# Emergence as Mesoscopic Coherence

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## Abstract

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## 1. Introduction

- (a)
- Within the mesoscopic description level of complex processes, areas of continuous negotiations between micro and macro, it is impossible in principle—because the problem becomes intractable—to consider all the dynamic inter-relations between agents and between system and environment. We have to keep in mind that the concept of “mesoscopic variable” can mean many things. In the systems where it is possible to apply the ideal models—for ex Haken Synergetics [12]—identifying the “right” mesoscopic variables is suggested by the same system’s dynamics. In other cases, such as very fast dynamics between agents where the role of the hubs change rapidly, the identifying of mesoscopic variables is not so patent. Also for non-ideal models it is necessary to do some a priori assumptions on the system’s dynamics, for example: choosing rules for cellular automata and weights for neural nets. In many cases of natural or artificial populations it is impossible. However, they are phenomena showing their peculiar regularity. Therefore, we have changed the strategy to approach them by reducing, at minimum, the a priori hypotheses, to observe the system over time and so to acquire a wide sequence of historical data. On the basis of these data it will be possible to make a posteriori conjectures on the more suitable mesoscopic variables to describe the significant patterns of the system.
- (b)
- In these irreducible cases it is on the observer to define the mesoscopic variables, which could be the clue of dynamics, even if not so “clean” as in ideal models. It is necessary to underline that this approach is different from statistics—which gives an important contribution for the study of the single variables and their correlations—because the strategy is purely constructivist: it is the observer’s choice to decide which global configurations bring significant information and which have to be considered just as noise, on the basis of its own operative choices on system and environment. In another paper we called such strategy as “without Physics” [13] just for the switching of the a priori and a posteriori and for the accent on the observer’s role. We have to select from several families of possible observables and simultaneously, consider mesoscopically some of them as being clustered by the particular aspects the researcher defined when adopting a research strategy. We expect the identification of structural changes, similar in a way to phase transitions. As a matter of fact, the characteristic of collective systems, when adopting collective behaviors, is precisely their ability to rearrange through dynamic sequences of local emergent relationships and ways of interacting, based on the emergence of local and temporary constraints, which are an integral part of the system’s dynamics that make it unpredictable. The idea is to use and superimpose several mathematical grids, i.e., precise meta-structures, in order to detect emergence as mesoscopic coherence, for instance, by statistical regularities such as possible ergodicity or quasi-ergodicity [14,15].

## 2. Collective Systems

- (a)
- Examples of collective systems, given by the collective motion of living systems, provided with sufficiently complex cognitive systems, include flocking, swarming, anthills, herding, schooling, crowd, and traffic. In these cases, the interactions involve cognitive processing.
- (b)
- Examples of collective systems, given by the collective motion of living systems, provided with no cognitive systems, include amoeba, bacterial colonies, cells, and macromolecules. Examples of collective systems, given by the collective motion of non-living systems, include lasers, systems of boats, nano-swimmers, nematic fluids, networks, signaling traffic, rods on vibrating surfaces, shaken metallic rods—interaction involves reacting—and simple robots [10]. Interaction is given by simple artificial cognitive systems.
- (c)
- Examples of collective artificial systems, given by collective interactions of various natures other than physical motion include, communities of mobile phone networks, industrial districts, markets, morphological properties of cities and urban development, networks like the Internet, and queues. Interaction is given by systems of cognitive processing and reactions.

- The homogeneous and non-homogeneous cases where generic agents have or do not have the same properties used for interacting. In the first case, we will consider populations of generic agents, possibly labeled, and identical at a suitable level of description. For instance, they possess the same cognitive system, since they belong to the same species, or they use the same artificial ways of interacting. They interact in the same way, but at different times: using different values of variables, e.g., position; using different parameters, e.g., depending on density and topological position; or starting from different initial conditions. Cases (b) and (c) described above, apply as examples. In the second case, we consider populations of generic agents considered to be distinguishable, not only because they are labeled, but because they interact in different ways for reasons such as the use of structurally different rules (not just different values of variables, different parameters, and different initial conditions of the same analytical rule) but, for instance, different rules considering each value of direction, speed or altitude. Such rules are intended as if applied individually to any combination or sequence by each generic agent as introduced below.
- The number of generic agents may be fixed during the entire process or variable due to the entry or exit of generic agents for any reason.
- The way of interacting—called here the structure of interaction—may be the same, or not the same, during the entire process. In the first case—the same way of interacting during the entire process-generic agents interact using the same available library of rules. In the second case—different ways of interacting during the entire process—generic agents interact using different combinations of the same available library of rules as well as new rules established, for instance, by learning or environmental conditions.

#### What Happens in-between?

## 3. Modeling Mesoscopic Change

- ▪
- Mesoscopic variables transversally intercept and represent values adopted by aggregates of microscopic variables. Values of mesoscopic variables are considered to represent the effective application of interaction rules.
- ▪
- Properties of sets of such values represent the coherence of sequences of configurations, i.e., the collective behavior.

- (1)
- Mmx(t
_{i}) number of elements having the maximum distance at a given point in time; - (2)
- Mn(t
_{i}) number of elements having the minimum distance at a given point in time; - (3)
- Md(t
_{i}) number of elements having the same distance from the nearest neighbor at a given point in time; - (4)
- Ms(t
_{i}) number of elements having the same speed at a given point in time; - (5)
- Mdir(t
_{i}) number of elements having the same direction at a given point in time; - (6)
- Ma(t
_{i}) number of elements having the same altitude at a given point in time; - (7)
- Mt(t
_{i}) number of elements having the same topological position at a given point in time; - (8)
- Macroscopic variables such as measures of Vol(t
_{i}), volume of the collective entity over time (used to compute density) and Sur(t_{i}), surface of the collective entity over time.

_{1}, d

_{2}, ...., d

_{n}. Furthermore the same elements may belong to different clusters related to the same mesoscopic variable and to different mesoscopic variables.

_{k}establishing a collective system, and m is the mesoscopic property considered, for instance m

_{j:1-M}where M is the total number of mesoscopic variables available. In this exemplified description the numbers k and M are considered fixed during all the phenomenon in a discretized time t

_{i:1-T}where T is the total time.

_{m,j}(t

_{i}) = [m

_{1}(t

_{i}), m

_{2}(t

_{i}), …, m

_{M}(t

_{i})]

_{j}(t

_{i}) takes the value of 0 if no generic agents possess the mesoscopic property m

_{j}(t

_{i}) at time t

_{i}taking, in other cases, the number of generic agents possessing the mesoscopic property m

_{j}(t

_{i}) at time t

_{i}.

_{i}) given by generic agents e

_{k}for mesoscopic variables where the element KM

_{k}

_{,m}(t

_{i}) is equal to 0 if the generic agent e

_{k}does not possess the mesoscopic property m at time t

_{i}or 1 if the generic agent e

_{k}does possess the mesoscopic property m at time t

_{i}:

- (1)
- Properties of the values acquired by mesoscopic variables, single or crossed, such as any regularities including periodicity, quasi-periodicity, chaotic regularities possibly with attractors which characterize specific collective behaviors;
- (2)
- Possible statistical properties of sets of meta-elements detected by suitable techniques like Principal Components (PCs), Recurrence Quantification Analysis (RQA), Multivariate Data Analysis (MDA), Cluster Analysis, Principal Component Analysis (PCA), Time-Series Analysis, Pearson Product Moment Correlation Coefficient (PPMCC);
- (3)
- Properties, e.g., geometrical and statistical, of sets of generic agents constituting mesoscopic variables;
- (4)
- Properties related to the usage of degrees of freedom as introduced above;
- (5)
- Relationships between properties of sets of clustered generic agents and, macroscopic properties such as density, distribution, scale-freeness, numerical properties such as percentages;
- (6)
- Properties of the thresholds adopted for specifying the mesoscopic general vector;
- (7)
- Levels of ergodicity or quasi-ergodicity;
- (8)
- Properties of values of the vector and the matrix considered above over time;
- (9)
- Possible topological properties of network representations such as power laws and scale-freeness.

- (1)
- All of the generic agents simultaneously possess all the same single mesoscopic property which is constant over time;
- (2)
- All of the generic agents simultaneously possess the same subset, constant over time, of the mesoscopic properties available;
- (3)
- All of the generic agents simultaneously possess a subset, variable over time, of the mesoscopic properties available;
- (4)
- A significant percentage of the generic agents simultaneously possess all the same single mesoscopic property which is constant over time;
- (5)
- A significant percentage of the generic agents simultaneously possess the same subset, constant over time, of the mesoscopic properties available;
- (6)
- A significant percentage of the generic agents simultaneously possess a subset, variable over time, of the mesoscopic properties available;
- (7)
- Any combinations of the previous cases may occur regarding different or the same generic agents.

Mesoscopic Dynamics | |
---|---|

Structural cases | Meta-structural properties |

(1) All of the generic agents simultaneously possess all the same single mesoscopic property constant over time; | Trivial meta-structural properties Collective behaviors structurally “fixed” |

(2) All of the generic agents simultaneously possess the same subset, constant over time, of the mesoscopic properties available; | Trivial meta-structural properties Collective behaviors structurally at low variability |

(3) All of the generic agents simultaneously possess a subset, variable over time, of the mesoscopic properties available; | Significant meta-structural properties Collective behaviors structurally variable |

(4) A significant percentage of the generic agents simultaneously all possess the same single mesoscopic property constant over time; | Non-trivial meta-structural properties Collective behaviors mesoscopically fixed |

(5) A significant percentage of the generic agents simultaneously possess the same subset, constant over time, of the mesoscopic properties available; | Non-trivial meta-structural properties Collective behaviors mesoscopically variable |

(6) A significant percentage of the generic agents simultaneously possess a subset, variable over time, of the mesoscopic properties available; | Non-trivial meta-structural properties Collective behaviors mesoscopically at high variability |

(7) Any combinations of the previous cases may occur regarding different or the same generic agents. | Complex multiple meta-structural properties Topological correspondences |

Mesoscopic variables | Mesoscopic state variables are invented by the observer in a constructivist manner and represent clusters of agents taking on the same values at the same time. For instance, the value taken by a mesoscopic state variable at time t_{i} represents the number of elements which have the same (within a range of values) value of some microscopic state variables such as the same distance from their nearest neighbors, the same speed, the same direction or the same altitude over time. |

Meta-elements | Meta-elements are time-ordered sets of values in a discrete temporal representation adopted by mesoscopic variables over time and specifying mesoscopic state variables, e.g., values of the same distance, speed or altitude adopted by elements having this mesoscopic property. |

Meta-structural properties | Meta-Structural properties are given by the mathematical properties possessed by ordered sets of values establishing Meta-elements, e.g., statistical, periodic, or correlative. |

Meta-Structure | The term Meta-Structure relates to simultaneous multiple structures governing interactions between generic agents and their sequences establishing corresponding coherent sequences of spatial configurations. |

Dynamical Systems | Fixed analytical representations of the rules of interaction between microscopic or macroscopic state variables used to model a dynamic system are assumed to be the structures of the system, e.g., the Brusselator, Lorenz or Lotka-Volterra equations. |

#### Acting on Collective Behaviors

- Induce coherence within sets of elements interacting collectively giving rise to processes of emergence;
- Change properties of collective behaviors, allow merging processes;
- Maintain or restore the coherence of a collective behavior when possible changes or a loss of coherence occur for any internal and/or external reason;
- Destroy or prevent the coherence and related processes of emergence which should be avoided under any circumstances.

- Component generic agents of the PCB, may be the same generic agents of the collective behavior, mutated by another collective behavior, assuming at a certain instant, i.e., by using different interaction rules. Other component elements of the original collective behavior are invisible to the component elements of the PCB. The other component elements of the original collective behavior will interact following the constraints corresponding to the new muted agents. The PCB strategy can be applied in many different ways:
- (1)
- The distribution of PCB agents may be fixed or variable with some regularities;
- (2)
- The percentages of agents acquiring mutations may be fixed or variable and have a fixed duration or be properly distributed over time;
- (3)
- It is possible to obtain various simultaneous or subsequent PCBs.

- The possibility of considering feedback between PCB and SCB, allowing the former to change in number, properties and allowing some kind of intelligent learning regulating, according to the expected modification, to be induced.

## 4. Ideal, But Not Always Practical

## 5. Conclusions

## Conflicts of Interest

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Minati, G.; Licata, I.
Emergence as Mesoscopic Coherence. *Systems* **2013**, *1*, 50-65.
https://doi.org/10.3390/systems1040050

**AMA Style**

Minati G, Licata I.
Emergence as Mesoscopic Coherence. *Systems*. 2013; 1(4):50-65.
https://doi.org/10.3390/systems1040050

**Chicago/Turabian Style**

Minati, Gianfranco, and Ignazio Licata.
2013. "Emergence as Mesoscopic Coherence" *Systems* 1, no. 4: 50-65.
https://doi.org/10.3390/systems1040050