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We propose a social model of spontaneous self-organization generating criticality and resilience, called Self-Organized Temporal Criticality (SOTC). The criticality-induced long-range correlation favors the societal benefit and can be interpreted as the social system becoming cognizant of the fact that altruism generates societal benefit. We show that when the spontaneous bottom-up emergence of altruism is replaced by a top-down process, mimicking the leadership of an elite, the crucial events favoring the system’s resilience are turned into collapses, corresponding to the falls of the leading elites. We also show with numerical simulation that the top-down SOTC lacks the resilience of the bottom-up SOTC. We propose this theoretical model to contribute to the mathematical foundation of theoretical sociology illustrated in 1901 by Pareto to explain the rise and fall of elites.

The recent book of Haidt [

The new model of spontaneous organization [

Phase transitions and critical phenomena occur frequently in nature and have been widely studied by physicists; see, for instance, [

The Decision-Making Model (DMM) [

At criticality, namely, when the dynamics of the system are determined by the control parameter generating phase transition, the mean field

In the case of the brain dynamics there is wide consensus on the connection between consciousness and criticality. See, for instance, [

The crucial events are responsible for the information transport from one system at criticality to another system at criticality [

We may afford an intuitive interpretation of crucial (complex) events, using the example of a flock of birds flying in a given direction, as an effect of self-organization. A crucial (complex) event is equivalent to a complete rejuvenation of the flock that after an organizational collapse may freely select any new flying direction. An external fluctuation of even weak intensity can force the complex system to move in a given direction, if it occurs at the exact instant of the free will of the SOTC model system. It is important to stress that the organizational collapse is not the fall of an elite, which will be discussed subsequently, because the flock self-organization occurs spontaneously and does not rest on the action of a leader. The choice of a new flying direction is thus determined by an external stimulus of even weak intensity occurring at the same time as the collapse, thereby implying the property of

As mentioned earlier, the crucial events favor the transport of information from one complex system to another [

However, criticality may also be Achilles’ heel of a complex system, if criticality is generated by a fine tuning control parameter. In fact, committed minorities acting when a crucial event occurs in the case of DMM can make the system jump from the state

The model of [

For clarity in Sections

The decisions of single individuals in our model are made in accordance with the criterion of bounded rationality [

In this paper we use the DMM on a two-dimensional lattice of size

The unbiased transition rate is

This decision-making process is fast and emotional and does not involve any direct reasoning about the payoff. The connection with the self-interest, according to the slow thinking mechanism discussed by Kanheman [

We adopt the choice of parameter values made by Gintis [

The payoffs of Prisoner’s dilemma game. The first value of each pairs is the payoff of player

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Each individual receives a total payoff from the game with the four nearest neighbors and adjusts her imitation strength as follows:

The second term on the right-hand side of (

Note that in the limit of vanishing time intervals (

For the bottom-up case discussed in this section, the calculation is done with the parameters

The results of Figure

For the bottom-up SOCT: time evolution of the average social benefit

To stress the occurrence of crucial events in a social system resting on the bottom-up emergence of altruism, we have to extend the method used for criticality generated by the fine tuning of the control parameter

For bottom-up SOTC: the waiting-time PDF of the time interval between two consecutive crossings of the average value of the mean value of

Figure

In spite of the exponential truncation that may lead to the misleading conclusion that the Poisson-like character of the long-time regime quenches the manifestations of complexity, the transport of information is determined by the intermediate asymptotics. In the earlier work of [

The top-down approach to self-organization is done using again (

The second term on the right-hand side of (

For the top-down case discussed in this section, the calculation is done with the parameters

Black line: the time evolution

For bottom-up SOTC, the waiting-time PDF of the time intervals between two consecutive crossings of the horizontal line with

Under the leadership of an elite, see Figure

The falls to the small values of

It is interesting to notice that also the time interval between consecutive falls of elite is a complex dynamical process characterized by an IPL, with

To substantiate the arguments of the earlier section with the results of a numerical simulation we devote this section to illustrating some numerical experiments on the effects of a perturbation on the process of societal self-organization.

First of all let us define two different sources of perturbation, the

In the top-down SOTC model the independent influences the process through his vanishing contribution to

The other kind of perturbation, produced by committed minorities, has already been studied elsewhere [

Let us discuss first the strongest source of perturbation, the randomly moving independents. At any time step one of the

In the case of the top-down SOTC the effects of this perturbation are impressive. The black line of Figure

Important information on the lack of resilience of the top-down SOTC is afforded by Figure

The perturbing action of independents is weaker if the independent individuals do not move. This response to this form of perturbation is illustrated in Figure

Dependence of mean field (at time

We establish the perturbation of independent individuals in a different way. We assume that all the units are independent for a fraction

Dependence of mean field (at time

Finally in Figure

Dependence of mean field (at time

It is remarkable that according to SOTC the crucial events may be harmful as well as beneficial. If the global parameter

The main conclusion of this paper concerning resilience is that criticality is necessary for resilience, but it is not sufficient. The top-down SOTC model generates criticality, but it is not resilient. Therefore information transport from one top-down SOTC model system to another top-down SOTC model system is expected to occur by means of complexity matching, in spite of the fact that the two systems are not resilient and the information transport may be easily quenched by stray perturbing noises.

An attractive interpretation of the resilient nature of the bottom-up SOTC model is that the ideal condition of full democracy is the most robust form of social organization.

In this paper the social payoff is evaluated using the PDG [

The human inclination to cooperate is the result of biological evolution and of the spontaneous evolution towards criticality. The time appears ripe to unify the models of biology and physics made necessary to reach the ambitious goal of achieving a rigorous scientific foundation of this important human characteristic [

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Korosh Mahmoodi and Paolo Grigolini warmly thank ARO and Welch for support through Grants no. W911NF-15-1-0245 and no. B-1577, respectively.