Socio-configuration

In mathematical models of social dynamics, a socio-economic system is characterized on two levels, distinguishing the micro-aspect of individual decisions and the macro-aspect of collective dynamical processes in a society. The probabilistic macro-processes with stochastic fluctuations can be described by the master equation of human socio-configurations. 

In effecting transfers from the microlevel of individuals to the macrolevel of society the concept of "socio-configuration" will be introduced. The socio-configuration describes the distribution of attitudes over the subpopulations of a society and may be considered as an appropriate set of macrovariables for this society. 

The problem will be further simplified by assuming a homogeneous population whose members have the same individual behaviour probabilities of reactions and interactions in the opinion formation process. The socio-configuration ni, n at a given time t then consists of the numbers ni and 2 of people having opinion 1 and 2, respectively. If the total number of members of the population, say IN, is constant the socio-configuration is dependent on only one relevant variable, the integer n, thus ... 

The transition probabilities are assumed to be functions of the momentary socio-configuration defined by the variable n. Before considering specific transition probabilities in Sect. 2.4 the mathematical structure of the model in its general form will be discussed in Sect. 2.2 and 2.3, whereby in the following Sect. 2.2 the levels of description adopted for the motion of the system will be explained. 

First an ensemble (a statistical population) of societies - each of which consists of one homogeneous (human) population of 2 members - will be considered and the probability distribution over their socio-configurations will be defined. Thus the function... 

Correspondingly, the transition probabilities for the whole socio-configuration are given by... 

In reality there may exist a feedback between the socio-configuration and the trend parameters in which case the latter also become dynamic variables. This general problem will also be discussed in Chap. 3 and in later models. Here the problem is simplified by assigning a certain assumed time dependence... 

In the Figs. 2.12 a, b time has been eliminated and (x) is represented as a function of the preference parameter d. Illustrated are so-called hysteresis loops for which the Figs. 2.12a, b correspond to the results of Figs. 2.11a, b. These figures show that the motion of the socio-configuration (collective opinion configuration) in a liberal society follows closely and in smooth evolutionary way the development of the individual preference trends described by d. Further, the moderate values and variations in the variance (i.e. the dispersion of the distribution) lead to small fluctuations of sample paths around their mean... 

Summarizing, the model solutions of Figs. 13 and 14 can be interpreted as suggesting that totalitarian societies produce - even after delusively stable periods - hard, revolutionary and in their details unpredictable transitions in their socio-configuration. 

Attitude Space, Socio-Configuration and Situation Space... 

Introducing the C-dimensional socio-configuration space the time dependent socio-configuration of a sample society can be represented by a moving point n (t) = , (t) in the space This space plays a similar role to that of the phase space 2 for macrovariables in statistical physics. Under the assumption... 

The attitudes and the socio-configuration, which are indispensible for an understanding of a society s internal d5mamics, are mainly related to the... 

In terms of the concepts of socio-configuration and situation vector applied here the dynamics of the society to be described is generated as follows ... 

The definitions of socio-configuration and situation vector so far completely leave open which aspects of individual attitudes and which aspects of the material situation have to be taken into account. Of course, it cannot be expected that all individual and situational aspects can be treated simultaneously. It is initially a sufficient achievement to be able to describe the subdynamics of a sector of a given society represented in a subspace of SI and... 

It has been explained in Sect. 3.1, that the motions of the socio-configuration and the situation vector are coupled. Nevertheless it is reasonable in a first approximation to treat the equations for the socio-configuration n (t) separately. The justification for this is that the motion of n (f) is due to collective psychological processes which in general are much more flexible than the material situation y(t). This implies that n(t) often (but not always) changes much more quickly than y (t). In these cases y (t) can be regarded as quasi-constant and the couphng of its dynamics with n (t) need not be considered. This is the approximation adopted in the models of Chapters 2, 4. 

The most fundamental description of the dynamics of the socio-configuration (3.5) including its fluctuations is given by a master equation for its probability distribution. 

In general they depend on the socio-configuration n and on trend parameters K (e.g. the adaptation and preference parameters of Chap. 2). The latter are considered as constants here, but may also become dynamic variables in an extended treatment (see, for instance. Chap. 5). 

In the general case the composition formula for the transition probability between socio-configurations is ... 

Equations of Motion for the Socio-Configuration 63 is introduced and normalized to... 

For convenience from now on only one index i running from 1 to C instead of ai will be written for the components of the socio-configuration. Further, the following notation for moments of first and second order is introduced ... 

The relation between Langevin equations and the Fokker-Planck equation will now be discussed. As in Chap. 1 the starting point are the Langevin equations of the variables xi (t) for the socio-configuration (and eventually including y, (t) for the material situation) of the general form ... 

The equations of motion for the socio-configuration do not yet reflect the full complexity of the dynamics of a socio-system. Nevertheless they describe the motion of collective psychological and political processes (namely the changes... 

If they are constants with time, it can be shown (e.g. by using Boltzmann s famous H-theorem) that all distributions developing according to the master equation (3.14) or the Fokker-Planck equation (3.32) or (3.34) end up in the stationary distribution W (jc). After this the motion of a sample socio-configuration can only consist of stationary fluctuations around the probability peaks of the stationary distribution. While this kind of motion may describe a static society under stable psychological and situational conditions, it does not comprise the full scope of sociological phenomena. 

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