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Naming game: dynamics on complex networks A. Barrat, LPT, Université Paris-Sud, France

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A. Baronchelli (La Sapienza, Rome, Italy)

L. Dall’Asta (LPT, Orsay, France)

V. Loreto (La Sapienza, Rome, Italy)

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

-Europhys. Lett. 73 (2006) 969

-Preprint (2006)

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

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

Agents:

-can keep in memory different words

-can communicate with each other

Example of social dynamics or agreement dynamics

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

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

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

FAILURE

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

Possibility of giving weights to words, etc...

=> more complicate rules

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

Memory peak

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. 2005 (physics/0509075)

- 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. 2005 (physics/0509075)

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!)

N=1000 agents

MF=complete graph

1d, 2d: agents on a regular

lattice

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

- 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)

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)

Naming Game on a small-world

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.

p=0: linear chain

p À 1/N : small-world

p=0

increasing p

maximum memory:

/N

convergence time:

/N1.4

Better not to have

all-to-all communication,

nor a too regular network structure

What about other types of networks ?

Networks:Homogeneous and heterogeneous

1.Usual random graphs: Erdös-Renyi model (1960)

N points, links with proba p:

static random graphs

Poisson distribution

(p=O(1/N))

P(k) ~k-3

Networks:Homogeneous and heterogeneous

2.Scale-free graphs: Barabasi-Albert (BA) model

(1)GROWTH: At every timestep we add a new node with m edges (connected to the nodes already present in the system).

(2)PREFERENTIAL ATTACHMENT :The probability Π that a new node will be connected to node i depends on the connectivity ki of that node

/ ki

A.-L.Barabási, R. Albert, Science 286, 509 (1999)

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

Example: agents on a BA network:

Different behaviours

shows the importance

of understanding the role

of the hubs!

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

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

- Convergence time: tconv/ N1.5

Effects of average degree

larger <k>

- larger memory,
- faster convergence

Effects of enhanced clustering

larger clustering

C

increases

- smaller memory,
- slower convergence

- Hierarchical structures
- Community structures
- Other (more efficient?) strategies (i.e. dynamical rules)
- ...

Slow down/stop

the dynamics

- Importance of the topological properties for the
- processes taking place on the network
- Weighted networks
- Dynamical networks (e.g. peer to peer)
- Coupling (evolving) topology and dynamics
- on the network

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