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:

**MATHEMATICALS MODELS FOR IMMIGRATIONS**

**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 N_{1}
and N_{2}, with N=N_{1}+N_{2} a very large
number, interact within themselves with an interaction strength J_{1}
in group 1 and J_{2} in group 2. Moreover a cross-group
interaction with a tuneable strength J_{int} 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 N_{1}/N_{2}
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.

%%%

**MATHEMATICS FOR ECONOMICS: A STATISTICAL MECHANICS PERSPECTIVE**

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.