Saturday, March 14, 2009

anti-Social Ecology:

A Mathematical Model of Dysfunctional Family Dynamics Embedded in a Social Field

The purpose of this note is to map the attractors, and the phase transitions between them, of local and global potential surfaces and conformations of what are termed 'dysfunctional families' embedded in an (anti)social ecosystem. . A dysfunctional family can be characterized using several objects, including the Herder balance, the Karpov Drama Triangle, the nonoptimal metastable state of Huberman, the Levebre utility function, and the cognitive dissonance utility function. Some of these are graph theoretic or network models, while others are the non-simplical continuum variants familiar to studies of energy or fitness landscapes.

In this note, the dysfunctional family is embedded in a social field, which has its own topological and geometric structure based on all the families in the field. Essentially the picture is from general relativity theory, where the unified field is constructed from interactions between local massive bodies. There is actually a 'bootstrap' mechanism of co-creation (a hermeneutic circle, similar to models of neural interpretation of sensory stimuli). While the whole is determined by its parts, the parts in turn are determined by the whole.

Many of these ideas are essentially similar to models constructed by R Wallace and others, on crowd dynamics. Marxist theory, where consciousness is determined by class, and consiousness in turn stabilizes class via the process of alienation (or false consiousness), has essentially the same self-fullfilling, social constructionist bootstrap. Early models by Samuelson and Goodwin formulated in the newtonian and statistical mechanical formats of E Kerner, have been reformulated in the Feynman path integral and Einstein-Cartan differential manifold format in the more recent papers, and summarize that low-degree of freedom theory.

The mathematical questions involved here are 3 fold, beyond finding the metric and euler-lagrange equations from the lagrangian. First, what sort of Lagrangian is desired, either an optics-based average for fluid flow, or a complete lagrangian based on fisher information or some equivalent. Second, deciding whether one uses deterministic or ergodic (statistical) dynamics. And third, deciding how to relate finite, discrete network descriptions to continuous ones, in some limit, and whether any uniqueness properties of such a limit exist. The nature of the nonlinear coupling between the local metric and the global field is what is important.

The applicable questions involve stability of 'networks of networks', ie the time evolution of dysfunctional families embedded in some extended network of families of varying functional form. Often, a simplified 'ising' description is made, which is that there are only 2 kinds of families---functional and disfunctional. Otherwise one can use a Potts model and let there be n kinds of families. One can alternatively define a metric and assign a value or indicator to each family, which might be thought of as its 'elevation' or 'potential energy' in some social configuration. This leads to an n-body problem similar to problems in mechanics where one has a configuration of bodies attached to each other and on some general surface (eg a torus) by springs. Here, depending on the metric, the n-bodies likely will be embedded in a higher dimensional space, since there are many kinds of measures which can be assigned to families (eg class, race, education, history, ideology, etc.) One possible method is to assign families 'colors' or 'flavors', or temperatures. (Often the social field is assigned the temperature for mathematical conveniance, and from that family subsystems can be assigned local nonequilibrium temperatures.)

The question is the probability distribution (or attractor disctribution) for an ecosystem populated by different numbers of family types. One would like to characterize both the number of types, and their population, and their trajectory or history (from either a local 'lagrangian' view of a single family or 'tribe', and the ' families, and the 'eulerian' or fluid flow perspective of the evolution of the entire social ecology through time. In a discrete form, this is the problem of dividing a pie, with the additional format that the different divisions are seen to have different stability properties, as well as being path dependent (nonergodic). Another stipulation is that of distinguishability, which can change the manifold. (A very interesting question is whether quantum measurerment theory suggests that distinguishability is a convention so that transitions can occur simply by relabeling, just as Newton resolved light by filtering it through a prism. )

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