A. Kononovicius, I. Kazakevicius: Impact of the controlled agents on the dynamics of the Kirman model

Collective behavior of the individuals in the complex socio-economic systems is influenced by their herding, group, behavior tendencies and their individual preferences. The herding tendencies imply the possibility to control the collective behavior. In this text we discuss this possibility through the context of Kirman's agent-based herding model.

The possibility to control the collective behavior can be clearly seen by taking the social systems as a primary example. In this case we usually have a large uninformed population. Members of this population may not have the necessary skills (or information) to make certain decision, namely they cannot make independent decision on their own. So the uninformed individuals have to rely on the individuals with necessary skills (or information) for advice. They are usually a very small part of the society, yet they are able to shape the behavior on the collective level. Actually this is confirmed by the experiments [1]. Thus we see that the possibility to control the collective behavior is not very unrealistic idea, and therefore it is very interesting topic to be studied.

Introduction of the controlled agents

Previously we already wrote about Kirman's agent-based herding model. Now let us extend the original model by introducing \( M \) controlled agents, who occupy the selected state. Note that now we have a system composed of \( N+M \) agents. The only difference between the controlled agents and "normal" agents is that the controlled agents are only influenced by the external factors, while normal agents are influenced by internal interactions. Also note that the controlled agents are not taken into account then estimating the state of the system, namely \( X \) or \( x \). Which means that the controlled agents are only used to control the "normal" agents.

Mathematically these ideas can be expressed by postulating the one step transition probabilities of the following form:

\begin{eqnarray} P(X \rightarrow X-1) &=& X [ \sigma_2 +h(N-X) ] \Delta t, \ P(X \rightarrow X+1) &=& (N-X) [\sigma_1 +h(M+X) ] \Delta t . \end{eqnarray}

Note that the only difference between the one step transition probabilities in the original model and modified model is the \( (N- X ) M h \Delta t \) term, which is, evidently, responsible for the influence of the controlled agents.

Let us rewrite the one-step probability (which differs from the original model):

\begin{equation} P(X \rightarrow X+1) = (N-X) [(\sigma_1 + h M)+h X] \Delta t = (N-X) [\tilde{\sigma}_1 + h X] \Delta t. \end{equation}

As you can see the controlled agents have a very limited effect - they just serve as a perturbation of the \( \sigma_1 \). This is relatively similar to the case analyzed in [2], but in [2] the herding behavior parameter is perturbed. Our perturbation, in contrast to the one presented in [2], is more transparent in the microscopic sense.

By using the previous intuition we can easily rewrite the stochastic model, by using the previous results:

\begin{eqnarray} \mathrm{d} x &=& [\tilde{\sigma}_1 (1-x)- \sigma_2 x] \mathrm{d} t + \sqrt{2 h x (1-x)} \mathrm{d} W =\nonumber \ &=& [( \sigma_1 + h M ) (1-x) - \sigma_2 x]\mathrm{d} t + \sqrt{2 h x (1-x)} \mathrm{d} W. \end{eqnarray}

The influence of the controlled agents can be more transparently seen by obtaining the stationary PDF and mean for \( x \):

\begin{equation} p(x) =\frac{\Gamma(\varepsilon_1+\varepsilon_2+M)}{\Gamma(\varepsilon_1+M)\Gamma(\varepsilon_2)} (1-x)^{\varepsilon_2-1}x^{\varepsilon_1+M-1} , \quad \langle x \rangle =\frac{\varepsilon_1+M}{\varepsilon_1+\varepsilon_2+M} ,\label{xpdf} \end{equation}

where \( \varepsilon_i = \sigma_i / h \).

So, as we can see from \eqref{xpdf}, the controlled agents can shift the average system state towards the desired end. Note that we can have a fixed number of the controlled agents, \( M \), which can act on systems on any size.

Scaling of the collective behavior control

Kirman's agent-based herding model can be interpreted in two distinct ways [3, 4]. In one case the agents may interact with all other agents (this case was considered above), while in the other case agents may interact only with their neighbors (recall the comparison with Bass model). Note that if Kirman's model can be interpreted in two ways, then our model can be interpreted in four:

• "Normal" agents may interact with all "normal" agents (nonextensive interaction), controlled agents also interact nonextensively, (this case was considered in the previous section)
• "Normal" agents - only with their neighbors (extensive interaction), controlled agents - nonextensively,
• "Normal" agents - nonextensively, controlled agents - extensively,
• "Normal" agents - extensively, controlled agents - extensively.

Further we will use shorter notation to describe the interactions. We will use only two words - first one will correspond to the interaction between the "normal" agents themselves, while the second will describe how the controlled agents interact with the "normal" agents.

Extensive-extensive and nonextensive-extensive interactions

In the previous section we have considered the nonextensive-nonextensive case of the modified Kirman's model. We have seen that it is possible to control the mean behavior of the system. But what happens if the interactions occur only between neighbors? We know that the corresponding one-step probabilities would be given by:

\begin{eqnarray} P(X \rightarrow X-1) &=& X \left[\sigma_2+ \frac{h}{N} (N-X) \right] \Delta t , \ P(X \rightarrow X+1)&=& (N-X) \left[ \sigma_1 + \frac{h}{N} (M+X) \right] \Delta t.\end{eqnarray}

It is easy to see that in the limit of large \( N \) the herding terms disappear. Together with them disappears the term related to the controlled agents. Thus it should be evident that we cannot control the collective behavior in this case.

Note that if \( N \) is finite, then the herding interaction with the controlled agents might be kept constant by increasing \( M \). The problem is that \( M \) should grow linearly together with \( N \), which is not a very nice or practical feature. Thus we can retain the control, but in realistic terms it would very limited.

It should be evident that the interaction between the "normal" agents themselves doesn't matter that much in this case. So in the nonextensive-extensive case we result would be qualitatively similar.

Extensive-nonextensive interaction

A more realistic assumption would be that the "normal" agents interact only with their neighbors, while the controlled agents with all "normal" agents. In this case the "normal" agents may be seen as ordinary people who interact only with small number of people they know (friends, colleagues and etc.). It would be also very convenient if controlled agents were celebrities - people who are seen by whole or at least a majority of population. Due to being celebrities they (mostly indirectly) influence the behavior of majority. Thus the controlled agents, if seen as celebrities, may interact with "normal" agents nonextensively.

In such case the one-step transition probabilities are given by:

\begin{eqnarray} P(X \rightarrow X-1) &=& X \left[\sigma_2+ \frac{h}{N} (N-X) \right] \Delta t , \ P(X \rightarrow X+1)&=& (N-X) \left[ \sigma_1 + h M \frac{h}{N} X \right] \Delta t.\end{eqnarray}

Now once again in the limit of large \( N \) the terms related to the herding behavior disappear. Yet unlike before - the terms related to the controlled agents remain! It is worthwhile to note that the macroscopic model in this case is given by ordinary differential equation:

\begin{equation} \mathrm{d} x = \left[ (\sigma_1 + h M) (1-x) - \sigma_2 x\right] \mathrm{d} t , \end{equation}

solution of which exponentially fast converges to the desired value:

\begin{equation} x(t) =\frac{\varepsilon_1+M}{\varepsilon_1+\varepsilon_2+M} + \left(x_0 - \frac{\varepsilon_1+M}{\varepsilon_1+\varepsilon_2+M}\right) \exp(- h [\varepsilon_1+\varepsilon_2+M]) . \end{equation}

Thus in this case we have ideal control over the collective behavior!

Conclusions

It is realistic ambition to control large societies! And the herding behavior is responsible for it. It enables people to copy social norms, useful information or knowledge from the other people, but copying is indiscriminate process. Allowing us to control the collective behavior. The only significant limitation of the control mechanism discussed is the fact that the controlled agents must interact in the nonextensive manner (namely the should be seen or heard be all other agents). If the interaction is extensive (between the controlled and "normal" agents), then the control is very limited.

Interactive applet

Below you should find an interactive Kirman's agent-based model applet, which incorporates the controlled agents. We would like to draw your attention to the fact that in this applet we allow both negative and positive values of \( M \). The negative values just indicate that controlled agent choose the opposite state to the one represented by \( x \). The meaning behind and usage of the other parameters of the applet is standard.

On the left panel you can observe the time domain signal - red dots are the numerical results generated by the model, while blue curve corresponds to the theoretical expectations (the expected mean). On the right side of the applet you can see a plot of the probability density function of \( x \). Note that the right plot is cleared every time you press any of control buttons. Using the "Update" and "Stop"/"Resume" buttons you can continue evaluating old time series using the new parameters. "Restart" button always restarts modeling from scratch.

Acknowledgement

This text was prepared during the 2013 summer internship "Control of complex stochastic processes" (supervisor A. Kononovicius) financed by RCL. I. Kazakevicius acknowledges support by project "Promotion of Student Scientific Activities" (VP1-3.1-ŠMM-01-V-02-003) from the Research Council of Lithuania. This project is funded by the Republic of Lithuania and European Social Fund under the 2007-2013 Human Resources Development Operational Programme’s priority 3.

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