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Agreement dynamics on interaction networks: the Naming game

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Agreement dynamics on interaction networks: the Naming game

A. BarratLPT, Université Paris-Sud, FranceISI Foundation, Turin, Italy

A. Baronchelli (La Sapienza, Rome, Italy)

L. Dall’Asta (LPT, Orsay, France)

V. Loreto (La Sapienza, Rome, Italy)

http://www.th.u-psud.fr/

Phys. Rev. E 73 (2006) 015102(R)

Europhys. Lett. 73 (2006) 969

Phys. Rev. E 74 (2006) 036105

http://cxnets.googlepages.com

Introduction

Statistical physics: study of the emergence of global complex properties from purely local rules

“Sociophysics”:

Simple (simplistic?) models which may

allow to understand fundamental aspects

of social phenomena

=>Voter model, Axelrod model, Deffuant model….

Opinion formation models

Simplified models of interaction between N agents

Questions:

- Convergence to consensus without global external coordination?
- How?
- In how much time?

Most initial studies:

- “mean-field”: each agent can interact with all the others
- regular lattices

- Recent progresses in network science:
- social networks: complex networks
- small-world, large clustering, heterogeneous
- structures, etc…

- social networks: complex networks

Studies of agents on complex networks

Naming game

(Talking Heads experiment,Steels ’98)

Interactions of N agents who communicate on how to associate a name to a given object

=> Emergence of a communication system?

Agents:

-can keep in memory different words/names

-can communicate with each other

Example of social dynamics/agreement dynamics

Convergence? Convergence mechanism?

Dependence on N of memory/time requirements?

Dependence on the topology of interactions?

Naming game: dynamical rules

At each time step:

-2 agents, a speaker and a hearer, are randomly selected

-the speaker communicates a name to the hearer

(if the speaker has nothing in memory –at the beginning- it invents a name)

-if the hearer already has the name in its memory: success

-else: failure

Minimal naming game: dynamical rules

success =>speaker and hearer retain the uttered word as the correct one and cancel all other words from their memory

failure => the hearer addsto its memory the word given by the speaker

(Baronchelli et al, JSTAT 2006)

Minimal naming game: dynamical rules

FAILURE

Speaker

Hearer

Speaker

Hearer

ARBATI

ZORGA

GRA

REFO

TROG

ZEBU

ARBATI

ZORGA

GRA

REFO

TROG

ZEBU

ZORGA

SUCCESS

Speaker

Speaker

Hearer

Hearer

ZORGA

ZORGA

ARBATI

ZORGA

GRA

ZORGA

TROG

ZEBU

Speaker

Hearer

Speaker

Hearer

1.ARBATI

2.ZORGA

3.GRA

1.REFO

2.TROG

3.ZEBU

1.ARBATI

2.GRA

3.ZORGA

1.REFO

2.TROG

3.ZEBU

4.ZORGA

SUCCESS

Speaker

Speaker

Hearer

Hearer

1.ZORGA

2.ARBATI

3.GRA

1.TROG

2.ZORGA

3.ZEBU

1.ARBATI

2.ZORGA

3.GRA

1.TROG

2.ZEBU

3.ZORGA

Naming game: other dynamical rulesPossibility of giving weights to words, etc...

=> more complicate rules

Naming game:example of social dynamics

-no bounded confidence

( Axelrod model, Deffuant model)

-possibility of memory/intermediate states

( Voter model, cf also Castello et al 2006)

-no limit on the number of possible states

(no parameter)

Naming game:example of social dynamics

interactions among individuals create complex networks: a population can be represented as a graph on which

agents

nodes

interactions

edges

Simplest case: complete graph

a node interacts equally with all the others, prototype of mean-field behavior

Complete graph

Convergence

N=1024 agents

Total number

of words=total

memory used

Building of

correlations

Number of

different words

Success rate

Baronchelli et al. JSTAT 2006

Complete graph:Dependence on system size

- Memory peak: tmax/ N1.5 ; Nmaxw/ N1.5
average maximum memory per agent/ N0.5

- Convergence time: tconv/ N1.5

diverges as

N 1

Baronchelli et al. JSTAT 2006

Another extreme case:agents on a regular lattice

Baronchelli et al., PRE 73 (2006) 015102(R)

N=1000 agents

MF=complete graph

1d, 2d: agents on a regular

lattice

Nw=total number of words; Nd=number of distinct words; R=success rate

Another extreme case:agents on a regular lattice

Baronchelli et al., PRE 73 (2006) 015102(R)

Local consensus is reached very quickly through repeated interactions.

Then:

-clusters of agents with the same unique word start to grow,

-at the interfaces series of successful and unsuccessful interactions take place.

Few neighbors:

coarsening phenomena (slow!)

Another extreme case:agents on a regular lattice

The evolution of clusters is described as the diffusion of interfaces which remain localized i.e. of finite width

Diffusion equation for the probability P(x,t) that an interface is at the position x at time t:

Each interface diffuses with a diffusion coefficient D(N)» 0.2/N

The average cluster size grows as

tconv» N3

Regular lattice:Dependence on system size

- Memory peak: tmax/ N ; Nmaxw/ N
average maximum memory per agent: finite!

- Convergence by coarsening: power-law decrease of Nw/N towards 1
- Convergence time: tconv/ N3 =>Slow process!

(in d dimensions / N1+2/d)

Naming Game on a small-world

N nodes forms a regular lattice. With probability p, each edge is rewired randomly

=>Shortcuts

N = 1000

- Large clustering coeff.
- Short typical path

Watts & Strogatz,

Nature393, 440 (1998)

Dall'Asta et al., EPL 73 (2006) 969

1D

Random topology

p: shortcuts

(rewiring prob.)

(dynamical) crossover expected:

- short times: local 1D topology implies (slow) coarsening
- distance between two shortcuts is O(1/p), thus when a cluster is of order 1/p the mean-field behavior emerges.

Naming Game on a small-world

p=0: linear chain

p À 1/N : small-world

-slower at intermediate

times (partial “pinning”)

-faster convergence

p=0

increasing p

Better not to have

all-to-all communication,

nor a too regular network structure

What about other types of networks ?

Definition of the Naming Game on heterogeneous networks

Dall’Asta et al., PRE 74 (2006) 036105

recall original definition of the model:

select a speaker and a hearer at random among all nodes

=>various interpretations once on a network:

-select first a speaker i and then a hearer among i’s neighbours

-select first a hearer i and then a speaker among i’s neighbours

-select a link at random and its 2 extremities at random as hearer and speaker

- can be important in heterogeneous networks because:
- -a randomly chosen node has typically small degree
- -the neighbour of a randomly chosen node has typically large degree

(cf also Suchecki et al, 2005 and Castellano, 2005)

NG on heterogeneous networks

Example: agents on a BA network:

Different behaviours

shows the importance

of understanding the role

of the hubs!

NG on heterogeneous networks

Speaker first: hubs accumulate more words

Hearer first: hubs have less words and “polarize” the system,

hence a faster dynamics

NG on homogeneous and heterogeneous networks

-Long reorganization phase

with creation of correlations,

at almost constant Nw and

decreasing Nd

-similar behaviour for BA

and ER networks

(except for single node dynamics),

as also observed for Voter model

NG on complex networks:dependence on system size

- Memory peak: tmax/ N ; Nmaxw/ N
average maximum memory per agent: finite!

- Convergence time: tconv/ N1.5

Effects of enhanced clustering

(more triangles, at constant number of edges)

larger clustering

C

increases

- smaller memory,
- slower convergence

Bad transmissions/errors?

A. Baronchelli et al, cond-mat/0611717

Modified dynamical rules:

in case of potential successful communication:

- With probability : success
- With probability 1-: nothing happens (irresolute attitude)

=1 : usual Naming Game => convergence

=0 : no elimination of names => no convergence

Expect a transition at some c

Mean-field case

Stability of the consensus state ?

consider a state with only 2 words A, B

Evolution equations for the densities: nA, nB, nAB

> 1/3 : states (nA=nAB=0, nB=1), (nB=nAB=0, nA=1)

< 1/3 : state with nAB > 0 , nA=nB > 0

Mean-field case

- At c = 1/3,
- Consensus to Polarization transition
- tconv/ (-c)-1

The polarized state is active

( Axelrod model, in which the polarized state is frozen)

Mean-field case:numerics

Usual NG

NG with at most m different words

=>At least 2 different universality classes

Series of transitions

tm=time to reach a state with m different words

Transitions to more and more disordered active states

On networks, as in MF

At c ,

Consensus to Polarization transition

(c depends on strategy+network heterogeneity)

The polarized state is active

Other issues

- Community structures (slow down/stop convergence)
(cf also Castello et al, arXiv:0705.2560)

- Other (more efficient) strategies (dynamical rules)
(A. Baronchelli et al., physics/0511201; Q. Lu et al., cs.MA/0604075)

- Activity of single nodes
(L. Dall’Asta and A. Baronchelli, J. Phys A 2006)

- Coupling the dynamics of the network with the dynamics on the network: transitions between consensus and polarized states, effect of intermediate states…

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