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, 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 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 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-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-ElementsWe 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. We consider
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 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].
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.
· 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 ].
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 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. 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. |