Phase transitions in Phase transitions in social hierarchies with a social hierarchies with a distribution of resourcesdistribution of resources

(caciques phase…)(caciques phase…)

Gerardo G. Naumis, Marcelo del Gerardo G. Naumis, Marcelo del Castillo-Mussot,, Gerardo Vázquez, Castillo-Mussot,, Gerardo Vázquez,

Luis PérezLuis PérezInstituto de Física, UNAM. México D.F., Instituto de Física, UNAM. México D.F.,

MexicoMexico..

IntroductionIntroduction

Our interests: energy and societyOur interests: energy and society

Social hierarchiesSocial hierarchiesBonabeau model of social hierachies.Bonabeau model of social hierachies.

The role of resources.The role of resources.

ConclusionsConclusions

Hierarchies in animalsHierarchies in animals

C. Darwin, 1859, animal fitness: key to evolutionC. Darwin, 1859, animal fitness: key to evolution

Our abilities has planted:

“humans at the center of the universe”.

Plato differentiated two distinct souls,Man, with two souls: one, capable of reason. Non-human animals, only brute mortal soul.

Many animals held territories. Entry into foreign areas causes fighting. Thus animals are forced to spread out. Surplus animals may be expelled, or may stay on as non-breeders.

Gregarious animals develop hierarchies. Dominant animals often have first choices. Lowest members of the hierarchy suffer food shortages, This limits their reproduction rate.

Example: hens exhibit this type of hierarchy: the pecking order.

Inside species, there are social hierarchiesInside species, there are social hierarchies

Class formation in historic times did not take place through gradual differentiation in pacific economic competition, but was the result of

violent conquest and subjugation.

Sociological consensus:Sociological consensus:

Social stratification is relatively rigid; social position determined by ancestors. (example: slavery)

Everybody wants to rise (of course!!): role of education and productivity.

Bonabeau ModelBonabeau Model

Territory: a square lattice with Territory: a square lattice with NN sites sites MM agents are placed at random: agents are placed at random:

E. Bonabeau, G. Theraulaz, J.L. Deneubourg, Physica A 217, 1157 (1995).E. Bonabeau, G. Theraulaz, J.L. Deneubourg, Physica A 217, 1157 (1995).

Mp

N

Agents chose at random one of the 4 neighbouring sites and try to move there:Agents chose at random one of the 4 neighbouring sites and try to move there:1)1) Site empty: stay there.Site empty: stay there.2)2) Site not empty: FIGHTSite not empty: FIGHT

Fights are decided at random.Fights are decided at random.h(i) is the history (or fitness) of agent i, and his probability of winning is:h(i) is the history (or fitness) of agent i, and his probability of winning is:

( ( ) ( ))

1

1 e h j h iq

If the agresor wins, he and the defensor interchange places. If the agresor wins, he and the defensor interchange places.

For the winner h(i) h(i)+1, and the looser h(i) h(i)-1.For the winner h(i) h(i)+1, and the looser h(i) h(i)-1. is a parameteris a parameter

ORDER PARAMETER (DEGREE OF HIERARCHY)ORDER PARAMETER (DEGREE OF HIERARCHY)

2 2q q

Enhanced phase transition: D. Stauffer, Int. J. Mod. Phys. C 16, 8 (2003).Enhanced phase transition: D. Stauffer, Int. J. Mod. Phys. C 16, 8 (2003).

q

h(j)-h(i)

Our modifed rules of movementOur modifed rules of movement

At the beginning, a value is assigned to each site of the lattice at At the beginning, a value is assigned to each site of the lattice at random, following a certain distribution (natural resources) or with a random, following a certain distribution (natural resources) or with a certain geometry (México –USA border)certain geometry (México –USA border)

An agent always moves to the richest site in the neighbourhood. An agent always moves to the richest site in the neighbourhood.

Special cases arise when two or more sites have similar values. Special cases arise when two or more sites have similar values. Rule: decide at random among rich sites.Rule: decide at random among rich sites.

If the original site has the same value of the richest neighbour, jump If the original site has the same value of the richest neighbour, jump in order to recover Bonabeau model when all sites are rich or poor.in order to recover Bonabeau model when all sites are rich or poor.

ResultsResults

DiffusivityDiffusivity

Valuable sites per capita Valuable sites per capita

22

1

( ) (0)1lim

Ni i

ti

x t xD

N t

1

1 M

iiM

Percolation of rich sites: order parameter and diffusivity

Amplification of the diffusivity

Wealth or “money” per capita: percolation

1

1 1( )

M

ii

xxN

M M p

Non-Percolation of rich sites: order parameter and diffusivity

Distribution of valuable sites per capita: non-percolation of rich sitesDistribution of valuable sites per capita: non-percolation of rich sites

Caciques y cabezones

Cacique phase: Cacique phase: lack lack of global information and glassy behaviorof global information and glassy behavior

CONCLUSIONSCONCLUSIONS

•The introduction of valuable sites enhance hierarchies even at low populationsThe introduction of valuable sites enhance hierarchies even at low populations (except at high populations and high concentration of rich sites).(except at high populations and high concentration of rich sites).

•If there are few valuable sites, there is a high inequality due to local power: caciques.If there are few valuable sites, there is a high inequality due to local power: caciques.

•The system can be trapped in a “cacique phase”, as in a glass, due to lack of global The system can be trapped in a “cacique phase”, as in a glass, due to lack of global information about the available resources.information about the available resources.