Mathematics and physics applications in sociodynamics simulation: the case of opinion formation and diffusion Giacomo Aletti1 , Ahmad K. Naimzada2 , and Giovanni Naldi3 1 2 3 Department of Mathematics and ADAMSS Center, Università degli studi di Milano, via Saldini 50, 20133 Milano, Italy, giacomo.aletti@unimi.it Department of Quantitative methods for Business Economics, Università degli studi di Milano Bicocca, Piazza dell’Ateneo Nuovo 1, 20126 Milano, Italy, ahmad.naimzada@unimib.it Department of Mathematics and ADAMSS Center, Università degli studi di Milano, via Saldini 50, 20133 Milano, Italy, giovanni.naldi@unimi.it Summary. In this chapter, we briefly review some opinion dynamics models starting from the classical Schelling model and other agent-based modelling examples. We consider both discrete and continuous models and we briefly describe different approaches: discrete dynamical systems and agent-based models, partial differential equations based models, kinetic framework. We also synthesized some comparisons between different methods with the main references in order to further analysis and remarks. 1 Opinion dynamics Opinion dynamic models describe the process of opinion formation in groups of individuals: as the opinion behavior emerges, evolves, spreads, erodes, or disappears. Here we provide a brief overview of models and simulation tools for the opinion dynamics. Most people hold and exchange opinions about a lot of topics, from politics and sports to health, new products and the lives of others. These opinions can be either the result of serious reflection or as is often the case when information is hard to process or obtain, formed through interactions with others that hold views on given issues. The modelling of the opinion dynamics try to understand when the opinion formation leads to consensus, polarization or fragmentation within an interacting group. Opinion dynamics is one of the most widespread topics of Sociophysics (application of methods from physics to human relations) and Sociodynamics (the attempt to build up a modelling strategy allowing in principle of an integrative quantitative description of dynamic macro-phenomena in the society). Sociophysics covers numerous topics of social sciences and addresses many different problems G. Naldi et al. (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-0-8176-4946-3 8, c Springer Science+Business Media, LLC 2010  203 204 Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi including social networks, language evolution, population dynamics, epidemic spreading, terrorism, voting, and coalition formation. Among them the study of opinion dynamics has become a main stream of research. In fact, the public opinion is nowadays a feature of central importance in modern societies making the understanding of its underlining mechanisms a major challenge. The dynamics of agreement/disagreement among individuals is complex, because the individuals are. Physicists working on opinion dynamics aim at defining the opinion states of a population, and the elementary processes that determine transitions between such states. The main question is whether this is possible and whether this approach can shed new light on the process of opinion formation. Computer simulations play an important role in the study of social dynamics as they parallel more traditional approaches of theoretical physics and mathematical modelling, where a system is described in terms of a set of equations. One of the most successful methodologies used in social dynamics is agent-based modelling [1, 2]. In agent-based modelling (ABM), a system is modelled as a collection of autonomous decision-making entities called agents. Each agent individually assesses its situation and makes decisions on the basis of a set of rules. Agents may execute various behaviors appropriate for the system they represent. Repetitive interactions between agents are a feature of agent-based modelling, which relies on the power of computers to explore dynamics out of the reach of pure mathematical methods. At the simplest level, an agent-based model consists of a system of agents and the relationships between them. Even a simple agent-based model can exhibit complex behavior patterns and provide valuable information about the dynamics of the real-world system that it emulates. In addition, agents may be capable of evolving, allowing unanticipated behaviors to emerge. Sophisticated ABM sometimes incorporates neural networks, evolutionary algorithms, or other learning techniques to allow realistic learning and adaptation. ABM is a mindset more than a technology: it consists of describing a system from the perspective of its constituent units. The description of emerging collective behaviors and self-organization in multiagent interactions has gained increasing interest from various research communities in biology, ecology, robotics, and control theory, as well as sociology and economics. In the biological context, the emergent behavior of bird flocks, fish schools, or bacteria aggregations, among others, is a major research topic in population and behavioral biology and ecology [3–11]. Likewise, the coordination and cooperation among multiple mobile agents (robots or sensors) have been playing central roles in sensor networking, with broad applications in environmental control [12]. Emergent economic behaviors, such as distribution of wealth in a modern society [13–17], or the formation of choices and opinions [18–21], are also challenging problems studied in recent years in which emergence of universal equilibria is shown. Also, the development of a common language in primitive societies is yet another example of a coherent collective behavior emerging within a complex system [22, 23]. Opinion dynamics and diffusion 205 The social world that we observe reflects a lot of interdependent processes, with macro level structures of organizations, communities, and societies both emerging from and constraining the micro level interactions of individuals. Many social phenomena, such as the spread of epidemics, or the dissolution of organizations, are inherently time varying and depend on interactions between entities within a social system. Understanding the link between microlevel interactions and macrolevel dynamics promises to have profound impact on how human societies, organizations, and nations might be structured and how related policy decisions should be made. As we mentioned earlier, an increasing number of scientists are using mathematical and computational models to elucidate theoretical problems in social dynamics, often by applying general theories or methods that are well developed in the natural and physical sciences with a view to gaining insight into the underlying generative processes or the dynamic consequences of social relationships. It may be surprising, but the application of concepts from the natural sciences to social sciences is at least 25 centuries old. In fact, the Greek philosopher Empedocle stated that humans are like liquids: some mix easily like wine and water, and others refuse to mix. The discovery of quantitative laws in the collective properties of a large number of people was one of the pushing factors for the development of statistics. Many scientists and philosophers called for some quantitative understanding on how such precise regularities arise. Among many others, Hobbes, Laplace, Comte, Stuart Mill shared this line of thought. More recently, Majorana [24, 25] in the 1940s suggested to apply quantum-mechanical uncertainty to socio-economic questions. Weidlich [26] studied similar questions since 1971. In the same year, Thomas Schelling (Nobel Prize for Economics in 2005) published [27] his highly acclaimed model for for urban segregation in the first issue of Journal of Mathematical Sociology. Moreover, in the same issue of the journal Sakoda [28] presented a closely related work whose basic design was already present in his unpublished dissertation of 1949. In [33] Galam gave a personal testimony of Sociophysics going back to his 1982 publication. Other references may be found in the books of Arnopoulos [34] and Schweitzer [35], while some review articles can be found for example in [36–38]. 1.1 Schelling model We present here a significant and historical segregation model proposed by Schelling in 1971 [27] for the study of ethnic segregation in the United States. Schelling supposed that people had a threshold of tolerance of other ethnic groups based on the neighbored people. If, for instance, the threshold of tolerance was 40%, people were content to stay in the place they live provided that at least four in ten of their neighbors were from the same ethnic group. If this were not so, they would try to move to another neighborhood in which at least 40% were of their own group. The conventional assumption is that ethnic segregation in the USA is at least partly due to the fact that whites 206 Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi are prejudiced and have a tolerance threshold of over 50%. They therefore moved out of urban neighborhoods that had a majority of blacks, leaving the neighborhood with a still higher proportion of black people and, thus, accelerating the tendency towards complete segregation. Schellings point was that tolerance thresholds much lower than 50% could lead to the same result. Even a threshold as low as 30% could result in almost complete segregation. Thus, although people might be quite content with being in the minority in a neighborhood, so long as they demanded that some small proportion of their neighbors were of the same ethnic group as themselves, segregation could emerge. The original Schelling’s model was very simple. Take a large chessboard, and place a certain number of black and white counters on the board, leaving some free places. A counter prefers to be on a square where a certain fixed percentage of the counters in his Moore neighborhood (his eight nearest neighbors) are of its own color to the opposite situation. From the counters who wish to migrate one is chosen at random and moves to a preferred location. This model, when simulated, yields complete segregation even though people’s preferences for being with their own color are not strong. In Fig. 1 we show some simulations by using different threshold of tolerance, each chessboard refers to the steady state when there are no more unhappy. Schelling’s result has become famous precisely because the preferences of individuals for Fig. 1. The role of preferences on the end pattern of segregation that emerges in the Schelling’s model Opinion dynamics and diffusion 207 segregation were not particularly strong. Note that the Schelling result is of interest to economists because it illustrates the emergence of an aggregate phenomenon that is not directly foreseen from the individual behavior. The Schelling model is based on the idea that an individual agent makes decisions based on his preferences or utility function. Then the agent’s satisfaction is equivalent to the energy stored in him. An increase in happiness is a decrease in internal energy. An agent, therefore, wants to minimize his energy, which is generated either by taking some action or through the interaction with his environment. The Schelling model assumes that the agent’s utility depends on his local environment and that he moves if the utility falls below a certain threshold. Such a process is easily simulated through the twodimensional Ising model [29–31]. In the simplest case, as black and white, we have two Ising spin orientations. In this Ising model, each site i on a square lattice carries a variable Si with value +1 or −1. For each pair (i, k) of nearest neighbors produces an energy contribution Eik equal to Eik = −JSi Sk with some proportionality constant J. The total energy E is proportional to the total unhappiness is the sum of this pair energy over all neighbor pairs of the lattice Eik . In statistical physics, different distributions of the spins Si are realized with a probability proportional to e(−E/KB T ) , where T is the absolute temperature and KB the Boltzmann constant. We point out that the relevant quantity is the ratio (KB T /J). The Metropolis kinetics of the system is simulated by flipping a spin if and only if a random number between 0 and 1 is smaller than the probability e(−∆E/(KB T )) , where ∆E is the variation of the total energy. When the Glauber kinetics [32] is used the flipping probability is e(−∆E/(KB T )) /(1 + e(−∆E/(KB T )) ). For Glauber or Metropolis, after very long times (measured by the number of sweeps through the lattice) one of the two possibilities dominates at the end which depend on the temperature T . If T is not larger than the critical temperature Tc , for Kawasaki dynamics the fraction of black sites remains constant, and we get two large domains. For higher temperatures above Tc only small clusters and no large domains are formed. In this Ising model, two neighboring spins have due to their interaction JSi Sk a higher probability to belong to the same group than to belong to the two different groups. Schelling’s model avoided probabilistic rules and thus counted neighbors Si = ±1 at zero temperature. As Schelling moved only one person at a time, and made no exchange of two people simultaneously as in Kawasaki kinetics, he introduced a large fraction of empty residences. Thus at each step, one unhappy person or family moves into the closest vacancy where life would be happy. For a study of the model from this physical point of view and a link between Schellings socio-economic model of segregation and the physics of clustering we refer to [39] and [40]. There are several variants on Schelling’s original model modifying the form of the utility function used by Schelling, the size of neighborhoods, the rules for moving, and the amount of unoccupied space. Many of these fail to give large domains: only small clusters are seen. In many real cities, there are huge cluster which extends over many square kilometers. Thus, Schelling’s model 208 Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi does not give always the desired results. In general, the critique social systems models are too simplified is most of the times very well grounded. However, in most situations qualitative (and even some quantitative) properties of large scale phenomena can be simulated and reproduced. A discussion about the application of self-organising segregation and migration models can be found in [41–43]. 2 Binary opinion dynamics and beyond Scientists working on opinion dynamics aim at defining the opinion states of a population, and the processes that determine transitions between such states. In a mathematical model, opinion has to be a variable, or a set of variables. This may appear too reductive, thinking about the complexity of a person and of each individual position. Rightly, Humans do not like to be treated like a number, and indeed the human brain is much more complex than a set of variables. But one can observe that the value of an opinion or a decision could be represented by a numerical vector. As example binary opinions and binary choices are frequent in the everyday life: Windows/Linux, buying/selling, Coca Cola/Pespi Cola, etc. Moreover, appear to act on the opinion some kind of social forces even though they may appear of little relevance to any important decisions. For example we can consider the following simple experiment, first performed four decades ago and reproduced many times since. Stand on a busy street corner and look up at the sky. The crowd will part around you, indifferent to whatever it is you may be looking at. Now enlist the help of a friend to stand beside you and also look skyward. Soon, others will stop and gaze up as well. A similar thing would happen if you and your friend boarded an empty elevator and faced the rear wall. As more passengers boarded many would face the back wall too. Then a kind of peer pressure seems to appear. 2.1 The Sznajd model There are several models used in modelling consensus formation and we tried to name a few historical introduction listed earlier. Here we consider in more detail a recent Sznajd model only as an example just to illustrate some issues and problems in the simulation of the opinion formation. The general framework of the model is as follows: at any given time a distribution of opinions exists in a given population; at each time step some group of individuals interact and as a result, one or several opinions are shifted (generally towards some consensus among the group); as a result of these interactions, some form of pattern of opinions is observed. An important aspect always present in social dynamics is topology which is the structure of the interaction network describing who is interacting with whom, how frequently and with which intensity. In the Sznajd model [20, 46], as in the traditional statistical physics and in the Ising model, the geometry is a one-dimensional regular lattice. Opinion dynamics and diffusion 300 209 1 time m(t) 200 0.5 100 0 0 0 50 X 0 100 100 200 300 time 500 1 300 m(t) time 400 0.5 200 100 0 0 0 50 X 100 0 200 time 400 Fig. 2. Two examples of the dynamics of the Sznajd model, in the first row we have the behavior for corresponding to the original rules of the model while in second row the behavior for the modified rules in order to avoid the antiferromagnetic final state. In the left column we reported the evolution of the social state in time and space, in the right column we considered the evolution of the magnetization variable m(t) with respect to the time Agents are, thus, supposed to sit on vertices (nodes) of the lattice, and each agent can interact with the two adjacent agents. More generally, it is possible to consider a d-dimensional lattice of N d sites carries opinion variables Si , i = 1, 2, . . . , N d . The lattice may be square (four nearest neighbours), triangular (six nearest neighbours), many other choices are also possible. The opinion variables could be scalar or vector real variables. Usually, they are scalar variables and are integers between P1 and P2 or take values in a finite set of opinions {P1 , P2 , . . . , Pk }. The individual opinions Si in the Sznajd model are represented in our model by Ising spins (“yes” or “no,” −1 or 1), that is to say the opinions are binary opinions. Such an approach is not new but it had been used earlier (see for example [33] or [47]). At each time step, each Si is calculated from a rule according the spin dynamics [20]: • A pair of opinions (spins) Si , Si+1 is chosen to change their nearest neighbors, i.e., the opinions (spins) Si−1 and Si+1 . 210 Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi • If Si Si+1 = 1 (the opinions are equal) then Si−1 = Si and Si + 2 = Si (social validation). • If Si Si+1 = −1 then Si−1 = Si+1 and Si+2 = Si . The model was originally called USDF after the trade union maxim “United we Stand, Divided we Fall,” in other words: If you see two or more people sharing an opinion on certain issue, you are tempted to join. The crucial difference of the model compared to other Ising-type models is that information flows outward. The dynamic rules of the model lead to two possible steady states: the complete consensus (ferromagnetic state) in which all Si = 1 or all Si = −1; the stalemate (antiferromagnetic state) in which 50% of opinions Si = 1 and 50% are equal to −1. However, the last 50 − 50 state is realized in a very special way: every member of the community disagrees with his nearest neighbor. Even if the Ising model with only next nearest neighbor interaction has such kind of fixed point (ferro and antiferromagnetic state) it is unrealistic as a social state. Then, new dynamic rules were proposed [46], for example in the case of the disagreements, Si Si+1 = −1, then a new update is computed: Si−1 = Si and Si+2 = Si+1 . Also a reasonable rule is to assume that the individual keeps its opinion in such a case, rather than opposing its neighbor. That is if Si = −Si+1 , nothing happens at that time step, the system is not altered. A global index which corresponds to the general opinion of the society is the magnetization m(t) at the time t defined as: 1  (number of opinions 1) − (number of opinions − 1) = Si , m(t) = N N i where N is the number of people in the community. In the Fig. 2 we show the behavior of the opinions in a community and the evolution of the magnetization in the case of the original Sznajd model and for the modified model with the different active update rule. In [20] Sznajd-Weron and Sznajd have investigated some statistical properties of the magnetization function m(t) and some interesting behavior of the opinion changes of one particular individual. In particular they observed that if an individual changes the opinion at time t, he (she) will probably change it also at time t + 1. Moreover, if τ denotes the time needed by an individual to change the opinion its distribution P (τ ) follows a power law with an exponent −3/2. It is well known that the changes of opinion in a community are not only determined by the individual contacts between neighbors but also the social impact has a role. It is possible to introduce a noise p, some kind of social temperature, which is the probability that an individual, instead of following the dynamic rules, will make a random decision. In [20] a computational study is performed to study the influence on the distribution P (τ ) of the parameter p. Sznajd model has been modified and applied in marketing, finance, and politics (see for example [38]). In agreement with social theory assigning four nearest neighbors to an individual rather than only two is a more realistic simplification of a real society. So in search for a more realistic model, two dimensional version of Sznajd Opinion dynamics and diffusion 211 model was proposed by Stauffer et al. [48] for two-dimensional regular lattice. In accordance with the dimensionality of the model, six different versions were proposed. Here, we focus on two of the most realistic of the six: • A 2 × 2 square of four neighbors are chosen at random. If all four opinions of the square are the same then all eight neighbors to this square follow the same value. Otherwise, the system is not altered, the neighbors are unchanged. • A bond is chosen at random. If the two opinions forming the bond are the same, then all six neighbors to this bond follow the same value. Otherwise, the system is not altered, the neighbors are unchanged. In Fig. 3 we show one example of the behavior of the two-dimensional Sznajd model following the two rules described earlier, simulations were performed on a N × N square lattice, where individuals were placed on the lattice as spins in the two dimensional Ising model. Periodic boundary conditions are used in both directions. As for the original Sznajd model simulation, random sequential updating was used as in Monte-Carlo simulation. In [48] the emergence of a phase transition phenomena with respect to the size of the system was studied: this is the only important difference between the onedimensional model and the two-dimensional model. A natural extension of the binary Sznajd model developed in order to fit the experimental observation in some elections is to allow more than two opinions for each individual. So, if there are M candidates running for the post then each individual can have M possible opinions. This is an example of a many-opinion modification to the Sznajd model, and also equivalent to extending the Ising model to M -state Potts model [49]. Other steps toward more realism is to investigate the model on a complex network with different topology, adding noise or diffusion (see for examples [46]). In a short note [50] Ochrombel suggested a drastic simplification of the Sznajd model. In the Ochrombel version it is not necessary to have a cluster of identical opinions to change the neighbors. Any individual is capable to convince her neighbors to select the same opinion. We choose an agent i at random. Then, choose j randomly among neighbors and set Sj = Si . In the case of a fully connected network the Ochrombel simplification is equivalent to voter model, whose dynamical properties were studied, e.g., in [51]. In this last case for binary opinions the state is described by the magnetisation variable m(t) only. It can be found in the thermodynamic limit N → ∞ that the probability density Pm (m, t) of the magnetization at time t evolves according to the partial differential equation [52]  ∂ ∂2  (1 − m2 )Pm (m, t) . Pm (m, t) = ∂t ∂m2 (1) For the original binary Sznajd model but in the case of a fullyconnected network a different time scaling is needed in order to get sensible thermodynamic 212 Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi 0 0 50 50 100 0 50 t=1 100 100 0 0 0 50 50 100 0 50 t=10 100 100 0 50 t=5 100 50 t=50 100 0.52 0.515 0.51 m(t) 0.505 0.5 0.495 0.49 0.485 0.48 0 10 20 30 40 50 t Fig. 3. The time evolution for a two-dimensional Sznajd model as proposed by Stauffer et al. [48] and the behavior of the corresponding magnetization function m(t) limit [52]. Then the equation for the probability density is the following Fokker–Planck equation  ∂  ∂ Pm (m, t) = − (1 − m2 )mPm (m, t) . ∂t ∂m (2) The dynamics is those of a pure deterministic drift and no diffusion ever smears out the evolving probability packet. The possibility to have the Fokker– Planck equation provides a great tool for the theoretical study of the opinion dynamics model. Opinion dynamics and diffusion 213 One issue of interest in the opinion dynamics concerns the importance of the binary assumption: what would happen if opinion were a continuous variable such as the worthiness of a choice or some belief about the adjustment of a control parameter? Continuous opinions invalidate some of the concepts adopted in models with discrete choices so they require a different framework. The initial state is usually a population of N agents with randomly assigned opinions, represented by real numbers within some interval. The opinion clusters could be one (consensus), two (polarization), or more (fragmentation). In principle, each agent can interact with every other agent. In practice, there is a real discussion only if the opinions of the people involved are sufficiently close to each other. This realistic aspect of human communications is called bounded confidence. Usually, it is expressed by introducing a real number ε, the uncertainty or tolerance, such that an agent, with opinion x, only interacts with those of its peers whose opinion lies in the interval ]x − ε, x + ε[. As example consider the model introduced by Deffuant et al. [53]. There are, again, N individuals with opinions Si . In each update step, two of them, say, i and j, are chosen randomly. Then, we check to see whether their opinions differ less than (or equally to) the confidence bound ε . In the positive case, their opinions are slightly shifted towards each other, Si (t + 1) = (1 − µ)Si (t) + µSj (t), Sj (t + 1) = (1 − µ)Sj (t) + µSi (t), for |Si (t) − Sj (t)| ≤ ε, where ε is a parameter. For very large numbers of individuals, N → ∞, the dynamics can be expressed in terms of the continuous distribution of opinions P (s, t) which satisfies an equation with a fairly complex behavior [54]. 3 Agents based model and discrete dynamical systems In [45], the authors consider a micro-scale approach. The population is composed by n agents. At each discrete time t = 0, 1, . . . the opinion of the ith agent is given by the quantity xi (t) and therefore the opinion profile at time t is given by the vector x(t) = (x1 (t), . . . , xn (t)). The ith agent takes into account the opinion of the jth agent by means of the weight aij (t, x(t)). Accordingly, the opinion formation evolves in the following way: xi (t + 1) = ai1 (t, x(t))x1 (t) + ai2 (t, x(t))x2 (t) + · · · + ain (t, x(t))xn (t) or, in matrix form, the general model is given by: x(t + 1) = A(t, x(t))x(t). The only assumption on the matrix A is that it is a stochastic matrix (each row sums to 1). Obviously, further assumptions on the weight matrix A lead to different studied models. The review article [45] presents models which can be analytically studied and others that are computationally faced. The consensum property is given when all the opinion converge to the same opinion 214 Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi that depends only on the initial opinion profile: given x(0), there exists c = c(x(0)) such that xi (t) → c for any 1 ≤ i ≤ n. When x(t) converges to a vector with different components, we face an opinion fragmentation. The first part of [45] is devoted to a review of the linear models, while the second part refers to some computational results on some nonlinear models. 3.1 Linear models If A(t, x(t)) ≡ A, (3) reduces to the classical model of fixed weights x(t + 1) = Ax(t). A generalization of the previous model is given by Friedkin and Johnsen in [44] x(t + 1) = Gx(0) + (I − G)Ax(t), where G is a diagonal matrix containing the degree of adhesion to the initial opinion: G = diag(g1 , g2 , . . . , gn ). Obviously, when G ≡ 0, (3.1) reduces to (3.1). Note that in (3.1), x(t) = V (t)x(0), where V (t) = M t + t−1  i=0  Mi G with M = (I − G)A. Analytical conditions on the matrix A ensure limit properties as consensum or fragmentation. As an example, if A is irreducible, limt x(t) = (I − M )−1 Gx(0) when G = 0. Similar results, even if less sharp, may be found with the last model presented in the review part of [45]. This model is still linear but timevariant (or, in the theory of Markov chains, inhomogeneous), that is x(t + 1) = A(t)x(t), where the entries of matrix A(t), i.e., the weights, are dependent on time only. The time variant model (3.1) portrays, e.g., the so called “hardening of positions” where agents put in the course of time more and more weight on their own opinion and less weight on the opinion of others. 3.2 Nonlinear models The models proposed in [45] depend on opinions itself and hence the models are nonlinear. Analytical insights are not so easy to obtain. Therefore, computer simulations were used to investigate these models. An agent i takes only those agents j into account whose opinions differ from his own not more than a certain confidence level. More precisely, if  xi (t + 1) = |I(i, x(t))|−1 xk (t), where k I(i, x(t)) = #{1 ≤ k ≤ n : ǫl ≤ xk − xi ≤ ǫr }, Opinion dynamics and diffusion 215 the confidence level for each agent is the interval [ǫl , ǫr ]. We have the nonlinear model x(t + 1) = A(x(t))x(t), where aij (x) = 1 . #{1 ≤ k ≤ n : ǫl ≤ xk − xi ≤ ǫr } Asymptotic configurations are numerically studied both in the symmetric case (i.e., ǫl = ǫr ), and in the asymmetric one. 4 The kinetic approach The goal of the forthcoming kinetic model of opinion formation, is to describe the evolution of the distribution of opinions in a society by means of microscopic interactions among individuals which exchange information. Opinion is represented as a continuous variable w ∈ I, with, as example, I = [−1, 1]. The extremes w = ±1 represent extreme opinions. Toscani has recently studied [55], by means of kinetic collision-like models, the distribution of opinion among individuals in a simple, homogeneous society. This model is based on binary interactions. When two individuals with preinteraction opinion v and w meet, then their posttrade opinions (postcollisional) v ∗ and w∗ are given by: v ∗ = v − γP (|v|)(v − w) + ηv D(|v|) w∗ = w − γP (|w|)(w − v) + ηw D(|w|) where the coefficient γ ∈ (0, 1/2) is a given constant, while ηv and ηw are random variables with the same distribution with variance σ 2 and zero mean. The constant γ measures the compromise propensity while the variance σ 2 is related to the modification of opinion due to diffusion. The functions 0 ≤ P (·) ≤ 1 and 0 ≤ D(·) ≤ 1 describe the local relevance of the compromise and diffusion for a given opinion. In absence of the diffusion contribution, η = 0, if P (·) is not constant the total momentum is not conserved and it can increase or decrease depending on the opinions before the interaction. While if P is assumed constant, the interaction correspond to a granular gas like interaction [56]. Let f (w, t) denote the distribution of opinion w ∈ I at time t ≥ 0. Standard methods [57] of kinetic theory allow to describe the time evolution of f as a balance between bilinear gain and loss of opinion terms    1 ∂f = (3) β ′ f (v ′ )f (w′ ) − βf (v)f (w) dwdηv dηw , ∂t J B2 I where (v ′ , w′ ) are the preinteraction opinions that generate the couple (v, w) of opinions after the interaction, J is the Jacobian of the transformation of (v, w) into (v ′ , w′ ), while the kernels β ′ and β are related to the details of the binary interaction, finally B is the space of the random variables η. Following 216 Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi Toscani [55] different Fokker–Planck equations can be obtained by considering different limits and simplifications. In the following section we consider only one example. For the case of a kinetic model for the the evolution of the opinion in a closed group with respect to a choice between multiple options, we refer to [58]. In some sense intermediate between the Kinetic approach and agent-based model we can also consider an alternative approach based on active Brownian particles which interact via a communication field (see [59] and references therein). This scalar field considers the spatial distribution of the individual opinions; further, it has a certain lifetime, reflecting a collective memory effect, and it can spread out in the community, modelling the transfer of information. Also the active particle approach [60] leads to the derivation of evolution equations for a one-particle distribution function over the microscopic state. Two types of equations, to be regarded as a general mathematical framework for deriving the models, are derived corresponding to short- and long-range interactions. The active particle approach gives a general mathematical framework in order to link microscopic and macroscopic descriptions of the dynamics. 4.1 An example of nonlinear continuous models of opinion formation In [61], the large-time behavior of solutions of the equation ∂ ∂f =γ (1 − x2 )(x − m(t))f ∂t ∂x (4) is studied, where the unknown f (x, t) is a time-dependent probability density which may represent the density of opinion in a community of agents. This opinion varies between the two extremal opinions represented by ±1, so that x ∈ [−1, 1]. The constant γ is linked to the spreading (γ = −1) or to the concentration (γ = +1) of opinions. In (4) m(t) represents the mean value of f (·, t),  x f (x, t) dx, m(t) = (5) [−1,1] and its presence introduces a nonlinear effect into its evolution. Equation (4) describes the evolution of a probability density which represents the density of opinions in a community. For all values of the constant γ, the time-evolution driven by this equations leads the density toward a equilibrium state that is described in terms of two Dirac masses (γ = −1) or to a unique Dirac mass (γ = 1), see [61]. The convergence result toward equilibrium holds in weak-measure sense. A suitable way of treating convergence results of (4) is based on a rewriting of this equation in terms of pseudo-inverse functions. In fact, let F (x) denote the probability distribution induced by the density f (x),  f (y) dy. (6) F (x) = (−∞,x] Opinion dynamics and diffusion 217 As F (·) is not decreasing, we can define its pseudo inverse function (also called quantile function) by setting, for ρ ∈ (0, 1), X F (ρ) = inf{x : F (x) ≥ ρ}. Equation (4) for f (x, t) takes a simple form if written in terms of its pseudo inverse X(ρ, t). [61, Theorem 3.1] shows in fact that the evolution equation for X(ρ, t) reads ∂X(ρ, t) = −γ (X(ρ, t) − m(t)) (1 − X 2 (ρ, t)), ∂t (7) 1 where now ρ ∈ (0, 1) and m(t) = 0 X(ρ, t) dρ. Existence and uniqueness of solutions is studied. Then results on their large-time behavior are derived. First, it is noted that the initial masses in +1 (called p+1 ), and in −1 (called p−1 ) remain unchanged in time. For what concerns concentration (γ = 1), the steady state is characterized by the distribution p−1 δ−1 + (1 − p1 − p−1 )δa + p1 δ+1 . Obviously, the mean value m∞ = p1 −p−1 +a(1−p1 −p−1 ) characterize it. In [61], it is proved that if (1 − p1 )(1 − p−1 ) < 1 (i.e., if there are masses in ±1 at time t = 0) then, m∞ = p1 − p−1 . Otherwise, if log (1 + X(ρ, 0))/(1 − X(ρ, 0)) ∈ L1 (0, 1) then m∞ = exp {T (0)} − 1 , exp {T (0)} + 1 (8)   1 where T (0) = 0 log 1+X(ρ,0) 1−X(ρ,0) dρ. The spreading case is difficult to handle analytically due to the nonlinearity present through the term m(t). Therefore, in [61], numerical methods are discussed to capture the behavior for large t. In Fig. 4 we show the evolution of numerical and quantile solution for a benchmark case as reported in [61]. 1 0.8 f(x;0) concentration T = 2 spreading T = 2 0.6 0.4 f(x;·) X(½;·) 0.2 exact initial numerical 0 −0.2 −0.4 −0.6 −0.8 −1 −0.5 0 x 0.5 1 −1 0 0.5 ½ 1 Fig. 4. A benchmark case from [61]: evolution of density function (left hand side figure) and comparison between analytical and numerical solution for the quantile function (right hand side figure) 218 Giacomo Aletti, Ahmad K. Naimzada, and Giovanni Naldi 5 New opportunities: an opinion Different computational techniques are useful for developing and adapting social dynamics models. Differential Equations Systems of both ordinary and partial differential equations are useful tools for developing models at the aggregate level, but are more limited when considering the interdependent behavior of individuals in a population. Agent-based methods are particularly useful for studying the microlevel behavior of individual objects in a system. The interactions between agents may be stochastic according to defined interaction and choice probabilities, and may be subject to exogenous shocks. Given the limited applicability of systems dynamics models using differential equations to populations of heterogeneous and interdependent agents, agentbased models may be used in conjunction with such systemlevel models or by coupling agents at different resolutions to serve as a basis for developing multilevel models. Markov chains and similar probabilistic models have a wide spectrum of applicability in social and economic process even if the independence assumptions that they make limit their modeling power. General theoretical frameworks provide the conceptual underpinnings for a variety of modeling techniques. Examples include Interacting Particle Systems and game theory. The fundamental complexity of the dynamics of social phenomena and the complications associated with representing agents who engage in rational, responsive, and realistic interactions present challenges. We point out only few opportunities and the list is certainly not exhaustive. • • • • In order to study the effectiveness of possible interventions in social and economical problems, agent based methods must be simulated numerous times to compare alternative intervention scenarios, which is a limitation of method. Results of agent based methods are not optimal solutions, but rather scenarios with various assumptions. Thus, one possible improvement may affect the possibility of developing control-theoretic approaches for agent-based methods, which could be applied in studying interventions. More realistic topologies must be taken into account in the opinion formation models. 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