A Biased Review of Sociophysics

Abstract

Various aspects of recent sociophysics research are shortly reviewed: Schelling model as an example for lack of interdisciplinary cooperation, opinion dynamics, combat, and citation statistics as an example for strong interdisciplinarity.

Appendix: Critique of Mean Field Theories

This section explains mean field theory for readers from outside statistical physics, as well as its dangers.

If you want to get answers by paper and pencil, you can use the mean field approximation (also called molecular field approximation), which in economics corresponds to the approximation by a representative agent. Let us take the Ising model on an L×L square lattice, with spins (magnetic moments, binary variables, Republicans or Democrats) S _i =±1 and an energy

$$E = -J \sum_{ \langle ij \rangle} S_i S_j - H \sum_i S_i $$

where the first sum goes over all ordered pairs of neighbour sites i and j. Thus the “bond” between sites A and B appears only once in this sum, and not twice. The second sum proportional to the magnetic up field H runs over all sites of the system. We approximate in the first sum the S _j by its average value, which is just the normalised magnetisation m=M/L ²=∑ _i S _i /L ². Then the energy is

$$E = -J \sum_{ \langle ij \rangle} S_i m - H \sum_i S_i = -H_{\mathit{eff}} \sum_i S_i $$

with the effective field

$$H_{\mathit{eff}} = H + J\sum_j m = H + Jqm $$

where the sum runs over the q neighbours only and is proportional to the magnetisation m. Thus the energy E _i of spin i no longer is coupled to other spins j and equals ±H _eff . The probabilities p for up and down orientations are now

$$p(S_i=+1) = \frac{1}{Z} \exp(H_{\mathit{eff}}/T) ; \qquad p(S_i=-1) = \frac{1}{Z} \exp(-H_{\mathit{eff}}/T) $$

with

$$Z = \exp(H_{\mathit{eff}}/T) + \exp(-H_{\mathit{eff}}/T) $$

and thus

$$m = p(S_i=+1) - p(S_i=-1) = \tanh(H_{\mathit{eff}}/T) = \tanh\bigl[(H + Jqm)/T\bigr] $$

with the function $\operatorname{tanh}(x) = (e^{x} - e^{-x})/(e^{x}+ e^{-x})$. This implicit equation can be solved graphically; for small m and H/T, $\operatorname{tanh}(x) = x-x^{3}/3 + \cdots$ gives

$$H/T = (1-T_c/T)m + \frac{1}{3} m^3 + \cdots ; \quad T_c = qJ $$

related to Lev Davidovich Landau’s theory of 1937 for critical phenomena (T near T _c , m and H/T small) near phase transitions.

All this looks very nice except that it is wrong: In the one-dimensional Ising model, T _c is zero instead of the mean field approximation T _c =qJ. The larger the number of neighbours and the dimensionality of the lattice is, the more accurate is the mean field approximation. Basically, the approximation to replace S _i S _j by an average S _i m takes into account the influence of S _j on S _i but not the fact that this S _i again influences S _j creating a feedback.

Thus, instead of using mean field approximations, one should treat each spin (each human being, …) individually and not as an average. Outside of physics such simulations of many individuals are often called “agent based” [22], presumably the first one was the Metropolis algorithm published in 1953 by the group of Edward Teller, who is historically known from the US hydrogen bomb and Strategic Defense Initiative (Star Wars, SDI).

Of course, physicists are not the only ones who noticed the pitfalls of mean field approximations. For example, a historian [109] years ago criticised political psychology and social sciences: “There is no collective individual” or “generalised individual”. And common sense tells us that no German woman gave birth to 1.4 babies, even though this is the average since about four decades. A medical application is screening for prostate cancer. Committees in USA, Germany and France in recent months recommended against routine screening for PSA (prostate-specific antigen) in male blood, since this simple test is neither a sufficient nor a necessary indication for cancer. However, I am not average, and when PSA concentration doubles each semester while tissue tests called biopsies fail to see cancer, then relying on PSA warnings is better than relying on averages.