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Fragmentation transitions in multi-state voter models Gesa A. Böhme∗ 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 fragmentation 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 different final states: fully active (leading eventually to 12(1−p1) + global consensus), full fragmentation, and partial fragmentation 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 particular 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 disconnected 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 active 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 respect 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) Dî2(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,corresponding 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. ∀ ∀ ∀ [1] R. Albert and A. Baraba´si, Rev.Mod.Phys. 74, 47 [15] B.KozmaandA.Barrat,Phys.Rev.E77,016102(2008) (2002) [16] G. A. Bo¨hme and T. Gross, Phys. Rev. E 83, 035101 [2] M. E. J. Newman, SIAM Rev. 45, F00 (2003) (2011) [3] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and [17] G. Zschaler, G. A. Bo¨hme, M. Seißinger, C. Huepe, and D. U. Hwang, Physics Reports 424, 175 (2006) T. Gross, Arxiv preprint arXiv:1110.1336(2011) [4] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, [18] I.D.Couzin,C.C.Ioannou,G.Demirel,T.Gross,C.J. Rev.Mod.Phys. 80, 1275 (2008) Torney, A. Hartnett, L. Conradt, S. A. Levin, and N. E. [5] T.GrossandB.Blasius,J.R.Soc.Interface5,259(2008) Leonard, Science 334, 1578 (2011) [6] T. Gross and H. Sayama, Adaptive Networks (Springer, [19] F. Vazquez, V. M. Egu´ıluz, and M. San Miguel, 2009) Phys.Rev.Lett. 100, 108702 (2008) [7] M.G.Zimmermann,V.M.Egu´ıluz,M.SanMiguel,and [20] D. H. Zanette and S. Gil, Physica D 224, 156 (2006) A. Spadaro, Applications of Simulations in Social Sci- [21] C. Nardini, B. Kozma, and A. Barrat, Phys.Rev.Lett. ences (Hermes Science Publications, 2000) pp. 283–297 100, 158701 (2008) [8] J.ItoandK.Kaneko,Phys.Rev.Lett. 88,028701(2001) [22] S. Gil and D. H. Zanette, Phys.Lett.A 356, 89 (2006) [9] A.Do,L.Rudolf,andT.Gross,NewJ.Phys.12,063023 [23] D. Kimura and Y. Hayakawa, Phys.Rev.E 78, 016103 (2010) (2008) [10] S. Bornholdt and T. Rohlf, Phys.Rev.Lett. 84, 6114 [24] G. Demirel, R. Prizak, P. N. Reddy, and T. Gross, to (2000) appear in Eur. Phys. J B [11] C. Meisel, A. Storch, S. Hallmeyer-Elgner, E. Bullmore, [25] G. Weisbuch, G. Deffuant, F. Amblard, and J. Nadal, and T. Gross, PLoS Comput Biol 8, e1002312 (01 2012) Complexity 7, 55 (2002) [12] http://adaptive-networks.wikidot.com [26] G. Demirel, F. Vazquez, G. A. Bo¨hme, and T. Gross, in [13] F.VazquezandV.M.Egu´ıluz,NewJ.Phys. 10,063011 preparation(2011) (2008) [27] I.Kuznetsov,Elementsofappliedbifurcationtheory,Vol. [14] P. Holme and M. E. J. Newman, Phys.Rev.E 74, 56108 112 (Springer Verlag, 1998) (2006) [28] E. Deutsch, Pacific J. Math. 92, 49 (1981) [29] M. Keeling, Proc. Roy. Soc. Lond. B 266, 953 (1999)