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Citation: Dall'Asta, L., Baronchelli, A., Barrat, A. and Loreto, V. (2006). Agreement
dynamics on small-world networks. Europhysics Letters, 73(6), pp. 969-975. doi:
10.1209/epl/i2005-10481-7
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arXiv:cond-mat/0603205v2 [cond-mat.stat-mech] 9 Mar 2006
Europhysics Letters PREPRINT
Agreement dynamics on small-world networks
L. Dall’Asta
1
, A. Baronchelli
2
, A. Barrat
1
and V. Loreto
2
1
Laboratoire de Physique Th´eorique (CNRS UMR8627) - Bˆatiment 210, Universit´e
Paris-Sud, 91405 Orsay cedex, France
2
Dipartimento di Fisica, U niversit`a “La Sapienza” and SMC-INFM, P.le A. Moro 2,
00185 ROMA, (Italy)
PACS. 89.75.Fb – .
PACS. 05.65.+b – .
Abstract. – In this paper we analyze the effect of a non-trivial topology on the dynamics
of the so-called Naming Game, a recently introduced model which addresses the issue of how
shared conventions emerge spontaneously in a population of agents. We consider in particular
the small-world topology and study the convergence towards the global agreement as a function
of the population size N as well as of the parameter p which sets the rate of rewiring leading
to the small-world network. As long as p ≫ 1/N there exists a crossover time scaling as
N/p
2
which separates an early one-dimensional-like dynamics from a late stage mean-field-like
behavior. At the beginning of the process, the local quasi one-dimensional topology induces a
coarsening dynamics which allows for a minimization of th e cognitive effort (memory) required
to the agents. In the late stages, on the other hand, the mean-field like top ology leads to a
speed up of the convergence process with respect to the one-dimensional case.
The recent past has witnessed an important development of the activities of statistical
physicis ts in the area of social sciences (for a recent collection of papers see [1]). Indeed,
statistical physics is the natural field to study how global complex properties can emerge from
purely local rules. Social interactions have thus been described and studied by statistica l
physics models and tools, in particular models of opinion formation in which agents update
their internal state, or opinion, through an interaction with its neighbors. An interesting
point concerns whether and how a population of agents converg e towards a c ommon and
shared state (consensus) without external global coordination [2]. Early studies have mostly
dealt with age nts either able to interact with all the other agents (mean-field case), or sitting
on the nodes of regular lattices. Such situations, although not always realistic, have the
advantage to be accessible to the usual methods of statistical mechanics.
Even more rece ntly however, the growing field of complex networks [3–5] has allowed to
obtain a better knowledge of social networks [6], and in particular to show that the topology of
the network on which agents interact is not regular. A natural step has then been to consider
various models embedded on more realistic networks and to study the influence of various
complex topologies on the corresponding dynamical behavior.
In particular, social networks are typically “small-worlds” in which, on the one hand,
the average distance between two agents is small [7], growing only logarithmically with the
c
EDP Sciences
2 EUROPHYSICS LETTERS
network’s size, and, on the other hand, ma ny triangles are present, unlike totally random
networks. In order to reconcile bo th properties, Watts and Strogatz have introduced the
now famous small-world network model [8] which allows to interpolate between regular low-
dimensional lattices and random networks, by introducing a certain amount of random long-
range connections into an initially regular network.
Subsequently, a number of papers have fo c us ed on the influence of these long-range “short-
cuts” on the behavior of various models defined on the network: from the Ising model [9] to
the spreading of epidemics [10], or the evolution of random walks [11]. Dynamics of models
inspired by social sciences are no exception, such as the Voter model [12, 13] or Axelrod’s
model of culture dissemination [14 , 15].
In this letter, we consider the effect of a small-world topology on the so-called Naming
Game model, which was inspired by the field of semiotic dynamics, a new emerging area fo-
cusing on the development of shared communication sy stems (languages) among a population
of agents. Such a process can indeed be considered as arising through self-organization o ut
of local interactions. The language is then seen as constantly r eshaped by its users in order
to maximize co mmunicative success and ex pressive power while minimizing the co gnitive ef-
fort [16, 17]. In addition, there are recent developments in Information Technology in which
new forms of s e miotic dynamics begin to appear. One example are social tagging sites (such
as del.icio.us or www.flickr.com), through which tens of thousands of web users share infor-
mation by tagging items like pictures or web-sites and thus develop ”folksonomies” [18, 19].
In this context, simplified models or “Language Games” have been defined and studied in the
theoretical community [16, 17]. As for opinion formation models, it is interesting to under-
stand if a common state for all agents can be reached and, in the positive answer case, how
the s ystem converges towards such a state.
The model. – The original model [20] is related to an artificial intelligence experiment
called Talking Heads [21], in which embodied software agents observe a set of objects through
digital cameras , assign them randomly chosen names and c ommunicate these na mes to each
other. Since different agents can invent different names for the same object, the final e mergence
of a common dictionary for all the agents is not granted from the start. However, it turns out
that such a consensus is in fact experimentally re ached. In order to try to capture the essential
relevant features of such a dynamics, Baronchelli et al. [22] have proposed a minimal model
of Naming Game that reproduces the phenomenology of the experiments, despite the age nts
of the model are far from the complicate software effectively used as “Talking Heads”. Such
a model is however amenable to both analy tical and extensive numerical treatment [22, 23],
allowing for a better understanding of the mechanisms at work.
The model considers N identical individuals (o r agents) which observe the same object
and try to communicate its name one to the other. Each agent is endowed with an internal
inventor y or memory in which it can store an a priori unlimited number of different names
or opinions. Initially, each agent has an empty inventory. The dynamics proceeds as follows:
at each time step, two individuals are chosen at random for a pairwis e interaction (or “com-
munication”). One of these agents acts as the “speaker” and the other one as the “hearer”. If
the speaker does not know a name for the object (its inventory is empty), it invents a new
name and r e c ords it. Else, if it already knows one or more synonyms (stored in the inventory),
it chooses one of them randomly. The invented or selected word is then transmitted to the
hearer. If the hearer already has this term in its memory, the interaction is a success, and
both agents retain that term as the right one, cance ling all the other terms in their inventories;
otherwise, the inter action is a failure, and the new name is included in the inventory of the
hearer, without any cancellation.
L. Dall’Asta, A. Baronchelli, A. Barrat and V. Loreto: Agreement dynamics on small-world networks3
The way in which agents may interact with e ach other is determined by the topology of the
underlying contact network. The mean-field case corresp onds to a fully connected network,
in which all agents are in mutual contact. In this case, studied in [22], each agent rarely
interacts twice with the same partner, so that the system initially accumulates a large number
(O(N/2)) of different names (syno nyms) for the object, invented by different agents (speaker s)
and O(N
3/2
) total words in the whole population. Interestingly however, this profusion of
different names leads in the end to an asymptotic absorbing state in which all the agents share
the s ame name.
As a sec ond step towards the understanding of the model from a statistical physics point
of view, we have considered in [23] the case of agents sitting on the nodes of a regular lattice
in dimension d. In this ca se, each agent is connected to a finite number of neighbors (2d) so
that it may possess only a finite number of different words in its inventory at any given time.
As a result, the total amount of memory used by the whole system grows as N instead of
N
3/2
. Local consensus appears at very early stag e s of the evolution, since neighboring agents
tend to share the same unique word. The dynamics then proceeds through the coarsening of
such clusters of agents sharing a common name; the interfaces be tween clusters are composed
by agents who still have more than one possible name, and diffuse randomly. Because of
this particular coarsening process, the average cluster size grows as
p
t/N, and the time to
convergence corresponds to the time needed for one cluster to reach the system size, i.e. a
time N
1+2/d
for d ≤ 4. In one dimension in particular, the convergence is thus dramatically
slowed down from O(N
3/2
) to O(N
3
).
In the following, we investigate the effect of long-range connections which link agents that
are far from each other on the reg ular lattice. We use the small-world model of Watts and
Strogatz: starting from a one-dimensional lattice of N sites, with periodic boundary conditions
(i.e. a ring), each vertex being connected to its 2m nearest neighbors, a stochastic rewiring
procedure is applied. The vertices are vis ited one after the other, and each link connecting a
vertex to one of its m nearest neighbors in the clockwise sense is left in place with probability
1 − p, and with probability p is reconnected to a randomly chosen other vertex. For p = 0
the network retains a purely one-dimensional topology, while the r andom network structure
is approached as p goes to 1. At small but finite p (1/N ≪ p ≪ 1), a small-world structure
with short distances between nodes, together with a large clus tering, is obtained.
Global pictu re: Expected behavior. – For p = 0, it has been shown in [23] that the
dynamics proceeds by a slow coarsening of clusters of agents sharing the same state or word.
At small p, the short-cuts are typically far from e ach other, with a typical distance 1/p
between short-cuts so that the early dynamics is not affected and procee ds as in dimension
1. In particular, at very short times many new words are invented since the success rate is
small. After a time of order N , each agent ha s played typically once, and therefore O(N)
different words have been invented: the number of different words reaches a peak which scales
as N. Since the number of neighbors of each site is bounded (the degree distribution decreases
exp onentially [9]), each agent has access only to a finite number of different words, so that the
average memory per agent used remains finite, as in finite dimensions and in contrast with
the mean-field case. The interaction of neighboring agents first leads to the usual coarsening
phenomena as long as the clusters are typically one-dimensional, i.e. as long as the typical
cluster size is smaller than 1/p. However, as the average cluster size reaches the typical
distance between two short-cuts ∼ 1/p, a crossover phenomena is bound to take place; since
the cluster size grows as
p
t/N [23], this corresponds to a crossover time t
cross
= O(N/p
2
).
For times much larger than this crossover, one expects that the dynamics is dominated by the
existence of short-cuts and enters a mean-field like behavior. The convergence time is thus
4 EUROPHYSICS LETTERS
10
0
10
1
10
2
10
3
10
4
t/N
10
-3
10
-2
10
-1
10
0
N
w
/N -1
A)
Increasing p
10
0
10
2
10
4
t/N
1
1,2
1,4
1,6
1,8
2
N
w
(t)/N
p=0.01
p=0.08
B)
Increasing N
Fig. 1 – A) Average number of words per agent in the system, N
w
/N as a function of the rescaled
time t/N, for small-world networks with hki = 8 and N = 10
3
nodes, for various values of p. The
curve for p = 0 is shown for reference, as well as p = 5.10
−3
, p = 10
−2
, p = 2.10
−2
, p = 4.10
−2
,
p = 8.10
−2
, from bottom to top on the left part of the curves. B) (Color Online) N
w
/N for p = 10
−2
and p = 8.10
−2
and increasing system sizes: N = 10
3
, N = 10
4
, N = 10
5
. Larger system sizes yield
larger plateau lengths.
exp ected to scale as N
3/2
and not as N
3
. In order for this picture to be possible, the crossover
time N/p
2
needs to be much larg e r than 1, a nd much smaller than the consensus time fo r the
one-dimensional case N
3
; these two conditions read p ≫ 1/N, which is indeed the necessary
condition to obtain a small-world networ k.
It is therefore expected that the small-world topology allows to combine advantages from
both finite-dimensional lattices and mean-field networks: on the one hand, only a finite mem-
ory per node is needed, in opposition to the O(N
1/2
) in mean-field; on the other hand the
convergence time is expected to be much shorter than in finite dimensions.
Numerical study. – Various quantities of interest can be monitored in numerical studies
of the Naming Game model in order to verify and quantify the qualitative expected picture.
Among the most relevant ones are the average number of words in the agents inventory, N
w
(t),
which co rresponds to the average memory used, and the total number of distinct words in the
system, N
d
(t).
Figure 1 displays the evolution of the average number of words per agent as a function of
time, for a small-world network with average degree hki = 8, and various values of the rewiring
probability p and size N. While N
w
(t) in all cases decays to N (Fig. 1A), after an initial peak
whose height is proportional to N (Fig. 1B), the way in which this convergence is obtained
depends on the parameters. At fixed N, for p = 0 a power-law behavior N
w
/N −1 ∝ 1/
√
t is
observed due to the one-dimensional coarsening process [23]. As soon as p ≫ 1/N however,
deviations are observed and get s tronger as p is increased: the decrease of N
w
is first slowed
down after the peak, but lea ds in the end to an exponential convergence. The intermediate
slowing down and the faster convergence are both enhanced as p increases. On the other hand,
a system size increase at fixed p corresponds, as shown in Fig. 1B, to a slower convergence
even on the rescaled time t/N, with a longer and longer plateau at a lmost constant average
used memory.
As mentioned previously, a crossover phenomenon is expected when the one-dimensional
clusters reach sizes of order 1/p, i.e. at a time of order N/p
2
. By definition, in the interior of
each cluster, sites have only one word in memory, w hile the sites with more than one word are
localized at the interfaces between clusters, whos e number is then of order Np. The average
excess memory per site (with respect to global consensus) is thus of order p, so that one
