Strategic Research Projects, University of Bologna, 2007-2012

  Mathematical Methods in Social and Economical Science

The project has 4 permanent members of University of Bologna:

Vittorio Capecchi
Pierluigi Contucci, Coordinator.
Stefano Ghirlanda
Sandro Graffi

Collaborations with:

Elena Agliari, University of Parma
Adriano Barra, University la Sapienza, Roma.
Raffaella Burioni, University of Parma
Micaela Fedele, PhD student, Bologna.
Ignacio Gallo, Pd.D student, Bologna.
Cristian Giardina, Technical University, Eindhoven.
Claudio Giberti, University of Modena.
Francesca Romiti, University of Bologna
Cecilia Vernia, University of Modena.

Related organised activities:

- Complexity in Life and Socio-Economical Sciences, Dipartimento di Matematica, Universita di Bologna, May 2010

- Statistical Mechanics and Applications V, Dipartimento di Matematica, Universita di Modena, march 2010

- Statistical Mechanics and Applications IV, Weierstrass Institute, Berlin, July-August 2009

- Statistical Mechanics on Random Structures, International Research Station, Banff, November 2009

- Statistical Mechanics and Applications III, Dipartimento di Matematica, Bologna, February 2009

- Statistical mechanics and applications II, EURANDOM, Eindhoven, July 2008

- Statistical Mechanics and Applications I, Dipartimento di Matematica, Bologna, June 2008

- Statistical Mechanics on Random Structures, EURANDOM, Eindhoven, March 2008

Two Research Topics Examples


Does modern science have the possibility to study social matters like those related to immigration phenomena on solid mathematical grounds? Can we for instance determine cultural robustness and what causes abrupt changes from cultural legacies? Can we predict, cause or avoid swings? A novel approach is under investigation which uses the statistical mechanics formalism deviced for the study of phase transitions in physics. 

From the European immigration rate growth it is nowadays clear that in a few decades foreigners (people born outside of Europe) will represent a large percentage of Europe’s total population. Although immigration is often perceived as a threat from the emotional point of view it represents, to a large extent, it is an opportunity on economical bases. When two cultures are merged together issues like the survival of each own cultural identity play a major role in determining a proper and functional mutual integration. History has several times displayed occurrences in which a cultural trait, regardless of how small the fraction of people carrying it, overcomes another one in a relatively short time and with associated dramatic changes. Some other times two different cultural traits may coexist peacefully for long period of times. 

Do we have the possibility to study those phenomena on solid scientific grounds? Can we establish for instance what determines cultural robustness and what causes sudden changes from pre-existing cultural legacies? Can we predict them or avoid them? In modern epistemological perspective: can we build a “simple” mathematical model that in terms of a few measurable parameters would provide a predictive description of the observed phenomenology at a social level? What is the idea the teams are hunting after? People do interact, they exchange information, and they tend to imitate in average each other when belonging to the same community. While a handful of people have to be studied on all their possible decision strategies, a million of them have a well defined social average status largely independent from individual details. The science that learned how to infer the macroscopic properties of a large number of particles starting from rules governing mutual interaction of small groups is called Statistical Mechanics, born with the work of Boltzmann and used to derive the laws of Thermodynamics. In the last decades a Statistical Mechanics formalism has proven to be an excellent method to study the typical problems in which a system is described by a large number of individuals and the investigated properties are the averages. With this perspective it has been introduced a statistical mechanics model by Contucci and Ghirlanda aiming at the description of the interaction of two groups, for instance immigrants and residents. The model assumes that the elements of the two populations of sizes N1 and N2, with N=N1+N2 a very large number, interact within themselves with an interaction strength J1 in group 1 and J2 in group 2. Moreover a cross-group interaction with a tuneable strength Jint is present between individuals of different groups. The model is of mean-field type: it is assumed that individuals are nodes of a fully connected graph. By means of parameters that measure the strength of the interactions and by considering the original cultures prior to cultural meeting it is possible to provide a quantitative description of the system. The model considered is rich of structure and able to predict, as the ratio N1/N2 of the population is varied, both coexistence of cultures but also and especially sudden changes acting with the features of phase transitions (Contucci, Gallo and Menconi).

The future developments of the present research project, in collaboration with the EU project CULTAPTATION, will evolve in two directions (I. Gallo, PhD thesis, in preparation). The first is to bridge theory and experiment by quantifying the predictive value of the model by statistical estimation of parameters starting from poll data and using the maximum likelihood methods. Second is to extend to realistic random interaction networks the formalism used so far. There is indeed rather clear evidence that the social interaction network among people has several topological features appearing in random networks of “small world” and “scale-free” type. The necessity to extend the statistical mechanics methods to complex network environment is of fundamental importance. With this aim, the conference YEP-V addressed to Young European Probabilists has been organised by Contucci and Giardina in March 2008 at EURANDOM. That initiative is going to be continued and developed in a forthcoming conference at Banff International Research Station, Canada, in November 2009.



Is the noble (and sometimes snobbish) queen of sciences mathematics going to have a role in the future studies of Economics? Will its role (if any) be as crucial as the one it had in hard sciences like physics? We argue that mathematics is very likely going to have a pivotal beneficial mutual exchange with Economics especially through the study of complex system statistical mechanics models.

In recent times we have witnessed a large scale economic turmoil whose future is undoubtedly hard to predict. The crisis has so deeply involved the world population that it is constantly on the newspapers and apparently in the everyday government agendas. People reactions and opinions are as diversified as their experiences on the difficult matters discussed.

How can mathematics be of help for all that? The spectacular success that mathematics had within the hard sciences like physics is based on a long interaction between theory and experiments with trial and errors procedures and several feedbacks. Eventually a portion of reality is “understood” and quantitatively “described” by a theory whose language is mathematics and has the capability to deduce the observed phenomena from a small number of simple principles and “predict” the output of new experiments. When even a single experiment contradicts the theoretical predictions the whole machinery must be modified at the cost of giving up some of the principles and replacing them with new ones.

Unlike physics Economics has followed an apparently different path. On one side the large amount of available data has started to be seriously taken only in the last century. The discovery that the tails of the probability distribution of price changes are generally non-Gaussian is a quite recent achievement. On the other side the axiomatic method of deductive science has been applied without a real feedback check with observations: the principles of rationality of economic agents, the market efficiency, etc. have prospered with some school of Economics more like religious precepts than scientific hypotheses. Yet testable and predictive theories have appeared in Economics. The study by D. Mc Fadden (2000, Nobel Laureate in Economics) on the use of S. Francisco BART transportation system is a celebrated example. It is interesting to notice that from the mathematical point of view that work is equivalent to the Langevin theory of a small number of types of independent particles. When applied to cases in which peer-to-peer effects play a more substantial role that theory turns out to be inefficient.

The delay in the advent of the scientific method within Economical sciences has several causes. The intrinsic difficulty of its topics and the gap from available mathematical techniques is one of them: until a few decades ago in fact mathematics only treated models with translation or permutation invariance. From the statistical mechanics point of view only uniform interactions were understood. But, as the physicist Giorgio Parisi like to phrase it, science has become more robust and the theory of complex systems has made enormous progresses. Among the things that have been learned there is how to treat systems in which imitative and counter-imitative interactions play and where in general interactions themselves are random variables and are related to novel topological properties.

The challenge we face now is to fill the gap between phenomenological and theoretical approach. Data analysis must increase in depth and especially must follow a theoretical guide. An extensive search of data without having an idea of what to hunt for is an illusion no less dangerous than the search for principles regardless of experiments. In the same way the refinement of the suitable theoretical background in Economics must work in parallel to data search and analysis. The group of Strategic Research Project in Social and Economical Sciences of University of Bologna is working on those themes. Among the followed approaches there is the extension of the Mc. Fadden theory to interacting systems using the formalism of statistical mechanics (Gallo-Barra-Contucci). There are good indications that a similar approach could lead to interesting results. First it has the potentialities to include sudden changes in aggregate quantities even for small changes of the external parameters like it happens in an economical crisis. Second it may eventually make use of the complex systems theory of spin-glasses whose versatility for economical sciences is by now well understood. Third it has built-in the capability to include the acquaintance topologies, especially those that have been observed in network theory like the small-world and scale-free (Agliari-Burioni-Contucci).

The contribution that mathematics itself could provide is substantial in the paramount phase of checking the well posedness and successively solving. It is clearly to be expected that new mathematical instruments will be necessary and that the process of developing them is going to be long. A first phase in which mathematics is going to be involved is the so called “inverse problem”. Unlike physics, where most of the times the interaction between agents is established by pre-existing theories, in the realm of Economics effective interactions should be deduced from data, possibly at a non-aggregate level. From the mathematical point of view the computation of interaction coefficients from real data is a statistical mechanics inverse problem, a research setting in which many fields of science are turning their attention. The inverse problem solution is structurally linked to the monotonic behaviour of observed quantities with respect parameters (Contucci-Lebowitz).

At the time being the simple models for Economics considered in mathematics and derived from theoretical physics look like rough metaphors of reality. Still they are able to describe the main features of the observed phenomena and in any case they are a necessary step to get closer to reality by more refined approximations.

Last but not least the attempt of mathematics to provide solvable or treatable models for studies in Economics is going to be an important opportunity to fertilize mathematics itself with the entrance of new paradigms and their pressure to develop new parts of the Galileo language of nature.