# On the Complex Interaction between Collective Learning and Social Dynamics

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## Abstract

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## 1. Plan of the Paper

- (1)
- The level of learning of a certain knowledge is modeled, at the microscopic level, by a scalar variable $u\in [0,1]$, where $u=0$ represents the lowest level of achieved knowledge and $u=1$ the highest admissible level.
- (2)
- The whole system can be subdivided into functional subsystems, in which each one of them plays a different role in the learning process.
- (3)
- The overall state of the system is delivered, i.e., a probability distribution function over the microscopic state defines the collective state of the system, while macroscopic states are provided by weighted moments of this probability distribution.
- (4)
- The dynamics develops through encounters which can be either individual-based, or between individuals and the whole system.
- (5)
- The output of the interactions is modeled by developments of the theoretical tools of the evolutionary game theory, in which players are probability distributions or their averaged quantities.
- (6)
- The probability distribution, mentioned in Item 3, is the dependent variable of a differential system. Its dynamics is obtained by a balance of “living” particles in the elementary volume of the microscopic states.

## 2. State of the Art

#### 2.1. Phenomenological Description of Collective Learning

#### 2.2. A Brief Survey of Mathematical Approaches

#### 2.3. Critical Analysis

- The individual’s role in the collectivity. Collective learning modifies the individual’s ability to develop interaction rules that evolve over time.
- Learning dynamics. Living entities can learn not only by micro-scale interactions, but also from the whole population. The key problem consists of referring the dynamics at a micro-scale to treat at the macro-scale.
- Role of networks. Networks can have an important influence on the learning dynamics, as observed in [9] both exogenous and endogenous networks have to be considered corresponding, respectively, to physically localized nodes and nodes generated by aggregations due to affinity principles.

## 3. Mathematical Tools

- (1)
- Characterization of the soft variables and representation of the system;
- (2)
- Modeling of interactions;
- (3)
- Derivation of a general structure suitable to describe the collective dynamics of interactions between learning and social dynamics.

#### 3.1. Soft Variables and Representation

- The system is composed by living entities which interact in a spatially homogeneous dynamics;
- The microscopic scale corresponds to the state of each entity modeled as an a-particle [4];
- Each a-particle interacts with the others at both the microscopic level and the collective set of all the particles.

#### 3.2. Modeling Interactions

Evolutionary game theory deals with an entire population of players, all programmed to use the same strategy (or type of behavior). Strategies with higher payoff will spread within the population (this can be achieved by learning, by copying or inheriting strategies, or even by infection). The payoffs depend on the actions of the co-players and hence on the frequencies of the strategies within the population. Since these frequencies change according to the payoffs, this leads to a feedback loop. The dynamics of this feedback loop is the object of evolutionary game theory.

- Test particles of the r-FS with microscopic state ${u}_{i},{v}_{j}$ and probability ${f}_{ij}^{r}\left(t\right)$. These particles are assumed to be representative of the whole system.
- Field particles of the r-FS with microscopic state ${u}_{p},{v}_{q}$ and probability ${f}_{pq}^{r}\left(t\right)$ which are generic particles for each r-FS.
- Candidate particles, of the k-FS with microscopic state ${u}_{h},{v}_{k}$ and probability ${f}_{hk}^{k}\left(t\right)$ which are generic particles for each k-FS deemed to take, due to interactions, the state of the test particles.

- ${\eta}_{hk}^{pq}$ is the interaction rate of a $hk$-particle with a $pq$-particle, namely between particles with states $hk$ and $pq$.
- ${\mu}_{hk}$ is the interaction rate of a $hk$-particle with the mean state $\mathbb{E}$ of the whole system.
- ${\mathcal{A}}_{hk}^{pq}(hk\to ij)$ models the transition of a candidate $hk$-particle into the state of the test $ij$-particle due to the interaction with a field $pq$-particle.
- ${\mathcal{M}}_{hk}(hk\to ij)$ models the transition of a candidate $hk$-particle into the state of the test $ij$-particle due to the interaction with the mean value within the system.

## 4. Case Studies towards Perspectives

#### 4.1. Social Conflicts and Radical Opposition

- Modeling the role of the individual learning from the whole population to understand how a certain trend, which can be conditioned by the wealth distribution, can lead to radicalization of the political contrast.
- Modeling the transition, which can be viewed as a Darwinist mutation from a radical, however democratic, opposition, to a segregation of individuals ready to break laws and bring their opposition up to extreme levels.
- Modeling the selection following mutations by an approach which might require the addition of a new FS corresponding to security forces/actions to act against the aforementioned radicalization.

#### 4.2. Collective Learning and Dynamics of Crowds

- The development of a learning dynamics in a crowd, where individuals communicate and learn through small group awareness. Communications propagate stress, which in turn can generate unsafe dynamics.
- An additional interesting example is that of swarm’s dynamics where communications and learning induce a collective motion with flocking properties [60]. Such specific interaction between learning and mechanics might generate a motion suitable to optimize the swarm defense against the attack of predators [61].

#### 4.3. Multicellular Systems and Immune Competition

- A possible research program would be a deep analysis of the learning dynamics and the related influence over the immune competition.
- The approach with an additional difficulty consists of the modeling of a system with a variable number of interacting living entities as proliferative and/or destructive encounters that appear in this dynamics.
- The number of interacting functional subsystem grows in time due to the onset, by mutations, of new subsystems.

#### 4.4. Critical Analysis towards Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

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Burini, D.; De Lillo, S.
On the Complex Interaction between Collective Learning and Social Dynamics. *Symmetry* **2019**, *11*, 967.
https://doi.org/10.3390/sym11080967

**AMA Style**

Burini D, De Lillo S.
On the Complex Interaction between Collective Learning and Social Dynamics. *Symmetry*. 2019; 11(8):967.
https://doi.org/10.3390/sym11080967

**Chicago/Turabian Style**

Burini, Diletta, and Silvana De Lillo.
2019. "On the Complex Interaction between Collective Learning and Social Dynamics" *Symmetry* 11, no. 8: 967.
https://doi.org/10.3390/sym11080967