Content uploaded by Cristian Huepe
Author content
All content in this area was uploaded by Cristian Huepe
Content may be subject to copyright.
arXiv:1110.1336v1 [nlin.AO] 6 Oct 2011
Early fragmentation in the adaptive voter model on directed networks
Gerd Zschaler,1, ∗Gesa A. B¨ohme,1Michael Seißinger,1Cristi´an Huepe,2and Thilo Gross3
1Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, 01187 Dresden, Germany
2614 N. Paulina Street, Chicago IL 60622-6062, USA
3University of Bristol, Department of Engineering Mathematics,
Merchant Venturers Building, Woodland Road, Clifton, Bristol BS81TR, UK
We consider voter dynamics on a directed adaptive network with fixed out-degree distribution. A transition
between an active phase and a fragmented phase is observed. This transition is similar to the undirected case
if the networks are sufficiently dense and have a narrow out-degree distribution. However, if a significant
number of nodes with low out-degree are present, then fragmentation can occur even far below the estimated
critical point due to the formation of self-stabilizing structures that nucleate fragmentation. This process may
be relevant for fragmentation in current political opinion formation processes.
I. INTRODUCTION
The defining feature of complex systems is the emergence
of collective phenomena from the interaction of many parts
[1]. A vivid example is provided by social (swarm) intelli-
gence [2]. Crowds of humans, shoals of fish, and even swarms
of insects are known to solve problems more efficiently than
any individual on its own [3,4]. However, only in humans
social intelligence is also used on a higher level. Among all
species, only we have evolved the ability to discuss (and de-
bate) future problems and opportunities, and formulate long-
term policy.
Recent media reports have pointed out that a central func-
tion of the political opinion formation process, the debunking
of counter-factual opinions, may be starting to fail (see [5] for
an example from U.S. politics).
In the political discourse our opinions are exposed to close
scrutiny and criticism. Well-founded criticism may cause us
to change counter-factual beliefs and thus promote rational
decision making. However, because of the stresses involved,
humans tend to favor discussing with others who share similar
beliefs.
With the increasing diversity of offline and online media
[6,7] and new media technologies [8], it is becoming easier
to avoid opposing opinions altogether. In particular, the In-
ternet enables people not only to access but also to publish
information easily. One of the best examples is perhaps the
micro-blogging service Twitter, which currently is approach-
ing a billion user posts per week. Among this flood of in-
formation, it is easy to find sources supporting almost every
conceivable opinion, while avoiding contradicting evidence.
It thus seems likely that situations develop where a given
subset of the society (and the media by which it is represented)
only pay attention to information sources with the same be-
lief system, thus reinforcing and perpetuating myths that are
never confronted with opposing views. In this light, one may
ask whether we are heading for a society that is fractionated
into groups adhering to internally consistent but mutually ex-
clusive belief systems.
∗zschaler@pks.mpg.de
The question when fragmentation in opinion formation pro-
cesses can occur on a social scale is beyond the scope of clas-
sical social research, because it focuses on an emergent phe-
nomenon that may require new conceptual approaches from
physics. A minimal yet paradigmatic physical model of opin-
ion formation is the voter model [9,10]. It describes a network
of nodes representing agents, and links representing the social
contacts among them. The agents hold one of two possible
opinions, which they can change by adopting the opinion of
their topological neighbors. Due to its similarity to interact-
ing spin models, the voter model has attracted considerable
attention in the physics literature and has been studied using
different interaction geometries, such as regular lattices or het-
erogeneous networks [11–14].
An important extension of the voter model is achieved by
including homophily, the agents’ propensity to discard links to
opposing neighbors and establish new links to agents holding
the same opinion [15]. Thus, the interaction network is not
static but co-evolves with the agent dynamics, as the agents
rewire their links depending on their opinions. Such networks,
in which the node and link dynamics co-evolve, are called
adaptive networks [16–18].
The long-term dynamics of the adaptive voter model can
reach one of two absorbing states: A consensusstate, in which
all agents hold the same opinion, or a fragmented state, in
which the network breaks into at least two components, which
are internally in consensus. The adaptive voter model there-
fore provides a simple framework in which the fragmentation
of opinion formation processes can be studied [19–23]. In
this model, fragmentation occurs in a phase transition that
was identified as a generic absorbing transition [20]. It can
be computed analytically using a recently proposed motif ex-
pansion approach [24].
The adaptive voter model has been studied so far on undi-
rected networks. In the context of opinion formation, how-
ever, the underlying interactions are often asymmetric. It is
therefore reasonable to encode “who pays attention to whom”
as directed links in the interaction network. Directed links
were considered in previous studies on static networks [25–
29], but directed adaptive networks have only been investi-
gated in a generic threshold model for boolean networks [30].
In this paper, we investigate voter dynamics in a directed
adaptive network, in which both the opinion dynamics and
2
the topological change are influenced by the directionality of
the interactions. The agents can avoid disagreeing with neigh-
bors by rewiring their outgoing links, whereas they cannot af-
fect their incoming links. The overall network topology thus
changes adaptively, while the out-degree of each agent, i.e.,
its number of outgoing links remains unchanged. This enables
us to study the influence of different realistic out-degree distri-
butions. For sufficiently dense Poissonian out-degree distribu-
tions, fragmentation occurs if the rewiring of network connec-
tions exceeds a critical rate, which is consistent with previous
results on undirected networks [19–23]. However, for scale-
free out-degree distributions and Poissonian distributions with
small mean degree, we find that fragmentation can already be
observed at much lower rewiring rates than in undirected net-
works. We show that this behavior is due to the nodes of low
out-degree, which can form self-stabilizing topological struc-
tures that nucleate fragmentation.
II. MODEL
We consider a network of Nnodes representing agents and
Kdirected links representing social contacts. Each node i
holds a binary opinion σi∈ {A,B}. The direction of links
indicates the flow of attention between the agents. In other
words, in our notation we draw links in the direction that one
would draw the “follows”-links on Twitter.
We initialize the network as a random directed graph with
randomly assigned equiprobable strategies. The node states
and the network topology are then left to evolve according to
the following rules: In sequential updates, a link i→jis picked
at random from the network [31]. If σi=σj, the link is said
to be inert and nothing happens. Otherwise, the link is said to
be active, and it is either rewired (probability p), or an opinion
update takes place (probability 1−p). In the former case, the
node icuts the link and reconnects to a random node kwith
σk=σi. In the latter case, node iswitches its opinion σito
σj. We note that the rewiring of links only changes the in-
degree distribution, whereas the out-degree distribution and
the average degree hki=K/N of the network remain fixed.
In contrast to previous studies of the voter model on static
directed networks, we do not need to restrict our model to
networks consisting of a single strongly connected compo-
nent [25,28,32], because the network’s component structure
is affected by the ongoing rewiring of links, which contin-
uously forms and re-routes paths between different strongly
connected components.
Below, we study the proposed model in terms of the density
nof A-nodes (corresponding to agents holding opinion A) and
the per-capita densities fand hof active links, f≡[A→B]
and h≡[B→A]. Following [20,21] we characterize the state
of the network by the magnetization m= 1 −2nand the
active link density ρ= (f+h)/hki.
In network simulations of the directed adaptive voter model
one observes qualitatively different types of trajectories: First
at sufficiently high rewiring rates the network rapidly ap-
proaches a fragmented state (|m|>0,ρ= 0), in which the
network breaks into at least two components, which are inter-
-1 -0.5 00.5 1
m
0
0.25
0.5
ρ
11500 12500
t
0
0.04
ρ
FIG. 1. Typical trajectories from network simulations. The state
of the network is characterized by the density of active links ρand
the magnetization m. The trajectories shown correspond to networks
with Poissonian out-degree distributions with hki= 4 (red), hki= 8
(black), and an out-degree distribution following pk∝k−2(blue).
The trajectories initially drift along a parabola of active states (dotted
lines in matching colors, analytical results from Eq. (10)). However,
only the black trajectory reaches a consensus state, whereas the oth-
ers eventually collapse to a fragmented state. The inset shows a time
series of ρfrom the scale-free network shortly before fragmentation.
N= 104,p= 0.1.
nally in consensus. Second, for sufficiently low rewiring rates
the network first approaches a stationary active state (|m| ≪
1,ρ > 0), in which the opinions and the topology change con-
tinually. Because such active states form a parabola in the ρ-
m-plane, the system can drift randomly along the parabola un-
til an absorbing consensus state (m=±1,ρ= 0) is reached.
These dynamics are closely reminiscent of the adaptive voter
model on undirected networks [20,22].
In addition to the trajectories described above, the directed
model can show a third type of behavior not observed in the
undirected case. Here, the systems drifts along the parabola
of active states for some time and thencollapses slowly to the
fragmented state (Fig. 1). This can lead to fragmentation sig-
nificantly below the critical rewiring rate found in undirected
networks. The delayed fragmentation after the drift along the
parabola of active states suggests that in the active state the
network undergoes some slow reorganization that eventually
leads to the destabilization of the active states. In the fol-
lowing we investigate the nature of this reorganization and its
implications for network fragmentation.
III. ANALYTICAL APPROACH
The main goal of this paper is exploring the impact of di-
rectionality of attention on the opinion formation process. For
this purpose we compare the dynamics of the directed adap-
tive voter model to well known results for the undirected adap-
tive voter model. In the following we refer to these two mod-
els simply as the directed model and the undirected model,
3
respectively. A direct comparison of different models is of-
ten difficult and may lead to misleading results. We therefore
compare simulation results of the directed model to two ana-
lytical approximations that are known to capture the dynamics
of the undirected model in different limits. In this compar-
ison an agreement between analytical and numerical results
indicates that the assumptions made for the undirected model
are still valid in the directed model, whereas a disagreement
points to new physics in the directed model that is not ob-
served in the undirected model.
A. Moment expansion
The undirected model was studied extensively by moment
expansions [20,22,33]. Following previous works, we de-
rive differential equations for the time evolution of so-called
network moments, namely the densities n,f, and hdefined
above. The change in the density of A-nodes, n, is given by
the balance between opinion adoption in B→A- and A→B-
links,
˙n= (1 −p) (h−f).(1)
The density of A→B-links, f, changes according to
˙
f=−pf + (1 −p)[A→A→B] −2[B←A→B]
+ [B←B→A] −[A→B→A] −f,(2)
where [X→Y→Z]and [X←Y→Z]denote the per-capita densi-
ties of triplets in the network (X, Y , Z ∈ {A,B}). In Eq. (2),
the first term corresponds the gain in fdue to rewiring,
whereas the remaining terms correspond to gains and losses
due to opinion adoption. In an opinion adoption event, a
node copies a neighbor’s opinion via one of its outgoing ac-
tive links, transforming it into an inert link. This also affects
all other links connected to the focal node, transforming ac-
tive links into inert ones and vice versa. The resulting indirect
change in the density of active links is accounted for by the
triplet variables.
The time evolution of the density of B→A-links, h, is de-
termined by the analogous equation
˙
h=−ph + (1 −p)[B→B→A] −2[A←B→A]
+ [A←A→B] −[B→A→B] −h.(3)
Equations (1)–(3) do not constitute a closed ODE sys-
tem, as they involve the triplet moments [X→Y→Z]and
[X←Y→Z]. In principle, the equation system could be com-
plemented by similar equations for the triplet moments. These
would, however, depend on higher moments, such as four-
node motifs. An appropriate truncation of the expansion is
thus necessary in order to obtain a closed system of equations.
This approximation is referred to as moment closure [33,34].
In the following, we express the triplet densities in terms of
node and link densities using the pair approximation [33–36],
[X→Y→Z]≈[X→Y]hki[Y→Z]
hki[Y],(4)
µXZ [X←Y→Z]≈[Y→X]hqoi[Y→Z]
hki[Y],(5)
where µXZ = 1 + δXZ accounts for the double-counting of
symmetric triplets. In the equations, the numbers of triplets
are approximated as the number of X→Y- or Y→X-links
times the average number of Y→Z-links connected to a Y-
node. Here, we assume that the probability of finding an
outgoing Z-neighbor of a Y-node is independent of the pres-
ence of an X-neighbor of the Ynode. Thus the probability
of finding a given triplet depends on the global link density
[Y→Z]/hki[Y]. In (4), each of the hkioutgoing links of the
Y-node is a Y→Z-link with this probability. In (5), on the
other hand, the Y-node has already been selected by follow-
ing one of its outgoing links. In this case, each of its remaining
hqoioutgoing links is a Y→Zlink with this probability. The
quantity hqoiis the mean excess out-degree in the network,
which can be computed from the out-degree distribution [37].
As a further simplification we assume that the mean degree
of both node types is equal, so that [A→A] = hkin−fand
[B→B] = hki(1 −n)−h[20]. We then obtain a closed set of
ODEs,
˙n= (1 −p) (h−f),(6)
˙
f=−pf + (1 −p)(hkin−f)f
n−f
−fh
1−n+κ(hki(1 −n)−h)h
1−n−κf2
n,(7)
˙
h=−ph + (1 −p)(hki(1 −n)−h)h
1−n−h
−fh
n+κ(hkin−f)f
n−κh2
1−n,(8)
where κ=hqoi/hki.
This system of differential equations has a trivial solution,
h∗=f∗= 0,(9)
corresponding to the absorbing states, in which no active links
are left. These states can be either fragmented states (0<
n∗<1) or consensus states (n∗= 0 or n∗= 1).
Additionally, there is a continuum of non-absorbing, active
stationary states,
h∗=f∗=n∗(1 −n∗)hki − 1
(1 + κ)(1 −p),(10)
in which neither consensus nor fragmentation is achieved.
These states form the parabola in the ρ-m-plane discussed
above (Fig. 1).
4
(min,mout,lin,lout)
lin
lin+loutp
lin
lin+lout(1−p)
lout
lin+loutp
lout
lin+lout(1−p)
(min,mout,lin−1,lout)
(min+1,mout,lin−1,lout)
+Pin(gin)Pout(gout+1)
(0,1,gin,gout)
(min,mout+1,lin,lout−1)
(lin,lout,min,mout)
FIG. 2. Transitions of a general active motif (min, mout, lin , lout ).
The transition probabilities follow from the link update rule of the
directed adaptive voter model. New active motifs are only created
when an opinion update occurs on an incoming active link (second
transition). In this case the number of incoming and outgoing active
links (dashed) of the new motif is estimated based on the in-degree
distribution Pin and out-degree distribution Pout of the underlying
network.
In the undirected model the moment expansion is known
to yield good results when pis far from the fragmentation
point, but to become less precise for pclose to fragmentation.
To estimate the fragmentation point we therefore resort to a
motif expansion described below. It is nevertheless instructive
to consider the critical rewiring rate
˜pc= 1 −[hki(κ+ 1)]−1,(11)
which is computed by a linear stability analysis of the ac-
tive states. For p < ˜pcthe active states are stable, whereas
the absorbing states are stable for p > ˜pc. This estimate of
the critical point closely resembles the analogous result for
the adaptive voter model using link update on undirected net-
works [36]. The main difference is that in the undirected case,
the parameter κis unknown and therefore usually set to unity.
This approximation is well justified, because in the undirected
model, the ongoing rewiring leads to an approximately Pois-
sonian degree distribution for which κ= 1 [37]. By con-
trast, in the directed model, the outgoing degree distribution
remains fixed, and κhas to be considered explicitly. In the
results presented here, we use values of κthat are explicitly
measured in realizations of the respective out-degree distribu-
tions.
B. Motif expansion
In the undirected model a precise estimate of the transition
point can be obtained by a motif expansion proposed in [24].
In this expansion we consider an almost fragmented network,
consisting of two almost isolated components which are inter-
nally in consensus, but are still connected by a low density of
“active motifs”. In the directed network the active motifs are
characterized by their numbers of inert incoming and outgo-
ing links (min , mout) and active incoming and outgoing links
(lin, lout ). Following [24], we derive a set of balance equa-
tions capturing the effect of all possible update processes on
the densities of active motifs,
˙ρ(min, mout, lin , lout) = −ρ(min, mout , lin, lout)
+lin + 1
lin + 1 + lout
pρ(min, mout , lin + 1, lout)
+lin + 1
lin + 1 + lout
(1 −p)ρ(min −1, mout, lin + 1, lout )
+lout + 1
lin +lout + 1pρ(min, mout −1, lin , lout + 1)
+mout
min +mout
(1 −p)ρ(lin, lout , min, mout)(12)
for min >0and mout >1, and
˙ρ(0,1, lin, lout) = −ρ(0,1, lin , lout)
+ (1 −p)Pin(lin )Pout(lout + 1)
×Xnin
nin +nout
(min, mout , nin, nout)
for min = 0, mout = 1. The summation runs over all active
motifs (min, mout , nin, nout)up to a maximum in- and out-
degree, i.e. over all possible 4-tuples with min +nin ≤ˆ
kin
and mout +nout ≤ˆ
kout, where ˆ
kin and ˆ
kout denote the cut-
offs. Note that the dimension of the transition matrix grows
with the cut-off faster than ˆ
k3, which has a significant effect
on the computation time.
A schematic representation of the transition probabilities is
shown in Fig. 2. We account for heterogeneous in- and out-
degree distributions (Pin, Pout ), but assume that the in- and
out-degree of a node are uncorrelated. In the balance equa-
tions, the fragmented state is obtained as the stationary solu-
tion containing zero active motifs. The critical rewiring rate pc
is then extracted from the linear stability analysis of this state
as the rewiring rate at which the fragmented state becomes
stable.
IV. NUMERICAL EXPLORATION OF EARLY
FRAGMENTATION
In the following we compare the estimated fragmentation
points, obtained from the approximations above, with results
from agent-based simulation of the networks. We first con-
sider the case of a network with Poissonian out-degree distri-
bution with hki= 8. As a second example we study a network
with a scale-free out-degree distribution, in which the frag-
mentation occurs much earlier. We conjecture that this early
fragmentation occurs due to the presence of a large number of
nodes with low out-degree, which is then verified in a network
with Poissonian degree distribution with hki= 4.
5
0
0.5
1
|m|
N = 103
N = 104
00.2 0.4 0.6 0.8 1
p
0
0.5
1
r
FIG. 3. Fragmentation of a network with Poissonian out-degree dis-
tribution and hki= 8. Shown is the absolute value of the magnetiza-
tion in the final frozen state (top) and the proportion rof simulation
runs that reach a fragmented state before tmax = 80000 (bottom) as
a function of the rewiring rate p. Each point is an average over 100
simulations. The critical point computed by the moment closure ap-
proximation (dotted) overestimates the critical rewiring rate, whereas
the motif expansion yields a better estimate of the true fragmentation
point. For the motif-expansion a cut-off of ˆ
kin =ˆ
kout = 10 was
used. For a higher cut-off the estimated critical rewiring rate is ex-
pected to shift to slightly higher values.
A. Poissonian out-degree distribution with hki= 8
We first consider networks with a Poissonian out-degree
distribution, because this distribution closely matches the dis-
tribution observed in the undirected model [20]. Starting from
a random graph with both in- and out-degrees drawn from a
Poisson distribution with mean hki, we simulate the full net-
work dynamics for systems of up to N= 104nodes until a
frozen state is reached or a maximum simulation time tmax is
exceeded. Time is measured in units of 1/K, so that Kupdate
events take place in one simulated time unit.
The results in Fig. 3show a relatively sharp fragmentation
transition at a critical rewiring rate pc≈0.79. For p < pc,
the network reaches a state of global consensus, in which
all nodes have the same state (|m|= 1). By contrast, for
p > pc, it separates into two disconnected components of ap-
proximately the same size, which hold opposing opinions but
are internally in consensus. These results are strongly remi-
niscent of the undirected model [20,22,24].
The analogy between the directed and undirected model
extends also to the analytical results. As in the undirected
model the moment expansion overestimates the transition
point, whereas the motif expansion yields a relatively precise
estimate.
The study of the directed model in Poissonian networks
with hki= 8 highlights the similarities between the directed
and undirected networks and provides a basic test for our an-
alytical approaches. For these networks the directed model
exhibits the same dynamics as the undirected model and the
0
0.5
1
|m|
00.2 0.4 0.6 0.8 1
p
0
0.5
1
r
100102
k
0.001
0.1
pk
FIG. 4. Early fragmentation in scale-free networks. The plots are
analogous to Fig. 3, but describe networks with the out-degree dis-
tribution pk∝k−2(inset). Fragmentation occurs far below the es-
timated transition points (dashed, dotted) and extends over a wider
range. N= 104,hki= 5.5665,ˆ
kin =ˆ
kout = 10.
analytical approaches capture the dynamics with similar pre-
cision as in the undirected case.
B. Scale-free out-degree distribution
We now ask how the model behaves for more realistic out-
degree distributions which cannot be realized in the previ-
ously studied undirected model. In the following, we consider
power-law distributions of the form pk∝k−2, which capture
the diversity that is observed in a wide variety of social ap-
plications [38]. For generating networks with power-law dis-
tributed out-degrees and Poissonian in-degrees, we first draw
an out-degree sequence of length Nfrom a power-law dis-
tribution. For each out-degree kiin this sequence, we then
connect the outgoing links of node ito kirandom nodes in
the network. We explicitly avoid creating nodes without out-
going links as these would never change their state and act as
“zealots”, trivially preventing the possibility of global consen-
sus [39].
The results in Fig. 4show that in scale-free networks the
fragmentation occurs much earlier than in the Poissonian case.
Moreover, the proportion rof networks reaching fragmen-
tation now increases gradually with increasing p. Consider-
ing individual simulation runs in detail one finds that the net-
works remain for some time in an active state before slowly
approaching fragmentation – a behavior not observed in the
undirected model or in the directed networks considered in
the previous section.
The observation that the networks spend some time in
the active state before fragmenting indicates that these states
are still feasible at least in the beginning of the simulation
runs. The mechanism by which fragmentation is reached must
therefore differ from the mechanism observed in the cases
6
FIG. 5. Almost fragmented network of N= 100 nodes with out-
degree distribution pk∝k−2. The two components are only con-
nected by the hub node with the largest out-degree and a single link
from the second-to-largest hub (black line). A self-stabilizing trian-
gle of nodes of out-degree one and a subsequently recruited stable
“chain” of nodes of out-degree one are marked in blue. Note that the
nodes with high out-degree have very low or zero in-degree.
studied so far, where fragmentation occurs due to the desta-
bilization of the parabola of active states in a transcritical bi-
furcation.
Notably both the moment and motif expansion seem not
to capture the different mechanism for fragmentation because
they significantly overestimate the fragmentation point. The
main assumption used in both approximations is the absence
of correlations between a node’s in- and out-degree and be-
tween nearest neighbors. Their failure thus indicates the ap-
pearance of correlations that are absent or not substantial in
the networks with Poissonian out-degree distribution. In the
following we call fragmentation well below the estimated
fragmentation point early fragmentation.
Network simulations suggest that early fragmentation is ini-
tiated by the formation of self-stabilizing structures among the
agents. For understanding the process leading to such struc-
tures, consider that the networks, even far from fragmentation,
are partially ordered. In average the number of in- and outgo-
ing neighbors of an agent that share the focal agent’s opinion
will be greater than the number of neighbors that oppose the
focal agent’s opinion, because the rewiring dynamics trans-
forms active links into inert ones. This implies that if an agent
changes her opinion, she is likely to experience a subsequent
loss of incoming links because the majority of her neighbors
now oppose her opinion and rewire their links with some prob-
ability. In the long run agents that frequently change their
opinion have lower in-degree than those who change their
opinion rarely. Therefore, the attention, measured in terms
of incoming links, focuses on the agents that have a low out-
degree and thus rarely change their opinion.
Focusing the attention on agents of low out-degree impedes
the propagation of opinions across the network. In particu-
lar, it can lead to the formation of small clusters which have
few outgoing links and hence have a very high resistance to
invasion of the opposing opinion. In an extreme case small
subgraphs can form in which all nodes are in consensus and
all outgoing links starting within the subgraph lead to other
nodes in the same subgraph. Because of this lack of outgoing
links, such subgraphs can never be invaded by the opposing
opinion. Further, no outgoing links leaving the subgraph can
be formed because none of the nodes in the subgraph will ever
rewire an outgoing link.
We call subgraphs that are hard or impossible to invade self-
stabilizing structures. The initial formation of such a struc-
ture is a stochastic event that occurs with a small probabil-
ity. However, once such a structure has been formed it can
grow as other nodes rewire their outgoing links into the struc-
ture. Nodes of low out-degree can be recruited rapidly be-
cause only few rewiring events are necessary to rewire all of
their outgoing links into the self-stabilizing structure. Recruit-
ment of nodes with more outgoing links takes longer as more
rewiring events are required. In simulations, networks ob-
served shortly before fragmentation are often found to consist
of two almost disconnected clusters, which are only connected
by a few nodes of high out-degree. Because of their frequent
changes of state, these connecting hubs have very few or no
incoming links.
For illustration of the mechanism described above an em-
bedding of a small network shortly before fragmentation is
shown in Fig. 5. The network has broken into two almost
disconnected clusters. The remaining connections are formed
by a single hub and one additional link. The fragmentation
has been nucleated by the formation of a self-referential cy-
cle consisting of three nodes of out-degree one. Subsequently,
almost half of the network has been recruited into this self-
stabilizing structure.
Given the observations above we can explain the shape of
the trajectories shown in Fig. 1. Because the formation of
a self-stabilizing structure is a rare event, they are generally
not present in the initial network. The system therefore ap-
proaches the parabola of active states, which is in agreement
with results from the undirected model and the analytical ap-
proximations for the directed model. However, while the sys-
tem drifts along the parabola of active states, self-stabilizing
structures are eventually formed due to the ongoing rewiring.
As the self-stabilizing structures grow, the permissible range
for the magnetization shrinks, effectively arresting mas al-
most all nodes are recruited into the self-stabilizing structures.
Because the last nodes to join the structures are “hub” nodes
with high out-degree, a relatively high density of active links
can be maintained for some time. Because the hub nodes un-
dergo rapid opinion switches, rewiring can only slowly sepa-
rate them from opposing neighbors, which explains the slow
fragmentation. The switching and rewiring of the hub nodes
are clearly visible in the time series of the active link den-
sity ρ. In the inset in Fig. 1, this is shown for the last 103
time units before fragmentation. Here, one of the two remain-
ing connecting hubs detaches from one of the componentsand
the final hub still switches several times before eventually also
separating.
Summarizing the observations above, we conjecture that
early fragmentation is initiated by the formation of self-
stabilizing structures among nodes of low out degree. We em-
phasize that contrary to most dynamical phenomena observed
in scale-free networks, the dynamics of interest is generated
7
primarily in the nodes of low degree. Nodes of high degree
still play an important role as they are the last nodes to con-
nect the separating components and thus determine the time
of fragmentation. This mechanism is not captured by cur-
rent analytical approaches, because it relies on the build-up of
negative correlations between the in-degree and out-degree of
nodes that is neglected in previously proposed approximation
schemes.
C. Poissonian out-degree distribution with low hki
Because the mechanism postulated above relies on the for-
mation of correlations, one can perform a simple test by con-
sidering a system in which these correlations are removed by
an additional rewiring process. However, such a test is for
two reasons difficult in scale-free networks: First, because of
the constraints in scale-free topology it is well-known that it
is difficult to remove correlations in scale-free networks com-
pletely, and second, because of the presence of nodes of very
high degree, fragmentation takes a long time, making numer-
ical studies of fragmentation tedious.
Our reasoning above predicts that early fragmentation
should be observed also in directed Poissonian networks with
sufficiently low mean degree. In the present section we there-
fore consider a Poissonian network with a mean degree of
4, which avoids the difficulties encountered in scale-free net-
works. In this section we show a) that this network exhibits
early fragmentation and b) that the early fragmentation can
be avoided by an additional rewiring mechanism that destroys
the correlations implicated in the formation of self-stabilizing
structures.
Simulation results for the network described above are
shown in Fig. 6. The figure shows clear evidence of frag-
mentation well below the estimated fragmentation point. Fur-
ther, this early fragmentation is accompanied by the build-up
of negative correlations between the in- and out- degrees of
the nodes. This confirms our previous observation that atten-
tion focuses on those nodes who pay little attention to others
themselves.
To verify that the correlation described above is the cause
and not a symptom of the early fragmentation, we now con-
sider a different variant of the model. This variant is identical
to the model used so far, except that when an inert link is cho-
sen, this link is also rewired to a randomly chosen target node
that is in the same state as the source.
The model variant in which the rewiring of inert links is
switched on shows no evidence for early fragmentation (see
Fig. 6). Fragmentation occurs in a relatively sharp transition
at a critical rewiring rate pcthat is consistent with the estimate
from the motif expansion. We emphasize that the rewiring
of inert links neither introduces nor destroys active links. It
therefore has no direct impact on fragmentation. However,
rewiring inert links prevents the build-up of correlation be-
tween the in-degree and the out-degree of nodes and thereby
inhibits the formation of self-stabilizing structures. The ab-
sence of early fragmentation in a model where these corre-
lations are removed confirms the causal relationships postu-
0
0.5
1
|m|
0
0.5
1
r
00.2 0.4 0.6 0.8 1
p
-0.3
0
0.3
θio
FIG. 6. Fragmentation of networks with Poissonian out-degree distri-
bution and low hki, in which the inert links are also rewired (squares)
or are not rewired (circles). Early fragmentation is clearly visible, al-
though the shown averages over 100 simulation runs are still rather
noisy due to the highly stochastic nature of early fragmentation.
Shown are the absolute value of the magnetization (top), the pro-
portion of fragmenting simulation runs (center), and the correlation
coefficient between the in- and out-degree of the nodes in the final
state (bottom). N= 104,hki= 4,ˆ
kin =ˆ
kout = 10.
lated above. We therefore conclude that in directed adaptive
networks the slow build-up of negative correlations between
in-degree and out-degree can initiate early fragmentation by
leading to the formation of self-stabilizing structures.
V. CONCLUSIONS
In the present article, we have investigated an extension of
the voter model on adaptive networks that takes the direction-
ality of the interactions among the agents into account. We
found that our model can transition to a fractionated state for
rewiring rates that lie much below the critical value estimated
using analytical approaches known to work well in undirected
models. We discovered that fragmentation occurs due to a
novel mechanism that depends inherently on the directed na-
ture of the links. This early fragmentation occurs when agents
focus their attention on those who are steady in their opinion
because they pay attention only to few sources of information.
In this case self-stabilizing structures can form that nucleate
fragmentation.
Our results illustrate that directed networks can exhibit new
physics not observed in their undirected counterparts. Espe-
cially in the investigation of opinion formation processes, the
often directed flow of attention should therefore be taken into
account in models.
8
In the context of real-world opinion formation processes the
mechanism described here may constitute a threat but also an
opportunity. Early fragmentation maintains diversity of opin-
ions, it may thus aid the survival of counter-factual myths, but
also of legitimate and well-founded views of minorities. It is
conceivable that this mechanism may be employed in the fu-
ture to adjust the perceived degree of controversy in controlled
environments. For instance, in online discussion boards or
music recommendation systems the underlying software can
in principle control which posts are displayed to whom. It
could thus encourage rewiring of attention that either facili-
tates or inhibits early fragmentation.
Even the adaptive directed voter model paints a highly sim-
plified picture of real-world opinion formation processes and
thus must be considered as a toy model. Therefore, investiga-
tion of more realistic models is an important goal for the fu-
ture. Based on the resultsand analysis presented in the present
paper we believe that the mechanism of early fragmentation
will be observed whenever directed attention is focused pref-
erentially on agents that change their opinions at less than
average rate. We therefore expect that early fragmentation
should be robust to future refinements of the model.
A key ingredient that is missing in our present model is
novelty. Here we considered only the exchange of opinions
regarding a single well defined question, whereas in reality
many discussions are enriched by the constant inflow of new
ideas. We have shown that homophily favors connecting to
poorly informed agents and thereby promotes early fragmen-
tation, whereas curiosity would favor connecting to well in-
formed agents and thereby hinder early fragmentation. In this
light, novelty, whether in the form of true innovation or arbi-
trarily changing fashions may play an important role in pre-
venting social fragmentation.
The authors thank G. Demirel for fruitful discussions. The
work of CH was partially supported by the National Science
Foundation under Grant No. PHY-0848755.
[1] P. W. Anderson, More is different,
Science 177, 393–396 (1972).
[2] Iain D. Couzin, Collective minds, Nature 445, 715 (2007).
[3] David J. T. Sumpter, Jerome Buhl, Dora Biro, and
Iain D. Couzin, Information transfer in moving animal groups,
Theory in Biosciences 127, 177–186 (2008).
[4] Cristi´an Huepe, Gerd Zschaler, Anne-Ly Do, and Thilo
Gross, Adaptive-network models of swarm dynamics,
New J. Phys. 13, 073022 (2011).
[5] Rachel Maddow, The Rachel Maddow show, MSNBC TV show
(2010), critical comment on U.S. news reports that supported
their claim that the president’s visit to India cost $200 million
per day by citing other reports in the news and on the Internet.
http://www.msnbc.msn.com/id/3096434/#40018314.
[6] Allen P. Webb, ed., Profiting from Proliferation (McKinsey &
Co., New York, 2006).
[7] Ilias Flaounas, Marco Turchi, Omar Ali, Nick Fyson, Tijl De
Bie, Nick Mosdell, Justin Lewis, and Nello Cristianini, The
structure of the EU mediasphere, PLoS ONE 5, e14243 (2010).
[8] Elena Hensinger, Ilias Flaounas, and Nello Cristianini, Learn-
ing the preferences of news readers with SVM and lasso rank-
ing, in Artificial Intelligence Applications and Innovations,
IFIP Advances in Information and Communication Technology,
Vol. 339, edited by Harris Papadopoulos, Andreas Andreou,
and Max Bramer (Springer Boston, Berlin, Heidelberg, 2010)
Chap. 25, pp. 179–186.
[9] Richard A. Holley and Thomas M. Liggett, Ergodic theorems
for weakly interacting infinite systems and the voter model,
Ann. Probab. 3, 643–663 (1975).
[10] Thomas M. Liggett, Stochastic interacting systems: contact,
voter, and exclusion processes, Vol. 324 (Springer, Berlin Hei-
delberg New York, 1999).
[11] V. Sood and S. Redner, Voter model on heterogeneous graphs,
Phys. Rev. Lett. 94, 178701 (2005).
[12] Claudio Castellano, Effect of network topol-
ogy on the ordering dynamics of voter models,
AIP Conf. Proc. 779, 114–120 (2005).
[13] V. Sood, Tibor Antal, and S. Redner, Voter models on hetero-
geneous networks, Phys. Rev. E 77, 041121 (2008).
[14] E. Pugliese and C. Castellano, Heterogeneous pair approxima-
tion for voter models on networks, EPL 88, 58004 (2009).
[15] George C. M. A. Ehrhardt, Matteo Marsili, and Fernando Vega-
Redondo, Phenomenological models of socioeconomic network
dynamics, Phys. Rev. E 74, 036106 (2006).
[16] Thilo Gross and Bernd Blasius, Adaptive coevolutionary net-
works: a review, J. R. Soc. Interface 5, 259–271 (2008).
[17] Thilo Gross and Hiroki Sayama, eds.,
Adaptive networks: Theory, Models and Applications, Un-
derstanding Complex Systems (Springer, New York, 2009).
[18] ANWiki, The adaptive networks wiki, (2010).
[19] Petter Holme and M. E. J. Newman, Nonequilibrium phase
transition in the coevolution of networks and opinions,
Phys. Rev. E 74, 056108 (2006).
[20] Federico Vazquez, V´ıctor M. Egu´ıluz, and Maxi San
Miguel, Generic absorbing transition in coevolution dynamics,
Phys. Rev. Lett. 100, 108702 (2008).
[21] C. Nardini, B. Kozma, and A. Barrat, Who’s talking first? con-
sensus or lack thereof in coevolving opinion formation models,
Phys. Rev. Lett. 100, 158701 (2008).
[22] Daichi Kimura and Yoshinori Hayakawa, Coevolu-
tionary networks with homophily and heterophily,
Phys. Rev. E 78, 016103 (2008).
[23] B. Kozma and A. Barrat, Consensus formation on co-
evolving networks: groups’ formation and structure,
J. Phys. A 41, 224020 (2008);Consensus formation on
adaptive networks, Phys. Rev. E 77, 016102 (2008).
[24] Gesa A. B¨ohme and Thilo Gross, Analytical calcula-
tion of fragmentation transitions in adaptive networks,
Phys. Rev. E 83, 035101(R) (2011).
[25] Alejandro D. S´anchez, Juan M. L´opez, and Miguel A.
Rodr´ıguez, Nonequilibrium phase transitions in directed small-
world networks, Phys. Rev. Lett. 88, 048701 (2002).
[26] Sung M. Park and Beom J. Kim, Dynamic behaviors in directed
networks, Phys. Rev. E 74, 026114 (2006).
[27] L. Jiang, D. Hua, J. Zhu, B. Wang, and T. Zhou,
Opinion dynamics on directed small-world networks,
Eur. Phys. J. B 65, 251–255 (2008).
[28] M. ´
Angeles Serrano, Konstantin Klemm, Federico Vazquez,
V´ıctor M. Egu´ıluz, and Maxi San Miguel, Conservation
laws for voter-like models on random directed networks,
9
J. Stat. Mech. 2009, P10024 (2009).
[29] Naoki Masuda and Hisashi Ohtsuki, Evolutionary dy-
namics and fixation probabilities in directed networks,
New J. Phys. 11, 033012 (2009).
[30] Renaud Lambiotte and Juan C. Gonz´alez-Avella, On
co-evolution and the importance of initial conditions,
Physica A 390, 392–397 (2011).
[31] In this work, we use the link update scheme. We have checked
that the results reported here do not change qualitatively when
using node update, i.e., selecting a random node and then one
of its outgoing neighbors.
[32] S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin,
Giant strongly connected component of directed networks,
Phys. Rev. E 64, 025101 (2001).
[33] Thilo Gross, C. J. D. D’Lima, and Bernd Bla-
sius, Epidemic dynamics on an adaptive network,
Phys. Rev. Lett. 96, 208701 (2006).
[34] Matt J. Keeling and K. T. D. Eames, Networks and epidemic
models, J. R. Soc. Interface 2, 295 (2005).
[35] Gerd Zschaler, Arne Traulsen, and Thilo Gross, A homoclinic
route to asymptotic full cooperation in adaptive networks and
its failure, New J. Phys. 12, 093015 (2010).
[36] G¨uven Demirel, Federico Vazquez, Gesa A. B ¨ohme, and Thilo
Gross, Moment closure approximations of discrete-state adap-
tive networks, (2011), in prep.
[37] Mark E. J. Newman, The structure and function of complex net-
works, SIAM Review 45, 167–256 (2003).
[38] Albert L´aszl´o Barab´asi and R´eka Albert, Emergence of scaling
in random networks, Science 286, 509–512 (1999).
[39] M. Mobilia, A. Petersen, and S. Redner, On the role of zealotry
in the voter model, J. Stat. Mech. 2007, P08029 (2007).
