An adaptive voter model on simplicial complexes
AAn adaptive voter model on simplicial complexes
Leonhard Horstmeyer
1, 2 and Christian Kuehn
3, 2 Section for the Science of Complex Systems, CeMSIIS,Medical University of Vienna, Spitalgasse 23, A-1090, Vienna, Austria Complexity Science Hub Vienna, Josefst¨adterstrasse 39, A-1090 Vienna, Austria Faculty of Mathematics, Technical University of Munich,Boltzmannstr. 3, 85748 Garching M¨unchen, Germany (Dated: Version September 13, 2019)Collective decision making processes lie at the heart of many social, political and economic chal-lenges. The classical voter model is a well-established conceptual model to study such processes.In this work, we define a new form of adaptive (or co-evolutionary) voter model posed on a sim-plicial complex, i.e., on a certain class of hypernetworks/hypergraphs. We use the persuasion rulealong edges of the classical voter model and the recently studied re-wiring rule of edges towardslike-minded nodes, and introduce a new peer pressure rule applied to three nodes connected via a2-simplex. This simplicial adaptive voter model is studied via numerical simulation. We show thatadding the effect of peer pressure to an adaptive voter model leaves its fragmentation transition,i.e., the transition upon varying the re-wiring rate from a single majority state into to a fragmentedstate of two different opinion subgraphs, intact. Yet, above and below the fragmentation transition,we observe that the peer pressure has substantial quantitative effects. It accelerates the transitionto a single-opinion state below the transition and also speeds up the system dynamics towards frag-mentation above the transition. Furthermore, we quantify that there is a multiscale hierarchy in themodel leading to the depletion of 2-simplices, before the depletion of active edges. This leads to theconjecture that many other dynamic network models on simplicial complexes may show a similarbehaviour with respect to the sequential evolution of simplicies of different dimensions.
PACS numbers: 89.75.Fb 02.60.Cb 05.45.a 64.60.De
I. INTRODUCTION
Contact and voter processes are a key theme of manydisciplines including economics, epidemiology, mathe-matics, physics, and social science [28]. One naturalsetting for these processes is an underlying (complex)network, or graph, on which a population of individu-als interacts [1]. For example, in the context of opin-ion formation, which we focus upon here, two individu-als/nodes/vertices may hold different opinions. For sim-plicity, let us assume that there are only two opinionspossible, e.g., we can only vote for two possible par-ties. The classical voter model [10, 17, 34] describes theevolution of opinion dynamics on a fixed network by aMarkov chain, either in discrete or continuous time; herewe use the discrete-time variant. At each time step, anedge is selected at random. If both vertices hold thesame opinion, nothing happens. If they hold oppositeopinions, then one adapts the opinion of the other withequal probability. Studying the long-time behaviour ofsuch a dynamical system is already highly non-trivial asit does depend crucially on the network structure; seee.g. [8, 12, 30, 33–36].However, an important element of realism is miss-ing in the classical voter model. Social interactions inlarge populations almost never take place on a fixednetwork. In fact, with whom we are in contact mayalso depend upon the difference or similarity of opin-ions [29]. This viewpoint has led to the development ofadaptive, or co-evolutionary, network models, in which there is interacting dynamics on and of the network; seee.g. [5, 13, 14, 18, 19, 27, 31]. More precisely, the mostcommon version of the adaptive voter model allows inaddition to persuasion events also for re-wiring events,where edges between opposite-minded vertices are re-wired to edges between like-minded vertices. This makesthe process much more realistic as it allows one to studyvia a relatively simple model complex self-adapting net-work structures. There is already a substantial literatureon adaptive voter models [2, 4, 11, 22, 23, 32, 37, 39].Yet, recent changes in communication and social net-work formation cast serious doubt on the assumptionthat only binary interactions should matter in opinionformation [20, 38]. These higher-order interactions haverecently started to appear as a new focus in the anal-ysis of complex network data sets [3]. Yet, there arecurrently no available standard adaptive/co-evolutionarynetwork dynamics models taking into account higher-order interactions. In this work, we propose and study aminimalistic extension of the adaptive voter model to in-clude higher-order interaction between individuals. Thismodel takes into account the well-known effect of peer-pressure, which has been studied widely in many scien-tific fields [7, 15, 21]. For example, if three individuals areconnected in a friendship, and there is a disagreement inopinions, it is very likely that the majority opinion withinthe group of three prevails, i.e., a peer-pressure effect hasoccurred. To model whether a fully connected subgraphof three vertices is in a close-enough friendship or not, weneed an additional structure beyond vertices and edges.A very general underlying structure would be to consider a r X i v : . [ n li n . AO ] S e p a hypergraph [6] instead of a usual graph/network. Al-though this is possible, we are looking to develop a min-imal and mathematically elegant formulation to capturethe essential effects of peer-pressure opinion dynamics.One natural choice in this context is to restrict ourselvesto simplicial complexes [16], where a triangle of connectednodes is in close friendship interaction if there is a 2-simplex between them in addition to the 1-simplices (theedges) connecting them. Of course, the model we de-velop here could be generalized very naturally beyond2-simplices but we postpone this more involved general-ization to future work.In this work, we define the simplicial adaptive votermodel and study its dynamics numerically by direct simu-lation. On the one hand, our model turns out to preservesome key features of the standard adaptive voter modelregarding metastability and diffusive absorption [9] intoa single opinion for low re-wiring as well as a fragmen-tation transition for high re-wiring. The quantitativestructure of these transitions turns out to be significantlyaffected by peer-pressure. In particular, absorption oc-curs faster, and the fragmentation transition occurs ear-lier with respect to the re-wiring frequency. From theviewpoint of opinion dynamics, this can be interpretedin the sense that societies are driven faster into mono-opinion/polarized or fragmented opinion states if peer-pressure effects occur. These effects could evidently beinduced from social network interactions, i.e., there is apotential danger that we are going to observe fragmentedor polarized societies much earlier/faster than classicalmodels would anticipate. Furthermore, we also find ahighly interesting mathematical effect in the simplicialadaptive voter model. It does happen frequently thatdue to re-wiring of edges, even with replacement of lostsimplices, that there are eventually no active simpliceswith different opinions left. This effect tends to occurbefore the final asymptotic dynamics of the voter processhas been reached, i.e., when there are no active edges leftwith different opinions. This leads to the conjecture thatdynamical models on simplicial complexes can displaya multiscale [26] hierarchy, where higher-order simplicesequilibrate before lower-order simplices do [9, 25]. II. THE ADAPTIVE SIMPLEX VOTER MODEL
In this section, we introduce the variants of the votermodels in more detail. We are going to provide some ba-sic background and references and then define the sim-plicial adaptive voter model.Consider a simplicial 2-complex S , which consistsof zero-dimensional 0-simplices (or vertices) V , one-dimensional 1-simplices (or edges) E and two-dimensional2-simplices T . Recall that for a simplicial complex onerequires that each face of a simplex is again in the simpli-cial complex, and that the non-empty intersection of twosimplices is a face for each of the two simplices. In our modelling context this makes particular sense since a tri-adic friendship does generally also contain friendships be-tween the respective three individuals. These friendshipsare represented by the faces of the 2-simplex, which arethe edges. We want to define a minimal adaptive votermodel on the space of simplicial complexes with verticeslabeled by two possible opinions. For notational simplic-ity and convenience we allow for two states − − , and = vertex of state 1.The possible edgesare either state-homogeneous ( inactive ) or state-inhomogeneous ( active ). The 2-simplices can occur inany of the following four configurationswhere the first and the last one are state- and edge-homogeneous, while the second and third are state- andedge-heterogeneous. Note that we use double-edges toindicate a 2-simplex in comparison to a triangle, i.e., afull subgraph on three nodes, which is not part of a 2-simplex. The interior of a 2-simplex is color-coded withcurrent majority opinion within the triangle.Let us recall the classical adaptive voter model on agraph ( V , E ). At each time step an edge e ∈ E is chosenat random and one of the following possibilities can thenoccur:(C1a) (“social avoidance”): If e is active, then with prob-ability p ∈ [0 ,
1] one of the vertices (chosen withequal probability) re-wires the edge to a vertex withits own opinion, which is chosen at random fromthe remaining vertices. We can represent this rulegraphically by: (cid:55)→ (cid:55)→ . Note that the probability p is a very crucial param-eter in the adaptive voter model.(C1b) (“personal discussion”): If e is active, then withprobability 1 − p one of the two vertices of the edge e is chosen at random with probability 1 / (cid:55)→ (cid:55)→ . (C2) (“inert situation”): If e is inactive, then nothinghappens.Of course, the rules imply a conservation law of thenumber of edges as well as the number of vertices, whichtends to be helpful to reduce dimensionality, simplify themathematical analysis, and to benchmark computationsby checking whether the conservation laws hold. Next,we describe a minimal model extension to include therole of 2-simplices, which is also meaningful for appli-cations. Consider the simplical complex ( V , E , T ) andanother probability q ∈ [0 , e at random at each eventtime step. Then we use the following rules:(R0) If e is not part of a 2-simplex, then the classicalrules apply.(R1) If e is part of at least one 2-simplex and of typethen with probability q a majority rule is imple-mented and with probability (1 − q ) the classicalrule (C1) is implemented. When the majority ruleapplies, one of the simplices attached to e is cho-sen and the majority persuades the minority withprobability p (cid:55)→ → . When the rewiring is chosen, irrespective of thepeer-pressure, all simplices attached to that edgebreak up, e.g., (cid:55)→ (cid:55)→ and triangles are chosen randomly for conversioninto a 2-simplex → → → → to preserve the total number of simplices.(R2) If e is part of at least one 2-simplex and not of typeit stays inert.The rules (R0)-(R2) are a very natural extension of theclassical adaptive voter model rules (C1)-(C2). Again,conservation of vertices and edges is guaranteed. Thenew rule (R1) tries to conserve simplices as long as pos-sible. However, due to re-wiring, one may eventuallynot have any triangles left that can be converted into2-simplices. A subsequent rewiring event within a 2-simplex would therefore reduce the overall content of 2-simplices and thus violate the conservation of simplices.This leaves two natural options: • We stop the simulation precisely at the first timewhen the triangles have been depleted. • We continue with the simulation despite the viola-tion of the simplex conservation until an absorbingstate of the Markov chain is reached.We will always indicate in our simulations, which op-tion we have chosen. Practically, all our main new resultsare just focusing on using the first option as it providesus with the regime, where the new 2-simplex rules arerelevant.
III. RESULTS
Before describing our simulation results for (R0)-(R2),we briefly recall the well-known results for the classicaladaptive voter model (C1)-(C2) on a graph.
A. The Clasical Co-evolving Voter Model
The classical co-evolving voter model [9, 37] corre-sponds to q = 0. There one observes two different phases— the active and the frozen phase — along the parame-ter p , separated by a fragmentation transition at p c . Inthe active phase ( p < p c ) the dynamics evolves towards aslow manifold of strictly positive active edge densities andthen follows a random walk along this manifold towardsa state with a giant component, all of whose membersare of the same state and all of whose active edges haveconsequently vanished; cf. Figure 1. In the frozen phase( p > p c ) the dynamics evolves towards a fragmented statein which two disconnected and internally state-uniformcomponents exist. These two phases still exist in thesimplical co-evolving voter model. This shows that ourmodel is really a minimal extension as several main ef-fects are preserved. We are interested in the behaviourof the transient and limiting behaviour as q is deformedaway from zero. When q > q >
0) by the peerpressure.It was observed in the classical case that it is helpful toview the dynamics within a compact region under a suit-able projection [37]. To this end let σ + and σ − denote therelative densities of the two opinions ± ≤ σ ± ≤ σ + + σ − = 1. Then we denotethe difference in opinion, i.e., the majority disparity by m = σ + − σ − . In a statistical physics context we can alsodraw the analogy of m to the magnetization, e.g., whenthinking of the classical Ising model. Furthermore, wedenote the active link density by ρ . It is very helpful touse the coarse-grained ( m, ρ )-coordinates to understandthe dynamics. The network initially looses active links.Either all active links are depleted directly without anyof the opinions becoming dominant in the process or theactive links reach a quasi-stationary density at a positivevalue. There it enters into a random walk on a neigh-bourhood of a parabolic-shaped region defined via therelation ρ = ξ p (1 − m ), for some constant ξ p indexedby p . In that case one opinion may gain the majority, as m deviates from zero along that region. Which of thesetwo scenarios happens depends on whether p < p c or p > p c ; see Figure 1. On the parabola the active edgedensity evolves much slower than initially, which is whyone may refer to this region as the slow or inertial mani-fold. Eventually the random walk hits either of the endpoints ( m, ρ ) = ( ± , ξ p is a characteristic -1 -0.5 0 0.5 1 m ; q=0.00q=0.75 ( = )q=0.75 (> = )fit parabola FIG. 1. We show two sample paths of the dynamics in the( m, ρ )-space at a rewiring probability of p = 0 .
55: The blackpath shows the scenario where peer-pressure is absent, i.e. q = 0. The light and dark green paths shows the scenariowhere the peer-pressure is at q = 3 /
4, respectively beforethe depletion of triangles at time τ and thereafter. The dot-ten grey lines are best fits of the paths to the parameterizedparabola ρ = ξ p (1 − m ), respectively for q = 0 and q = 0 . ξ p . The size of thenetwork is N = 500, its mean degree is µ = 8 and the simplex-per-edge degree is s = 0 .
2. There is an initial population oftriplets, such that the triplet-per-edge density is 0.8. p -dependent value of the slow manifold and correspondsto the quasi-stationary density of active links at vanish-ing majority disparity. In [37] they consider the time-average of all the quasi-stationary densities of survivingruns ρ surv , which is then taken as the order parameterof the model, however ξ p is another legitimate choice.In this work, we perform a numerical study of the q -deformed simplical co-evolving voter model. In the fol-lowing we describe the implementation and the results. B. Initialization
First we initialize a random simplicial complex( V , E , S ) and assign an equal amount of +1 and − N , of edges E and of 2-simplices S , or alternativelythe mean degree µ = 2 E/N and the 2-simplices-degreeper edge s = 3 S/E . An important aspect of the dynam-ics (R0)-(R2) is the simplex-preserving transformationof a triangle into a 2-simplex, whenever a 2-simplex is destroyed by a rewiring action. Due to these transfor-mations one also requires an initial population of trian-gles in the network, an amount T , so that a significantfraction of edges should be part of triangles. There areat least two ways to create a random simplicial com-plex from the data ( N, E, S, T ). One option is to pick E edges uniformly at random from the list of (cid:0) N (cid:1) pos-sible vertex pairs. However, there is a chance that notenough triangles are created to declare S of them as 2-simplices and T as triangles. Another method is to pick S + T triangles from the (cid:0) N (cid:1) combinations of unorderednon-repeating three-tuples. An amount S are declared2-simplices and the rest, i.e. T , remain triangles. Alledges that were thus created and are part of trianglesor 2-simplices become part of the edge set E . If thereare not yet E edges, ones picks uniformly at random theremaining edges from vertex pairs that are not yet partof the edge set. If more than E edges were introducedalready by forming triangles and simplices, the methodhas failed. The first method tries to reduce degree cor-relations and works well for large µ and very low s . Thesecond method aims to reduce correlations between thenumber of simplices per edge at the cost of degree cor-relations and is guaranteed to work when 3( S + T ) ≤ E .We choose the second method to allow for larger values of s . In the Appendix A we show the details of this method. C. Phase Portrait
For q > m, ρ )-space, on which the ac-tive edges evolve much slower. This is exemplified inFigure 1, where we compare two sample paths with peerpressures q = 0 and q = 0 .
75 at a rewiring rate p = 0 . −
1. It has two activeedges and suppose one is chosen for an update. If thenode with the minority opinion convinces the neighborwith the majority opinion, then there are still two activeedges in the simplex. If on the other hand the major-ity convinces the minority node there are none. Thus,the higher the probability of a majority rule, the higherthe tendency to reduce active edges. The same argumentholds of course for a 2-simplex with the opposite major-ity. This effect happens irrespective of system size, edge-or 2-simplex-densities, as long as they are positive.In Figure 2 we show the phase portraits for variousvalues of q with N = 500 at an edge-per-vertex degree µ = 8 and a simplex-per-edge degree s = 0 .
2. We ob-serve that the peer pressure shifts the critical thresholdto lower rewiring probabilities. This also follows fromthe previous observation. If the active edge densities arereduced by the majority rule, then the active edges willvanish at lower rewiring probabilities. This means that p p q =0.00q =0.25q =0.50q =0.75 FIG. 2. We compare the the phase portraits for various peer-pressures. The estimates of the order parameters ξ p , i.e. theapexes of the parabolic regions (c.f. Figure 1), are plotted fora range of rewiring probabilities between 0 and 1 and peer-pressures q ∈ { , . , . , . } , which are respectively color-coded with black, red, blue and green. Each data point isgenerated from 200 runs and shows the mean and the standarddeviations of the fitted values for ξ p . Here, the size of thenetwork is N = 500, its mean degree is µ = 8 and the simplex-per-edge degree is s = 0 .
2. We also have an initial populationof triplets, such that the triplet-per-edge density is 0.8. the slow manifold is never reached and consequently afragmentation takes place rather than a random walk to-wards any of the single-opinioned final states. We alsonote that the variance of the order parameter increasestowards the fragmentation threshold, as is expected.
D. Depletion of Triangles
In the classical co-evolving voter model there is a de-pletion of active edges. One crucial feature of the co-evolving voter model on a simplicial complex is the de-pletion of the higher-order structures, which in our caseare the 2-simplices and the triangles. Heterogeneous sim-plices are either homogenized by the peer pressure or de-stroyed via rewiring. As simplices are destroyed new onesare created by converting a randomly chosen triangle intoa 2-simplex. In some parameter regimes this process de-pletes triangles at a higher rate than their productionvia rewiring. Thus, in these regimes there is a finite firsttriangle-depletion time τ , at which no triangles are leftfor conversion into 2-simplices.In Figure 3 we show the average inverse triangle-depletion time (cid:104) τ − (cid:105) for a range of rewiring probabilitiesand peer pressures. A value of zero implies an infinite p h = i -4 q =0.00q =0.25q =0.50q =0.75 FIG. 3. We show the average inverse depletion time of tripletsfor rewiring probabilities in the entire range between 0 and1 and for peer pressures q ∈ { , . , . , . } . All otherparameters, i.e. N, µ, s are as above. depletion time, or one that is as long as the duration of asimulation. Of course for p = 0, in the absence of rewiringevents, the depletion time is infinite. As the rewiringprobability is increased, we observe a rise of (cid:104) τ − (cid:105) upto a point where a maximum is reached, succeeded by adrop back to zero around the fragmentation threshold.This qualitative behavior can be observed for all valuesof q , with the additional effect that (cid:104) τ − (cid:105) is lowered as q becomes larger. In the following we explain this be-haviour.First, we note that there is a net-decay of triangles onlywhen on average more of them are destroyed than pro-duced. The only way that a destruction or production oftriangles can occur is by way of rewiring events. We alsoremark that the chance for a rewiring to turn a triplet ofvertices with two edges into a triangle with three edgesrises as the mean degree of the network increases. If atthe same time the rewiring link, prior to its rewiring, hasa lower probability of being part of a triangle, then thereis a net production of triangles.With this in mind we can explain, both the rise andthe fall of the curves as p is increased. When p is get-ting close to the fragmentation transition, the densityof active edges ρ decreases. Consequently, the densityof ++-links ( −− -links), say ρ + (resp. ρ − ), is higherthan in uncorrelated networks and so is the mean degree µ ± = 2 ρ ± E/N ± of the subgraphs with +1 states (resp. − N ± are their respective sizes. An ap-proximation of the respective mean degrees µ ± via thesimplifying assumption ρ + /ρ − ≈ N /N − , correspond-ing to an uncorrelated scattering of edges on the twocomponents, and the identity N m = N + − N − yields µ ± ≈ µ (1 − ρ )(1 ∓ m ) to first order in m . See the ap-pendix B for a derivation. This relation supports theintuition that the minority component is becoming moredensely connected as active edges are depleting and | m | is increasing. Therefore, as the active link density de-creases a rewiring event has an increasing probability toproduce a triplet rather than to destroy one, given thesparseness of active edges. This effect is enhanced as p is approaching the fragmentation transition with lowquasi-stationary values of ρ . There, the production maycompete with the loss leading to a positive stationaryabundance of triangles and a diverging triangle deple-tion time τ . This effect is of course more pronouncedfor higher peer pressures as they tend to reduce ρ evenfurther.We now argue for the initial rise of (cid:104) τ − (cid:105) . As longas triangles are not produced to a level that compen-sates their loss, as it happens close to the transition,there is an exponential depletion of them after a finitetime. The rate of this depletion depends on the rewiringprobability. The higher p , the faster they deplete. Thiseffect is opposing the one mentioned before. One maythink of it as follows: The values of ρ and m determinehow many triangles plus simplex-declared triangles, i.e. T + S , the network can maintain on average. The arrivalat that quasi-stationary state happens at an exponentialrate that is of course proportional to p . If, however, thatstate can only support less than an amount S of trianglesand 2-simplices, then the triangles will die very likely andwill do so much quicker when p is higher. The effect ofpeer pressure is that ρ is decreased and the majority dis-parity in the network is enhanced. Both of these implyan increased production of triangles and thus a longertriangle depletion time.Finally we also note that beyond the fragmentationthreshold the dynamics hits the absorbing fragmented ρ = 0 state directly without entering the parabolic quasi-stationary region. Therefore it hits an absorbing stateever quicker as the rewiring probability p takes ever largervalues. In that regime one cannot make very meaningfulstatements about the triplet depletion time τ , since it isbounded by the extinction time of the active edges.We now discuss some direct consequences of the trian-gle depletion. The conservation of simplices is lost oncethere are no convertible triangles left. This implies alsothat the simplex-degree per edge s decreases and in turnresults in a weakening net effect of the peer pressure upto a point, where possibly all 2-simplices are gone. Thus, s ceases to be a fixed parameter of the model. In order toexamine the pure effect of the peer pressure q at a givenparameter set ( N, µ, s ) we therefore measure quantitiesonly as long as the number of simplices is conserved, i.e.,until the triangle depletion time τ .Consider a rewiring rate below the fragmentation tran-sition, i.e., p < p c . If τ is less than the time at whichthe slow manifold is reached, then one cannot take un-biased data from the slow manifold, as 2-simplices havealready diminished. When triangles are depleted after p h d ; /dt i -4 q =0.00q =0.25q =0.50q =0.75 FIG. 4. We show the average initial depletion rate of activeedges for rewiring probabilities in the entire range between 0and 1 and for peer pressures q ∈ { , . , . , . } . All otherparameters, i.e. N, µ, s are as above. reaching the slow manifold, but before reaching an ab-sorbing state, then one may take unbiased data from theslow manifold for times less than τ . This data is suit-able for computing the apex ξ p of the parabola that isfitted through the sample path in the ( m, ρ )-plane. It isnot suitable for extracting the average quasi-stationaryactive link density of surviving runs, ρ surv , because sys-tematic biases towards higher values would be introducedwhen data is only taken until some time τ . Consequentlywe plot ξ p rather than ρ surv . E. Depletion of Active edges
There is a fast evolution, before the dynamics reacheseither the slow parabolic manifold or a fragmented state.During this fast evolution active edges are converted intoinactive edges by persuasion until the quasi-stationarydensity is reached. In Figure 4 we plot the average initialdepletion rate of active edges for various values of p and q . Higher peer pressures enhance the depletion rate, asexpected. This effect is diminished for higher rewiringrates up to the point p = 1 where persuasion does notexist anymore and the peer pressure has no effect. F. Diffusion and Drift Velocity
We have seen that the peer pressure enhances the de-pletion rate of active edges during the fast dynamics ei-ther towards fragmentation, in the regime p > p c , orbefore reaching the slow manifold, for p < p c . Here, p sqrt h dm /dt i -6 q =0.00q =0.25q =0.50q =0.75 FIG. 5. This plot shows the diffusivity of the dynamicsalong the direction of m at m = 0. It is measured interms of the mean rate of change of m for rewiring prob-abilities in the range between 0 and 1 and for peer pressures q ∈ { , . , . , . } . All other parameters, i.e. N, µ, s areas above. we study the transient behavior towards the absorbingstates. As can be seen in Figure 1 the parabolic regionis not as well explored for q = 0 .
75 as it is for q = 0.For q = 0 .
75 it can be seen often, as in the figure, thatonly one side of the parabola is visited, indicating thata majority grows once a bias towards one opinion exists.We investigate this behavior by looking at the averagerate of change of the majority disparity and its squaredvalue, which are proxies for drift and diffusion respec-tively. A pure Brownian motion has a vanishing drift.Therefore, we are interested in the extend to which theprocess is not like a pure Brownian motion. A processwith a higher drift is expected to hit the absorbing statesat m = ± τ (c.f.Subsection III D).Consider a sample path ( m t , ρ t ) . The mean rate ofchange of the squared distance from m = 0 at time s isgiven by (cid:28) dd t (cid:12)(cid:12)(cid:12) s m t (cid:29) . (1)We are interested in this quantity not at any arbitrarypoint, but precisely as the dynamics is on the slow man- Formally a sample path ω t is a dicrete time-indexed path in thestate space Ω, which in this case is the space of {− , } -labelledsimplicial complices ( V , E , S ) and m and ρ are random variableson Ω. ifold and has vanishing m . Hence, we consider this aver-age, conditional on the event that m s = 0 for some time s at which the slow manifold has been reached. For aone-dimensional Brownian motion along the m -directionthis quantity would be twice the diffusion constant D . Inour simulations we measure it as follows. First we evolvethe dynamics until the quasi-stationary parabolic regionis reached. After that, whenever the dynamics passesthrough the m = 0 line we approximate the average rateof change (1) via finite differences at that time instance.This is then repeated for many runs and initializations,keeping the parameters fixed.In Figure 5 we show, how the diffusion (1) changesas the rewiring and peer pressure are varied. The firstobservation is that the peer pressure does not influencethe diffusion very much. There is even a slight tendencytowards lower diffusions for higher peer pressures. Asthe rewiring probability increases the diffusion decreaseslinearly for all peer pressures. There are two reasonsthat account for this effect, both of which are based onthe fact that only persuasion can change m . First ofall, given a certain amount of active edges one expectsless changes in m for lower persuasion probabilities, i.e.,higher rewiring probabilities. Secondly, higher rewiringprobabilities decrease the quasi-stationary level of activeedges, so that persuasions cannot yield as much changein majority disparity.We also look at the average drift velocity of the ma-jority disparity. Since the definition of the dynamics andhence its probability laws are invariant with respect tothe discrete symmetry of interchanging the opinions, weexpect on average no drift velocity at m = 0. Therefore,we study the average drift velocity at some non-zero ma-jority disparity m + (cid:54) = 0: (cid:28) dd t (cid:12)(cid:12)(cid:12) s m t (cid:29) , where s is such that m s = m + . We expect that thedrift velocities at − m + should be minus the drift at m + ,in the sense of their probabilitiy laws. This is due tothe discrete reflection symmetry. Hence the quantity ofinterested is the radial drift velocity, pointing away fromthe origin.In Figure 6 we plot the average radial drift velocityat m + = 0 .
2. We find that there is no drift velocityat q = 0. The majority disparity in the classical co-evolving voter model is therefore more like a pure diffu-sion process. When q > q the strongeris the average drift velocity towards the extreme points m = ±
1. This effect can be explained by the ratio ofheterogeneous 2-simplices. When m t = m + there aremore -simplices than -simplices. Thus the major-ity rule tends to further increase the amount of nodes.The argument holds mutatis mutandis for m t = − m + .These drifts also explain why it becomes more unlikelyto change majorities once there is a bias. p -0.500.511.522.533.544.5 h d m/dt i |m|=0.2 -5 q =0.00q =0.25q =0.50q =0.75 FIG. 6. We show the average radial drift velocity of m at m = ± . q ∈ { , . , . , . } . Thereare no data points for rewiring rates above the fragmentationtransition, because the dynamics does not enter the slow man-ifold on which it can explore the regions of higher majoritydisparity m . All other parameters, i.e. N, µ, s are as above.
In summary, we also see in the slow regime on theparabolic-shaped manifold a similar effect as in the fastregime. The peer pressure enhances the drift towards asingle majority opinion, i.e., polarization is enhanced.
IV. CONCLUSIONS & OUTLOOK
We have shown, how to naturally (from the viewpointof applications) and minimally (from the mathematicalperspective) extend the adaptive voter model to a modelon simplicial complexes. It seems now plausible as fur-ther steps to also extend other adaptive contact processesto simplicial complexes, e.g., epidemic spreading models.Then we demonstrated that the main structural featuresof the adaptive voter model remain in the simplicial ver-sion, which still yields a fragmentation transition uponvarying the re-wiring rate. Yet, the quantitative proper-ties are changed and we observe faster transitions to asingle-opinion absorbing state or towards a fragmentedtwo-opinion state. This is in line with heuristic argu-ments that peer pressure effects may lead to polarization;in fact, our model seems to be one of the simplest math-ematical manifestations of this effect. We expect thatsimilar effects do occur in far more complicated modelsinvolving very large-scale higher-order structures such asinteractions on social networks. Furthermore, we foundthat the simplicial adaptive voter model often displaysmultiple time scales, where the higher-order 2-simplices die out out before the active edges decay to zero. Thismultiscale effect leads one to the conjecture that simpli-cial dynamics models could be analyzed order-by-orderwith respect to the dimension of the simplices.
Acknowledgments:
LH thanks the Austrian Science(FWF) for support via the grant Project No. P29252.CK thanks the VolkswagenStiftung for partial supportvia a Lichtenberg Professorship and the EU for partialsupport within the TiPES project funded the EuropeanUnions Horizon 2020 research and innovation programmeunder grant agreement No 820970.
Appendix A: Algorithm for the Initialization of aRandom Simplicial Complex
Let there be N vertices. We would like to draw uni-formly at random T distinct elements from the set of allunordered K -tuples without repetition. For the purposeof this paper we are randomly sampling triangles fromthe vertex list, so that K = 3, but we expose the methodmore generally. This set has cardinality (cid:0) NK (cid:1) . Storingsuch a list is prohibitively large when N is large and K larger than 2 and smaller than N −
1. Thus it is ad-vantageous to find an algorithm that needs less storagecomplexity. The idea of the algorithm is to take a nat-ural number n and find the corresponding K -tuple in colex , which is one of the two canonical orderings definedon unordered tuples without repetitions. We recall that u = ( u , u , . . . , u K ) is less than v = ( v , v , . . . , v K ) incolex if and only if u k < v k for the last k where u k and v k are different. This mapping from n (cid:55)→ ( u , u , . . . , u K ) isdone by iteratively determining the entries, starting fromthe last one. Suppose the k + 1-th entry u k +1 was foundto be i + 1, where 0 ≤ i ≤ ( N − k -th entry. To this end we define F i,k asthe number of tuples whose k -th entry, u k , equals i , withall entries above the k th kept fixed, since they are alreadydetermined. It is given by F i,k = (cid:0) i − k − (cid:1) , because we haveto fill k − j − K -tuples whose k + 1-th entry is known to be u k +1 = i + 1 is denoted by G i,k . It is simply given by G i,k = (cid:80) i(cid:96) = k F (cid:96),k , which sums up all the tuples whose k -th entry is less than or equal to i , again keeping higher en-tries fixed. In other words G i,k = F k,k + F k +1 ,k + · · · + F i,k and clearly there cannot be any summands where (cid:96) < k Unordered K -tuples without repetitions are simply sets of theform u = { u , u , . . . , u K } of cardinality K , for which the orderof their elements doesn’t matter. When u i ∈ N , one may howeverassociate to it uniquely a particular choice of an ordered K -tupleˆ u = ( u , u , . . . , u K ) where u k < u (cid:96) if and only if k < (cid:96) . Thisis then a unique representation of the unordered tuple u in therealm of ordered tuples, which is of course a much larger space.In abuse of notation we denote the representation ˆ u of u in the setof ordered tuples again by u . The colex order is strictly speakingdefined via the representation ˆ u , rather than u . in colex. One may find F and G iteratively F i +1 ,k = F i,k ii − k + 1 G i +1 ,k = G i,k + F i,k . where we use the hockey-stick identity for binomial coef-ficients.The last entry, u K , is easy to determine as follows: Wenote that the first G K,K tuples end with K . There is lit-erally only one, namely (1 , , , . . . , K ). The first G K +1 ,K tuples end with K or K + 1 and there are 1 + K of them.The first G j,K tuples end with any number smaller orequal to j . Thus we find u K by searching for j such that G j,K ≥ n > G j − ,K , or equivalently min { j : G j,K ≥ n } .Now, that we know u K , our problem reduces to find-ing the right K -tuple in the range between G u K − ,K and G u K ,K . This is equivalent to finding the K − n (cid:48) = n − G u K − ,K , with the slight modifi-cation that these tuples must be taken from the smallerset { , . . . , u K − } . So now there are G j,K − such tu-ples whose ( K − th entry is j or less. We determinethe ( K − th entry again by finding the minimal j suchthat G j,K ≥ n (cid:48) . We repeat this procedure until all en-tries have been determined. So in general we have thefollowing iterative algorithm that uses an auxiliary set ofvariables n = ( n , n , . . . , n K ), initialized as n K = n : u k = min { j : n k ≤ G j,k } n k − = n k − G u k − ,k . We also define G k − ,k := 0 to resolve the problem thatarises when u k = k . Appendix B: Approximation of thesubgraph-connectivities µ ± in the ( m, ρ ) − space Let G be a graph with N vertices, E edges and meandegree µ = 2 E/N . Its nodes are either in the +1 or − G ± ⊆ G be the subgraph consisting of all thenodes in the ± G ± by µ ± . We also define three types oflinks (++),( −− ) and (+ − ) and their respective densities ρ + , ρ − and ρ as a fraction of E . Thus we have ρ + + ρ − + ρ = 1 . (B1)The number of nodes in state +1 or -1 are N + or N − andtheir respective fractions of the entire set of vertices is denoted by σ + = N + /N or σ − = N − /N . Their differenceis denoted by m = σ + − σ − . Therefore N ± = N (1 ± m ),which can be used to obtain the following expression forthe mean degree in the subgraphs G ± with E ± = ρ ± E edges: µ ± := 2 ρ ± E/N ± = 4 EN (1 ± m ) ρ ± = 2 µ ± m ρ ± (B2)So far no assumptions were made about the graph. Wewould like to approximate the link densities in terms ofthe effective coordinates m and ρ . When m = 0 a rea-sonable assumption is ρ + = ρ − due to symmetry, whichimplies by (B1) that ρ ± = (1 − ρ ) and consequentlythat µ ± (cid:12)(cid:12) m =0 = µ (1 − ρ ) (B3)at m = 0. One may obtain an approximation for the casewhen m deviates slightly from 0 by requiring that it satis-fies (B3) for m →
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