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Consensus formation on coevolving networks:
groups’ formation and structure
Balazs Kozma1 and Alain Barrat1,2
1LPT, CNRS, UMR 8627, and Univ Paris-Sud, Orsay, F-91405, France
2Complex Networks Lagrange Laboratory, ISI Foundation, Turin, Italy
E-mail: kozmab@th.u-psud.fr, alain.barrat@u-psud.fr
Abstract.
We study the effect of adaptivity on a social model of opinion dynamics and
consensus formation. We analyze how the adaptivity of the network of contacts
between agents to the underlying social dynamics affects the size and topological
properties of groups and the convergence time to the stable final state. We find that,
while on static networks these properties are determined by percolation phenomena,
on adaptive networks the rewiring process leads to different behaviors: Adaptive
rewiring fosters group formation by enhancing communication between agents of
similar opinion, though it also makes possible the division of clusters. We show how
the convergence time is determined by the characteristic time of link rearrangement.
We finally investigate how the adaptivity yields nontrivial correlations between the
internal topology and the size of the groups of agreeing agents.
PACS numbers: 89.75.-k, -87.23.Ge, 05.40.-a
Submitted to: J. Phys. A: Math. Gen.
1. Introduction
In the last years, many efforts have been devoted to the understanding of how the
behavior of models of interacting agents is affected by the topology of these interactions.
Various simple models have indeed been defined in order to study how local rules of
evolution can lead to the emergence of global phenomena. This topic, at the center
of the statistical physics, has been connected to the field of social sciences, leading to
the so-called sociophysics [1]. The statistical physics approach starts from the simplest
possible models, in which each agent is defined by an internal state, which evolves
according to certain rules of interactions with the neighboring agents. The internal
state can be an Ising variable taking only two possible values [2, 3, 4], a vector of traits
[5], a continuous variable [6, 7, 8, 9, 10], or have a more complex structure [11, 12, 13].
The local interaction rules may involve the simple imitation of a neighbor, an alignment
to a local majority, or more involved negotiation processes. The main question concerns

Consensus formation on adaptive networks 2
in all cases the possibility of the appearance of a global consensus, defined by the fact
that all agents’ internal states coincide, without the need for any central authority
to supervise such behavior. Alternatively, polarized states in which several coexisting
states or opinions survive can be obtained, and the parameters of the model drive the
transition between consensus and polarization.
The science of complex networks has recently led to an intense activity devoted to
the characterization and the understanding of the topology of man-made and natural
networks [14, 15, 16, 17, 18, 19, 20, 21], such as social networks, which determine how
social agents interact. Numerous studies have revealed the ubiquity of various striking
characteristics, such as the small-world effect: although each node has a number of
neighbors which remains small with respect to the total number of agents, only a small
number of hops suffices to go from any agent to any other on the network. This has
prompted the investigation of the effect of various interaction topologies on the behavior
of agents connected according to these topologies, highlighting the relevance of small-
world and heterogeneous structures (see for example [22, 23, 24, 25]).
More recently, the focus has shifted to take into account the fact that many
networks, and in particular social networks, are dynamical in nature: links appear and
disappear continuously in time, on many timescales. Moreover, such modifications of the
network’s topology do not occur independently from the agents’ states but as a feedback
effect: the topology determines the evolution of the agents’ opinions, which in its turn
determines how the topology can be modified [26, 27, 28, 29, 30, 31, 32]; the network
becomes adaptive. In this framework, we consider here the coevolution of an adaptive
network of interacting agents, and investigate how the final state of the system depends
on this coevolution. We focus on the Deffuant model in which opinions are continuous
variables and neighboring agents which have close enough opinions (as determined by
a tolerance parameter) can reach a local consensus. In our model, the rate of evolution
of the network’s topology is tunable and represents one of the parameters. We focus on
simple evolution rules of the topology, that do not require prior knowledge of the state of
agents to which new links are established. We study the role of the various parameters
such as the tolerance of agents and the rate of topology evolution. As already shown
in a previous investigation [32], the possibility of the interaction network to adapt to
the changes in the opinion of the agents has important consequences on the evolution
mechanisms and on the structure of the system’s final state. We focus here on the one
hand on the convergence time needed to reach this final state, and on the other hand on
the detailed characterization of this state, in which various groups of agents are formed,
each group with a distinct uniform opinion.
The paper is organized as follows. We define precisely the model in section 2. We
briefly recall in section 3 the results obtained in [32] concerning the comparison of the
Deffuant model on static and adaptive networks, and in particular how the rewiring
affects the transitions between consensus and polarized states. We focus in section 4 on
the issue of convergence time, i.e. on how the parameters determine the time to reach
a global or partial consensus. Finally, we consider in section 5 the detailed structure of

Consensus formation on adaptive networks 3
the final state of the agents. We investigate in particular how the size of the various
connected components, or groups of agents, is linked with their average number of
connections.
2. The model
We consider the Deffuant model for the evolution of opinions, in which N agents are
endowed with a continuous opinion o ∈ [0 : 1] [6, 7, 8, 9, 10]. Starting from random
values, the agents’ opinions evolve through binary interactions according to the following
rules: at each timestep t, two neighbouring agents are chosen at random. If their
opinions are close enough, i.e., if |o(i, t) − o(j, t)| ≤ d, where d defines the tolerance
range or threshold, they can communicate, and the interaction tends to bring them
closer, according to the rule
o(i, t + 1) = o(i, t) + µ(o(j, t)− o(i, t))
o(j, t + 1) = o(j, t) − µ(o(j, t)− o(i, t)) (1)
where µ ∈ [0, 1/2] is a convergence parameter. For the sake of simplicity and to avoid
a too large number of parameters, we will consider the case of µ = 1/2: i and j adopt
the same intermediate opinion after communication [7]. The tolerance parameter plays
a crucial role in the ability of the population of agents to reach a global consensus
or not. It is indeed intuitively clear that, for large tolerance values, agents can easily
communicate and converge to a global consensus. On the contrary, small values of d
naturally lead to the final coexistence of several remaining opinions [6, 7].
For large populations, it may be more realistic to consider that the interactions
between agents define a network with a finite average connectivity: each agent has
a limited number of neighbours and can not a priori communicate with all the other
agents. A typical example of such an interaction network structure is given by an
uncorrelated random graph in which agents have k¯ acquaintances on average, i.e. the
initial network corresponds to an Erdo˝s-Re´nyi network with average degree k¯. While
such a topology lacks many interesting features displayed by real social networks, such
as degree heterogeneity or community structures, it is nevertheless interesting to first
consider such a simple case as a reference frame. Moreover, we focus in this work on
another important aspect of real networks: the fact that their topology may evolve on
the same timescale as the agents’ opinions. Agents can indeed break a connection or
establish new ones, depending on the success of the corresponding relationship. The
rules defining the evolution of the network topology can be modeled in many different
ways. A possibility is to consider that links decay at a constant rate, independently from
the agents’ opinions [27]. In the case of opinion dynamics, we consider instead that only
neighbouring agents with far apart opinions (i.e., |o(i, t) − o(j, t)| > d) may terminate
their relationship [32]. In order to keep the average number of interactions constant,
a new link is then introduced between one of the agents having lost a connection and
another agent, chosen at random. The new link may of course break again if the newly

Consensus formation on adaptive networks 4
connected agents have too-far-apart opinions. The rewiring process thus occurs as a
random search for agents with close-enough opinions.
Even a simple rewiring process such as the one depicted above leads to the
introduction of two new parameters. The first one, w, quantifies the relative frequencies
of the two following processes: a local opinion convergence for agents whose opinions
are within the tolerance range, and a rewiring process for agents whose opinions differ
more. At each time step t, a node i and one of its neighbors j are chosen at random.
With probability w, an attempt to break the connection between i and j is made: if
|o(i, t)−o(j, t)| > d, the link (i, j) is removed and a new link is created. With probability
1 − w on the other hand, the opinions evolve according to (1) if they are within the
tolerance range. The second parameter concerns the creation of a new link whenever
a link (i, j) has been removed: a new node k is then chosen at random, and with
probability p ∈ [0 : 1] a link (i, k) is created, while with probability 1 − p the new link
instead connects j and k. Since j is chosen as a neighbour of a randomly chosen node
i, it will have on average a larger degree. Larger p therefore favors the removal of links
from larger degree nodes, while small p means that large degree nodes preferentially
keep their links. We will see in section 5 how the parameter p affects the final structure
of the agents groups.
Both for static and adaptive networks, the system is initialized in a state where the
agents have a random opinion between 0 and 1 and the network configuration is Erdo˝s-
Re´nyi. The averages were generated over 10 to 100 different networks, as specified in
the captions of the figures. The dynamics stops when no update is anymore possible. If
w > 0, this corresponds to a state in which no link connects nodes with different opinions.
This can correspond either to a single connected network in which all agents share the
same opinion, or to several disconnected clusters representing different opinions. For
w = 0 on the other hand, the final state is reached when neighboring agents either share
the same opinion or differ of more than the tolerance d.
3. Static versus adaptive
In this section, we focus on the case p = 0 and compare the results of the opinions
evolution on a static and a dynamically adaptive network. Figure 1 displays for both
cases the size of the largest (〈Smax〉/N) and second largest (〈S2〉/N)) opinion clusters in
the final state, where a cluster is defined as a connected group of agents sharing the same
opinion. In both cases, at large tolerance d, a global consensus is achieved, with a single
cluster containing all agents. A jump of 〈Smax〉/N from a value close to 1 to a value
close to 1/2 is observed at a critical value dc1(w). Interestingly, and as hypothesized
in [32], the structure of the largest cluster changes as d decreases towards dc1(w): the
average shortest path between two nodes increases as d → dc1(w) (data not shown),
because the cluster typically acquires a community structure where only a few nodes
or links keep the communities connected. Moreover, the critical value dc1(w) increases
with w (see Fig. 1). If the rewiring is more frequent, it allows more easily to break the

Consensus formation on adaptive networks 6
by the number of opinion clusters in the final state, Nclusters, shown in Fig. 1. For
dc2 ≤ d ≤ dc1, an extensive number of clusters is indeed obtained in the static case,
saturating at O(N) at dc2. The system presents therefore a “false”-polarized state,
with a coexistence of macroscopic opinion clusters with an extensive number of finite
size clusters. As d decreases, more and more macroscopic clusters appear, as in mean-
field [7], but there is also an extensive proliferation of finite size “microscopic” clusters.
For adaptive networks, the number of clusters is much smaller, and decreases as w
increases. In fact, the precise investigation of the cluster size distribution reveals that
the density of nodes in non-extensive clusters vanishes in the thermodynamic limit [32].
The polarized phase on adaptive networks differs therefore strongly from the one on
static networks: thanks to the possibility of link rewiring, agents who would remain
isolated (or in very small groups) on a static network may manage to find agents with
whom to communicate and thus enter a macroscopic cluster. Without rewiring on the
other hand, a macroscopic number of agents remain in fragmented components which
coexist with few macroscopic clusters.
4. Convergence time
On static networks, the time to converge to the final state of the system, tconv, is
determined by the topological properties of the opinion clusters. In turn, the behavior
of this topology is mostly determined by the distance of the tolerance of the agents
from that of the polarized-to-fragmented transition, dc2 (see Fig. 2): For dc2 < d, tconv
grows linearly with the system size and increases as d decreases: the clusters formed by
agents who can communicate become more and more tree-like, which slows down the
convergence to a common opinion. As d → dc2, tconv diverges as a signature of the phase
transition. For d < dc2, the clusters of agents with close enough opinions become small
and the convergence time decreases as d decreases.
On adaptive networks, two scenarios are possible: on the one hand, if the network
evolution is slow compared to the timescale of opinion formation, tconv is mostly
determined by the characteristic time of topological cluster formation, tl ‡. The scaling
of tl and therefore tconv can be estimated by considering a typical opinion cluster: its
size is proportional to the tolerance range of the agents, 2d; the number of its links
which need to be rewired is proportional to the total number of links (∝ k¯N), and to
the amount of opinions outside of the tolerance range (∝ (1 − 2d)); the probability to
rewire a link towards an agent with a close enough opinion is moreover ∝ d and the
time between two link updates is ∝ 1/w so that
tl ∝ k¯(1− 2d)/(wd). (2)
Figure 2B) shows the rescaled convergence time wtconv/k¯ as a function of d for different
parameter values, in good agreement with Eq. (2). If the network evolution is fast, on
‡ It is important to note that, even in this limit, the network topology cannot be considered static and
adaptivity plays an important role in the early time evolution of the system too, as shown in [32].

Consensus formation on adaptive networks 7
0 0.1 0.2 0.3 0.4 0.5
d
101
102
103
104
105
t c
o
n
v

/ N
N=500
N=1000
N=2000
N=5000
d
c2
A)
101
102
t c
o
n
v

w
/(k

N
) N=1000, k=10, w=0.1N=2000, k=10, w=0.1
N=5000, k=10, w=0.1
N=5000, k=5, w=0.1
N=5000, k=10, w=0.3
N=5000, k=10, w=0.5
0.01 0.1
d
102
103
t c
o
n
v
(1-
w)
/N N=1000, w=0.9N=2000, w=0.9
t c
o
n
v
(1-
w)
/N N=5000, w=0.9N=5000, w=0.7
B)
C)
Figure 2. (Color online) A) convergence time, measured in the number of simulation
steps, on static networks for different system sizes as a function of the tolerance, d,
when k¯ = 10 (The data points were generated averaging over 10 to 100 realizations). B)
rescaled convergence time on adaptive networks for small rewiring rates as a function of
d for different system sizes, average degrees, and rewiring rates (Data points averaged
over 30 realizations.). The dashed line is proportional to 1/d. C) Rescaled convergence
time on adaptive networks as a function of d for different system sizes and rewiring
rates (k¯ = 10) averaged over 100 realizations.
the other hand, it is possible for an agent to rewire most links towards agents with close
opinions in a short time. This scenario takes place when tl is less than or comparable to
1/(1−w), the average time between two fruitful discussions. In this case, the convergence
time is expected to scale simply as 1/(1− w), as indeed shown in Fig. 2C).
5. Group structure
Once the population of agents has reached its final state, an interesting question concerns
the structural differences between the various opinion groups that have been formed.
The most basic property one can investigate is the average degree of an agent. It turns
out that the average degree of an agent inside a group is strongly correlated with the
size of the group. On static networks, the average degree of a cluster is a linear function
of its size (left plot of Fig. 3), which can be explained as follows: for a cluster of size S,
the probability for a node to have a link pointing towards this cluster is simply S/N ,
and a node of degree k will have on average kS/N links pointing towards other nodes in
the cluster. The average “in-degree” of the nodes in a cluster of size S is therefore Sk¯/N
(assuming that there is no correlation between the degree of a node in the network and
the cluster to which it finally belongs) §.
On adaptive networks on the other hand, the linear relationship is no longer valid,
as shown in Fig. 3. At small tolerance values, many clusters are obtained, with very
§ Moreover, the average degree cannot be less than two, except for starlike clusters where the average
degree can be between 1 and 2, which explains the flattening of the curves at small cluster sizes.

Consensus formation on adaptive networks 8
different sizes. A power-law-like relationship appears between the clusters’ size and
degree. The behaviour depends on p, the parameter of the rewiring rule: a sublinear
relationship holds for small p values while a superlinear behaviour appears for p close to
1. Since these cases correspond to the situations when the development of the opinion
clusters takes place at a much shorter timescale than that of the topological clusters
[32], an analytical treatment of the problem is possible, investigating the diffusion of
the links between the clusters of constant opinions at a mean-field level [33]. For large
tolerance values, the cluster’s average degree becomes less correlated with its size (right
graph of Fig. 3): this is due to the fact that the clusters correspond to large fractions
of the original network and their average degree saturates at k¯.
1 10 100 1000 10000
S
1
10
100
k(S
)
d=0.01,0.05,0.1
k=300
k=100
k=50
1 10 100 1000
S
1
10
100
k(S
) 10 100 1000
1
10
p=0
p=0.7
p=1
d=0.01, 0.05, 0.1
k=30
k=20
k=10
k=5
Figure 3. (Color online) Left figure: average degree of the agents sharing the same
opinion as a function of the size of the uni-opinion groups, S, on static networks.
The dashed line corresponds to a linear behaviour. Right figure: same in the case of
adaptive networks when w = 0.5 and p = 0 for different average degrees and tolerance
values. The dashed line shows a power-law S0.6. Inset: the binned average of the
same measurement for k¯ = 20, d = 0.01, w = 0.5, and different values of p. The lines
correspond to the power laws S0.6, S, and S1.3. In both figures, N = 104 and data
points were collected from 10 realizations for each set of parameter values.
6. Conclusions
In this work we have investigated how adaptive network topology can influence the final
state and structure of the Deffuant model compared to its behavior on static Erdo˝s-
Re´nyi networks. While on static networks the model exhibits three different phases, the
fragmented phase, present at very small tolerance values and characterised by the lack
of extensive-size clusters, disappears on adaptive networks since rewiring allows small
groups to connect to each other and grow to macroscopic sizes for all tolerance values.
The consensus-to-fragmented transition is shifted to larger tolerance values on adaptive
networks since rewiring promotes the division of agents with too different opinions. We
have found that the convergence time of the system on static networks is determined
by the distance of the tolerance of the agents from that at the polarized-to-fragmented

Consensus formation on adaptive networks 9
transition, dc2. On adaptive networks instead, the convergence time is set either by the
time it takes to successfully rewire the links between disagreeing agents or simply by
the frequency of opinion updates, depending on which of these timescales is larger than
the other. Finally, probing the local structure of the groups by measuring the average
degree of them reveals nontrivial correlations between the size of a group and its average
degree: The average degree is either a sub or superlinear function of the size, determined
by the parameter of the rewiring rule, p. Adaptive dynamics moreover display robust
features with respect to small changes in the opinion-update rule contrarily to what
happens on static networks, on which for example the consensus-to-polarized transition
disappears when the tolerance threshold is not sharp anymore [32].
Acknowledgments
The authors are partially supported by the EU under contract 001907 (DELIS).
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