  
imply
Let denote the distribution of opinion at time . The time evolution of is recovered as a balance between bilinear gain and loss of opinion terms, described in weak form by the integrodifferential equation of Boltzmann type
(1.4)  
where are the postinteraction opinions generated by the pair in (1.1), represents a constant rate of interaction and the brackets denote the expectation with respect to the random variables and .
Equation (1.4) is consistent with the fact that a suitable choice of the function in (1.1) coupled with a small support of the random variables implies that both and . We do not insist here on further details on the evolution properties of the solution, by leaving them to the next Sections, where these properties are studied for the particular problems.
2 Optimal control of consensus
Different to the classical approach where individuals are assumed to freely interact and exchange opinions with each other, here we are particularly interested in such problems in a constrained setting. We consider feedback type controls for the resulting process and present kinetic models including those controls. This can be used to study the influence on the system dynamics to enforce emergence of non spontaneous desired asymptotic states.
Two relevant situations will be explored, first a distributed control, which models the action of an external force acting as a policy maker, like the effects of the media [6], next an indirect internal control, where we assume that the control corresponds to the strategies of opinion leaders, aiming to influence the consensus of the whole population [7]. In order to characterize the kinetic structure of the optimal control of consensus dynamics, we will start to derive it as a feedback control from a general optimal control problem for the corresponding microscopic model, and thereafter we will connect it to the binary dynamics.
2.1 Control by an external action
We consider the microscopic evolution of the opinions of agents, where each agent’s opinion , , evolves according the following first order dynamical system
(2.1) 
where has again the role of the compromise function defined in (1.1). The control models the action of an external agent, e.g. a policy maker or social media. We assume that it is the solution of the following optimal control problem
(2.2) 
with the space of the admissible controls. In formulation (2.2) a quadratic cost functionals with a penalization parameter is considered, and the value represents the desired opinion state. We refer to [3, 6, 36, 60] for further discussion on the analytical and numerical studies on this class of problems. The additional constraints on the pointwise values of given by and , are necessary in order to preserve the bounds of (see [35, 82]).
2.1.1 Model predictive control of the microscopic dynamics
In general, for large values of , standard methods for the solution of problems of type (2.1)–(2.2) over the full time interval stumble upon prohibitive computational costs due to the nonlinear constraints.
In what follows we sketch an approximation method for the solution of (2.1)–(2.2), based on model predictive control (MPC), which furnishes a suboptimal control by an iterative solution over a sequence of finite time steps, but, nonetheless, it allows an explicit representation of the control strategy [6, 35, 81, 80].
Let us consider the time sequence , a discretization of the time interval , where , for all and . Then we assume the control to be constant on every interval , and defined as a piecewise function, as follows
(2.3) 
where is the characteristic function of the interval . We consider a full discretization of the optimal control problem (2.1)(2.2), through a forward Euler scheme, and we solve on every time frame , the reduced optimal control problem
(2.4) 
subject to
(2.5) 
for all , and in the space of the admissible controls . Note that since the control is a constant value over the time interval , and depends linearly on through (2.5), the discrete optimal control problem (2.4) reduces to
(2.6) 
Thus, in order to find the minimizer of (2.4), it is sufficient to compute the derivative of (2.6) with returns us the optimal value expressed as follows
(2.7) 
Expression (2.7) furnishes a feedback control for the full discretized problem, which can be plugged as an instantaneous control into (2.5), obtaining the following constrained system
(2.8) 
A more general derivation can be obtained through a discrete Lagrangian approach for the optimal control problem (2.4)–(2.5), see [6].
Remark 1.
We remark that the scheme (2.8) furnishes a suboptimal solution w.r.t. the original control problem. In particular if is symmetric, only the average of the system is controlled. Let us set , and . Summing on equation (2.8) we have
(2.9) 
which implies . Thus, while the feedback control is able to control the mean of the system, it does not to assure the global consensus. We will see in the next Section how the introduction of a binary control depending on the pairs permits to recover the global consensus.
2.1.2 Binary Boltzmann control
Following Section 1.2, we consider now a kinetic model for the evolution of the density of agents with opinion at time , such that the total mass is normalized to one. The evolution can be derived by considering the change in time of depending on the interactions among the individuals of the binary type (1.1). In order to derive such Boltzmann description we follow the approach in [5, 59]. We consider the model predictive control system (2.8) in the simplified case of only two interacting agents, numbered and . Their opinions are modified according to
(2.10)  
where the feedback control term is derived from (2.7) and yields
(2.11) 
having defined . This formulation can be written as a binary Boltzmann dynamics
(2.12)  
All quantities in (2.12) are defined as in (1.1). The control , which is not present in (1.1), acts as a forcing term to steer consensus, or, in other words, it models the action of promoting the emergence of a desired status.
Thus we can associate the binary dynamics in (2.10) to the original dynamics in (2.12). Choosing , the control term for the arbitrary pair reads
(2.13) 
where
(2.14)  
(2.15) 
Note that and are both symmetric, which follows directly by (2.1)–(2.2), since is the same for every agent. Embedding the control dynamics into (2.12) we obtain the following binary constrained interaction
(2.16)  
with defined as follows
(2.17) 
In the absence of diffusion, from (2.16) it follows that
(2.18) 
thus in general the mean opinion is not conserved. Observe that the computation of the relative distance between opinions is equivalent to (1.2), since the subtraction cancels the control terms out, giving the inequality
(2.19) 
which tells that the relative distance in opinion between two agents diminishes after each interaction [99]. In presence of noise terms, we should assure that the binary dynamics (2.16) preserves the boundary, i.e. . An important role in this is played by functions , as stated by the following proposition.
Proposition 2.
Let assume that there exist and such that and . Then, provided
(2.20) 
are satisfied, the binary interaction (2.16) preserves the bounds, i.e. the postinteraction opinions are contained in .
Remark 3.
Observe that, from the modeling viewpoint, noise is seen as an external term which can not be affected by the control dynamics. A different strategy is to account the action of the noise at the level of the microscopic dynamics (2.1) and proceed with the optimization. This will lead to a different binary interaction with respect to (2.16), where the control influences also the action of the noise.
2.1.3 Main properties of the Boltzmann description
In general the time evolution of the density is found by resorting to a Boltzmann equation of type (1.4), where the collisions are now given by (2.16). In weak form we have
(2.21) 
where we omitted the time dependence for simplicity. Therefore the total opinion, obtained taking , is preserved in time. This is the only conserved quantity of the process. Choosing , we obtain the evolution of the average opinion, thus we have
(2.22) 
Indicating the average opinion as using (2.18) we get
(2.23) 
Since , , we can bound the derivative from below and above
Note that in the limit , the average converges to the desired state , if . This implies the following restriction A similar analysis can be performed for the second moment , showing the decay of the energy for particular choices of the interaction potential (cf. [99, 6, 7] for further details on the proprieties of the moment of (1.4)).
Remark 4.
In the symmetric case, , equation (2.23) is solved explicitly as
(2.24) 
which, as expected, in the limit converges to , for any choice of the parameters.
2.1.4 The quasi–invariant opinion limit
We will now introduce some asymptotic limit of the kinetic equation. The main idea is to scale interaction frequency and strength, and respectively, diffusion at the same time, in order to maintain at any level of scaling the memory of the microscopic interactions (2.16). This approach is refereed to as quasi–invariant opinion limit [99, 62, 102]. Given , we consider the following scaling
(2.25) 
The ratio is of paramount importance in order to show in the limit the contribution of both the compromise propensity and the diffusion . Other scalings lead to diffusion dominated or compromise dominated equations. In the sequel we show through formal computations how this approach leads to a Fokker–Planck equation type [93]. We refer to [99] for details and rigorous derivation.
After scaling, equation (2.21) reads
(2.26) 
while the scaled binary interaction dynamics (2.16) is given by
(2.27) 
In order to recover the limit for we consider the secondorder Taylor expansion of around ,
(2.28) 
where for some ,
Therefore the approximation of the interaction integral in (2.21) reads
(2.29)  
The term indicates the remainder of the Taylor expansion and is such that
(2.30) 
Under suitable assumptions on the function space of and the remainder converges to zero as soon as (see [99]). Thanks to (2.27) the limit operator of (2.29) is the following
Integrating back by parts the last expression, and supposing that the border terms vanish, we obtain the following Fokker–Planck equation
(2.31) 
where
As usual, indicates the mean opinion.
2.1.5 Stationary solutions
One of the advantages of the Fokker–planck description is related to the possibility to identify analytical steady states. In this section we will look for steady solutions of the Fokker–Planck model (2.31), for particular choices of the microscopic interaction of the Boltzmann dynamics.
The stationary solutions, say , of (2.31) satisfy the equation
(2.32) 
As shown in [7, 9, 99], equation (2.32) can be analytically solved under some simplifications. In general solutions to (2.32) satisfy the ordinary differential equation
(2.33) 
Thus
(2.34) 
where is a normalizing constant.
Let us consider the simpler case in which . Then the average opinion evolves according to
(2.35) 
which is obtained from the scaled equation (2.26) through the quasiinvariant opinion limit (cf. also equation (2.24) for a comparison).
In absence of control, i.e. for , the mean opinion is conserved, and the steady solutions of (2.31) satisfy the differential equation [99]
(2.36) 
In presence of the control the mean opinion is in general not conserved in time, even if, from (2.35), it is clear that converges exponentially in time to . Consequently
(2.37) 
Let us consider as diffusion function . Therefore the solution of (2.36) takes the form
(2.38) 
where is such that the mass of is equal to one. This solution is regular, and thanks to the presence of the exponential term . Moreover, due to the general non symmetry of , the initial opinion distribution reflects on the steady state through the mean opinion. The dependence on can be rendered explicit. It gives
(2.39) 
with the normalization constant.
We plot in Figure 1 the steady profile and for different choice of the parameters and . The initial average opinion is taken equal to the desired opinion . In this way we can see that for the steady profile of (2.37) approaches the one of (2.36). On the other hand small values of give the desired distribution concentrated around . It is remarkable that in general we can not switch from to only acting on the parameter , since the memory on the initial average opinion is lost for any . We refer to [6, 99] for further discussion about stationary solutions of (2.32).
2.1.6 Numerical Tests
Our goal is to investigate the action of the control dynamic at the mesoscopic level. We solve directly the kinetic equation (2.21) obeying the binary interaction (2.27), for small value of the scale parameter . We perform the numerical simulations using the Monte Carlo methods developed in [5, 89].
Sznajd’s model
Our first example refers to the meanfield Sznajd’s model [97, 11]
(2.40) 
corresponding to equation (2.31) in the uncontrolled case without diffusion. It is obtained choosing and assuming that the mean opinion is always zero. In [11] authors showed for concentration of the profile around zero, and conversely for a separation phenomena, namely concentration around and , by showing that explicit solutions are computable. We approximate the meanfield dynamics in the separation case, , through the binary interaction (2.27), sampling agents, with scaling parameter . In Figure 2 we simulate the evolution of in the time interval , starting from the uniform distribution on , , in three different cases: uncontrolled (), mild control () and strong control (). In the controlled cases the distribution is forced to converge to the desired state .
Bounded confidence model
We consider now the bounded confidence model introduced in [71], where every agent interacts only within a certain level of confidence. This can be modelled through the potential function
In Figure 3, we simulate the dynamics of the agents starting from a uniform distribution of the opinions on the interval . The binary interaction (2.27) refers to a diffusion parameter and . Here . The bounded confidence parameter is , and we consider both cases (without control and with control), letting the system evolve in the time interval , with . The figure to the left refers to the uncontrolled case, where three mainstream opinions emerge. On the right the presence of the control with leads the opinions to concentrate around the desired opinion .
2.2 Control through leadership
Several studies have been recently focused on the control of a large population through the action of a small portion of individuals, typically identified as leaders [3, 4, 24, 61]. In this section we are interested in the opinion formation process of a followers’ population steered by the action of a leaders’ group. At a microscopic level we suppose to have a population of followers and leaders. Their dynamics is modelled as follows
(2.41)  
(2.42) 
where , for all and are the followers’ and leaders’ opinions. As in the previous section, and are given compromise functions, measuring the relative importance of the interacting agent in the consensus dynamics. Leaders’ strategy is driven by a suitable control , which minimizes the functional
(2.43) 
where represents the final time horizon, is the desired opinion and is the average opinion of the followers group at time , and are such that . Therefore, leaders’ behavior is driven by a suitable control strategy based on the interplay between the desire to force followers towards a given state, radical behavior (), and the necessity to keep a position close to the mean opinion of the followers in order to influence them populistic behavior ().
Note that, since the optimal control problem acts only over the leader dynamics we can approximate its solution by a model predictive control approximation as in Section 2.1. Next we can build the corresponding constrained binary Boltzmann dynamic following [7].
2.2.1 Boltzmann constrained dynamics
To derive the system of kinetic equation we introduce a density distribution of followers and leaders depending on the opinion variables and time , see [7, 57]. It is assumed that the densities of the followers and the leaders satisfy is
The kinetic model can be derived by considering the change in time of and depending on the interactions with the other individuals and on the leaders’ strategy. This change depends on the balance between the gain and loss due to the binary interactions. Starting by the pair of opinions and , respectively the opinions of two followers and two leaders, the postinteraction opinions are computed according to three dynamics: the interaction between two followers; the interaction between a follower and a leader; the interaction between two leaders.

We assume that the opinions in the followerfollower interactions obey to the rule
(2.44) where as usual is the compromise function, and the diffusion variables are realizations of a random variable with zero mean, finite variance . The noise influence is weighted by the function , representing the local relevance of diffusion for a given opinion, and such that .

The leaderfollower interaction is described for every agent from the leaders group. Since the leader do not change opinion, we have
(2.45) where is the communication function and a random variable with zero mean and finite variance , weighted again by the function .

Finally, the postinteraction opinions of two leaders are given by
(2.46) where is the compromise function and, similar to the previous dynamics, are random variables with zero mean and finite variance , weighted by . Moreover the leaders’ dynamics include the feedback control, derived from (2.43) with the same approach of Section 2.1.1. In this case the feedback control accounts for the average values of the followers’ opinion
(2.47) and it is defined as
(2.48) In (2.48), has the same form of (2.17) and
(2.49) (2.50) Note that the control term, depends on two contributions, a steering force towards the desired state and one towards the average opinion of the followers , weighted respectively by the parameters and , such that .
2.2.2 Boltzmann–type modeling
Following [89], for a suitable choice of test functions we can describe the evolution of and via a system of integrodifferential equations of Boltzmann type