On the Dynamics of Social and Political Conﬂicts Looking ...community.dur.ac.uk/c.c.d.s.caiado/Bellomo.pdf · On the Dynamics of Social and Political Conﬂicts Looking for the

On the Dynamics of Social and Political Conflicts

Looking for the Black Swan

Nicola [email protected]

Politecnico di Torino

http://calvino.polito.it/fismat/poli/

“Modelling Social problems and Health”

University of Durham, 13-14-September, 2012

On the Dynamics of Social and Political Conflicts Looking forthe Black Swan– p. 1/83

THANKS

THANKS:

Giulia Ajmone Marsan , OECD, Paris, France

http://www.oecd.org/gov/regionaldevelopment/

Miguel A. Herrero , Universidad Complutense, Madrid, Spain

http://www.mat.ucm.es/imi/People/Herrero–Garcia–MiguelAngel.htm

Andrea Tosin, I.A.C - National Research Council, Roma, Italy

http://www.iac.rm.cnr.it/ tosin/


Lecture 1 - Index

1. Reasonings on Complex Systems

1.1. A brief historical excursus on complex systems

1.2. A Personal Bibliographic Search

1.3. Do Living, hence complex, systems exhibit common features?

1.4. What is the Black Swan?

1.5. Complexity through the ”Metamorphosis” by Escher

2. Mathematical Tools and Sources of Nonlinearity

3. Modeling Welfare Policy

4. Social Conflicts and Political Competition


Lecture 1 - Reasonings on Complex Systems

1.1 A brief historical excursus on complex systems

E. Kant , (1790),Critique de la raison pure, Traduction Francaise, Press Univ. de

France, 1967

Living systems: Special structures organized and with the ability to chase a purpose.

E. Schrodinger, (1943),What is Life?

Living systems have the ability to extract entropy to keep their own at low levels.

R. May, (2003), Science

In the physical sciences, mathematical theory and experimental investigation have

always marched together. Mathematics has been less intrusive in the life sciences,

possibly because they have been until recently descriptive, lacking the invariance

principles and fundamental natural constants of physics.




APRIL 2011 UNRAVELING COMPLEX SYSTEMS

The American Mathematical Society, the American Statistical Association, the

Mathematical Association of America, and the Society for Industrial and Applied

Mathematics announce that the theme of Mathematics Awareness Month 2011 is

”Unraveling Complex Systems.”

From www.mathaware.org

We live in a complex world. Many familiar examples of complexsystems might be

found in very different entities at very different scales: from power grids, to

transportation systems, from financial markets, to the Internet, and even in the

underlying environment to cells in our bodies. Mathematicsand statistics can guide us

in unveiling, defining and understanding these systems,in order to enhance their

reliability and improve their performance.




N.B. H. Berestycki, F. Brezzi, and J.P. Nadal, Mathematics and Complexity in Life

and Human Sciences, Mathematical Models and Methods in Applied Sciences, 2010.

• The study of complex systems, namely systems of many individuals interacting in a

non-linear manner, has received in recent years a remarkable increase of interest

among applied mathematicians, physicists as well as researchers in various other

fields as economy or social sciences.

• Their collective overall behavior is determined by the dynamics of their interactions.

On the other hand, a traditional modeling of individual dynamics does not lead in a

straightforward way to a mathematical description of collective emerging behaviors.

In particular it is very difficult to understand and model these systems based on the

sole description of the dynamics and interactions of a few individual entities.



1.2 A Personal Bibliographic Search

• E. Mayr,What Evolution Is , (Basic Books, New York, 2001).

• A.L. BarabasiLinked. The New Science of Networks, (Perseus Publishing,

Cambridge Massachusetts, 2002.)

• F. Schweitzer,Brownian Agents and Active Particles, (Springer, Berlin, 2003).

• M. Talagrand,Spin Glasses, a Challenge to Mathematicians, Springer, (2003).

• M.A. Nowak,Evolutionary Dynamics, Princeton Univ. Press, (2006).

• N.N. Taleb,The Black Swan: The Impact of the Highly Improbable, 2007.

• N.Bellomo,Modelling Complex Living Systems. A Kinetic Theory andStochastic Game Approach, (Birkhauser-Springer, Boston, 2008).

• N.Bellomo, On the Difficult Interplay Between Life (Complexity) and Mathematical

Sciences, (2012), to be published.


1 - Reasonings on Complex Systems

1.3 Do Living, hence complex, systems exhibit common features?

10 Selected Common Features

• 1. Ability to express a strategy:Living entities are capable to develop

specificstrategiesrelated to theirorganization abilitydepending on the state and

entities in their surrounding environment. These can be expressed without the

application of any principle imposed by the outer environment. In general, they

typically operateout-of-equilibrium. For example, a constant struggle with the outer

environment is developed to remain in a particular state, namely stay alive.

• 2. Heterogeneity: The said ability is not the same for all entities. Indeed,

heterogeneous behaviorscharacterize a great part of living systems. Namely, the

characteristics of interacting entities can even differ from an entity to another

belonging to the same structure. In biology, this is due to different phenotype

expressions generated by the same genotype. In some cases, these expressions induce

genetic diseases, or different evolution of epidemics.




• 3. Large number of components: Complexity in living systems is

induced by a large number of variables, which are needed to describe their overall

state. Therefore, the number of equations needed for the modeling approach may be

too large to be practically treated. For instance, biological systems are different from

the physical systems analyzed by statistical mechanics, which typically deals with

systems containing many copies of a few interacting components, whereas cells

contain from millions to a few copies of each of thousands of different components,

each with specific interactions.

• 4. Interactions: Interactions are nonlinearly additive and involve immediate

neighbors, but in some cases also distant particles, as living systems have the ability to

communicate and may possibly choose different observationpaths. In some cases, the

topological distribution of a fixed number of neighbors can play a prominent role in the

development of the strategy and interactions. Living entitiesplay a game at each

interactionwith an output that is technically related to their strategyoften related to

surviving and adaptation ability.




• 5. Stochastic games: Living entities at each interactionplay a gamewith an

output that is technically related to their strategy often related to surviving and

adaptation ability, namely to a personal search of fitness. The output of the game is not

generally deterministic even when a causality principle isidentified. This dynamics is

also related to the fact that they receive a feedback from their environments, which

modifies the strategy they express adapting it to the new environmental conditions.

The interaction modifies the outer environment, hence the dynamics of interactions

evolves in time.

• 6. Learning ability: Living systems have theability to learn from past

experience. Therefore their strategic ability and the characteristics of interactions

among living entities evolve in time. In fact, living systems receive inputs from their

environments and have the ability to adapt themselves to theenvironmental conditions

that also evolve in time.




• 7. Darwinian selection and time as a key variable: All living

systems are evolutionary. For instance birth processes cangenerate individuals more

fitted to the environment, who in turn generate new individuals again more fitted to the

outer environment. Neglecting this aspect means that the time scale of observation and

modeling of the system itself might not be long enough to observe evolutionary events.

In biology such a time scale can be very short for cellular systems and very long for

vertebrates. In some cases, such as the generation of daughter cells from mother cells,

new cell phenotypes can have origin from random mistakes during replication.

• 8. Multiscale aspects: The study of complex living systems always needs a

multiscale approach. For instance in biology, the dynamics of a cell at the molecular

(genetic) level determines the ability of cells to express specific functions. Moreover,

the structure of macroscopic tissues depends on such a dynamics. Moreover, living

systems are generally constituted by alarge number of components. In

general, we can state that one only observation and representation scale is not

sufficient to describe the dynamics of living systems.




• 9. Emerging behaviors: Large living systemsshow collective emerging

behaviorsthat are not directly related to the dynamics of a few interacting entities.

Generally, emerging behaviors are reproduced at a qualitative level given certain input

conditions, However, quantitative levels are not generally reproduced. In fact small

changes in the input conditions generate large deviations even when the qualitative

behavior is reproduced.

• 10. Large deviations: The observation of living systems should focus on the

emerging behaviors that appear in non-equilibrium conditions. Very similar input

conditions reproduce, in several cases, the qualitative behaviors. However, large

deviations can be observed corresponding to small changes in the input conditions. In

some cases, they break out the quality of the afore-said behaviors, namely substantial

modifications can be observed. Heterogeneity of individualstrategies, learning ability,

and interactions with the outer environment modify such phenomena.



1.4 What is the Black Swan?

It is worth detailing a little more the expressionBlack Swan, introduced in the

specialized literature for indicating unpredictable events, which are far away from

those generally observed by repeated empirical evidence. According to the definition

by Taleb a Black Swan is specifically characterized as follows:

“A Black Swan is a highly improbable event with three principal characteristics: It is

unpredictable; it carries a massive impact; and, after the fact, we concoct an

explanation that makes it appear less random, and more predictable, than it was.”

N. N. Taleb,The Black Swan: The Impact of the Highly Improbable, Random House,

New York City, 2007.

Since it is very difficult to predict directly the onset of a black swan, it is useful

looking for the presence of early signals

M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos,

H. Held, E. H. van Nes, M. Rietkerk, and G. Sugihara, Early-warning signals for

critical transitions,Nature, 461:53–59, 2009.



1.5 The Metamorphosis by Escher




Let us imagine, with the help of our fantasy, the time is on thex-axis. Some free

(imaginative) interpretations are as follows:

i) The landscape evolves in time going first from a geometrical village made of

similarly looking houses (looking like boxes) to a real village with heterogeneous

distribution of the shape of houses;

ii) The evolution is related to interactions. Definitely multiple ones; but are they

nonlinear?

iii) Does the dynamics shows multiscale interactions and isthe evolution selective in

some weak Darwinian sense?

iv) Suddenly the landscape changes from a village it becomesa chess plate, the only

connection is a Bridge and a Tower. Can this sudden change be interpreted as a “Black

Swan”?




Since the village exists in reality some further considerations can be developed:

v) The Tower is not yet the one of the chess plate. Can it be, however, interpreted as an

early signal of what is happening?

vi) Does the Tower exists in reality? If the answer is equallyYes and Not the same

time, why Yes and why Not?

vii) Referring to a Village, may even be the real one, can our imagination provide a

conceivable interpretation of the physical meaning of the sudden change?


Lecture 2 - Index


2. 2. Mathematical Tools and Sources of Nonlinearity

2.1. Functional Subsystems and Representation

2.2. Stochastic Games

2.3. Mathematical Structures

2.4. Sources of Nonlinearity


4. Modeling Social Conflicts and Political Competition


2 - Mathematical Tools and Sources of Nonlinearity

2.1 Functional Subsystems and Representation

Hallmarks of the kinetic theory of active particles:

• The overall system is subdivided intofunctional subsystemsconstituted by

entities, calledactive particles, whose individual state is calledactivity;

• The state of each functional subsystem is defined by a suitable, time dependent,

probability distribution over the activity variable;

• Interactions are modeled by games, more precisely stochastic games, where the

state of the interacting particles and the output of the interactions are known in

probability;

• Interactions are delocalized and nonlinearly additive;

• The evolution of the probability distribution is obtained by a balance of particles

within elementary volumes of the space of the microscopic states, where the

dynamics of inflow and outflow of particles is related to interactions at the

microscopic scale.




Let us consider a system constituted by a large number of interacting living entities,

calledactive particles. Their microscopic state includes, in addition to

geometrical and mechanical variables, also an additional variableu ∈ Du, called

activity, which is heterogeneously distributed.

Living systems are constituted by different types of active particles, which

express several different functions. This source of complexity can be reduced byby

decomposing the system into suitable functional subsystems,

where afunctional subsystem is a collection of active particles, which have the

ability to express cooperatively the sameactivity regarded as a scalar variable.

A particular case is when particles are distributed ina system of networks

and nodes in a way that the role of the space variable is not relevant in the nodes,

while transition from one node to the other has to be properlymodeled by migration

dynamics.




The description of the state of the system within each node isdelivered by a

probability distribution function over the microscopic state, labeled by the

indexi, which denotes the functional subsystem within each node.

fi = fi(t, u) , i = 1, . . . , n,

such thatfi(t, u) du denotes the number of active particles whose state, at timet, is in

the interval[u, u+ du]. If the number of active particles is constant in time, then the

distribution function can be normalized with respect to such a number and

consequently is aprobability density. Moreover, the physical meaning of the

activity variable differs for each subscript.

Consider now the wholenetwork, then the overall number of functional subsystems

is given by the sum of all of them in the nodes:

fij = fij(t, u) , i = 1, . . . , n, j = 1, . . . ,m, where functional subsystems

differ if are localized in different nodes although they express the same activity

variable.




Representation by Moments and Macroscopic Quantities

Macroscopic quantities can be computed by moments of the afore defined distribution

function. In particular, the number of particles in each functional subsystem is given

by:

nij [fij ](t) =

∫

Du

fij(t, u) du.

Higher order moments are computed as follows:

Epij [fij ](t) =

1

nij [fij ](t)

∫

Du

upfij(t, u) du.

Remark: The physical meaning of the moments depends on the specific system under

consideration.




Representation for space distributed systems in each node

Consider active particles in a node for functional subsystems labeled by the subscripti.

The description of the overall state of the system is delivered by thegeneralized

one-particle distribution function

fi = fi(t,x,v, u) : [0, T ] × Ω ×Dv ×Du →+,

such thatfi(t,x,v, u) dx dv du denotes the number of active particles whose state, at

time t, is in the interval[w,w + dw]. wherew = x,v, u is an element of thespace

of the microscopic states. Marginal densities are computed as follows:

fmi (t,x,v) =

∫

Du

fi(t,x,v, u) du , fai (t,x, u) =

∫

Dv

fi(t,x,v, u) dv.

and

fbi (t, u) =

∫

Dx×Dv

fi(t,x,v, u) dx dv.




Representation for space distributed systems in each node

Local density of theith functional subsystem is:

ν[fi](t,x) =

∫

Dv×Du

fi(t,x,v, u) dv du =

∫

Dv

fmi (t,x,v) dv.

Integration over the volumeDx containing the particles gives thetotal size of the

ith subsystem:

Ni[fi](t) =

∫

Dx

ν[fi](t,x) dx ,

which can depend on time due to the role of proliferative or destructive interactions.

First order moments provide eitherlinear mechanical macroscopic

quantities, orlinear activity macroscopic quantities.




• Theflux of particles, at the timet in the positionx, is given by

Q[fi](t,x) =

∫

Dv×Du

v fi(t,x,v, u) dv du =

∫

Dv

fmi (t,x,v) dv.

• Themass velocity of particles, at the timet in the positionx, is given by

U[fi](t,x) =1

νi[fi](t,x)

∫

Dv×Du

v fi(t,x,v, u) dv du.

Theactivity terms are computed as follows:

a[fi](t,x) =

∫

Dv×Du

ufi(t,x,v, u) dv du ,

while thelocal activation density is given by:

ad[fi](t,x) =

aj [fi](t,x)

νi[fi](t,x)=

1

νi[fi](t,x)

∫

Dv×Du

ufi(t,x,v, u) dv du.



2.2 Stochastic Games

• Test particles with microscopic stateu, at the timet: fij = fij(t, u).

• Field particles with microscopic state,u∗ at the timet: fij = fij(t, u∗).

• Candidate particles with microscopic state,u∗ at the timet: fij = fij(t, u∗).

The interaction rule is as follows:

i) Thecandidate particle interacts withfield particles and acquires, in probability,

the state of thetest particle.Test particles interact with field particles and lose their

state.

ii) Interactions can: modify the microscopic state of particles; generate proliferation or

destruction of particles in their microscopic state; and can also generate a particle in a

new functional subsystem.

Loss of determinism: Modeling of systems of the inert matter is developed

within the framework of deterministic causality principles, which does not any longer

holds in the case of the living matter.




C ΩT

F F

F

F

F

Figure 1: – Active particles interact with other particles in their action domain


2 Mathematical Tools and Sources of Nonlinearity


Interactions with modification of activity and transition: Generation of particles

into a new functional subsystem occurs through pathways. Different paths can be

chosen according to the dynamics at the lower scale.

F F F

F C FF

T

T

FF FF FF T

i + 1

i

i − 1

Figure 2: – Active particles during proliferation move fromone functional subsystem

to the other through pathways.




Cooperative/Competitive Games

Cooperative behaviorThe active particle with stateh < k improves its state by

gaining from the particlek , which cooperates loosing part of its state.

Competitive behavior The active particle with stateh < k decreases its state by

contributing to the particlek, which due to competition increases part of its state.



2.3 Mathematical Structures

Balance within the space of microscopic states




The Spatially Homogeneous CaseThe evolution equation follows from a balance equation for net flow of particles in the

elementary volume of the space of the microscopic state by transport and interactions:

∂tfi(t, u) + Fi(t) ∂ufi(t, u) = Ci[f ](t, u) + Pi[f ](t, u),

where

Ci[f ] =

n∑

j=1

η0ij

∫

Du×Du

ψij [f ]Bij [f ] (u∗ → u|u∗, u∗)

− fi(t, u)n

∑

j=1

η0ij

∫

Du

ψij [f ] fj(t, u∗) du∗

.

and

Pi[f ] =n

∑

h=1

n∑

k=1

η0hk

∫

Du

∫

Du

ψhk[f ]µihk[f ](u∗, u

∗)fh(t, u∗)fk(t, u∗) du∗.


Lecture 2 - Mathematical Tools and Sources of Nonlinearity


Models with Space Dynamics

H.1. Candidate or test particles inx, interact with the field particles inx∗ ∈ Ω located

in the interaction domainΩ. Interactions are weighted by theinteraction rate

ηhk[f ](x∗) supposed to depend on the local density in the position of thefield

particles.

H.2. Candidate particle modifies its state according to the term defined as follows:

Ahk[f ](v∗ → v, u∗ → u|v∗,v∗, u∗, u

∗), which denotes the probability density that

a candidate particles of theh-subsystems with statev∗, u∗ reaches the statev, u after

an interaction with the field particlesk-subsystems with statev∗, u∗.

H.3. Candidate particle, inx, can proliferate, due to encounters with field particles in

x∗, with rateµihk, which denotes the proliferation rate into the functional subsystemi,

due the encounter of particles belonging the functional subsystemsh andk.

Destructive events can occur only within the same functional subsystem.




(∂t + v · ∂x) fi(t,x,v, u) =

[ n∑

j=1

Cij [f ] +n

∑

h=1

n∑

k=1

Sihk[f ]

]

(t,x,v, u) ,

where

Cij [f ] =

∫

Ω×D2u×D2

v

ηij [f ](x∗)Ahk[f ](v∗ → v, u∗ → u|v∗,v

∗, u∗, u

∗)

× fi(t,x,v∗, u∗)fj(t,x∗,v

∗, u

∗) dv∗ dv∗du∗ du

∗dx

∗,

− fi(t,x,v)

∫

Ω×Du×Dv

ηij [f ](x∗) fj(t,x

∗,v

∗, u

∗) dv∗du

∗dx

∗

Sihk[f ] =

∫

Ω×D2u×Dv

ηhk[f ](x∗)µihk[f ](u∗, u

∗)

× fh(t,x,v, u∗)fk(t,x∗,v

∗, u

∗) dv∗du∗ du

∗dx

∗.



Conjecture 1. Interactions modify the activity variable according to topological

stochastic games, however independently on the distribution of the velocity variable,

while modification of the velocity of the interacting particles depends also on the

activity variable.

Ahk(·) = Bhk(u∗ → u, |u∗, u∗) × Chk(v∗ → v |v∗,v

∗, u∗, u

∗) ,

whereA, B, andC are, for positive definedf , probability densities.

Conjecture 2. Stochastic velocity perturbation

(

∂t + v · ∇x

)

fi = νi Li[fi] + ηi Ci[f ] + ηi µi Si[f ],

where

Li[fi] =

∫

Dv

(

Ti(v∗ → v)fi(t,x,v

∗, u) − Ti(v → v

∗)fi(t,x,v, u))

dv∗,

and whereTi(v∗ → v) is, for theith subsystem, the probability kernel for the new

velocityv ∈ Dv assuming that the previous velocity wasv∗.



2.4 Sources of Nonlinearity

Encounter rate: In general,ηij is supposed to decay with the distance between the

interacting active particles, which depends both on their state and on the distance of

their functional subsystems. The following sources of nonlinearity can be possibly

identified:

Hierarchic distance: The rate of interactions within the same functional subsystem

depends on the distance of the interacting microstates. Generally, it decays with the

distancedhd = |u∗ − u∗|. However, when two active particles belong to different

functional subsystems, the rate differs depending on the distance between the

functional states. A conceivable numbering criterion consists in selecting the first

subsystem by a certain selection (for instance in the animalworld the “dominant”) and

in numbering the others by increasing numbers depending on the decreasing rate.

Namely:k ↑⇒ ηhk ↓.




Affinity distance: The distance between the distribution function interacting active

particles can be source of nonlinearity according to the general idea that two systems

with closed distribution areaffineand hence they interact with higher frequency. In

this case the distance isdad = ||fh − fk||, where a uniform, e.g.L∞, or mean

approximation, e.g.L1, can be adopted as the most appropriate according to the

physics of the system under consideration. The rate decays with increasingdd.

An example is:

ηhk = η

hk[f ](u∗, u∗) = η

0hk exp

− α|u∗ − u∗| − β|h− k| − γ||fh − fk||

,

whereη0hk andα, β, andγ are positive constants, and

||fh − fk||(t) =

∫

R+

|fh(t, u) − fk(t, u)| du.




Transition probability density:

Threshold: The occurrence of a type of interaction with respect to the other is ruled,

in several models, by a threshold on the distance between theinteracting particles. On

the other hand, the threshold evolves in time and depends on the whole dynamics of of

the overall system, hence onf .

Domain of influence: The communication is effective only within a certain domainof

influence of the micro-state. In the simplest case it is fixed,more in general it depends

on the density of active particles which can be captured in the communication, namely

there exists a critical density that defines the maximum number of particles, which can

be included in the interaction (see Parisi’s conjecture andthe application to swarm

modeling).




Not sufficient visibility zone: An additional difficulty occurs when the visibility zone

is reduced and the candidate particle does not receive sufficient information. In this

case, the output of the interaction might be different from the usual one.

Output of the interaction: An active particle interacts with all particles in its

interaction domain. The output of the interaction depends on f within the interaction

domain. For instance it can depend on the moments computed for the particles within

the interaction domain as shown again on the modeling of swarms. A simple case

occurs when it depends on low order moments:

Proliferation rate: Sources of nonlinearity in the case of proliferative and/or

destructive interactions, in addition to those induced by the interaction rate, are

generally the same affecting conservative interactions ruled by games.


Lecture 3 - Index


2. 2. Mathematical Tools and Sources of Nonlinearity


3.1 Bibliography and Preliminary Reasonings

3.2 Modeling of the Dynamics of Welfare Distribution

3.3 Simulations



3 - Modeling Welfare Policy

3.1. Bibliography and Preliminary Reasonings

M.A. Nowak, Evolutionary Dynamics, Princeton Univ. Press, Princeton, (2006).

A.-L. Barabasi, R. Albert, and H. Jeong, Mean-field theory for scale-free random

networks,Physica A, 272(1–2):173–187, 1999.

F. Vega-Redondo, Complex Social Networks, Cambridge University Press,

Cambridge, (2007).

D. Helbing, Quantitative Sociodynamics. Stochastic Methods and Models of Social

Interaction Processes. Springer Berlin Heidelberg, 2nd edition, 2010.

D. Acemoglu and J. Robinson, Economic Origins of Dictatorship and Democracy,

Cambridge University Press, Cambridge, (2006).

D. Acemoglu, D. Ticchi, and A. Vindigni, A theory of military dictatorship.

American Economic Journal: Macroeconomics, 2(1):1–42, 2010.

D. Acemoglu, D. Ticchi, and A. Vindigni, Emergence and persistence of inefficient

states.J. Eur. Econ. Assoc., 9(2):177–208, 2011.




G. Ajmone Marsan New Paradigms and Mathematical Methods for Complex Systems

in Behavioral Economics. PhD thesis, IMT Lucca, Italy, 2009.

G. Ajmone Marsan, N. Bellomo, and M. Egidi, Towards a mathematical theory of

complex socio-economical systems by functional subsystems representation.Kinet.

Relat. Models, 1(2):249–278, 2008.

N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems

focusing on developmental biology and evolution,Physics of Life Reviews, 8 (2011)

1-18.

N. Bellomo, M.A. Herrero, and A. Tosin, On the dynamics of social conflicts looking

for the black swan, submitted, (2012).

N.N. Taleb, The Black Swan: The Impact of the Highly Improbable, 2007.

K. Sigmund, The Calculus of Selfishness, Princeton Univ. Press, (2011).



3.1. Bibliography and Preliminary ReasoningsSome conceivable fields of

application

G. Aletti, G. Naldi, and G. Toscani, First-order continuous models of opinion

formation,SIAM J. on Applied Mathematics67, 837-853, (2007).

F. Bagarello and F. Olivieri, An operator description of interactions between

populations with application to migrations,Mathematical Models and Methods in

Applied Sciences, 22, (2012), to appear.

U. Bastolla, M.A. Fortuna, A. Pacual-Garcia, A. Ferrera, B.Luque, and

J. Bascompte, The architecture of mutualistic networks minimizes competition and

increases biodiversity,Nature, 458:1018–1021, 2009.

M. Olson, Dictatorship, Democracy, and Development,The American Political Science

Review, 87, 3, 567-576, (1993).

G. Mulone and B. Straughan, A note on heroin epidemics,Mathematical

Biosciences, 218, (2009), 138-141.

G. Mulone and B. Straughan, Modeling binge drinking,Int. J. Biomathematics5,

(2012), no.1250005On the Dynamics of Social and Political Conflicts Looking forthe Black Swan– p. 41/83



The dynamics of social and economic systems are necessarilybased on individual

behaviors, by which single subjects express, either consciously or unconsciously, a

particular strategy, which is heterogeneously distributed. The latter is often based not

only on their own individual purposes, but also on those theyattribute to other agents.

However, the sheer complexity of such systems makes it oftendifficult to ascertain the

impact of personal decisions on the resulting collective dynamics. In particular,

interactions need not have an additive, linear character. As a consequence, the global

impact of a given number of entities (“field entities”) over asingle one (“test entity”)

cannot be assumed to merely consist in the linear superposition of any single field

entity action.

This nonlinear feature represents a serious conceptual difficulty to the derivation,and subsequent analysis, of mathematical models for that type of systems.




In the last few years, a radical philosophical change has been undertaken in social and

economic disciplines. An interplay among Economics, Psychology, and Sociology has

taken place, thanks to a new cognitive approach no longer grounded on the traditional

assumption of rational socio-economic behavior. Startingfrom the concept of bounded

rationality, the idea of Economics as a subject highly affected by individual (rational or

irrational) behaviors, reactions, and interactions has begun to impose itself. In this

frame, the contribution of mathematical methods to a deeperunderstanding of the

relationships between individual behaviors and collective outcomes may be

fundamental.

A key experimental feature of such systems is that interaction among

heterogeneous individuals often produces unexpected outcomes, which wereabsent at the individual level, and are commonly termed emergent behaviors.




In fields ranging from Economics to Sociology and Ecology, the last decades have

witnessed an increasing interest for the introduction of quantitative mathematical

methods that could account for individual, not necessarilyrational, behaviors. Terms

as game theory, bounded rationality, evolutionary dynamics are often used in that

context and clearly illustrate the continuous search for techniques able to provide

mathematical models than can describe, and predict, livingbehaviors. The new point

of view promoted the image of Economics as an evolving complex system, where

interactions among heterogeneous individuals produce unpredictable emerging

outcomes. In this context, setting up a mathematical description able to capture the

evolving features of socio-economic systems is a challenging, however difficult, task,

which calls for a proper interaction between mathematics and social sciences.

Mathematical models might focus, in particular, on the prediction of the so called

Black Swan. The latter is defined to be a rare event, showing up as an irrationalcollective trend generated by possibly rational individual behaviors.




Socio-economic systems can be described as ensembles of several living entities able

to developbehavioral strategies, by which they interact with each other. For this

reason, such entities are also calledactive particles. Typically, their strategies are

heterogeneously distributedandchange in timein consequence of the interactions,

since active particles can update them bylearning from past experiences.Therefore the mathematical representation calls fordistribution functions over the

strategies, whose time evolution depends on the dynamics ofinteractions. According

to the KTAP approach, the latter are regarded asstochastic games, whose payoff is the

new strategy that each player will use in the next interaction.

Remarkably, payoffs are supposed to be known only in probability because irrational

behaviors in individual choices cannot be excluded. Deterministic payoffs would

imply instead a kind of rational behavior of active particles, as if they reacted the same

when placed in the same conditions.




Let u be a variable denoting the strategy expressed by active particles, that in the

present context can be identified with their wealth status. For the kind of applications

we are interested in it is customary to speak ofwealth classes, which implies that the

strategyu is a discrete variable taking values in a lattice

Iu = u1, . . . , ui, . . . , un.

The number of individuals that, at a certain timet, are expressing the strategyui, or, in

other words, that belong to thei-th wealth class, is given by a distribution function

fi = fi(t) : [0, Tmax] →+, i = 1, . . . , n,

Tmax > 0 being a certain final time (possibly+∞). Notice that, up to normalizing with

respect to the total number of active particles,f = (f1, . . . , fi, . . . , fn) is a

time-evolving discrete probability distribution over thelatticeIu.




• The testparticle, with strategyui, is the representative entity of the system.

Studying interactions means studying how the test particlecan loose its strategy or

other particles can gain it.

• Candidateparticles, with strategyuh, are the particles which can gain the test

strategyui in consequence of the interactions.

• Field particles, with strategyuk, are the particles whose presence triggers the

interactions of the candidate particles.

• The interaction rateηhk, modeling the frequency of interaction between auh

candidate particle and auk field particle.

• The transition probabilities, denoted byBhk(i), expressing the probability that a

candidate particle with strategyuh changes the latter toui after interacting with a

field particle with strategyuk.

n∑

i=1

Bhk(i) = 1, ∀h, k = 1, . . . , n.




By letting all particles play together for a time∆t > 0, the expected number of

particles shifting to and losing, respectively, the strategy ui during the time interval

[t, t+ ∆t] can be estimated, at first order, as:

∆tn

∑

h, k=1

ηhkBhk(i)fh(t)fk(t) + o(∆t), ∆tfi(t)n

∑

k=1

ηikfk(t) + o(∆t).

The variationfi(t+ ∆t) − fi(t) of the number of active particles expressing the

strategyui can then be equated with the net balance of these two terms, which, in the

limit ∆t→ 0+, leads to the equation:

dfi

dt=

n∑

h, k=1

ηhkBhk(i)fhfk − fi

n∑

k=1

ηikfk, i = 1, . . . , n. (1)




The modeling approach adopts the following qualitative paradigm of

consensus/dissensus dynamics:

• Consensus– The candidate particle sees its state either increased, byprofiting

from a field particle with a higher state, or decreased, by pandering to a field

particle with a lower state. After mutual interaction, the states of the particles

become closer than before the interaction.

• Dissensus– The candidate particle sees its state either further decreased, by facing

a field particle with a higher state, or further increased, byfacing a field particle

with a lower state. After mutual interaction, the states of the particles become

farther than before the interaction.

A critical distance triggers either cooperation or competition among the classes. If the

distance is lower than the critical one then a competition takes place, which causes a

further enrichment of the wealthier class and a further impoverishment of the poorer

one. Conversely, if the actual distance is greater than the critical one then the social

organization forces cooperation.



Modeling Consesus/dissensus Games


3 - Modeling Social Conflicts and Political Competition


The characteristics of the present framework are summarized as follows.

• Interaction rate. Two different rates of interaction are considered, corresponding

to competitive and cooperative interactions, respectively.

• Strategy leading to the transition probabilities. When interacting with other

particles, each active particle plays a game with stochastic output. If the difference

of wealth class between the interacting particles is lower than a critical distance

γ[f ] (where, here and henceforth, square brackets indicate a functional

dependence on the probability distributionf ) then the particles compete in such a

way that those with higher wealth increase their state against those with lower

wealth. Conversely, if the difference of wealth class is higher thanγ[f ] then the

opposite occurs. The critical distance evolves in time according to the global

wealth distribution over wealthy and poor particles.




1. Interaction between equally wealthy particles, viz. h = k:

Bhh(h) = 1

Bhh(i) = 0, ∀ i 6= h(2)

2. Cooperative interaction, viz. h 6= k and|h− k| > γ[f ]:

h < k

Bhk(h) = 1 − αhk

Bhk(h+ 1) = αhk

Bhk(i) = 0, ∀ i 6= h, h+ 1

h > k

Bhk(h− 1) = αhk

Bhk(h) = 1 − αhk

Bhk(i) = 0, ∀ i 6= h− 1, h,

(3)

where it is assumed that interactions within the same class produce no effect.



3. Competitive interaction, viz. h 6= k and|h− k| ≤ γ[f ]:

h = 1, n

Bhk(h) = 1

Bhk(i) = 0, ∀ i 6= h

h 6= 1, n

h < k

k < n

Bhk(h− 1) = αhk

Bhk(h) = 1 − αhk

Bhk(i) = 0, ∀ i 6= h− 1, h

k = n

Bhn(h) = 1

Bhn(i) = 0, ∀ i 6= h

h > k

k = 1

Bh1(h) = 1

Bh1(i) = 0, ∀ i 6= h

k > 1

Bhk(h) = 1 − αhk

Bhk(h+ 1) = αhk

Bhk(i) = 0, ∀ i 6= h, h+ 1

(4)




The parameterαhk ∈ [0, 1] has the following meaning:

• In case of competition, it is the probability that the candidate particle further

increases (or decreases) its wealth if it is richer (or poorer) than the field particle;

• In case of cooperation, it is the probability that the candidate particle gains (or

transfers) part of its wealth if it is poorer (or richer) thanthe field particle.

This probability may be constant or may depend on the wealth classes, e.g.,

αhk =|k − h|

n− 1,

in such a way that the larger the distance between the interacting classes the more

stressed the effect of cooperation or competition. Moreover, for wealth conservation

purposes, the transition probabilities are such that the extreme classes never take part

nor trigger social competition.




Among the possible choices, we select a uniformly spaced wealth grid in the interval

[−1, 1] with odd numbern of classes:

Iu = u1 = −1, . . . , un+1

2

= 0, . . . , un = 1,

ui =2

n− 1i−

n+ 1

n− 1, i = 1, . . . , n,

agreeing thatui < 0 identifies a poor class whereasui > 0 a wealthy one.

We next assume that theinteraction rate ηhk ≥ 0 is piecewise constant over the

wealth classes:

ηhk =

η0 if |k − h| ≤ γ[f ] (competition),

µη0 if |k − h| > γ[f ] (cooperation),

whereη0 > 0 is a constant to be hidden in the time scale and0 < µ ≤ 1.




Thetransition probabilities Bhk(i) ∈ [0, 1] are required to satisfy additional

conditions such as conservation of the average wealth status of the population:

n∑

i=1

uifi(t) = constant, ∀ t ≥ 0. (5)

This means that the interaction dynamics cause globally neither production nor loss of

wealth, but simply its redistribution among the classes.

U0 :=

n∑

i=1

uifi(0).

It turns out that sufficient conditions for the fulfillment ofthis condition are:

symmetric encounter rate, i.e.,ηhk = ηkh, ∀h, k = 1, . . . , n; and

n∑

i=1

uiBhk(i) = uh + σhk, ∀h, k = 1, . . . , n,

whereσhk is an antisymmetric tensor, i.e.,σhk = −σkh, ∀h, k = 1, . . . , n.



Welfare Policy and Political Competition

Thecritical distance γ[f ] is here assumed to depend on the instantaneous distribution

of the active particles over the wealth classes, such that the time evolution ofγ[f ]

should translate the following phenomenology of social competition:

• in general,γ[f ] grows with the number of poor active particles, thus causinglarger

and larger gaps of social competition. Few wealthy active particles insist on

maintaining, and possibly improving, their benefits;

• in a population constituted almost exclusively by poor active particlesγ[f ] attains

a value such that cooperation is inhibited, for individualstend to be involved in a

“battle of the have-nots”;

• conversely, in a population constituted almost exclusively by wealthy active

particlesγ[f ] attains a value such that competition is inhibited, because

individuals tend preferentially to cooperate for preserving their common benefits.




Introduce the number of poor and wealthy active particles attime t:

N−(t) =

n−1

2∑

i=1

fi(t), N+(t) =

n∑

i=n+3

2

fi(t).

Notice that, by excluding the middle classun+1

2

= 0 from bothN− andN+, we

implicitly regard it as economically “neutral”. Up to normalization over the total

number of active particles, we have0 ≤ N± ≤ 1 with alsoN− +N+ ≤ 1, hence the

quantity

S[f ] := N− −N

+,

which provides a macroscopic measure of thesocial gapin the population, is bounded

between−1 and1. Given that, we now look for a quadratic polynomial dependence of

γ[f ] onS[f ] taking into account the following conditions, which bring to a quantitative

level the previous qualitative arguments:




• S[f ] = S0 ⇒ γ[f ] = γ0, whereS0, γ0 are a reference social gap and the

corresponding reference critical distance, respectively;

• S[f ] = 1 ⇒ γ[f ] = n, which implies that when the population is composed by

poor particles only (N− = 1,N+ = 0) the socio-economic dynamics are of full

competition;

• S[f ] = −1 ⇒ γ[f ] = 0, which implies that, conversely, when the population is

composed by wealthy particles only (N− = 0,N+ = 1) the socio-economic

dynamics are of full cooperation.

Only integer values ofγ[f ] are meaningful, for so are the distances between pairs of

indexes of wealth classes, the resulting analytical expression ofγ[f ] yields

γ[f ] =2γ0(S[f ]2 − 1) − n(S0 + 1)(S[f ]2 − S0)

2(S20 − 1)

+n

2S[f ],

where· denotes integer part (floor).



Simulations

0

1

2

3

4

5

6

7

8

9

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

γ

S

γ0=3

γ0=7

Figure 3: The critical distanceγ[f ] vs. the macroscopic social gapS[f ] for the two

casesγ0 = 3, 7 andn = 9 social classes. Empty bullets indicate the reference value

γ0 corresponding to the reference social gapS0 = 0. Filled bullets indicate instead the

actual initial critical distanceγ[f ](t = 0) corresponding to the actual initial social gap

S[f ](t = 0). Dotted lines are plotted for visual reference.



Simulations

The evolution of the system predicted by the model depends essentially on the four

parametersn (the number of wealth classes),µ (the relative encounter rate for

cooperation,U0 (the average wealth of the population), andγ0 (the reference critical

distance). In more detail:

• n = 9 andµ = 0.3 are selected;

• two case studies forU0 are addressed, namelyU0 = −0.4 < 0 andU0 = 0, in

order to compare, respectively, the economic dynamics of a society in which poor

classes dominate with those of a society in which the initialdistribution of active

particles encompasses uniformly poor and rich classes;

• in addition, in each of the case studies above the asymptoticconfigurations for

both constant and variableγ[f ] are investigated, assuming, for duly comparison,

that in the former the critical distance coincides withγ0. Particularly,γ0 = 3,

corresponding to a mainly cooperative attitude, andγ0 = 7, corresponding instead

to a strongly competitive attitude, are chosen.



Simulations



Simulations

!

"#$!


Lecture 4 - Index


2. Mathematical Tools and Sources of Nonlinearity



4.1 Welfare Policy and Political Competition

4.2 Simulations

4.3 Early Signals of a Black Swan




The mathematical framework presented does not account for apossible interplay

between different interaction dynamics due to concomitantsocial and economic issues.

Let us consider the case of a multiple strategy(u, v), with u representing the wealth

status of the active particles andv their level of support/opposition to a Government

policy. Let us investigate how the welfare dynamics can induce changes of personal

opinions in terms of support/opposition to a certain political regime.

Assume that alsov is a discrete variable taking values in a lattice

Iv = v1 = −1, . . . , vm+1

2

= 0, . . . , vm = 1,

vr =2

m− 1r −

m+ 1

m− 1, r = 1, . . . , m,

agreeing thatv1 = −1 corresponds to the strongest opposition whereasvm = 1 to the

maximum support.




The mathematical structure is as follows:

dfri

dt=

m∑

p, q=1

n∑

h, k=1

ηpq

hkBpq

hk(i, r)fp

hfq

k − fri

m∑

q=1

n∑

k=1

ηrq

ik fq

k ,

for i = 1, . . . , n andr = 1, . . . , m, and where

fri = f

ri (t) : [0, Tmax] → IR +

is the number of individuals that, at timet, are expressing the strategy(ui, vr), or, in

other words, that belong to the wealth classui and to the opinion classvr. In addition:

• ηpq

hk is theinteraction rate between candidate particles with strategy(uh, vp) and

field particles with strategy(uk, vq). The same model fur the welfare policy is used,

according to the assumption that interactions among activeparticle are mainly driven

by the wealth status rather than by the difference of political opinion. Thusηpq

hk is

independent of the opinion classes that candidate and field particles belong to,

ηpq

hk = ηhk.




• Bpq

hk(i, r) is theprobability density that a candidate particle changes its strategy to

the test one(ui, vr) after interacting with a field particle. Analogously to the one

dimensional case, it is required to fulfill the probability density property:

m∑

r=1

n∑

i=1

Bpq

hk(i, r) = 1, ∀h, k = 1, . . . , n, ∀ p, q = 1, . . . , m.

The following factorization on the output test state(ui, vr) is proposed, relying

simply on intuition:Bpq

hk(i, r) = Bpq

hk(i)Bpq

hk(r),

• Bpq

hk(i) encodes the transitions of wealth class, which are further supposed to be

independent of the political feelings of the interacting pairs: Bpq

hk(i) = Bhk(i).

• Bpq

hk(r) encodes the changes of political opinion resulting from interactions.

Coherently with the observation made above that political persuasion is neglected,

so that political feelings originate in the individuals in consequence of their own

wealth condition, this term is assumed to depend on the wealth status and political

opinion of the candidate particle only:Bpq

hk(r) = Bp

h(r).




In view of the special structure

Bpq

hk(i, r) = Bhk(i)Bp

h(r),

it turns out that sufficient conditions ensuring the conservation in time of both the total

number of active particles and the average wealth status of the system are:

∑n

i=1Bhk(i) = 1, ∀h, k = 1, . . . , n

∑n

i=1uiBhk(i) = uh + σhk, ∀h, k = 1, . . . , n, σhk antisymmetric

∑m

r=1B

p

h(r) = 1, ∀h = 1, . . . , n, ∀ p = 1, . . . , m.

Mathematical are obtained by prescribing the interaction rateηpq

hk and the transition

probabilitiesBpq

hk(i, r).




As far as the modeling ofBp

h(r) is concerned, the following set of transition

probabilities is proposed:

1. Poor individual (uh < 0) in a poor society (U0 < 0):

p = 1

B1h(1) = 1

B1h(r) = 0, ∀ r 6= 1

p > 1

Bp

h(p− 1) = 2β

Bp

h(p) = 1 − 2β

Bp

h(r) = 0, ∀ r 6= p− 1, p



Welfare Policy and Political Competition 2. Wealthy individual (uh ≥ 0) in a poor

society (U0 < 0) or poor individual (uh < 0) in a wealthy society (U0 ≥ 0):

p = 1

B1h(1) = 1 − β

B1h(2) = β

B1h(r) = 0, ∀ r 6= 1, 2

p 6= 1, m

Bp

h(p− 1) = β

Bp

h(p) = 1 − 2β

Bp

h(p+ 1) = β

Bp

h(r) = 0, ∀ r 6= p− 1, p, p+ 1

p = m

Bmh (m− 1) = β

Bmh (m) = 1 − β

Bmh (r) = 0, ∀ r 6= m− 1, m




3. Wealthy individual (uh ≥ 0) in a wealthy society (U0 ≥ 0):

p < m

Bp

h(p) = 1 − 2β

Bp

h(p+ 1) = 2β

Bp

h(r) = 0, ∀ r 6= p, p+ 1

p = m

Bmh (m) = 1

Bmh (r) = 0 ∀ r 6= m,

whereβ ∈ [0, 1

2] is a parameter expressing the basic probability of changingpolitical

opinion.




Therefore, changes of political opinion are triggered jointly by the individual wealth

status, and the average collective one of the population:

• poor individuals in a poor society tend to distrust markedlythe Government

policy, sticking in the limit at the strongest opposition;

• wealthy individuals in a poor society and poor individuals in a wealthy society

exhibit, in general, the most random behavior. In fact, theymay trust the

Government policy either because of their own wealthiness,regardless of the

possibly poor general condition, or because of the collective affluence, in spite of

their own poor economic status. On the other hand, they may also distrust the

Government policy either because of the poor general condition, in spite of their

individual wealthiness, or because of their own poor economic status, regardless

of the collective affluence;

• wealthy individuals in a wealthy society tend instead to trust earnestly the

Government policy, sticking in the limit at the maximum support.



Simulations



Simulations

This Figure refers to the caseU0 = 0, and shows that:

• In an economically neutral society with uniform wealth distribution not only do

wealthy classes stick at an earnest support to the Government policy, but also poor

ones do not completely distrust them, especially in a context of prevalent

cooperation among the classes (γ0 = 3).

• Therefore, this example does not suggest the development ofsignificant

polarization in that society, although a greater polarization is observed for higher

values ofγ.



Simulations

!

!

"

!

"

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""##

!



Simulations

The Figure refers to the caseU0 = −4, and shows that:

• A strong radicalization of the opposition. The model predicts indeed that, in such

a poor society, poor classes stick asymptotically at the strongest opposition,

whereas wealthy classes spread over the whole range of political orientations,

however with a mild tendency toward opposition for the moderately rich ones

(say,u5 = 0, u6 = 0.25, andu7 = 0.5.

• The growth of political aversion is especially emphasized under variable critical

distanceγ[f ], when the marked clustering of the population in the lowest wealth

classes, due to a more competitive spontaneous attitude, entails in turn a clustering

in the highest distrust of the regime.



Simulations

Remark 1. The case studies discussed above indicate that an effectiveinterpretation

of the social phenomena under consideration requires a careful examination of the

probability distribution over the microscopic states. Indeed, show entirely different

scenarios, that might not be completely caught simply by average macroscopic

quantities. An important role is given by the initial wealthin addition to the

distribution.

Remark 2. As a general concluding remark, we notice that, even with a variable

critical distance, none of the asymptotic configurations ofthe system seems to be

properly identifiable as a Black Swan. On the other hand, the simple case studies

addressed here show that radical situations can be generated from the complex

interplay between these two social aspects so that that Black Swans are mostly

expected to arise. Therefore it is worth looking for early signals.



Early Signals of a Black Swan

Let us details the expressionBlack Swan, introduced in the specialized literature for

indicating unpredictable events, which are far away from those generally observed by

repeated empirical evidence. A Black Swan is specifically characterized as follows:

(H. Taleb) “A Black Swan is a highly improbable event with three

principal characteristics: It is unpredictable; it carries a massive impact;

and, after the fact, we concoct an explanation that makes it appear less

random, and more predictable, than it was.”

In the author’s opinion, mathematical models usually rely on what is already known,

thus failing to predict what is instead unknown. It is a rare example of research moving

against the main stream of the traditional approaches, generally focused on

well-predictable events. His book had an important impact on the search of new

research perspectives: for instance, it motivated appliedmathematicians to propose

formal approaches to study the Black Swan, in an attempt to forecast conditions for its

onset. Individual behavioral rules and strategies are not,in most cases, constant in time

due to the evolutionary characteristics of living complex system.



Early Signals of a Black Swan

Bearing in mind the previous remarks, we now provide some suggestions for the

possible detection of a Black Swan within our current mathematical framework.

Let us assume that a specific model, derived from the mathematical structures

presented above, has a trend to an asymptotic configuration described by stationary

distributionsfri i=1, ..., n

r=1, ..., m:

limt→+∞

‖fr· − f

r· (t)‖ = 0, ∀ r = 1, . . . , m,

where‖ · ‖ is a suitable norm inn over the activityu ∈ Iu, for instance the1-norm:

‖fr· (t)‖1 =

n∑

i=1

|fri |(t).

Alternative metrics can also be introduced depending on thephenomenology of the

system at hand, which may need, for instance, either uniformor averaged ways of

measuring the distance between different configurations.



Early Signals of the Black Swan

In addition, let us assume that the modeled system is expected to exhibit a stationary

trend described by some phenomenologically guessed distributionsfri i=1, ..., n

r=1, ..., m

. In

principle, such expected distribution have to be determined heuristically for each

specific case study, as we will see in the following.

Accordingly, we define the following time-evolving distancedBS (the subscript “BS”

standing for Black Swan):

dBS(t) := maxr=1, ..., m

‖fr· − f

r· (t)‖,

which, however, will generally not approach zero as time goes by for the heuristic

asymptotic distribution does not translate the actual trend of the system. This function

can be possibly regarded as one of theearly-warning signalsfor the emergence of

critical transitions to rare events, because it may highlight the onset of strong

deviations from expectations.




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 Tmax

dB

S

t

γ0=3

γ0=7

The mappingt 7→ dBS(t) computed in the case studies with variableγ, taking as

phenomenological guess the corresponding asymptotic distributions obtained with

constantγ.




Remark 3. Specific applications may suggest other distances. For instance, linear or

quadratic moments might be taken into account. We consider that extreme events are

likely to be generated by the interplay of different types ofdynamics, a fact that should

be reflected in the choice of appropriate metrics.

Remark 4. The Figure shows the (common) qualitative trends of the mapping

t 7→ dBS(t): an initial decrease of the distance, which may suggest a convergence to

the guessed distribution, is then followed by a sudden increase (notice the singular

point in the graph ofdBS) toward a nonzero steady value, which ultimately indicatesa

deviation from the expected outcome. Such a turnround is possibly a macroscopic

signal that a Black Swan is about to appear. However, in orderto get a complete

picture, the average gross information delivered bydBS(t) has to be supplemented by

the detailed knowledge of the probability distribution over the microscopic states,

which is the only one able to properly distinguish between lower (γ0 = 3) and higher

(γ0 = 7) radicalization of political feelings when welfare dynamics are ruled by a

variable critical thresholdγ[f ].


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