2. The Project

A semi-classical approach, i.e., non-quantistic [Sewell, 1986], using Meta-Structures specified below, has been introduced [Minati 2008a, 2009a 2009b] for the study and modelling of the phenomena of emergence, particularly collective behaviours. The approach is considered as a semi-classical one because it is not based on microscopic or macroscopic models as in classic approaches for studying systems  as, for example, in thermodynamics, engineering and cybernetics, but considers mathematical properties of multiple dynamics clusters of mesoscopic variables, their relationships and Meta-Elements as introduced below. The approach cannot be considered as ‘non-classic’ because it is not based, for instance,  on quantistic models of the physics of emergence used to study and model radical emergence [Licata, 2008; Pessa, 2002; 2006], as previously discussed[Pessa, 2008; Politi, 2006]. The Meta-Structures approach was introduced as a suitable tool to study and model, at a mesoscopic level of description,  general processes of change [Pessa, 2008] acquiring emergent properties, i.e., processes of emergence.

 In order to outline the approaches and assumptions used in this project, we first of all need to specify that the approach considered is based on assuming a mesoscopic level of description because of its effectiveness in providing representations of change in configurations of interacting elements taking into account simultaneous multiple, partial, different, microscopic aspects over time. In this way we consider multiple dynamic clusters of interacting elements corresponding to multiple microscopic variables taking the same values over time, such as the speed, altitude and direction of boids composing a flock establishing collective behaviour.

Secondly, we introduce the concept of  Meta-Element. Meta-Elements are given, for instance, by sets of time-ordered vectors such as: a) values of mesoscopic state variables, e.g., how many elements belong to clusters over time; b) values specifying mesoscopic state variables such as thresholds and parameters, e.g., distance, speed and altitude of microscopic interacting boids; c) values adopted by the mesoscopic general vector over time  specifying how mesoscopic properties diffuse and change over time, e.g., how many elements possess one or more properties over time; d)  levels of usage of  microscopic degrees of freedom; and values taken by suitable macroscopic variables such as volume and surface.

Thirdly, in analogy with the mathematical structures considered here on Meta-Elements, we introduce the concept of Meta-Structure. In our project we use the term Meta-Structure with reference to simultaneous coherent multiple structures over time and their sequences, as previously introduced, suitably represented by mesoscopic variables and Meta-Elementsas mentioned above.

 Meta-Structural properties or, in short, Meta-Structures tout-court, are given by their mathematical properties, such as: statistical; due to correlation or auto-correlation; represented by interpolating functions; given by quasi-periodicity; ergodicity; and possible relationships between them; possessed by sets of time-ordered vectors constituting Meta-Elements. Meta-Structures are thus considered to be given by the properties and relationships between Meta-Elements

2.1 Outlines   


We introduce here an experimental protocol [Minati, 2010] for research into Meta-Structural properties in simulated Collective Behaviours, such as flocks of boids, to model their coherence with reference to the theoretical approaches mentioned above. Several models have been introduced in the literature to simulate collective behaviours for instance of swarms and flocks, e.g. , [Bajec et al., 2005; 2007; Bonabeau et al., 1999; 2000; Heppner and Grenander, 1990; Huth and Wissel, 1992; Inada and Kawachi, 2002 ; Kunz and Hemelrijk, 2003; Millonas, 1992; 1994; Reynolds, 1987; 1999; 2006; Theraulaz and Deneubourg, 1994; Theraulaz and Gervet, 1992; Vabo and Nottestad, 1997

We initially turn our experimental research towards simulated collective behaviours because all values taken by microscopic and mesoscopic variables and thresholds, with respect to time, are available to researchers. 
Simulated collective behaviours should be at a sufficient level of complexity, for instance at Class 4 with reference to Wolfram's classes of cellular automata. 
The purpose is to research meta-structural properties in simulated collective behaviours. 
 That is, to test and experiment the approach by dealing with a simple framework, namely computational emergence where the processes of interaction occur through fixed, learning evolutionary rules. 
 We consider finding Meta-Structures in this simplified framework as a necessary condition before the approach can be tentatively applied to non-simulated emergence such as real swarms and flocks, industrial districts, markets and traffic. 

2.2 The approach


 We conside

2.2.1 Values assumed by mesoscopic variables such as:

1. Mx(ti),  number of elements having the maximum distance at a given point in time;

2. Mn(ti), number of elements having the minimum distance at a given point in time;

3. N1(ti) number of elements having the same distance from the nearest neighbour at a given point in time; 

4. N2(ti) number of elements having the same speed at a given point in time;

5. N3(ti) number of elements having the same direction at a given point in time;

6. N4(ti) number of elements having the same altitude at a given point in time;

7. N5(ti) number of elements having the same topological position at a given point in time

8. Vol(ti), volume of the collective entity at a given point in time;

9. Sur(ti) surface of the collective entity at a given point in time.


 2.2.2 Degrees of ergodicity given by indexes of ergodicity related to properties considered to set mesoscopic variables, such as considering same distance:

    x1 (ti)  =  average percentage of elements having the same distance from the nearest neighbour at time ti;

    y1 (ti) = average percentage of time spent by a single element having the same distance from the nearest neighbour.

    The index of ergodicity considered on the total observational time is E1  = 1/[1 + (x1%  – y1%  )2].                                                

2.2.3 Values assumed by the mesoscopic general vector

    When considering the mesoscopic general vector  Vk,m(ti)  = [ek,1(ti)  , ek,2(ti)  , …, ek,m(ti)] 

    where:

    k  - identifies one of the k elements ek ; 

    i  -  is the computational step or instant in a discretised time; 

    m - identifies one of the m mesoscopic or ergodic properties possessed 

      by the element ek at instant ti; 

    ek,m -  takes the  value 0 if the element ek does not have the                         

                                     m-mesoscopic  or ergodic properties at time t; 

                                    or 1 if the element ek does possess the                     

                                     m-mesoscopic or ergodic properties at time ti.

    for instance:          Vk,m(ti)  = [1, 0, 1,1, 0,1 0, 0].

    It is possible to detect properties such as:

      · Multiple repetitiveness, i.e., how many times a specific mesoscopic general vector  is repeated during the total     observational time. It is also considered its occurring over time.

      · periodicity and  quasi-periodicity of values assumed over time by the mesoscopic general vector ;

       ·statistical properties of sets  of numbers of steps, elements and times.

2.2.4 Properties of sets of degrees of freedom
Consider the sets of degrees of freedom for the microscopic behavior of each elements, such as:

· min and max distance

· min and max speed

· min and max change of direction

· min and max altitude

At each computational step it is possible to consider that the value of the speed Vk (t) of a k-boid, component of the flock under study, at  time t must not only respect the degree of freedom, but is also considered to set the degree of respect of that degree of freedom. An introductory example is given by considering the percentage  [100 * Vk (t)] / [Vmax - Vmin ].

2.2.5 Topological properties

We consider topological properties [ Ballerini et al., 2008; Hildenbrandt, 2010] such as how many and which elements possess a topological position. Topological positions may be:

§ Belonging to the geometrical surface or to a specific area of interest;

§ Having a specific topological distance from one of the elements belonging to the geometrical surface or a specific area of interest;

§ Be at the topological centre of the system, i.e., all topological distances between the element under study and all the element belonging to the geometrical surface are equal. This element may be virtual and be considered as a topological attractor for the system. Its trajectory may represent the trajectory of the system.


2.2.6 Meta-structural properties as properties of ordered sets of values of Meta-Elements

A Meta-structure is given by the mathematical properties possessed by ordered sets of values establishing Meta-Elements, values assumed by Mesoscopic State variables, sets of degrees of respect of the degrees of freedom and values eventually assumed by suitable macroscopic variables such as Vol(ti), volume of the collective entity at a given point in time and Sur(ti), surface of the collective entity at a given point in time.

 Mathematical properties may be statistical such as the Multivariate data Analysis (MDA) -Cluster Analysis, Principal Components Analysis (PCA) and  Principal Components (PCs)-, Recurrence Plot Analysis (RPA) and Recurrence Quantification Analysis (RQA); due correlation or auto-correlation; represented by interpolating functions; given by quasi-periodicity; levels of ergodicity –degrees of ergodicity (as introduced below in this Section), or possible relationships between them in an N-space, suitable for modelling a kind of entropy of correlations.  

2.3 Purposes  

The purpose is to research meta-structural properties in simulated collective behaviours. That is, to test and experiment the approach by dealing with a simple framework, namely computational emergence where the processes of interaction occur through fixed, learning evolutionary rules. 

We consider finding Meta-Structures in this simplified framework as a necessary condition before the approach can be tentatively applied to non-simulated emergence such as real swarms and flocks, industrial districts, markets and traffic. 

Experimental issues may involve the following aspects: 

Continuity

How many and which elements have the same, different or no mesoscopic properties over time;

The total number of and which elements possessed at least one mesoscopic property and total number of and which properties possessed by elements at the end of the global observational computational time;

Multiple repetitiveness, coherence and local, partial quasi-periodicity; 

Cognitive topological distance, topological position and properties of elements and mesoscopic properties. 

Diffusivity

The set of all values adopted by the mesoscopic general vector allows detection, for instance, of:

Number of computational steps, i.e., Computational Distance (CD), occurring before all elements have been at least once in the  on state (indicated as general meso-state on), repetitiveness;

How many times the general meso-state on occurs, i.e., how many times did it take the on state;

Properties of the sets of numbers of steps: statistical, periodic, quasi-periodic, etc. 

The availability of meta-structural properties is expected to allow researchers, as specified at point 1 -Theoretical Basis-  to:

a) Recognise a phenomenon as emergent such as collective behaviours acquiring emergent properties;

b) Induce emergence of collective behaviour in populations of agents collectively interacting;  

c) Act on collective emergent phenomena with the purpose to change, regulate and maintain acquired properties;

d) Merge different  collective emergent phenomena.

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