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information and Phase Transitions in Socio-Economic Systems
Terry Bossomaier, Dan Mackinlay, Lionel Barnett...
\section{Introduction}
\label{sec:intro}
Diamonds are not a good very long term investment! They are steadily turning into graphite. It will take millions of years, but the most stable form of carbon at room temperature and pressure is graphite. Thus diamonds will undergo a phase transition to graphite, albeit over a very long timescale.
When we normally think of phase transitions we think of the states of matter, ice melting to water or water turning to steam. They are order/disorder transitions. In graphite the carbon atoms are linked together in layers. The layers can slide over one another giving graphite its excellent lubricant properties. In diamond the carbon atoms are linked together in a three dimensional structure with each carbon at the centre of a tetrahedron linked to carbons at all four corners. Thus carbon has to go through a major structural reorganization to change from diamond to graphite.
We can easily see the outcomes of a phase transition of diamond to graphite or a solid turning into a liquid. But are there properties which we can measure which go through a minimum or maximum at the transition? This turns out to be a surprisingly difficult question to answer. It gets even more difficult if we look for measures which can apply to systems in general, not just to the physical systems above. Organsiations, societies, economies all go through radical reorganization, but should these changes be called phase transitions. This paper explores a metric based on information theory~xc{shannon} which is quite general if not universal. It then considers a much newer metric which, on examples to date, might be a predictor of impending transitions: since stock markets exhibit such transitions, early warning would be highly desirable.
sec#p-t discusses the characteristics of phase transitions and the peak in mutual information that accompanies them. It also introduces the idea of transfer entropy, recent extension to mutual information, which in some cases is known to peak before a transition to synchronous behaviour. With this background the first example, considered in section#wick considers a computational example derived from physics. The computation of mutual information from discrete data is tricky, involving difficult decisions on bin sizes and other stiatistical issues, which are also considered in this section. The next two sections discuss phase transitions in two areas in the social/humanities domain. sec#econo
discusses how peaks in mutual information occur around stock market crashes. sec#cog discusses the reorganization of strategy in the human brain during the acquisition of expertise. sec#t-e-net discusses how transfer entropy is calculated in practice using the example of inferring social networks from time series data. Finally we conclude in sec#conclusion with some opportunities for further work in the application of these techniques in stock markets and brain science.
sec:p-t:Overview of Phase Transitions and Metrics:
The simple physical notion of a phase transition, such as ice melting to water, is surprisingly hard to transfer to non-physical systems, such as society and organisations. This section will try to first look at the physical intuition behind the transition and then move on to look at some of the possible metrics.
The essential feature of a phase transition is a structural reordering of some kind, usually an order-disorder change. To get this involves some sort of long range order -- everything gets connected so that things can be reconnected in a different way. We next consider two more formal ways of looking at this: using random graphs; and by the use of an order parameter.
Random graphs
Order parameter in general
Mutual information
Transfer entropy
Fisher information
sec:wick:Computing Mutual Information for Phase Transitions in Simple Physics Models:
sec:econo:Phase Transitions in Socio-Economic Systems
sec:cog:Phase Transitions in the Acquisition of Human Expertise:
sec:t-e-net:Inferring Social Networks with Transfer Entropy:
sec:conclusion:Conclusions:
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