Title: Naming game: dynamics on complex networks A' Barrat, LPT, Universit ParisSud, France
1Naming game dynamics on complex networksA.
Barrat, LPT, Université Paris-Sud, France
A. Baronchelli (La Sapienza, Rome, Italy) L.
DallAsta (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/
2Naming game
- 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
3Minimal 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
4Minimal naming game dynamical rules
- success gt speaker and hearer retain the uttered
word as the correct one and cancel all other
words from their memory - failure gt the hearer adds to its memory the
word given by the speaker
5Minimal naming game dynamical rules
FAILURE
Speaker
Speaker
Hearer
Hearer
ARBATI ZORGA GRA
ARBATI ZORGA GRA
REFO TROG ZEBU
REFO TROG ZEBU ZORGA
SUCCESS
Speaker
Speaker
Hearer
Hearer
ZORGA
ZORGA
ARBATI ZORGA GRA
ZORGA TROG ZEBU
6Naming game other dynamical rules
Possibility of giving weights to words, etc... gt
more complicate rules
7Naming gameexample 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
8Complete graph
Convergence
N1024 agents
Total number of wordstotal memory used
Building of correlations
Number of different words
Success rate
Baronchelli et al. 2005 (physics/0509075)
9Complete graphDependence 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. 2005 (physics/0509075)
10Another extreme caseagents 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!)
11Another extreme caseagents on a regular lattice
N1000 agents MFcomplete graph 1d, 2d agents
on a regular lattice
Nwtotal number of words Ndnumber of distinct
words Rsucess rate
12Regular latticeDependence 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 gtSlow process!
(in d dimensions / N12/d)
13Two extreme cases
14Naming Game on a Small-world
N nodes forms a regular lattice. With probability
p, each edge is rewired
randomly gtShortcuts
N 1000
- Large clustering coeff.
- Short typical path
Watts Strogatz, Nature 393, 440 (1998)
15Naming 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.
16Naming Game on a small-world
p0 linear chain p À 1/N small-world
p0
increasing p
17Naming Game on a small-world
maximum memory / N
convergence time / N1.4
18Better not to have all-to-all communication, nor
a too regular network structure
What about other types of networks ?
19NetworksHomogeneous and heterogeneous
1.Usual random graphs Erdös-Renyi model (1960)
N points, links with proba p static random graphs
Poisson distribution
(pO(1/N))
20NetworksHomogeneous 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 ? 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)
21Definition of the Naming Game on heterogeneous
networks
- recall original definition of the model
- select a speaker and a hearer at random among all
nodes - gtvarious interpretations once on a network
- -select first a speaker i and then a hearer among
is neighbours - -select first a hearer i and then a speaker among
is 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
22NG on heterogeneous networks
Example agents on a BA network
Different behaviours
shows the importance of understanding the role of
the hubs!
23NG on heterogeneous networks
Speaker first hubs accumulate more words Hearer
first hubs have less words and polarize the
system, hence a faster dynamics
24NG 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
25NG on complex networksdependence on system size
- Memory peak tmax / N Nmaxw / N
- average maximum memory per agent finite!
- Convergence time tconv / N1.5
26Effects of average degree
larger ltkgt
- larger memory,
- faster convergence
27Effects of enhanced clustering
larger clustering
C increases
- smaller memory,
- slower convergence
28Other issues
- Hierarchical structures
- Community structures
- Other (more efficient?) strategies
(i.e. dynamical rules) - ...
Slow down/stop the dynamics
29Conclusions and (Some) Perspectives
- 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
30- Alain.Barrat_at_th.u-psud.fr
- http//www.th.u-psud.fr/