Table Of ContentFragmentation transitions in multi-state voter models
Gesa A. B¨ohme∗
Max-Planck Institute for the Physics of Complex Systems, Dresden, Germany
Thilo Gross
University of Bristol, Department of Engineering Mathematics, Bristol, UK
(Dated: January 26, 2012)
Adaptive models of opinion formation among humans can display a fragmentation transition,
whereasocialnetworkbreaksintodisconnectedcomponents. Here,weinvestigatethistransitionin
aclassofmodelswitharbitrarynumberofopinions. Incontrasttopreviousworkwedonotassume
that opinions are equidistant or arranged on a one-dimensional conceptual axis. Our investigation
2 reveals detailed analytical results on fragmentations in a three-opinion model, which are confirmed
1 by agent-based simulations. Furthermore, we show that in certain models the number of opinions
0 can be reduced without affecting the fragmentation points.
2
n
I. INTRODUCTION from their topological neighbors. The network topology,
a
i.e. the specific pattern of nodes and links, changes as
J
5 In many different fields networks have been used to individuals break up relationships with dissenting neigh-
2 describe and analyze complex systems consisting of in- bors and/or establish new relationships to those holding
teracting subunits. The applications of networks range similar opinions.
] from biological systems to technical devices and social The simplest adaptive-network model for opinion for-
h
p communities [1–4]. Accordingly, the building blocks of a mation is the adaptive voter model [13, 19–21]. In this
- network, the network nodes, can correspond to different modelandinmanyofitsvariants[14,22–24]therelative
c entities, such as genes, neurons, computers, websites or rate of change of the topology compared to the change
o
individuals. The interactions among them, the links be- of node states is controlled by a single parameter, p, the
s
. tweenthenodes, represente.g. chemicalreactions,phys- rewiring rate. Depending on this parameter, the net-
s
c ical connections, or social bonds. In the applications the work either reaches a global homogeneous state, where
i temporalevolutionofanetworkisoftengovernedbytwo all nodes hold the same opinion or a fragmented state,
s
y different types of dynamics: the dynamics on the net- consisting of two disconnected components which are in-
h work, describing the evolution of the internal degrees of ternally homogeneous. The transition separating these
p freedom,andthedynamicsof thenetwork,capturingthe two regimes is called a fragmentation transition.
[ evolution of the network topology. Whilemostofthevoter-likemodelsonlyconsiderabi-
1 Adaptive networks are characterized by an interplay nary choice of opinions, many real world situations offer
v of the dynamics on the network and the dynamics of the alargernumberofchoices. Inthephysicsliteraturesome
8 network, whereneitherofbothtypesofdynamicscanbe models for opinion formation, which consider arbitrary-
9 neglected [5, 6]. It has been shown that this interplay many opinions have been studied [14, 23]. In these mod-
1 gives rise to the emergence of complex topologies and els all opinions are “equidistant” in the sense that all in-
5
dynamics[7],spontaneousappearanceofdifferentclasses teractions between any given pair of (different) opinions
.
1 ofnodesfromaninitiallyhomogeneouspopulation[8,9], followthesamedynamicalrules. Modelsrecognizingthat
0 and robust self-organization to critical states associated the outcome of interactions may depend on a measure of
2 with phase transitions [10, 11]. The self-organization of similarity (or distance) between opinions are often con-
1
adaptive networks is believed to be of importance in the sideringanuncountablesetofopinionsandaretherefore
:
v evolution of cooperation, opinion formation processes, hard to treat analytically [15, 25]. For instance in [25]
Xi epidemicdynamics,neuralnetworks,andgeneregulation opinions are placed on a 1-dimensional axis, resembling
[12]. e.g.the political spectrum.
r
a In the adaptive networks literature, opinion dynam- Here, we consider a natural extension of the original
ics has currently attracted particular attention [13–18]. adaptive voter model, where we allow for an arbitrary
Typically,inthesemodelsasocietyisdescribedasanet- countable set of opinions. In the proposed model the
work, where nodes correspond to individuals and links rewiring rate that governs the interaction of conflicting
to social relationships. The internal state of the indi- agents is assumed to depend on the specific pairing of
vidualsindicatetheirpositionregardingsomeissue,such opinions held by the agents. The model can thus ac-
as political opinion, religious affiliation, or musical taste. countforheterogeneous“distances”betweenopinions. A
Theindividualschangetheirstatesbyadoptingopinions large distance, characterized by a high rewiring rate, in-
dicates a controversial pairing, whereas a small distance,
andcorrespondinglylowrewiringrate,indicatesthatthe
respective opinions are almost in agreement.
∗ [email protected] TheproposedmodelisdescribedindetailinSec.II.We
2
thencalculatethefragmentationdiagramofathree-state
model in Sec. III and derive the corresponding fragmen-
tation thresholds for systems with an arbitrary number p B
1
of states in Sec. IV. Finally, in Sec. V, we show that in
certain systems a reduction is possible such that the dy- A p
3
namicscanbecapturedbyasystemwithalowernumber
of opinions. p
2 C
II. MULTI-STATE VOTER MODEL
FIG. 1. State-network of a three-state voter model. The
We consider a network of N nodes, corresponding to
nodes in this network are the three opinions A, B and C.
individuals, and L links, corresponding to social con- The rewiring rates p ,p or p , encode the degree of contro-
1 2 3
tacts. Each node α holds a state sα, indicating the versy between agents holding the respective pairings of opin-
opinion held by the corresponding individual. The net- ions AB, BC or CA. The ellipse and dashed lines illustrate
work is initialized as a random graph with mean degree theexampledescribedinthetext: partialfragmentationwith
k = 2L/N. The initial node states are drawn ran- respecttoA,i.e. thefragmentationofthecorrespondingnet-
(cid:104)do(cid:105)mlyandwithequalprobabilityfromthesetofallstates works of agents in a component where all individuals hold
Γ= g ,g ,g ,...,g ,wherethetotalnumberofstates opinionAandasecondcomponentwheretheindividualshold
1 2 3 G
{ } opinions B and C. The ongoing dynamics in the latter com-
is Γ =G N.
| | (cid:28) ponent may later lead to the disappearance of either opinion
Thesystemisthenupdatedasfollows: Ineachupdate
B or C.
step a random link (α,β) is chosen. If s = s the link
α β
is said to be inert and nothing happens. If s = s ,
α β
(cid:54)
the link is said to be active and an update occurs on
the link. A given update is either a rewiring event or an Because the G-state model contains several different
opinion adoption event, decided randomly depending on types of active links (corresponding to all possible pair-
the similarity of the respective opinions. For individuals ings in Γ), regimes can occur where active links of a cer-
α, β with opinions sα = gi and sβ = gj, the update tain type vanish while others prevail. This can lead to
is an rewiring event with probability pij and an opinion configurations where a certain subset of the states only
adoption event otherwise (probability 1 pij). In the appears in one component of the network in which no
−
following the parameters pij are called rewiring rates. state not belonging to this subset is present. In the fol-
Inarewiringeventthefocallink(α,β)issevered, and lowingwecallthissituationapartially fragmented state.
a new link is created either from α to a randomly chosen In contrast to the fully fragmented state where every
node γ with s = s , or from β to a randomly chosen component is internally in consensus, the dynamics in
γ α
node γ with s = s . The choice between the two out- thepartiallyfragmentedstatecancontinueinsomecom-
γ β
comes is made randomly with equal probability. In an ponents while others may be frozen in internal consen-
opinion update, either node α changes its state to s sus. The partial fragmentation cannot be undone, so
β
or node β changes its state to s , where the choice be- that achieving global consensus is impossible after par-
α
tweenbothoutcomesisagainmaderandomlywithequal tial fragmentation has occured. However, the ongoing
probability. In the following we assume symmetric in- dynamics in the active components will eventually lead
teractions, which implies p =p such that the specific tointernalconsensusineverycomponent. Theabsorbing
ij ji
model is characterized by a set p of G(G 1)/2 pa- statewhichisultimatelyreachedafterapartialfragmen-
ij
{ } −
rameters. tationthereforeconsistsof1<γ <Gmajorcomponents,
The model proposed above preserves the symmetry holdingtheγ survivingopinions,respectively. Forγ =G
of the direct interaction of two opinions postulated in we recover full fragmentation and the case where γ = 1
the adaptive voter model, i.e. in direct comparison no we denote as the fully active regime where all types of
opinion is stronger then the other. However, it breaks activelinksprevail. Onlyinthelattercase(duetofinite-
thesymmetrybetweendifferentpairingsofopinionssuch size effects) global consensus can be reached eventually.
that rewiring is more likely in certain pairings than in
others.
From the adaptive voter model [13, 19] it is known III. FRAGMENTATION TRANSITIONS IN A
that the fragmentation transition separates a so-called THREE-STATE VOTER MODEL
active regime from a fragmented regime. In the active
regimeafinitedensityofactivelinkspersistsinthelong- We start our exploration of the proposed multi-state
termdynamicssuchthatthereisongoingdynamicsuntil voter model by considering the case G = 3, which is
fluctuations drive the system eventually to an absorbing the simplest case which is not trivial (G = 1) or exten-
consensus state. In the fragmented regime, disconnected sively studied (G = 2) [16, 19, 26]. Let us consider the
components emerge, which are internally in consensus. set of opinions Γ = A,B,C , giving rise to three dif-
{ }
3
ferent rewiring rates p ,p ,p which we denote
as p1,p2,p3 accordin{gAtBothAeCstatBeC-n}etworkdepictedin 12(1−p1)
{ }
Fig. 1. A B
staLteet-neutswoemrkphaansdizethaeganinetwthoerkdioffferienndcievidbueatwlse.enTthhee 12(1−p1) (1−ρ3) +ρ3
state-network is a complete, weighted graph with G
nodes which represents the relationships between differ- 12(1−p1)
ent states, such that states connected by small rewiring
rates are similar to each other, whereas states which are 12(1−p1) +(1−ρ3) +ρ3
connectedbylargerewiringratesdiffersignificantlyfrom
eachother. Thenetworkofindividuals,incontrast,isan p1
unweighted graph with N nodes and mean-degree k ,
(cid:104) (cid:105)
which represents the interactions between individuals. 12(1−p1)
In principle, the three-state system can reach five dif-
ferent final states: fully active (leading eventually to 12(1−p1) +
global consensus), full fragmentation, and partial frag-
mentation with respect to A, B, or C. Here, partial p1
fragmentation with respect to a certain state refers to a
situation where a component of nodes in that particu-
lar state fragments from an active component (a mixed 12(1−p2)
component of nodes in the other states).
A C
Forthecalculationoffragmentationthresholdswefol-
low the approach given in [16]. We determine the evolu- 12(1−p2) (1−ρ3) +ρ3
tion equations for the number of active motifs (network
motifs containing active links) starting from a situation 12(1−p2)
close to the fragmentation threshold. For simplicity, the
ptirveesemntotairftibcaleseussepsroopnolysetdheinsi[m16p]l.erWofetwemopdhiffaseirzeentthaact- 12(1−p2) +(1−ρ3) +ρ3
all calculations below can also be carried out using more p2
elaborate motif bases, but at the price of having to deal
with considerably larger matrices. 12(1−p2)
Following [16], we define q-fans as a bundle of q active
links of one type, say AB-links, connected to a single 12(1−p2) +
A- or B-node. We do not account for the number of
inert links connecting to this focal node. For the sake p2
of simplicity we also do not consider mixed active motifs
containing all three states. We confirmed that effects of
mixed motifs can be suitably captured by the procedure FIG. 2. Transitions of AB- and AC-fans for a degree-regular
described below. network with k =3 and equiprobable states for the scenario
We start by calculating the condition for partial frag- of partial fragmentation with respect to A. Black, white and
mentation with respect to A (see Fig. 1). The A-cluster greynodescorrespondtoagentsholdingopinionA,B,andC,
respectively. The active link densities ρ and ρ are assumed
fragments from the rest of the network when all AB- 1 2
tovanishclosetothepartialfragmentationpoint,whereasρ ,
motifs and all AC-motifs vanish. Because in general 3
the densitiy of BC-links can be finite.
p =p ,wehavetotreatAB-andAC-motifsseparately.
1 2
(cid:54)
We start by considering a network with two almost dis-
connected clusters, one of which is composed purely of
A-nodes and the other of B- and C-nodes and then ask
whether the fragmented state is stable, such that frag- k 1-fan[16]. Thisfancansubsequentlyloseactivelinks
−
mentation is reached, or unstable, such that the system due to subsequent opinion updates and rewiring events.
avoids fragmentation. We account for a finite density of active BC-links, ρ3, in
the active component by creating an AC-fan (AB-fan)
In the almost fragmented state the expected effect of
insteadofanAB-fan(AC-fan)withprobabilityρ when
network updates on the active motifs is captured by a 3
procedure proposed in [16]. For the case of k = 3 we anewfaniscreatedbyaB-node(C-node)adoptingopin-
ion A.
obtain the transitions rules shown in Fig. 2. New ac-
tive motifs are created when an opinion update occurs. If we start with equal distribution of states, the rela-
We approximate the degree of the focal node k by the tion ρ = [BC]/(k[B]) = [BC]/(k[C]) holds, where [B]
3
networks mean degree k . Because of the clusters be- and[C]denotethenumbersofB-nodesandC-nodes,re-
(cid:104) (cid:105)
ing almost-separated the newly formed active motif is a spectively. Note that ρ differs from the global BC-link
3
4
density ρ(G) =[BC]/L.
3
The set of transitions for k = 3 (see Fig. 2) defines a
dynamical system, describing the time evolution of the
densities of active motifs close to the partial fragmen-
I II III
tation with respect to A. The stability of the partially
fragmented state in this system is then governed by the
fragmentation
block-structured Jacobian,
(cid:18) (cid:19)
D X (ρ ) X (ρ )
J(p1,p2,ρ3)= p1X− (ρp1) 3 D p1X3(ρ ) , (1)
p2 3 p2 − p2 3
2
where p pc r =0
3
Dp =−111 12(211(−10−p)p−) 1 1(121(10−p)p) 1 (2) pmin no fragmentation/ r 3=r max
2 − − consensus
and
0 1(1 p)ρ 0
Xp(ρ)=0 212(1−−p)ρ 0. (3) pmin pc
0 0 0 p
1
The diagonal blocks in the Jacobian given in Eq.(1)
FIG. 3. Fragmentation diagram for a three-state system.
can be interpreted as “self-interaction” terms, captur-
Fragmentation with respect to opinion A occurs always in
ing contributions from the same motif-type, and the off-
the black region and never in the grey region. In the dashed
diagonaltermsas“exchange”terms,capturingcontribu-
region, fragmentation can occur depending on the value of
tions from different motif-types. The structure of this
ρ ,thedensityofBC-linksintheBC-component. Ifp >p
3 1 3
Jacobian remains unchanged for any partial fragmenta- and p > p then this fragmentation diagram characterizes
2 3
tion of a three-state system, while the matrices (2) and thefinalstateofthewholesystem. InthiscaseregionIcoin-
(3) change when the motif set is altered. In particular, cideswiththeglobalconsensusregimeandfullfragmentation
thedimensionofthesematricesincreaseswithincreasing isreachediffp ≥p ,wherep isthefragmentationthreshold
3 c c
mean degree and/or number of motifs considered. for the adaptive two-state voter model.
In a dynamical system a steady state is stable if all
eigenvalues of the corresponding Jacobian have negative
real parts [27]. For the present system this means that
For studying the case ρ > 0 we first note that every
thefragmentedstateisstableifalleigenvaluesoftheJa- 3
matrix-valued row of J sums to D , where i=1,2. Fol-
cobian J are negative and the fragmentation transition pi
lowing [28], as will be discussed below, λ(J) is bounded
occurs as at least one of the eigenvalues acquires a pos-
by λ(D ) and λ(D ). Therefore, fragmentation with
itive real part. Therefore, demanding λ(J) = 0, where p1 p2
respect to A is guaranteed when p > p and p > p ,
λ(J) is the leading eigenvalue of J, yields a condition for 1 c 2 c
butcanalreadyoccurwhenonlyeitherp >p orp >p
thefragmentationtransition,dependingonthethreepa- 1 c 2 c
is satisfied (see Fig. 3 region II).
rametersp ,p , andρ . ThephasediagraminFig.3isa
1 2 3
The maximum extension of region II is observed when
projection of this fragmentation condition on the p -p -
1 2 p = 0, the corresponding maximal value of ρ , ρ ,
plane for the extreme values of ρ . 3 3 max
3
can be determined to good approximation by a moment
Let us first consider the case where ρ = 0, which is
3
closure approach (see Appendix), yielding
encounteredifp exceedsp ,thefragmentationthreshold
3 c
of the adaptive two-state voter model. In this case, X k 1
becomes zero and the set of eigenvalues of the Jacobian ρmax = 2−k . (4)
J is the conjunction of the eigenvalues of the matrices
D and D . Thus, λ(J) is negative iff λ(D ) and Solving the condition λ(J(p ,p ,ρ )) = 0 numerically
p1 p2 p1 1 2 max
λ(D ) are negative. Indeed matrices D and D are yields the curve separating regions I and II in Fig. 3.
p2 p1 p2
theJacobiansofthetwouncoupledtwo-statesystemsA Moreover, from the diagram in Fig. 3 it is clear that
−
B andA C. ThusfragmentationofArequiresthatthe this curve implies the existence of a minimal rewiring
−
two-state fragmentation condition is met separately for rate p , such that for p < p or p < p partial
min 1 min 2 min
the AB and AC subsystems. In other words, if the links fragmentation with respect to A becomes impossible.
between B and C nodes vanish (ρ = 0), fragmentation Let us emphasize, that calculations of fragmentation
3
occurs when both p >p and p >p (see Fig. 3 region thresholds for partial fragmentations build on the esti-
1 c 2 c
III). mation of the active link densities from given rewiring
5
A C B
1 0.3 1
0.3
if
l l l
0.2 0.2 p3>pc p1 p1 p3 l l l
p2pc p2pc
0.1 0.1 p2 p3 p2 full fragm.
else if
0.0 0.0 l l l
00 pcp1 1 00 pcp1 1 p2>pc p1 p1 p3 l l
l
p2 p3 p2 part. fragm. (A)
FIG. 4. Numerical phase diagram for the three-state model.
Color-codedisthedensityoflinks, connectingtheAandthe else if
l l l
BgrCey-crleugsitoenrs(ρc1or+reρs2p)onovdertothfreargemweinritnagtiroantewsipth1 arensdpepc2t. tDoaArk. p1>pc p1 p1 p3 l l
The left panel shows the case ρ = 0(p = 0.5). This corre- l
3 3 part. fragm. (A) /
spondstoanuncoupledsystem: thecriticalrewiringratesfor p2 p3 p2 cons.
p and p are the same as for the two-state voter model, p .
1 2 c else
In the right panel, ρ is maximal (p = 0). Here, the active l
3 3
link density in the active cluster leads to an extension of the p1 l p1 l p3 l l
fragmentationregion. Blacklinesrepresentanalyticalresults.
l
N = 10000, (cid:104)k(cid:105)=4, averaged over 20 realizations.
p2 p3 p2 cons.
FIG. 5. Fragmentation diagrams for partial fragmentation
with respect to state A, C or B for p ≥ p ≥ p . There
rates. As there is no analytical expression for ρ(p) in 1 2 3
are four different cases, according to the conditions given on
the whole p-range ([26]) the long-term behavior can only
the left-hand side of the chart. The positions of the points
be predicted with certainty in regions I and III of the P =(p ,p ),Q=(p ,p ),R=(p ,p ) indicate for each case
2 1 3 1 2 3
fragmentation diagram. whether partial fragmentation is reached for the respective
Insummary,evaluatingthepartialfragmentationcon- state. The final state of the whole system can be deduced
dition with respect to A, i.e. λ(J) = 0, leads to a phase from the outcomes for all three states and is given on the
diagramasshowninFig.3, wherethreedifferentregions right-hand side of the chart. For the first, second and forth
case the final state can be predicted without ambiguity. In
can be distinguished: In regions I and III partial frag-
the third case, either partial fragmentation with respect to
mentation occurs or is avoided regardless of p , whereas
3
A or consensus can be reached, depending on the specific
inregionIIpartialfragmentationdependsonρ andcon-
3 parameter setting (the two possibilities are indicated by two
sequently on the setting of the related rewiring rate p .
3 different symbols ◦ and × in the first diagram and a solid
We found that these results are in very good agreement
andadashedlineintheseconddiagramofthethirdrow). It
with data obtained from agent-based simulation of large can be seen that partial fragmentation is only possible with
networks (Fig. 4). respect to the state with the largest rewiring rates (here A).
Until now, we studied partial fragmentation with re-
spect to one specific state (state A). In order to predict
thefinalstateofthewholesystem,onehastoanalyzethe
correspondingpartialfragmentationdiagramsforeachof
In case 3) the point P lies either in region I or II of the
the three states. Let us assume p p p . Then,
1 ≥ 2 ≥ 3 corresponding fragmentation diagram for A, whereas Q
there are four cases to distinguish (see Fig. 5):
is always in region I, because p p and ρ ρ . This
2 3 3 2
≥ ≥
1. p >p meansthatinthiscaseeitherpartialfragmentationwith
3 c
respect to A or consensus is reached depending on the
2. p <p and p >p specific values of p ,p ,p . In case 4) P, Q and R lie
3 c 2 c 1 2 3
in the region I of their respective diagrams and global
3. p2 <pc and p1 >pc consensus is reached. Note that this shows that partial
fragmentation can only occur with respect to that state,
4. p <p .
1 c which is connected via the largest rewiring rates to the
two other states in the state network.
Incase1)fullfragmentationisreached,becauseallpoints
P = (p ,p ), Q = (p ,p ) and R = (p ,p ) lie in the In summary, we showed that in the three-state voter
2 1 3 1 2 3
region III of their respective diagrams. In case 2) P lies modeleitherconsensus,partialfragmentationorfullfrag-
inIIIandQandRinI.Thus,partialfragmentationwith mentation occurs. Full fragmentation is only reached
respecttoAoccurs,whileB-andC-nodesformanactive when all rewiring rates exceed p . Analyzing the phase
c
clusterandinafinitesystemeventuallyreachconsensus. diagram with respect to the state which is connected by
6
the largest rewiring rates to the other states suffices for where ∆ and ξ are matrices of s s matrix-valued en-
i i
×
the prediction of the final state of the whole system. For tries,
quantitative predictions in region II of the diagram the
(cid:0) (cid:1)
active link density corresponding to the lowest rewiring ∆i ρij,ρij =
rate has to be known. Qualitatively, one can say that Dˆ (cid:0)ρ ,ρ (cid:1) X (cid:0)ρ (cid:1) ··· X (cid:0)ρ (cid:1)
i1 1j ij i1 12 i1 1s
i“fmpoasrttidailffferraegnmt”ensttaattieo.n occurs, then with respect to the Xi2(cid:0)ρ21(cid:1) Dˆi2(cid:0)ρ2j,ρij(cid:1) ... ...
... ... ... Xis−1(ρ(s−1)s)
X (cid:0)ρ (cid:1) ··· X (ρ ) Dˆ (cid:0)ρ ,ρ (cid:1)
IV. FRAGMENTATION TRANSITIONS IN A is s1 is−1 s(s−1) is sj ij
G-STATE VOTER MODEL
and
Let us now consider a general system of G states. In X(cid:48)i1(cid:0)ρij(cid:1) 0 ··· 0
ccmoaunnlttarila-sssottaottceocnuterhtwewioptrhrkervceiasopnuestchtsuytsostfareamggrmopueapnrttoifianslttofartaseegsvm.eAreangltaeantcietoirnvaesl ξi(cid:0)ρij(cid:1)= 0... X(cid:48)i2.(cid:0).ρ.ij(cid:1) ...... 0... .
components. Let us therefore calculate the condition for 0 ··· 0 X(cid:48)is(cid:0)ρij(cid:1)
a system to fragment into two components containing s
Here, we introduced the abbreviation
andG sstates,respectively(seeFig.6). Thisisinprin-
−
ciple no restriction, as a fragmentation into more than s s
two components can be treated as a fragmentation into Dˆ (cid:0)ρ ,ρ (cid:1)=D (cid:88) X (ρ ) (cid:88) X(cid:48) (ρ ).
ii ij ij ii ii ij ii ij
two components where the active components in their − −
j=1,j(cid:54)=i j=1,j(cid:54)=i
turn fragment.
For clarity we only use one level of indices from now The matrices D and X for k = 3 were already given in
on: we write Dij and Xij instead of Dpij and Xpij. (2) and (3) and the matrix X(cid:48) for k =3 is,
Furthermore, in order to distinguish indices which re-
fceormtpoonoennetcwomepuosneenintdficroesmithose1,r.e.fe.r,isngftoor tthhee octohmer- X(cid:48)ii(ρij)=00 00 12(1−0pii)ρij. (6)
ponent with s states and ind∈ices{ i 1},...,s for the 0 0 1(1 p )ρ
∈ { } 2 − ii ij
component with s = G s states. For example, the
inter-component rewiring−rates are then denoted as p Thelattermatrixappearsforfragmentationswhereboth
ii
and intra-component active link densities as ρ and ρ , ofthefragmentingcomponentsareactive,i.e. for1<s<
ij ij
G 1.
respectively.
−
Note that every matrix-valued row of the Jacobian
In analogy to the treatment of the three-state model,
sums to D and refers to one specific type of inter-
weconsiderasituationwherethetwoclustersarealmost ii
component link with rewiring rate p in the state-
fragmented. We then determine the evolution equations ii
network. Onesuchrowthusrepresentsthetransitionsfor
forasetofactivemotifsconnectingthetwocomponents.
one motif-type. Entries Dˆ on the diagonal capture the
In the three-state case these were of two types, AB- and
creationofmotifsofthesametype,whileoff-diagonalen-
AC-fans, which led to a Jacobian of 2 2 matrix-valued
× triesXandX(cid:48)denotetransitionstodifferentmotif-types,
entries and a fragmentation condition which was a func-
which arise from the intra-component link densities ρ
tion of the rewiring rates p and p and the active link ij
1 2
density ρ3. In the general case the Jacobian contains and ρij, respectively. For example, the entries in one
ss ss matrix-valued entries, according to the number rowdescribingthetransitionsofg1g2-fansdependonthe
ofi×nter-componentlinksinthestate-network(seeFig.6) rewiring rate between the states g1 and g2, which is p12,
andthefragmentationconditionisafunctionofallinter- the active link densities between g1 and all other states
caocmtivpeonliennktdreenwsiirtiinegs rρaitjes a{npdii}ρainjd.all intra-component iancttihvee filirnsktdcoenmspitoiensenbte,twρ1exe,nwgi2thanxd∈a{ll2r,e.m..a,isn}i;ngansdtatthees
{ } { } in the second component, ρ , with y 1,3,...,s .
Following the same procedure as for the three-state 2y ∈{ }
Inanalogytothethree-statemodeltheactivelinkden-
system, one finds that the general Jacobian exhibits a
block-structure of s s submatrices, sities ρij entering in the Jacobian relate to the global
× active link density ρ(G) as
(cid:0) (cid:1) ij
J p ,ρ ,ρ = (5)
ii ij ij
∆1(cid:0)ρij,ρ1j(cid:1) ξ1(cid:0)ρ12(cid:1) ··· ξ1(cid:0)ρ1s(cid:1) ρij = [kg[iggj]] = G2 [giLgj] = G2ρ(ijG). (7)
ξ2(cid:0)ρ...21(cid:1) ∆2(cid:0)ρ.i.j.,ρ2j(cid:1) ...... ξ(s−1)(cid:0)ρ...(s−1)s(cid:1), ThSitsahboillidtsyaannaalloygsoisusoliyf tfhoer ρgiejn.eral Jacobian in (5) is in
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
ξ ρ ··· ξ ρ ∆ ρ ,ρ
s s1 s s(s−1) s ij sj principlepossible,butleadstoafragmentationcondition
7
ces M , such that
ij
M M M
{ρij} {pii} {ρij} M=M...1211 M...2122 ··.··.··. M...21NN.
M M M
N1 N2 NN
···
We define
N
(cid:88)
S = M , k =1,...N
k kj
j=1
s states G s states as generalized, matrix-valued rowsums of M. Then, the
− following inequality holds [28]:
FIG. 6. schematic representation of a state-network with G λ(minSk) λ(M) λ(maxSk). (8)
k ≤ ≤ k
states. Ellipses illustrate a fragmentation into two (possibly)
activeclustersofsandG−sstates. Dashedlinescorrespond The expressions min and max have to be understood
k k
tointer-clusterlinkswithrewiringrates{pii},connectingev- element-wise,i.e. thematrixminkSk isthematrixwhich
ery state in one cluster with every state in the other cluster. isobtainedwhenwetakeelement-wisetheminimumover
The number of inter-cluster links, s(G−s), determines the all S and analogously for the maximum.
k
dimension of the (matrix-valued) Jacobian. Solid lines corre-
Inthefollowingweapplythetheoremquotedaboveto
spond to intra-cluster links within both clusters with active
the Jacobian given in (5). This is possible because J can
link densities {ρ } and {ρ }, respectively.
ij ij be written as J=T 1, where T is a nonnegative irre-
−
duciblematrixand1istheidentitymatrixofappropriate
dimension. We will consider two different partitions.
which depends directly or indirectly (through the active First, let us consider a partition (P1) of the Jacobian
link densities) on all G(G 1)/2 different rewiring rates. into ss submatrices. Then, the matrix-valued rowsums
−
Incontrasttotheestimationoftheactivelinkdensityin corresponding to this partition yield
atwo-statesystem,inamulti-statesystemtheactivelink
s s
densityofacertainlink-typedoesnotonlydependonthe S(P1) =Dˆ (cid:0)ρ ,ρ (cid:1)+ (cid:88) X (ρ )+ (cid:88) X(cid:48) (ρ )
rewiring rate of that specific link-type, but also on the k ii ij ij ii ij ii ij
j=1,j(cid:54)=i j=1,j(cid:54)=i
rewiring rates and active link densities of the neighbor-
inglinksinthestate-network. Inferringthelinkdensities =Dii, k =1,...,ss.
analytically from the rewiring rates is presently an un-
The matrices D only depend on p and it can be seen
solved challenge. So, even for given rewiring rates it is ii ii
from(2)thatallnon-constantentriesinD increasewith
in general not possible to make quantitative predictions p
decreasing p. Therefore, we get for the upper and lower
aboutfragmentationthresholds. Nevertheless, thestruc-
bounds of λ(J)
ture of the Jacobian allows for qualitative predictions,
which will be shown in the next section. λ(D ) λ(J) λ(D ), (9)
pmax ≤ ≤ pmin
where
p =maxp , p =minp .
max ii min ii
V. REDUCTION PRINCIPLES FOR SPECIAL i,i i,i
STATE-NETWORK TOPOLOGIES
From (9) we deduce the following statements:
Inthissectionweusetheoremsaboutupperandlower When all inter-cluster rewiring rates pii are below
•
bounds of the largest eigenvalue λ(M) of a nonnega- the threshold pc, no fragmentation occurs, because
tive irreducible matrix M. The well-known Frobenius λ(Dpmax)>0.
inequality states
When all inter-cluster rewiring rates p exceed
ii
•
the threshold p , fragmentation occurs, because
c
mkinSk ≤λ(M)≤mkaxSk, λ(Dpmin)<0.
Ifp =p ,necessarilyallinter-clusterrewiring
min max
•
where S is the rowsum of the i-th row of M. A general- rates must be equal. In that case, the fragmenta-
i
ization of the above inequality for a partitioned nonneg- tion condition is the classical condition of the two-
ativeirreduciblesquarematrixMisgivenin[28]. Letus state voter model, λ(J)=λ(D )=0, which yields
p
assume that M can be partitioned into square submatri- the critical rewiring rate p .
c
8
Now we consider another partition, (P2), of the Jaco-
bian, which is a partition into s submatrices. Then, the
corresponding generalized rowsums yield
1
p
5 states s
nks 2 states r3 S(iP2) =∆i(cid:0)ρij,ρ1j(cid:1)+ (cid:88) ξi(ρij)
li r1 r2 j=1,j(cid:54)=i
ter r4 D˜i1(cid:0)ρ1j(cid:1) Xi1(cid:0)ρ12(cid:1) ··· Xi1(cid:0)ρ1s(cid:1)
er−clus =Xi2(cid:0)...ρ21(cid:1) D˜i2.(cid:0).ρ.2j(cid:1) ...... Xis−1(ρ...(s−1)s),
nt Xis(cid:0)ρs1(cid:1) ··· Xis−1(ρs(s−1)) D˜is(cid:0)ρsj(cid:1)
f i p=0.2
o
where
y
t
si s
n p=0.4 D˜ (cid:0)ρ (cid:1)=D (cid:88) X (ρ ).
e ii ij ii ii ij
−
d
j=1,j(cid:54)=i
p=0.6
0
First, we observe that every matrix S(P2) corresponds
0 1000 i
time to a partial fragmentation in a system of s+1 states.
More precisely, the set S(P2) describes a collection of
{ i }
s single-state-fragmentations where for every i a single
FIG. 7. Numerical test of the reduction principle. We run stateg istakenseparatelyfromthes-cluster. Thissingle
i
simulations for a 5-state system assigning random values to state (i.e. now s=1) then forms the first component of
the rewiring rates r (inset). Plotted is the density of inter-
i thepartialfragmentation,whilethesecondcomponentis
cluster links (dashed links) for three different values of p.
given by the whole s-cluster.
For each p-value we run ten independent simulations with
randomly chosen ri and compare the corresponding inter- Now, building the element-wise extrema of {S(iP2)}
cluster link density to the active link density of a two-state means to compare all the single-state-fragmentations by
model with the same rewiring rate and a ratio 2:3 for the comparing every matrix-entry of the corresponding gen-
number of nodes in opposite states. It can be seen that eralized rowsums. Taking min S(P2) (max S(P2)) yields
i i i i
the inter-cluster density does not depend on r and that the
i thereforeineverematrix-entrytheminimum(maximum)
steady-statevalueequalsthecorrespondinginter-clusterden-
value, i.e. that one which comprises the maximal (min-
sity(activelinkdensity)intheoriginaltwo-statevotermodel.
imal) rewiring rate. The resulting matrix corresponds
N =10000,(cid:104)k(cid:105)=4.
to a partial fragmentation with respect to a single state
wheretheinter-clusterrewiringratesarechosenextremal
according to the described comparison. We will refer to
Thefirsttworesultsrepresentanintuitivegeneralization such a system as bounding system (see Fig. 8 for exem-
of our findings for the three-state case. The last result plary bounding systems).
implies that if all inter-cluster rewiring rates are equal For a partition of type (P2) the leading eigenvalue of
then the value of these rewiring rates, p, is the only pa- the general Jacobian satisfies
rameterparameteronwhichthefragmentationtransition
depends. In this case a precise analytical estimation of λ(cid:0)minS(P2)(cid:1) λ(cid:0)J) λ(maxS(P2)(cid:1). (10)
the fragmentation point is possible because the active i i ≤ ≤ i i
link densities arising from the intra-cluster links, do not
The lower bound corresponds to the fragmentation of a
enter. Furthermorenotethatthisresultisindependentof
system where the largest inter-cluster rewiring rate of
the number of opinions. This implies that in the special
each state in the second component is connected to a
case of equal inter-cluster rewiring rates systems of any
single state. The upper bound corresponds to the frag-
size behave identically to a properly-initialized adaptive
mentation of a system where the smallest inter-cluster
two-state voter model.
rewiring rate of each state in the second component is
We test the latter result in a five-state system, con- connected to a single state.
sidering a fragmentation into two components of 2 and As in the Jacobian in (5) the matrices X and X(cid:48) can
3 states, respectively (inset in Fig. 7). Simulations show be interchanged, one can consider a corresponding parti-
thatforrandomlychosenrewiringratesr withinthetwo tion where the second component is reduced to a single
i
clusters, the inter-cluster link density reaches the same state, i.e. s = 1 and the first component remains as a
steady-state value (Fig. 7). A further comparison shows, whole. This leads to a different set of bounding systems,
that the behavior of a five-state model closely matches as illustrated in Fig. 8 a) and b).
that of a two-state model. From (10) we can draw the following conclusions:
9
a) The latter case is realized if every state in one compo-
p nent is connected via equal rewiring rates to every state
1
p1 p p3 intheothercomponent(seeFig.8c)). Forstate-network
p 2
< 3 < topologies which display this property the dimension of
q
q 1q2 q the Jacobian reduces significantly and thus the fragmen-
1 3
q
3 tation condition becomes much more tractable.
p > p > p ; q > q > q Insummary,theresultsfromthesecondpartitionshow
1 2 3 1 2 3
thatfortheleadingeigenvalueofaJacobian,correspond-
ing to a partial fragmentation into two active clusters,
b)
upper and lower bounds can be given, which correspond
p
1 to single-state-fragmentations in (properly constructed)
p q
1 p 1
p2 < p3 2 < q2 lower-dimensional systems. In particular, the leading
q1q eigenvalue of the full Jacobian can be exactly calculated
p 2 q
3 3 astheleadingeigenvalueofalower-dimensionalJacobian
q
3
if special state-network topologies are given. Otherwise,
p > q , i = 1,2,3 when such a reduction is not possible, the bounding sys-
i i
tems provide necessary conditions for a partial fragmen-
tationtooccur. So,calculatingfragmentationconditions
c)
p for the much simpler bounding systems in some cases
p
pp suffices to predict the ocurrence or absence of fragmen-
=
q tations in the full system.
q
q
q
FIG. 8. Schematic representation of the generalized Frobe-
niusinequalitiesusingtheexampleofafive-statesystemand VI. CONCLUSION
a specific partial fragmentation into a 2-state- and a 3-state-
cluster. The inequalities in a) and b) represent two different
sets of bounding systems obtained from a partition of type In the present paper we extended recent work on the
(P2), as described in the text. In a) the bounding systems adaptivetwo-statevotermodelstoafamilyofmulti-state
arethree-statesystems,obtainedbyreducingthe3-clusterto models. For the three-state model our analysis revealed
a single state. The remaining inter-cluster rewiring rates are
a phase diagram in which three distinct types of long-
the respective maximum (minimum) values of the rewiring
term behavior are observed. Depending on the param-
rates between each state in the 2-cluster and all states in
eters the system either approaches a consensus state, a
the 3-cluster. In b) the bounding systems are four-state sys-
partially fragmented state ultimately leading to two sur-
tems, obtained by reducing the 2-cluster to a single state.
The remaining inter-cluster rewiring rates are the maximum viving opinions or a fully fragmented state in which all
(minimum) values of the rewiring rates between each state three opinions survive.
in the 3-cluster and all states in the 2-cluster. In c) a special
Inageneralscenariowithanarbitrarynumberofstates
caseisshownwheretheupperandlowerthree-statebounding
making precise predictions is more difficult. In particu-
systems coincide. In this case, the leading eigenvalue of the
lar, the computation of fragmentation points generally
five-state system equals the leading eigenvalue of the associ-
requires the estimation of active link densities inside the
ated three-state system and the considered fragmentation of
clustersbetweenwhichthefragmentationoccurs. Byex-
theoriginalsystemisfullycapturedbythelower-dimensional
one. ploiting the specific structure of transition rates in the
system,onecanneverthelessgainanalyticalinsightsinto
thefragmentationdynamics. Forexampleweidentifieda
If the lower bounding system does not fragment class of special cases in which adaptive multi-state voter
• (λ(cid:0)min S(P2)(cid:1) > 0) the original system does not models exactly recover the behavior of the adaptive two-
i i
fragment. state voter model.
While the ultimate goal of understanding opinion for-
If the upper bounding system fragments mation in the human population is still far away, the
• (λ(maxiS(iP2)(cid:1) < 0) the original system frag- presentprogressshowsthatanalyticalunderstandingcan
ments. be pushed to more complex models. Recent studies have
shownthatalreadytodayvariantsofthevotermodelcan
If upper and lower bounding systems are the same betestedinexperimentswithswarminganimals. Anim-
• (min S(P2) =max S(P2)) the fragmentation of the portantgoalforthefutureistocontinuetherefinementof
i i i i
original system is exactly captured by the bound- models and analysis techniques in order to describe real-
ing system, i.e. the full Jacobian J reduces to the world situations. We hope that the approach presented
Jacobian of the bounding system. here will make a contribution to this ongoing process.
10
VII. APPENDIX the number of nodes in the active component. The total
number of links in the active component l is then given
Weuseamomentclosureapproach[29]forthecalcula- by
tionofthemaximalactivelinkdensityinanactivecluster
of s states in a G-state system. The evolution equation s(s 1)
l=sη+ − ζ
for the number of active links of type xy is given by 2
[x˙y]= [xy]+1(cid:0)(cid:88)[xzy]+(cid:88)[xzy] (cid:88)[xyz] (cid:88)[zxy](cid:1), and (11) yields
− 2 − −
z(cid:54)=x z(cid:54)=y z(cid:54)=y z(cid:54)=x
ζ2 ζη ζ2
ζ = s(s 2)+2 s s(s 1)
assuming p = 0 for all rewiring rates within the active n − n − n −
ij
cluster. Then, using the pair-approximation, we get for ζ2 ζl ζ2
s(s 2)+2 2 s(s 1).
the steady-state, n − n − n −
1(cid:16) (cid:88) [xz][zy] [xy][yy] [xx][xy] Using kn=2l, we get for the maximum active link den-
[xy]= 2 +2 +2
2 [z] [y] [x] sity in a cluster of s states:
z(cid:54)=x,y
(cid:88)[xy][yz] (cid:88) [zx][xy](cid:17)
. (11) s(s 1)ζ (s 1)(k 1)
−z(cid:54)=y [y] −z(cid:54)=x [x] ρmax = −2l = − sk − ,
For equal distribution of states we can write [x] = which gives ρ = (k 1)/(2k) for s = 2, as provided
max
−
n/s x,[xy]=ζ (x,y) and [xx]=η x, where n denotes in the text.
∀ ∀ ∀
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