Naming game: dynamics on complex networks A' Barrat, LPT, Universit ParisSud, France

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Naming 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/
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Naming 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
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Minimal 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

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Minimal 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

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Minimal 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
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Naming game other dynamical rules
Possibility of giving weights to words, etc... gt
more complicate rules
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Naming 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
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Complete 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)
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Complete 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)
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Another 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!)
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Another 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
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Regular 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)
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Two extreme cases
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Naming 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)
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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.

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Naming Game on a small-world
p0 linear chain p À 1/N small-world
p0
increasing p
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Naming Game on a small-world
maximum memory / N
convergence time / N1.4
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Better not to have all-to-all communication, nor
a too regular network structure
What about other types of networks ?
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NetworksHomogeneous and heterogeneous
1.Usual random graphs Erdös-Renyi model (1960)
N points, links with proba p static random graphs
Poisson distribution
(pO(1/N))
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NetworksHomogeneous 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)
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Definition 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
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NG on heterogeneous networks
Example agents on a BA network
Different behaviours
shows the importance of understanding the role of
the hubs!
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NG on heterogeneous networks
Speaker first hubs accumulate more words Hearer
first hubs have less words and polarize the
system, hence a faster dynamics
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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
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NG on complex networksdependence on system size

• Memory peak tmax / N Nmaxw / N
• average maximum memory per agent finite!
• Convergence time tconv / N1.5

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Effects of average degree
larger ltkgt

• larger memory,
• faster convergence

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Effects of enhanced clustering
larger clustering
C increases

• smaller memory,
• slower convergence

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Other issues

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

Slow down/stop the dynamics
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Conclusions 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

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• Alain.Barrat_at_th.u-psud.fr
• http//www.th.u-psud.fr/