Interacting opinion and disease dynamics in multiplex networks: discontinuous phase transition and non-monotonic consensus times
aa r X i v : . [ phy s i c s . s o c - ph ] M a y Interacting opinion and disease dynamics in multiplex networks:discontinuous phase transition and non-monotonic consensustimes
F´atima Vel´asquez-Rojas and Federico Vazquez
IFLYSIB, Instituto de F´ısica de L´ıquidos y SistemasBiol´ogicos (UNLP-CONICET), 1900 La Plata, Argentina (Dated: October 8, 2018)
Abstract
Opinion formation and disease spreading are among the most studied dynamical processes oncomplex networks. In real societies, it is expected that these two processes depend on and affecteach other. However, little is known about the effects of opinion dynamics over disease dynamicsand vice versa, since most studies treat them separately. In this work we study the dynamics of thevoter model for opinion formation intertwined with that of the contact process for disease spreading,in a population of agents that interact via two types of connections, social and contact. These twointeracting dynamics take place on two layers of networks, coupled through a fraction q of linkspresent in both networks. The probability that an agent updates its state depends on both, theopinion and disease states of the interacting partner. We find that the opinion dynamics has strikingconsequences on the statistical properties of disease spreading. The most important is that thesmooth (continuous) transition from a healthy to an endemic phase observed in the contact process,as the infection probability increases beyond a threshold, becomes abrupt (discontinuous) in thetwo-layer system. Therefore, disregarding the effects of social dynamics on epidemics propagationmay lead to a misestimation of the real magnitude of the spreading. Also, an endemic-healthydiscontinuous transition is found when the coupling q overcomes a threshold value. Furthermore,we show that the disease dynamics delays the opinion consensus, leading to a consensus time thatvaries non-monotonically with q in a large range of the model’s parameters. A mean-field approachreveals that the coupled dynamics of opinions and disease can be approximately described by thedynamics of the voter model decoupled from that of the contact process, with effective probabilitiesof opinion and disease transmission. . INTRODUCTION The formation of opinions and the propagation of an epidemic disease on a population ofindividuals are among the most studied dynamical processes on complex networks [1, 2]. Thebehavior of each of these two processes has been explored independently of one another forthe last decades, and many of their propagation properties on diverse complex topologies arewell established already (see [2] and [3] for recent reviews on opinion formation and epidemicspreading, respectively). However, less attention has been paid to a possible case scenariowhere the dynamics of opinions interact with that of the disease spreading. In fact, it ishardly expected that these two dynamics are isolated in real societies but rather depend onand affect each other, since they both run at the same time on the same population: anindividual can transmit a disease to a colleague while having a conversation and exchangingideas or opinions on a given topic. Then, the following questions arise: does the dynamicsof opinion formation have an impact on the extent and prevalence of the epidemic? Doesthe disease spread facilitate the ultimate dominance of one opinion, or does it rather hinderthe consensus of opinions?In an attempt to explore these questions, we study in this article how opinion formationand disease spreading processes affect each other, using two simple models as a proxy ofeach process: the voter model (VM) and the contact process (CP). The VM was originallyintroduced as the simplest system of interacting particles that can be exactly solvable inany dimension [4–6], and is one of the most studied models for opinion consensus. In thismodel, individuals can take one of two possible positions or opinions on a given issue, andare allowed to update them by adopting the opinion of a randomly chosen neighbor. The CP,on its part, has been extensively studied to explore the spread of an infection in a systemof interacting agents [7], where infected agents can transmit the infection to susceptibleneighbors in a lattice [8] or a complex network [9], and they can also recover at a givenrate. The CP exhibits a continuous transition from a healthy to an endemic phase when theinfection rate exceeds a threshold value. To model the interaction between the two dynamicswe implement the framework of multilayer complex networks [10–12] that consists of a set ofcomplex networks interrelated with one another, which allows to study systems composed bymany interdependent processes. In the present study we consider that the opinion dynamicstakes place on a network of social relations –formed by individuals that influence each other2n a social issue, while the disease spreads on a network of physical contacts –formed bypeople having daily face-to-face contacts. All individuals are in both layers of networks,but the pattern of connections between them may be different in each layer. The overlap ofconnections is taken as a measure of the coupling between the two networks.The bilayer network system described above may represent a simple case scenario wherethe social network supports a process that involves peer pressure, like the adoption of newbehaviors or opinions, while the contact network supports the spreading of a contagious viralinfection like flu, which is transmitted by proximity or direct contact between individuals.The different combinations of connections may correspond to different types of relationshipsbetween two individuals. For instance, two close friends can have both a contact and socialtie, as they can see each other at work every day and also interchange ideas on a politicalissue. But it can also happen that individuals are connected by only one type of tie; e.g.,two colleagues having a contact or proximity relation because they work in the same placebut never talk about politics; or two friends that never meet but discuss political ideas byelectronic means (phone, Facebook, Twitter, email, etc).Some related works on multilayer networks [13–19] have also explored the interrelationbetween two information spreading processes. For instance, in references [13, 14] the authorsanalyzed how the awareness of a disease affects the epidemic spreading on a multiplex net-work, by using the unaware-aware-unaware and the susceptible-infected-susceptible cyclicmodels, respectively. The interplay between opinion formation and decision making pro-cesses was studied in [15] using two interconnected networks. Another work considered twopolitical parties (two interacting networks) that compete for votes in a political election[16]. In a recent article [17], the authors studied the dynamics of the voter model on bilayernetworks with coevolving connections, while in [18] the same authors explored whether isappropriate to reduce the dynamics of the voter model from a two-layer multiplex networkto a single layer. A recent work [20] considers a complex threshold dynamics that competeswith a simple Susceptible-Infected-Susceptible dynamics on two interconnected networks.All the works listed above explore the interplay between two social or two epidemiologicalprocesses that are alike. However, there is a lack of specific studies on the interplay betweenopinion and disease dynamics.In this article we show that the dynamics of opinions has striking consequences on thedisease spreading and vice versa. The nature of the healthy-endemic transition observed in3he CP, as the infection probability increases, is largely modified by the dynamics of theVM. The transition changes from continuous to discontinuous when the disease and opinionsare coupled, showing a jump in the disease prevalence at the transition point, where themagnitude of the jump increases with the coupling. Also, a discontinuous transition froman endemic to a healthy phase is found when the coupling overtakes a threshold value. Inaddition, we find that the dynamics of the CP has important consequences in the dynamicalproperties of the VM. The diffusion of opinions is slowed down by the disease in a non-trivial manner as the coupling increases. This leads to consensus times that vary eithermonotonically or non-monotonically with the coupling, for a large range of the model’sparameters. We develop a mean-field approach to study the time evolution of macroscopicquantities, which takes into account state correlations between neighbors in the same network(pair approximation). This approach reveals that the interdependent system of opinions anddisease can be thought of as two independent systems, with external parameters that dependon the coupling. Specifically, the opinion dynamics can be approximated as the dynamicsof the VM on an isolated network, with an effective probability of opinion transmissionthat decreases with the coupling and the prevalence. Analogously, the disease spreadingis approximately described by the CP dynamics on an isolated network, with an effectiveinfection probability that decreases with the coupling and the fraction of neighbors withdifferent opinions.The article is organized as follows. In section II, we introduce the multiplex frameworkand the dynamics of the model on each layer. We present simulations results in section IIIand develop an analytical approach in section IV. Finally, in section V we give a summaryand conclusions.
II. THE MODEL
We consider a bilayer system composed by a contact and a social network layer of meandegree h k i = µ and N nodes each. These two layers are interrelated through their nodes,which are the same in both networks, while links connecting nodes may not necessarilybe the same. That is, both layers have the same number of nodes N and links µN/ q (0 ≤ q ≤
1) of links shared by both networks. In our model,4he extreme values q = 0 and q = 1 correspond to totally uncoupled and totally couplednetworks, respectively. To build this particular topology, we start by connecting the samepairs of nodes at random in both networks until the number of links reaches the overlapvalue q µN/
2. Then, the rest of the links (1 − q ) µN/ i and j are connected by both a social and a contact link, representing individualsthat have a daily face-to-face conversation, where they interchange opinions and also onecan infect its partner. Nodes j and k are only connected by a social link: they do not haveface-to-face contacts but still exchange ideas electronically or by phone. Nodes i and k areonly connected by a contact link: they have face-to-face or proximity contacts but they donot discuss and interchange opinions about the given issue.To mimic the spreading of opinions and the disease we use the voter model (VM) andthe contact process (CP) on each layer, respectively. Each node is endowed with an opinionstate O that can take two possible values O = + , − (see top layer of Fig. 1), and a diseasestate D = 0 , . p o ≤ . IG. 1: Schematic diagram showing a small part of a two-layer multiplex network. The top layerrepresents a social network supporting the propagation of opinions, while the middle layer describesa network of physical contacts on which a disease spreads. The bottom layer is the collapse of bothlayers, showing nodes connected by social (solid lines) and contact (dashed lines) links. Nodestates are susceptible (1) and infected (0) in the contact network, and follow the contact processdynamics, while + and − states in the social network are updated according to the voter modeldynamics. supposed to affect the probability of social interactions between them. Therefore, the socialinteraction probability is not reduced by the disease and, for simplicity, is set to 1 . . p d ≤ . β , leading to effective infection probabilities β and β p d ≤ β in each respective case. In a situation where there is a contact but not a socialconnection between two neighbors (they see each other but they don’t talk about politics),opinions are not expected to affect (neither increase or decrease) the contact probability.Therefore, this can be considered as an intermediate situation respect to the two casesmentioned above, where the contact probability should be smaller than 1 . p d , leading to an infection probability between β and β p d . However, for simplicity weassume that the contact probability in the absence of a social relation is the same as thatin homophilic relations (1 . β . Thisapproximation and the one mentioned above for the social interaction probability have theadvantage of reducing the number of free parameters, allowing for a deeper analysis of themodel which already exhibits a very rich behavior as we shall see.We now define the dynamics of the model according to the interaction properties discussedabove. In a single time step ∆ t = 1 /N an opinion and a disease update attempt take placein each network, as we describe below (see Fig. 2). Opinion Update [Fig. 2(a)]: a node i with opinion O i and one of its neighbors j withopinion O j are randomly chosen from the social network. If O i = O j nothing happens. If O i = O j , then i copies the opinion of j ( O i → O i = O j ) with probability p o if there isa contact link between i and j , and at least one of the two nodes is infected ( D i = 1 or D j = 1). Otherwise, i e., if there is no contact link or D i = D j = 0, then i copies j ’s opinionwith probability 1 . Disease update [Fig. 2(b)]: a node i with disease state D i is chosen at random from thecontact network. If D i = 0 nothing happens. If D i = 1, then i recovers with probability 1 − β or, with the complementary probability β node i tries to infect a randomly chosen neighbor j ,as long as it is in the susceptible state ( D j = 0). The infection happens ( D j = 0 → D j = 1)with probability p d if there is a social link between i and j , and O i = O j . Otherwise, i e., ifthere is no social link or O i = O j , then node j is infected with probability 1 . . p o ≤ β between two7 IG. 2: Update rules in the coupled opinion-disease system. (a) Opinion update. Node i adoptsthe opinion of its neighbor j with probability 1 . . p o ≤ . i recovers with probability 1 − β or transmits thedisease to a susceptible neighbor j with probability β when both nodes are only connected by acontact link, or when they are connected by both types of links and they share the same opinion.In case they hold opposite opinions the transmission happens with probability β p d ≤ β . neighbors, which is reduced to β p d ≤ β only in the case they are attached by a social linkand they share different opinions [see Fig. 2(b)]. III. NUMERICAL RESULTS
The CP and the VM are two of the most studied dynamical processes [6]. A relevantfeature of the CP is the existence of a transition from a healthy phase to an endemic phaseas the infection probability overcomes a threshold value β c . The healthy phase is static, as8ll nodes are susceptible and infection events cannot occur. The endemic phase is active,where each node undergoes an infected-susceptible-infected cycle and the total number ofinfected nodes fluctuates around a stationary value. The healthy-endemic transition is con-tinuous, and the critical value β c depends on the topological properties of the network [9].For its part, the VM has been extensively used to explore opinion consensus on differentnetwork topologies [24–29]. It was found that the diffusion properties of opinions dependon the heterogeneity of the network. This is reflected in the mean consensus time, which isproportional to the ratio µ /µ [28, 29], where µ and µ are the first and second momentsof the network’s degree distribution.The behavior described above is particular of each model on single isolated networks.In order to explore how the properties of these two processes are affected when they arecoupled through a multiplex network, we run extensive Monte Carlo (MC) simulations ofthe model described in section II, using two Erd¨os-R´enyi (ER) networks of mean degree h k i = µ = 10 each. Initially, each node in the system is infected with probability 1 /
2, andadopts either opinion state + or − with equal probability 1 /
2. That is, the system startsfrom a symmetric initial condition with roughly 1 / (cid:2) +0 (cid:3) , (cid:2) +1 (cid:3) , (cid:2) − (cid:3) and (cid:2) − (cid:3) .In the next two subsections we study separately the effects of one dynamics over theother. A. Effects of opinion formation on disease prevalence
We start the analysis of the model by describing the results related to the effects of opinionformation on the properties of disease spreading. In Fig. 3 we show the stationary fractionof infected nodes averaged over many independent realizations of the dynamics, h ρ stat i , as afunction of the infection probability β . For this first set of simulations we used p o = p d = 0,which corresponds to the extreme case scenario where opinions cannot be transmitted acrosscontact neighbors (nodes connected by a contact link) that are infected, and infectionsare not allowed between social neighbors (nodes connected by a social link) with differentopinions. Different curves correspond to different values of the coupling parameter q andnetwork size N , as indicated in the legend. We observe that, for q = 0 . q = 0 . h ρ stat i decreases smoothly with β until a point β cq that depends9 .52 0.56 0.6 0.64 0.68 β <ρ > q=0.0, N=10 q=0.0, N=3x10 q=0.0, N=3x10 q=0.4, N=10 q=0.4, N=3x10 q=0.4, N=3x10 q=0.7, N=10 q=0.7, N=3x10 q=0.7, N=3x10 s t a t FIG. 3: Average stationary fraction of infected nodes h ρ stat i vs infection probability β on twocoupled ER networks of mean degree µ = 10 and N nodes each, for p o = p d = 0 and couplingparameters q = 0 (circles), q = 0 . q = 0 . N = 10 , × and 3 × , as indicated in the legend. Theaverage was done over 5000 independent realizations starting from configurations consisting on afraction close to 50% of infected nodes uniformly distributed over the contact network and 50% of+ opinions uniformly distributed over the social network. on q , where it suddenly decays to a value close to zero. The sudden decrease in h ρ stat i becomes more abrupt as N increases, leading to a discontinuous change of h ρ stat i at β cq in thethermodynamic limit ( N → ∞ ). This behavior is reminiscent of a discontinuous transition.We also see that the jump in h ρ stat i decreases with q and vanishes for the uncoupled case q = 0, where the transition becomes continuous, in agreement with the known behavior ofthe CP on isolated networks. The critical point β c ≃ .
53 for q = 0 agrees very well withthe one found in previous numerical and analytical works [9].These results show that the dynamics of opinions has a profound effect on the statisticalproperties of disease spreading, changing the type of phase transition in the CP from acontinuous transition in the absence of coupling (when the two dynamics are independent)to a discontinuous transition when the dynamics are coupled.In order to achieve a deeper understanding of the nature of this transition we studied thetime evolution of the fraction of infected nodes ρ ( t ) for the case q = 0 .
4, where the transitionpoint is β c . ≃ .
58 (see Fig. 3). Solid lines in Fig. 4 correspond to results for networks ofsize N = 10 . As we can see, for β > β c . ≃ .
58 the average value of ρ ( t ) over many10 time <ρ > <ρ > stat (a) ρ stat time time ρρ (0) ρ (0.4) ρ (b) (c) (0.4) ρ + - T (0) (0.4) β=0.6 β=0.57 (0.4) + - ρ FIG. 4: (a) Time evolution of the average fraction of infected nodes h ρ i on a bilayer system withcoupling q = 0 .
4. Solid lines correspond to networks of size N = 10 , while open circles are fornetworks with N = 10 nodes. Curves correspond to infection probabilities β = 0 .
60, 0 .
59, 0 . .
57 (from top to bottom). Horizontal dashed lines indicate the stationary values for β = 0 . ρ , and thefraction of + − social links, ρ + − , in two distinct realizations for q = 0 . β = 0 . β = 0 .
57 (c). The evolution of ρ is also shown for q = 0 in both panels. realizations, h ρ ( t ) i , varies non-monotonically with time and asymptotically approaches astationary value h ρ stat i that depends on β , while h ρ ( t ) i decays to zero for β < β c . . Thatis, this non-monotonicity in h ρ ( t ) i makes h ρ stat i jump from a value close to zero for β <β c . ( h ρ stat i ≃ . β = 0 .
57) to a much larger value for β > β c . ( h ρ stat i ≃ . β = 0 . h ρ stat i is in the dynamicnature of the infection probability during each single realization, which can take two possiblevalues: either the value β p d = 0 a cross a contact link that overlaps with a + − social link,or the value β otherwise (simulations correspond to p d = p o = 0). In other words, theinfectivity across a given link i − j may switch between 0 and β over time, depending onthe opinion states of nodes i and j . This gives an average infection rate over the entiresystem that fluctuates according to the evolution of the fraction of + − links, ρ + − ( t ), inone realization. We shall exploit this observation in section IV to develop a mean-field11 .5 0.6 0.700.10.20.30.40.5 fr ac ti on o f i n f ec t e d nod e s β ρ ∞ <ρ > (a) (b) (c) statstat FIG. 5: Stationary fraction of infected nodes vs β , for couplings q = 0 (a), q = 0 . q = 0 . N = 10 , whilefilled symbols correspond to an average over 5000 realizations on networks of N = 10 nodes.In panel (a), open symbols overlap with filled symbols. Dashed curves represent the theoreticalapproximation from Eq. (11). (MF) approach for the evolution of the system. In panels (b) and (c) of Fig. 4 we plot ρ and ρ + − in a single realization of the dynamics, for q = 0 . β . For β = 0 . > β c . [panel (b)] we observe that ρ displays large variations up to a time T ≃ ρ + − becomes 0, after which ρ fluctuates around a stationaryvalue ρ stat ≃ .
255 (horizontal dash-dotted line), while for β = 0 . < β c . [panel (c)] ρ rapidly decays to zero, before ρ + − reaches zero. When ρ + − becomes zero [panel (b)] only++ or −− links remain and, therefore, the disease dynamics behaves as the one of thestandard CP with infection probability β = 0 . ρ stat ( q = 0 , β = 0 . ≃ . T the disease dynamicsuncouples from the opinion dynamics. Indeed, panel (b) also shows ρ in a single realizationon an isolated network ( q = 0) with β = 0 .
6, where we observe a very quick decay to astationary value that overlaps with the one for the coupled case q = 0 .
4. Therefore, as wecan see in Fig. 3, the value of ρ stat in the endemic phase of the coupled system ( β > β c . ) isthe same as in the uncoupled case. Then, at the transition point β c . ≃ . > β c ≃ . ρ stat jumps from the value ρ stat ( q = 0 , β c . ) ≃ .
22 corresponding to the uncoupled system,to the small value ρ stat ≃ . ρ stat is the origin of the discontinuous transitions for q > ρ [see Eq. (11)]. The theoretical approximation from Eq. (11), shown asa dashed curve in each panel of Fig. 5, describes a continuous transition with β , in contrastwith the discontinuity found in numerical simulations (solid symbols). This is because theMF approach assumes an infinitely large system ( N = ∞ ) where finite-size fluctuations areneglected, while simulations correspond to the limit of very large (but still finite) systems( N ≫ ρ = 0) and have either opinion + or − ( ρ + − = 0), ie., an opinion consensus on a completely healthy population. Therefore, fluctuations playa fundamental role in the discontinuous nature of the transition because, as previouslydiscussed, the stationary value of ρ in a single realization depends on whether ρ + − becomeszero before ρ does. To gain a better understanding of the results obtained from the MFtheory we run simulations on very large networks. Open circles in Fig. 4 correspond tosingle realizations on a network of N = 10 nodes, for the same values of β as for networkswith N = 10 nodes (solid lines). We observe that curves for N = 10 decay monotonicallywith time to a stationary value denoted by ρ stat , ∞ (only shown for β = 0 . N = 10 . We need to note that thesestates are not truly stationary, in the sense that ρ exhibits a very long plateau (outside theshown scale) but eventually increases and reaches the same stationary value h ρ stat i of thecurves for N = 10 . We have checked that the length of the plateau diverges with N , andthus is infinitely large when N = ∞ . Therefore, we take ρ stat , ∞ as the stationary value when N = ∞ . In Fig. 5 we observe that the numerical values ρ stat , ∞ (open symbols) agree reasonablewell with the theoretical approximation from Eq. (11) (dashed curves) for the three valuesof q , even though the agreement worsens as q gets larger. We also see that ρ stat , ∞ decayscontinuously as β decreases and becomes zero at the same value β cq of the transition in thethermodynamic limit corresponding to ρ stat (filled symbols). That is, the healthy-endemictransition is continuous in an infinite system.Up to here we studied the response of the system when the infection probability is varied,for a fixed coupling. We now explore the effects of having a varying coupling on diseaseprevalence. In Fig. 6(a) we plot h ρ stat i on two coupled networks of N = 10 nodes (circles),and ρ stat in a single realization on networks of size N = 10 (squares), as a function of thecoupling q , for β = 0 .
6. The upper curve for N = 10 shows an abrupt transition from anendemic to a healthy phase as the coupling overcomes a threshold value q c . ≃ .
5. To explore13 q fr ac ti on o f i n f ec t e d nod e s ρ ∞ < ρ > time < ρ > (a) (b) statstat q=00.20.4 0.5250.6 FIG. 6: (a) Stationary fraction of infected nodes ρ stat vs coupling q , for β = 0 .
6. Solid circlescorrespond to the average of ρ stat over 5000 realizations on networks with N = 10 nodes. Someof the values of ρ stat in a single realization are shown by dots, for q = 0 .
2, 0 . .
8. Opensquares represent results of ρ stat in a single realization on a network of size N = 10 . The dashedcurve is the theoretical approximation from Eq. (11). (b) Time evolution of h ρ i for couplings q = 0 , . , . , . , . , . , . . N = 10 . this behavior in more detail, we show with dots the value of ρ stat in every single realizationfor three values of q . For q = 0 .
2, all dots fall around its mean value h ρ stat i ≃ .
26, while for q = 0 . ρ stat = 0. At the transition point q c . the distribution of dots is bimodal,i e., dots are around ρ stat ≃ .
26 and at ρ stat = 0, giving an average value h ρ stat i ≃ . h ρ i [see Fig. 6(b)], similarly to what happens when β isvaried, as shown before. The only difference with this previous studied case is that, as β isfixed, the stationary value of ρ in single realizations does not change with q , but is either ρ stat = 0 or ρ stat ≃ .
26, in agreement with the binomial distribution. The former situationhappens in realizations where ρ hits zero before ρ + − does, while the later corresponds torealizations where ρ + − becomes zero and thus the two dynamics get uncoupled, after which ρ reaches a stationary value similar to 0 .
26 corresponding to q = 0. Figure 6(a) showsthat the transition with q is continuous in an infinitely large system (squares). One canalso check that the stationary value ρ stat for a given q in an infinite system agrees with theminimum of the corresponding h ρ i vs time curve of Fig. 6(b). This behavior is akin to theone shown in Fig. 4(a). 14 .54 0.57 0.6 0.63 0.66 0.69 β q β q (0.0 , 0.0)(1.0 , 0.0)(0.0 , 0.5)(1.0 , 0.5)(0.0 , 1.0) ENDEMICHEALTHY PHASEPHASE (p o , p d ) FIG. 7: Phase diagram of the contact process coupled to the voter model, showing the healthy andendemic phases in the β − q space, with p o = p d = 0. The dashed curve represents the analyticalapproximation of the transition line from Eq. (13). The inset shows the transition lines for the setof values ( p o , p d ) indicated in the legend. For p d = 1 . β ≃ .
53 for all values of p o (only p o = 0 . The β − q phase diagram of Fig. 7 summarizes the results obtained in this section, on howthe coupling between the contact and social networks affects the prevalence of the disease.By increasing the coupling q it is possible to bring an initially uncoupled system from theendemic to the healthy phase (vertical arrow). Also, as the coupling increases, a largerinfection probability β is needed to pass from the healthy to the endemic phase (horizontalarrow).Finally, we reproduced the phase diagram for various values of the probability p d ofhaving a successful infection across + − links, and the probability p o of opinion imitationbetween infected neighbors (inset of Fig. 7). We see that the orientation of the transition linethat separates the healthy from the endemic phase becomes more vertical as p d increases,enlarging the endemic phase, as we might expect. And when p d = 1 .
0, the transition becomesindependent of the coupling q and p o (the curve is the same for all values of p o ). We alsoobserve a slight decrease of the healthy phase when p o increases while keeping β fixed. Asthe fraction of + − links decreases faster when opinions are copied at a higher rate, oneexpects an increase of the effective infection rate and, consequently, an enlargement of theendemic phase. 15 . Effects of disease spreading on opinion consensus In this section we explore how the spreading of the disease affects the dynamics of opin-ions. As the transmission of opinions between neighboring nodes is more difficult when atleast one of them is sick, we are particularly interested in studying up to what extent thedisease slows down opinion diffusion over the social network, and how that depends on q , β , p o and p d . A way to quantify this is by looking at the time to reach opinion consensus.In Fig. 8 we show how the mean consensus time τ varies with the coupling q , for infectionprobability β = 0 . p o and p d . For a better comparison with the votermodel on an isolated network, τ is normalized by the mean consensus time τ when thenetworks are uncoupled ( q = 0). Symbols correspond to MC simulations, while solid linesare the analytical approximations from Eq. (17) obtained in section IV. Here we presentresults for β above the critical point of an isolated network β c ≃ .
53 because for β < β c the effects of disease on consensus times are negligible. This happens because for β < β c and any value of q the disease quickly disappears on the contact network and, as all nodesare susceptible, the dynamics of opinions is decoupled from the disease dynamics, reachingconsensus in a time very similar to the one in the uncoupled case ( τ ≃ τ ).We observe in Fig. 8 that the q -dependence of τ is quite diverse, showing monotonicas well as non-monotonic behaviors. This is a consequence of the competition betweentwo different mechanisms that directly affect opinion transmission. One is the link overlapbetween the two networks that is proportional to q , and the other is the disease prevalencethat decreases with q , as we explain below. The opinion transmission through a social linkthat overlaps with a contact link is slowed down when at least one of the two nodes is infectedand p o <
1. Therefore, the overall delay in opinion transmission caused by the total overlaptends to increase with q , and so does τ . This effect explains the initial monotonic increase of τ as q increases from 0, in all curves. However, as q becomes larger a second effect becomesimportant: the fraction of infected nodes decreases with q [see inset of Fig. 8(a)], due tothe coupling with the opinion dynamics that reduces the effective infection probability asdiscussed in section III A. Then, lower disease prevalence translates into fewer social linksaffected by the disease and, therefore, into a smaller opinion delay. This effect tends toreduce τ with q .With these two mechanisms at play, the shapes of curves in Fig. 8 for different values16 .0 0.2 0.4 0.6 0.8 1.0 q τ/τ (0.0 , 1.0)(0.0 , 0.5)(0.0 , 0.0)(1.0 , 1.0)(0.0 , 1.0)(0.0 , 0.5)(0.0 , 0.0)(1.0 , 1.0) q ρ (a) stat q τ/τ (0.5 , 1.0)(0.5 , 0.5)(0.5, 0.0)(0.5 , 1.0)(0.5 , 0.5)(0.5 , 0.0)(1.0 , 1.0) (b) FIG. 8: (Color online) Mean time τ to reach opinion consensus on the social network as a function ofthe coupling q with the contact network, normalized by the mean consensus time in the absence ofcoupling τ . The infection probability in the contact network is β = 0 .
6. Each network has N = 10 nodes and mean degree µ = 10. The average was done over 5000 independent realizations. Differentsymbols correspond to numerical results for the set of values ( p o , p d ) indicated in the legends [(a)for p o = 0 . p o = 0 . p o , p d ) = (1 . , . ρ stat vs q from Eq. (10) for p d = 1 .
0, 0 . . of p o and p d can be qualitatively explained in terms of the combined effects of overlap andprevalence. For instance, in Fig. 8(a) we observe that the three curves for p o = 0 . p d = 1 . q , giventhat ρ stat is independent of q [inset of Fig. 8(a)]. Then, τ increases monotonically with q asthe overlap increases. For p d = 0 . q ( ρ stat decreases),becoming dominant for q above 0 . τ decays, and leading to a non-monotonic behaviorof τ ( q ). Finally, for p = 0 . p d = 0 . τ becomes very similar to τ for all values of q > . ρ stat becomes zero above q ≃ .
583 and thus the disease has no effect onopinions, leading to consensus times similar to the ones measured in isolated networks.These behaviors for the p o = 0 . p o , as we showin Fig. 8(b) for p o = 0 .
5. We see that the shape of the curves for p d = 0 .
0, 0 . . p d . However, consensus17 .5 0.6 0.7 0.8 0.9 1 β τ/τ (0.0 , 1.0)(0.0 , 0.5)(0.0 , 0.0)(0.5 , 1.0)(0.5 , 0.5)(0.5 , 0.0)(1.0 , 1.0) p o = 0.0p o = 0.5 (a) p o = 1.0 β τ/τ -2 -1 β τ/τ p o = 0.0p o = 0.5 (b) p o = 1.0 FIG. 9: (Color online) Normalized mean consensus time τ /τ on the social network as a functionof the infection probability β on the contact network for the same network parameters as in Fig. 8,coupling q = 0 . q = 1 . p o , p d ) with the same color code as in Fig. 8, and indicated in the legend of panel (a). Curves forthe sets (1 . , .
0) and (1 . , .
5) overlap with the curve for (1 . , .
0) shown as the horizontal line τ /τ = 1. The inset of panel (b) shows the divergence of τ as β approaches 1 .
0, when q = 1 . p o = 0 .
0, and p d = 1 . p d = 0 . times are smaller for the p o = 0 . p o increases.In Fig. 9 we plot the normalized mean consensus time τ /τ as a function of the infectionprobability β obtained from Eq. (17). Panels (a) and (b) correspond to couplings q = 0 . q = 1 .
0, respectively. To analyze these plots we recall that, as explained above, consensustimes increase with the level of disease prevalence in the contact network, given that alarger disease prevalence translates into a larger delay in opinion propagation and in thesubsequent consensus. A first simple observation is that τ increases with β and also with p d ,as we expect from the fact that a larger value of β and p d implies a larger disease prevalence.A second observation is that τ decreases with the likelihood of opinion transmission p o , asexplained before when we compared τ in Fig. 8(a) with Fig. 8(b). A third observation isthat τ approaches a value independent of p d when β goes to 1 .
0. This is because for β = 1 . q and p o all nodes are infected at thestationary state, independently on the value of p d , and thus consensus times are the same18or all p d . As we see in Fig. 9(b), the case q = 1 and p o = 0 is special because τ diverges as β approaches 1 .
0. This happens because in this situation the transmission of opinions is onlypossible between connected nodes that are both susceptible, which vanish in the β → . τ scales with β canbe obtained by assuming that τ is proportional to the time scale associated to the opiniontransmission across two given neighboring nodes in the social network, i and j , with opinions+ and − , respectively. As q = 1, i and j are also neighbors in the contact network. Startingfrom a situation where i and j are infected for high β , the opinion transmission happensafter both nodes recover. Therefore, τ is determined by the time it takes the 1–1 contact linkto become a 0–0 link, which scales as (1 − β ) − . In section IV we derived a more accurateexpression for τ that exhibits this quadratic divergence in the β → IV. ANALYTICAL APPROACH
In order to gain an insight into the behavior of the two-layer system described in sectionIII, we develop here a MF approach that allows to study the time evolution of the system interms of the global densities of nodes and links in different states. We denote by ρ + and ρ − the fractions of nodes with + and − opinion in the social network, respectively, and by ρ and ρ the fractions of infected and susceptible nodes in the contact network, respectively.The fractions of social links between + and − opinion nodes are denoted by ρ + − , while ρ represents the fraction of contact links between infected and susceptible nodes. Ananalogous notation is used for ++ and −− social links and for 1–1 and 0–0 contact links.The fractions of nodes ρ + and ρ are normalized with respect to the number of nodes N in each network, while the fractions of links ρ + − and ρ are normalized by the number oflinks µN/ µ = h k i . Given that the number of nodesand links are conserved in each layer, the following conservation relations hold at any time19or the social layer 1 = ρ + + ρ − , (1a)1 = ρ ++ + ρ −− + ρ + − , (1b) ρ + = ρ ++ + 12 ρ + − , (1c) ρ − = ρ −− + 12 ρ + − , (1d)and analogously for the contact layer1 = ρ + ρ , (2a)1 = ρ + ρ + ρ , (2b) ρ = ρ + 12 ρ , (2c) ρ = ρ + 12 ρ . (2d)In appendices A, B, C and D we develop a mean-field approach that allows to obtain thefollowing system of coupled differential equations for ρ + , ρ + − , ρ and ρ , respectively: dρ + dt = 0 , (3a) dρ + − dt = 2 ωρ + − µ (cid:20) ( µ − (cid:18) − ρ + − ρ + (1 − ρ + ) (cid:19) − (cid:21) , (3b)with ω ≡ − q (1 − p o ) (cid:16) ρ + ρ (cid:17) , (4)and dρ dt = γβρ − (1 − β ) ρ , (5a) dρ dt = γβρ µ (cid:20) ( µ − (cid:18) − ρ − ρ (cid:19) − (cid:21) + 2(1 − β )( ρ − ρ ) , (5b)with γ ≡ − q (1 − p d ) ρ + − . (6)These equations represent an approximate mathematical description of the time evolution ofthe model on infinitely large networks, where finite-size fluctuations are neglected. We note20hat Eqs. (3) and Eqs. (5) are coupled through the prefactors ω and γ , which depend onthe coupling q and describe the opinion and disease dynamics, respectively. The interestedreader can find in the appendices the details of the derivation of these equations. For thesake of simplicity, we assumed in the derivation that all nodes have the same number ofneighbors k = µ chosen at random, which is equivalent to assuming that networks aredegree-regular random graphs. However, we expect this approximation to work well innetworks with homogeneous degree distributions like the ER networks we used in the MCsimulations. We also implemented an homogeneous pair approximation [29] that takes intoaccount correlations between the state of neighboring nodes within the same layer (intralayerpair approximation), but neglects correlations between opinion and disease states of bothlayers (interlayer annealing approximation). That is, we considered that the opinion stateof each node is uncorrelated with its own disease state and with its neighbors’ disease statesand, conversely, that its disease state is uncorrelated with its own and its neighbors’ opinionstates.It is instructive to analyze the structure of Eqs. (3) and (5). Equations (3) describe theevolution of opinions on the social layer. From Eq. (3a) we see that the fraction of + nodesis conserved over time: ρ + ( t ) = ρ + ( t = 0) for all t ≥
0. This behavior is reminiscent of thatof the VM on isolated topologies, where opinion densities are conserved at each time step.It seems that the disease dynamics is not able to break the intrinsic symmetry of opinionstates induced by the voter dynamics. Equation (3b) for the evolution of ρ + − has an extraprefactor ω compared to the corresponding equation for the VM on isolated networks [29],which reveals that the disease affects the dynamics of opinions through its prevalence level,expressed by ρ and ρ [see Eq. (4)]. As discussed in appendix A, ω can be interpretedas the “effective probability” that a node i adopts the opinion of a chosen social neighbor j with opposite opinion, which depends on the disease state of both i and j . Within aMF approach, we can assume that the probability that i copies j ’s opinion depends on thedisease state of an “average pair” of contact neighbors, and that this probability is the samefor all social neighbors. In these terms, ω becomes the average copying probability overthe entire social network. Indeed, we can check that the average value of ω over the three21ossible connection and disease state configurations of a contact paircopying probability = − q (no contact link) , q ρ (00 contact link) ,p o with prob. q (1 − ρ ) (10, 01 or 11 contact link),gives ω = 1 − q + q [ ρ + (1 − ρ ) p o ], which is reduced to Eq. (4) by using the relation1 − ρ = ρ + ρ / ω the rate at whichopinions change in each node. This effect slows down the propagation of opinions throughthe social network, but it does not seem to alter the properties of the voter dynamics.Equations (5) describe the evolution of the disease on the contact layer. These equationshave the same form as the corresponding equations for the CP on an isolated networkwithin the homogeneous pair approximation [9], but with a probability of infection given by γβ ≤ β . In analogy to the case of ω described above, γβ can be interpreted as the “effectiveprobability” that a given infected node i transmits the disease to a susceptible neighbor j on the contact layer, which depends on the opinions of both i and j . Indeed, the expression γβ = [1 − q (1 − p d ) ρ + − ] β from Eq. (6) is the average infection probability on the contactnetwork, calculated over the three possible connection and opinion state configurations of asocial pair:infection probability = β with prob. 1 − q (no social link) ,β with prob. q (1 − ρ + − ) (either ++ or −− social link) ,β p d with prob. q ρ + − (+ − social link) . Thus, our MF approach assumes that this “effective infection probability” from i to j de-pends on the opinion states of an “average pair” of neighbors on the social layer, and that isthe same for all contact neighbors. We can say that, at the MF level, the disease dynamicsfollows the standard CP on a single isolated network with homogeneous infection probability γβ and recovery probability 1 − β in each node. Therefore, the dynamics of opinions has aneffect on the disease dynamics equivalent to that of an external homogeneous field acting oneach node of the contact network, reducing the probability of infection between neighborsby a factor γ , while keeping the same recovery probability.22n the next two subsections we derive analytical expressions for the disease prevalence ρ stat and the mean consensus time τ , from the system of Eqs. (3-6). A. Disease prevalence
In order to study how the opinion dynamics affects the disease prevalence, we find thefraction of infected nodes at the stationary state ρ stat from Eqs. (3) and (5). We start bysetting the four time derivatives to zero, substituting ρ by 2(1 − β ) ρ /γβ from Eq. (5a)into Eq. (5b), and solving for ρ . After doing some algebra we obtain two solutions, butonly one is stable depending on the values of the parameters. The non-trivial solution ρ stat = [( µ − γ + µ ] β − µ [( µ − γ + 1] β − λ ≡ [( µ − γ + µ ] β − µ is larger than zero. For λ < ρ stat = 0, corresponding to the healthy phase where all nodes are susceptible,while λ = 0 indicates the transition point between the endemic and the healthy phase. Theexpression for ρ stat from Eq. (7) is still not closed because it depends on ρ + − , through theprefactor γ . From Eq. (3b) we see that the fraction of + − social links reaches a stationaryvalue given by the expression ρ + − stat = 2( µ − µ − ρ + (0)[1 − ρ + (0)] , (8)where we used ρ + = ρ + (0) given that ρ + remains constant over time, as mentioned before.We notice that ω does not affect the stationary value of ρ + − , which remains the same as inthe original VM [29]. For a symmetric initial condition on the social layer ( ρ + (0) = 1 / ρ + − stat = ( µ − / [2( µ − ρ + − stat in Eq. (6) we obtain the following expression for γ : γ = 1 − q (1 − p d )( µ − µ − . (9)Finally, plugging Eq. (9) into Eq. (7) we arrive to the following approximate expression forthe stationary fraction of infected nodes in the endemic phase: ρ stat = [2(2 µ − − q (1 − p d )( µ − β − µ [2 µ − q (1 − p d )( µ − β − . (10)23or a network of mean degree µ = 10 and p d = 0, Eq. (10) is reduced to the simple expression ρ stat = (19 − q ) β − − q ) β − , (11)which is plotted in Figs. 5 and 6 (dashed curves). As the MF theory is meant to work forinfinitely large systems, we also plot for comparison the numerical results obtained fromsimulations for very large networks (open symbols). We observe that, in all cases, theestimated theoretical value of the fraction of infected nodes from Eq. (11) is larger than thatfrom simulations. As we explain below, this due to the fact that correlations between opinionand disease states are neglected by the MF approach. We first notice that an infection event0 → (cid:2) +1 (cid:3) and (cid:2) +0 (cid:3) (a (cid:2) ++1 0 (cid:3) pair) or (cid:2) − (cid:3) and (cid:2) − (cid:3) ( a (cid:2) −− (cid:3) pair), because p d = 0 in Figs. 5 and 6. Then, it is expected that ++ social links are negatively correlatedwith 10 contact links and positively correlated with 11 and 00 contact links, given that same-opinion neighbors tend to infect each other and thus, at a given time, they are more likely tobe either both infected or both susceptible. However, the theoretical approximation assumesthat ++ social links are uncorrelated with 10 contact links (see appendix A) and, therefore,the estimated probability of finding a (cid:2) ++1 0 (cid:3) pair is larger than that obtained when negativecorrelations are considered. The same conclusion also holds for (cid:2) −− (cid:3) pairs. This leads toa theoretical overestimation of the number of (cid:2) ++1 0 (cid:3) and (cid:2) −− (cid:3) pairs and, consequently, to alarger rate of infections which increases the disease prevalence respect to numerical results,as we see in Figs. 5 and 6.Figure 5 shows that ρ stat form Eq. (11) continuously decreases and vanishes as β decreasesbeyond a threshold value, as it happens in the standard CP. This shows that the transitionto the healthy state is continuous within the MF approach, which assumes that the systemis infinitely large. In Fig. 6 we see that ρ stat decreases with q , reducing the prevalence andinducing a transition to the healthy phase. That is, Eq. (10) predicts a healthy-endemiccontinuous transition as β and q are varied, which happens at the point where ρ stat vanishes,leading to the relation [2(2 µ − − q c (1 − p d )( µ − β c − µ = 0 . (12)The transition line q c = 19 β c − − p d ) β c (13)24btained from Eq. (12) for µ = 10 is plotted in Fig. 7 for p d = 0 (dashed curve). Wecan see that the agreement with simulations is good for small values of the coupling q , butdiscrepancies arise as q increases, where the theoretical prediction from Eq. (13) overesti-mates numerical values. Another simple observation that follows from Eq. (13) is that for β > / [19 − − p d )] we obtain the nonphysical value q c >
1. This means that, in thenetwork model, it is possible to induce a transition by increasing the coupling only when β is lower than a given value, as we see in Fig. 7 for β < . B. Opinion consensus times
In this section we study the quantitative effects of the disease on the time to reach opinionconsensus. For that, we find an analytical estimation of the mean consensus time τ as afunction of the model parameters.As mentioned in section IV, in infinitely large systems ρ + remains constant over time [seeEq. (3a)]. However, in finite systems ρ + fluctuates until it reaches either value ρ + = 1 (+consensus) or ρ + = 0 ( − consensus), with both configurations characterized by the absence of+ − social links ( ρ + − = 0). A typical evolution of ρ + − towards the absorbing state can be seenin Fig. 4 (b) for q = 0 . N = 10 nodes. That is, consensus is eventuallyachieved in finite systems due to the stochastic nature of the opinion dynamics, which leadsthe social network to a state where all nodes share the same opinion. In a single opinionupdate ρ + may increase or decrease by 1 /N with the same probability ωρ + − /
2, calculated asthe probability ρ + − / ω of opinion adoption. Therefore, the stochastic dynamics of the VMcan be studied by mapping ρ + into the position of a symmetric one-dimensional randomwalker on the interval [0 , ωρ + − stat / /N . Starting from a symmetric configuration with N/ ρ + (0) = 1 / ρ + = 1 or ρ + = 0 in an averagenumber of steps that scales as N . Then, given that the walker makes a single step in anaverage number of attempts that scales as 1 /ωρ + − stat , and that the time increases by 1 /N in25ach attempt, we find that the mean consensus time scales as τ ∼ Nωρ + − stat . (14)As we see in Eq. (8), ρ + − stat is independent of the disease prevalence ρ and, therefore, theprevalence affects τ only through the effective copying probability ω , which sets the timescale associated to opinion updates. From Eq. (4) we see that ω equals 1 . q = 0) or when p o = 1 .
0, and thus the dynamics of opinions is exactly thesame as that of the original VM. However, ω is smaller than 1 . q >
0) and p o < .
0, and thus the evolution of the dynamics is “slowed down” –in average–by a factor 1 /ω > .
0, given that opinions are copied at a rate that is ω times smaller thanin the uncoupled case. As a consequence, τ increases by a factor 1 /ω respect to the meanconsensus time in the uncoupled case τ = τ ( q = 0) ∼ N/ρ + − stat , that is ττ ≃ ω . (15)To obtain a complete expression for the ratio τ /τ as a function of the model’s parameterswe express ω in terms of ρ stat , by substituting into Eq. (4) the stationary value of ρ thatfollows from Eq. (5a), ρ stat = 2(1 − β ) ρ stat /γβ . This leads to ω = 1 − q (1 − p o ) (cid:18) − βγβ (cid:19) ρ stat . (16)In the healthy phase is ρ stat = 0, thus ω = 1 . τ = τ . In this case, the theory predictsthat the disease has no effect on the time to consensus because there are no infected nodesthat can affect the opinion dynamics. However, having a value ρ stat > ω or, equivalently, increasing τ respect to τ .Plugging into Eq. (16) the expressions for γ and ρ stat from Eqs. (9) and (10), respectively,and reordering some terms, we obtain the following expression that relates τ and τ in theendemic phase: ττ ≃ " − q (1 − p o ) (cid:8) µ − − q (1 − p d )( µ − β (cid:9)(cid:8) [2(2 µ − − q (1 − p d )( µ − β − µ (cid:9) β (cid:8) µ − − q (1 − p d )( µ − (cid:9)(cid:8) [2 µ − q (1 − p d )( µ − β − (cid:9) − . (17)In Fig. 8 we plot in solid lines the ratio τ /τ vs q from Eq. (17) for µ = 10.We observe that the theoretical values of τ /τ are smaller than those obtained fromnumerical simulations (symbols) for all combinations of p o and p d shown. A possible expla-nation of these discrepancies can be given by analyzing how correlations affect the estimated26umber of different types of connected nodes, as we have done in section IV A for diseaseprevalence. If we take the p = 0 case, we see that an opinion change due to an interactionbetween two social and contact neighbors happens only if both nodes are susceptible, thatis, when they have states (cid:2) +0 (cid:3) and (cid:2) − (cid:3) . Then, as the theory assumes that + − social linksand 00 contact links are uncorrelated (appendix A), the estimated probability of finding a (cid:2) + − (cid:3) pair is larger than that in simulations, given that + − social links are expected to benegatively correlated with 00 contact links. This negative correlation is due to the fact thatsusceptible neighbors tend to align their opinions and, therefore, they are more likely to bein the same opinion state at a given time. This leads to an overestimation of the numberof (cid:2) + − (cid:3) pairs and, therefore, to a larger rate of opinion transmission. This has the overalleffect of speeding up consensus, decreasing the theoretically estimated mean time to reachconsensus respect to the mean consensus time measured in simulations, as we see in Fig. 8.Even though discrepancies with numerical results increase with the coupling q , the an-alytic expression (17) is able to capture the different qualitative behavior of the consensustime for several combinations of p o and p d , as we describe below. For low values of p d , thereis a transition to the healthy phase when q overcomes a value q c < τ = τ for all q > q c [see p d = 0 curves in the main plot and the inset of Fig. 8(a)].As a consequence, τ /τ exhibits a non-monotonic behavior with q , as we described in sectionIII B. For higher values of p d , the transition to the healthy phase does not happen for thephysical values q ≤ q c > τ may either increase monotonically with q for large p d values (see p d = 1 . p d values (see p d = 0 . q – andthe disease prevalence –which decreases with q . This competition can be seen quantitativelyin Eq. (16) for ω , which has three factors that depend on q and affect τ . Besides the factorproportional to q , the factor 1 /γ also increases with q , as seen from Eq. (9). But these twofactors are balanced by ρ stat , which decreases with q .An interesting case is the one for full coupling q = 1 . p o = 0, because τ from Eq. (17)diverges as β approaches 1 .
0. This happens in the model because when β = 1 . − opinions and consensus is neverachieved. By doing a Taylor series expansion of expression (17) up to to second order in thesmall parameter ǫ = 1 − β ≪ ττ ≃ [9 − − p d )] − β ) , (18)where we used µ = 10. Equation (18) shows that τ diverges as (1 − β ) − in the β → . β = 1 . p o = 0, we can check from Eq. (17)that τ /τ ≃ / (1 − q ), which shows the divergence of τ as the system approaches the fullycoupled state q = 1 . V. SUMMARY AND CONCLUSIONS
We proposed a bilayer network model to explore the interplay between the dynamics ofopinion formation and disease spreading in a population of individuals. We used the votermodel and the contact process to simulate the opinion and the disease dynamics running ona social and contact network, respectively. These two networks share the same nodes andthey are coupled by a fraction q of links in common. We showed that, when the networksare coupled, the opinion dynamics can dramatically change the statistical properties of thedisease spreading, which in turn modifies the properties of the propagation of opinions, ascompared to the case of isolated networks.The VM dynamics is able to change the order of the healthy-endemic phase transitionobserved in the CP as the infection probability β exceeds a threshold value β c , from acontinuous transition for the uncoupled case to a discontinuous transition when the coupling q is larger than zero. The magnitude of the change in the disease prevalence at the transitionpoint β c increases with q . The discontinuity is associated with the non-monotonic timeevolution of the fraction of infected nodes. This non-monotonicity is as a consequence of thetime-varying nature of the effective infection probability, which varies over time accordingto the stochastic evolution of the fraction of + − social links. The system also exhibits adiscontinuous transition from an endemic to a healthy phase when the coupling overcomesa value q c , for a fixed value of β . The origin of this discontinuity is the same as that of thediscontinuous transition with β , that is, the non-monotonicity in the time evolution of thefraction of infected nodes. We also obtained a phase diagram in the β − q space showing the28ealthy and endemic phases for different values of the probabilities p d and p o . In all cases,we observed that the transition point β c increases with q .We need to mention that changes in the order of topological and dynamical transitionswere already observed in multilayer networks [31–37]. In real populations, the implicationsof having continuous in contrast to discontinuous transitions are very different. Indeed,starting from a hypothetical situation that consists on a population of individuals withan infection rate just below the critical value (in the healthy phase), a small incrementin β would lead to a small number of infected individuals in the former case, but a largenumber of infections in the later case. Therefore, disregarding the effects of social dynamicson epidemics propagation could lead to an underestimation of the real magnitude of thespreading.We developed a mean-field approach that allowed to estimate with reasonable precisionthe healthy-endemic transition line ( β c , q c ) as a function of the model’s parameters. Thisapproach reveals that the disease dynamics is equivalent to that of the standard CP onan isolated network, with an effective infection probability that is constant over time andthat decreases with the coupling and the stationary fraction of + − social links, for a fixedvalue of β . This means that, at the mean-field level, the overall effect of the VM on the CPis to decrease the effective infection probability as the coupling increases. Therefore, as q increases, a larger value of β is needed to bring the system to the endemic phase, leading toan increase of the transition point β c with q .On its part, the CP dynamics has the overall effect of slowing down the propagationof opinions, delaying the process of opinion consensus compared to the one observed in anisolated network. The MF approach reveals that the opinion dynamics corresponds to thatof the standard VM model on an isolated network, with a probability of opinion transmissionthat decreases with q and the disease prevalence. Depending on the parameters values, themean consensus time τ can show a monotonic increase with q , as well as a non-monotonicbehavior. An insight on these results was given by the MF approach, which allowed to obtainan approximate mathematical expression that relates τ with the parameters. This approachshows that the behavior of τ with q is the result of two different mechanisms at play: theoverlap of social and contact links that tends to increase τ with q , which is counterbalancedby the fraction of infected nodes that tends to decrease τ with q . Therefore, the non-trivialdependence of τ with q is a consequence of the competition between these two mechanisms.29t is interesting to note that, despite the nontrivial interplay between the CP and theVM, the coupled interdependent system of opinions and disease can be roughly seen astwo systems that evolve independently of one another, where each system has an effectiveparameter that depends on the other dynamics and the coupling. Specifically, the opiniondynamics corresponds to that of the VM with an effective opinion transmission probabilitythat decreases with the disease prevalence and the coupling, while the disease spreadingis well described by the dynamics of the CP with an effective infection probability thatdecreases with the fraction of + − social neighbors and the coupling. However, this is onlyan approximation that comes from the MF analysis, which neglects correlations betweenopinion and disease states.The results presented in this article correspond to a particular initial state that consistson even fractions of + and − opinion states and even fractions of infected and susceptiblestates, uniformly distributed over the networks. As a future work, it might be worth studyingthe behavior of the system under different initial conditions, and with uneven fractions ofopinion and disease states. For example, one can simulate a population with initial polarizedopinions based on the disease, by correlating the opinion of each node with its disease state(for instance by infecting all nodes with − opinion and leaving all + opinion nodes in thehealthy state). Finally, it would be interesting to study the behavior of the present modelunder different update rules. For instance, we have checked a simple rule in which theconnection condition –connected or disconnected– between two nodes in one layer is nottaken into account for the update in the other layer. This is an ongoing work with somepreliminary results that suggest that the critical behavior of this new model is quite differentfrom that of the original model. ACKNOWLEDGMENTS
We thank Gabriel Baglietto and Didier Vega-Oliveros for helpful comments on themanuscript. We acknowledge financial support from CONICET (PIP 0443/2014).30 ppendix A: Derivation of the rate equation for ρ + We denote by (cid:2) OD (cid:3) the state of a given node, where O = + , − and D = 1 , (cid:2) +0 (cid:3) , (cid:2) +1 (cid:3) , (cid:2) − (cid:3) and (cid:2) − (cid:3) . In a single time step of the dynamics, the transitions from state (cid:2) −D (cid:3) to state (cid:2) + D (cid:3) when a node switches opinion from − to + lead to a gain of 1 /N in ρ + , while the transitions (cid:2) + D (cid:3) → (cid:2) −D (cid:3) when there is a − → + opinion change lead to a loss of 1 /N in ρ + . Consideringthese four possible transitions, the average change of ρ + in a single time step of time interval∆ t = 1 /N is described by the rate equation dρ + dt = dρ + dt (cid:12)(cid:12)(cid:12) −→ + + dρ + dt (cid:12)(cid:12)(cid:12) + →− = 11 /N h ∆ ρ + (cid:12)(cid:12) − → +0 + ∆ ρ + (cid:12)(cid:12) − → +1 + ∆ ρ + (cid:12)(cid:12) +0 → − + ∆ ρ + (cid:12)(cid:12) +1 → − i , (A1)where for instance the term ∆ ρ + (cid:12)(cid:12) − → +0 represents the average change of ρ + in a time stepdue to (cid:2) − (cid:3) → (cid:2) +0 (cid:3) transitions. In turn, ∆ ρ + (cid:12)(cid:12) − → +0 has four contributions corresponding tothe different social interactions that lead to the (cid:2) − (cid:3) → (cid:2) +0 (cid:3) transition. Thus, we can write∆ ρ + (cid:12)(cid:12) − → +0 = ∆ ρ + (cid:12)(cid:12)(cid:12) − +0 0 → ++0 0 + ∆ ρ + (cid:12)(cid:12)(cid:12) − +0 0 → ++0 0 + ∆ ρ + (cid:12)(cid:12)(cid:12) − +0 1 → ++0 1 + ∆ ρ + (cid:12)(cid:12)(cid:12) − +0 1 → ++0 1 , (A2)and similarly for the (cid:2) − (cid:3) → (cid:2) +1 (cid:3) transition corresponding to the second term in Eq. (A1)∆ ρ + (cid:12)(cid:12) − → +1 = ∆ ρ + (cid:12)(cid:12)(cid:12) − +1 0 → ++1 0 + ∆ ρ + (cid:12)(cid:12)(cid:12) − +1 0 → ++1 0 + ∆ ρ + (cid:12)(cid:12)(cid:12) − +1 1 → ++1 1 + ∆ ρ + (cid:12)(cid:12)(cid:12) − +1 1 → ++1 1 . (A3)Third and fourth terms in Eq. (A1) are obtained by interchanging symbols + and − inEqs. (A2) and (A3), respectively, due to the symmetry between + and − opinion states.We notice that disease states remain the same after the interactions, as only a change inthe social layer can lead to a change in ρ + . The first term in Eq. (A2) represents theaverage change in ρ + due to interactions in which a node i in state (cid:2) − (cid:3) copies the opinionof one its social neighbors j in state (cid:2) +0 (cid:3) , changing the state of i to (cid:2) +0 (cid:3) . This interactionis schematically represented by the symbol (cid:2) − +0 0 (cid:3) , where the horizontal line over the opinionsymbols describes a social link between i and j . In the same way, the symbol (cid:2) − +0 0 (cid:3) representsan interaction between a (cid:2) − (cid:3) node and a neighboring (cid:2) +0 (cid:3) node connected by both a socialand a contact link that are indicated by horizontal lines on top of the respective symbols.The second, third and fourth terms in Eq. (A2) describe, respectively, the transitions due31o an interaction of node i with a (cid:2) +0 (cid:3) social/contact neighbor, a (cid:2) +1 (cid:3) social neighbor and a (cid:2) +1 (cid:3) social/contact neighbor.We now illustrate how to build an approximate expression for each term of Eq. (A2) for∆ ρ + (cid:12)(cid:12) − → +0 . For the sake of simplicity, we assume that all nodes have the same number ofneighbors k = µ chosen at random, which is equivalent to assuming that networks are degree-regular random graphs. However, we expect this approximation to work well in networkswith homogeneous degree distributions like ER networks. The first term in Eq. (A2) can bewritten as ∆ ρ + (cid:12)(cid:12)(cid:12) − +0 0 → + +0 0 = P (cid:0) − (cid:1) µ X {N − } M (cid:0)(cid:8) N − (cid:9) , µ (cid:1) N [ − +0 0 ] µ N , (A4)which can be understood as the product of the different probabilities associated to each ofthe consecutive events that lead to the (cid:2) − +0 0 (cid:3) → (cid:2) ++0 0 (cid:3) transition in a time step, as we describebelow. A node i with state (cid:2) − (cid:3) is chosen at random with probability P (cid:0) − (cid:1) . If node i has N [ − +0 0 ] social neighbors in state (cid:2) +0 (cid:3) , then one of these social neighbors j is randomly chosenwith probability N [ − +0 0 ] /µ , after which i copies j ’ opinion with probability 1 . i and j . Finally, ρ + increases by 1 /N when i switches opinion.In order to consider all possible scenarios of having N [ − +0 0 ] = 0 , , .., µ social neigh-bors we sum over all possible neighborhood configurations represented by (cid:8) N − (cid:9) ≡ (cid:8) N [ − +0 0 ] , N [ − +0 0 ] , N [ −− ] , N [ −− ] , N [ − +0 1 ] , N [ − +0 1 ] , N [ −− ] , N [ −− ] (cid:9) , weighted by the probability ofeach configuration M (cid:0)(cid:8) N − (cid:9) , µ (cid:1) . Here we denote by N [ − O D ] the number of (cid:2) OD (cid:3) social neigh-bors and by N [ − O D ] the number of (cid:2) OD (cid:3) social/contact neighbors [see Fig. (10)]. The numberof each type of neighbor is between 0 and µ , and thus the sum in Eq. (A4) include eightsummations µ X {N − } ≡ X O =+ , −D =0 , µ X N [ −O D ] =0 + µ X N [ − O D ] =0 over all combinations subject to the constraint X O =+ , −D =0 , ( N [ −O D ] + N [ −O D ]) = µ. In order to carry out the summation in Eq. (A4) we only take into account correlationsbetween first neighbors, and neglect second and higher neighbor correlations (pair approxi-mation). Thus, we define the probability P [ −O D ] ≡ P (cid:0) OD | − (cid:1) that a given neighbor of node32 − µ + − N + − social links−N + − social linksprob = µ N + − 0 + + + − − −+ − − − − + + − − social links+ − social links + − µ −N − − social links + − N + − social links − +
N[ ]+ + −
N = + − + + − − + N[ ]+ − +
N[ ]+ − +
N[ ] i i + −+ − ∆ρ = 2(µ−2Ν )/µΝ
FIG. 10: (Color online) Schematic illustration of an opinion update in which a node i in state (cid:2) − (cid:3) changes to state (cid:2) +0 (cid:3) by copying the opinion + of a randomly chosen neighbor (green dashed links).The change in the density of + − links is denoted by ∆ ρ + − . i is a social neighbor with state (cid:2) OD (cid:3) , and consider P [ −O D ] to be conditioned to the state (cid:2) − (cid:3) of i only, and not on the other neighbors of i . Similarly, we denote by P [ −O D ] = P (cid:0) OD | − (cid:1) the conditional probability that a node connected to i is a social/contact neighbor withstate (cid:2) OD (cid:3) , given that i has state (cid:2) − (cid:3) . Therefore, M becomes the multinomial probabilitydistribution defined as M (cid:0)(cid:8) N − (cid:9) , µ (cid:1) ≡ µ ! Q O =+ , −D =0 , N [ OD ]! N [ OD ]! Q O =+ , −D =0 , P [ OD ] N [ OD ] P [ OD ] N [ OD ] when P O =+ , −D =0 , ( N [ OD ] + N [ OD ]) = µ ;0 otherwise,where we have used the symbols (cid:2) OD (cid:3) and (cid:2) OD (cid:3) as short notations for (cid:2) − O D (cid:3) and (cid:2) − O D (cid:3) , respec-tively. Then, performing the summation in Eq. (A4) we arrive to∆ ρ + (cid:12)(cid:12)(cid:12) − +0 0 → + +0 0 = P (cid:0) − (cid:1) hN [ − +0 0 ] i µN = P (cid:0) − (cid:1) P [ − +0 0 ] N , (A5)where we have used the identity hN [ − +0 0 ] i = µ P [ − +0 0 ] for the mean value of N [ − +0 0 ]. The otherthree terms in Eq. (A2) can be obtained following an approach similar to the one above for∆ ρ + (cid:12)(cid:12)(cid:12) − +0 0 → + +0 0 , leading to the expression∆ ρ + (cid:12)(cid:12) − → +0 = P (cid:0) − (cid:1) N ( P [ − +0 0 ] + P [ − +0 0 ] + P [ − +0 1 ] + p o P [ − +0 1 ]) , (A6)where the prefactor p o in the last term accounts for the probability of copying the opinionof an infected contact neighbor. Keeping in mind that we aim to obtain a closed system of33ate equations for ρ + , ρ , ρ + − and ρ , we now find approximate expressions for the differentprobabilities of Eq. (A6) in terms of the fractions of nodes and links in each layer. Westart by assuming that correlations between opinion and disease states of a given node arenegligible, and thus we can write P (cid:0) − (cid:1) ≃ ρ − ρ . (A7)Then, to estimate the conditional probabilities P [ − +0 D ] and P [ − +0 D ] it proves convenient to spliteach of them into two conditional probabilities P [ − +0 D ] = P (cid:0) + D | − (cid:1) = P (cid:0) | − (cid:1) P (cid:0) + D | − (cid:1) ,P [ − +0 D ] = P (cid:0) + D | − (cid:1) = P (cid:0) | − (cid:1) P (cid:0) + D | − (cid:1) , using the relation P ( a, b | c ) = P ( a | c ) P ( b | a, c ) and interpreting the entire event of connectinga given type of link to a [ + D ] node as two separate events. Assuming that the type of linkconnected to node i is uncorrelated with the state of i , we have P (cid:0) | − (cid:1) ≃ P (cid:0) (cid:1) = 1 − q and P (cid:0) | − (cid:1) ≃ P (cid:0) (cid:1) = q, and that opinion and disease states are uncorrelated, we have P (cid:0) + D | − (cid:1) ≃ P (+ | − ) P ( D|
0) and P (cid:0) + D | − (cid:1) ≃ P (+ | − ) P ( D| . Within an homogeneous pair approximation [29], the probability P (+ | − ) that a socialneighbor j of a node i with opinion O i = − has opinion O j = + can be estimated as theratio between the total number µN ρ + − / − to + nodes and the total number µN ρ − of links connected to − nodes, that is P (+ | − ) ≃ ρ + − / ρ − . Similarly, we estimatethe probability that a contact neighbor j of a susceptible node has disease state D j = 0 as P (0 | ≃ ρ /ρ , and disease state D j = 1 as P (1 | ≃ ρ / ρ . And if j is not aneighbor of i on the contact layer then P ( D| ≃ ρ D . Assembling all these factors weobtain P [ − +0 0 ] ≃ (1 − q ) ρ + − ρ ρ − , P [ − +0 0 ] ≃ q ρ + − ρ ρ − ρ ,P [ − +0 1 ] ≃ (1 − q ) ρ + − ρ ρ − , P [ − +0 1 ] ≃ q ρ + − ρ ρ − ρ . (A8)34inally, plugging into Eq. (A6) the approximate expressions for the conditional probabilitiesfrom Eqs. (A8) and for P (cid:0) − (cid:1) from Eq. (A7) we arrive to∆ ρ + (cid:12)(cid:12) − → +0 = ρ + − N h ρ − q (1 − p o ) ρ i , (A9)where we have used the conservation relations Eqs. (2a) and (2d).We now calculate the second gain term in Eq. (A1), ∆ ρ + (cid:12)(cid:12) − → +1 , which represents theaverage change in ρ + due to (cid:2) − (cid:3) → (cid:2) +1 (cid:3) transitions, following the same steps as above forthe term ∆ ρ + (cid:12)(cid:12) − → +0 . From Eq. (A3) we obtain∆ ρ + (cid:12)(cid:12) − → +1 = P (cid:0) − (cid:1) µ X {N − } M (cid:0)(cid:8) N − (cid:9) , µ (cid:1) µ n N [ − +1 0 ] + p o N [ − +1 0 ] + N [ − +1 1 ] + p o N [ − +1 1 ] o N = P (cid:0) − (cid:1) µN n hN [ − +1 0 ] i + hN [ − +1 1 ] i + p o (cid:0) hN [ − +1 0 ] i + hN [ − +1 1 ] i (cid:1)o = P (cid:0) − (cid:1) N n P [ − +1 0 ] + P [ − +1 1 ] + p o (cid:0) P [ − +1 0 ] + P [ − +1 1 ] (cid:1)o , and using the approximations P [ − +1 0 ] ≃ (1 − q ) ρ + − ρ ρ − , P [ − +1 0 ] ≃ q ρ + − ρ ρ − ρ ,P [ − +1 1 ] ≃ (1 − q ) ρ + − ρ ρ − , P [ − +1 1 ] ≃ q ρ + − ρ ρ − ρ , (A10)for the conditional probabilities we arrive to∆ ρ + (cid:12)(cid:12) − → +1 = ρ + − ρ N [1 − q (1 − p o )] , (A11)where we have used the conservation relations Eqs. (2a) and (2c).Adding Eqs. (A9) and (A11) we obtain the following expression for the average gain ofa + node in single time step, corresponding to the sum of the first and second terms ofEq. (A1) dρ + dt (cid:12)(cid:12)(cid:12) −→ + = 11 /N h ∆ ρ + (cid:12)(cid:12) − → +0 + ∆ ρ + (cid:12)(cid:12) − → +1 i ≃ ωρ + − , (A12)with ω ≡ − q (1 − p o ) (cid:16) ρ + ρ (cid:17) . (A13)The prefactor ω plays an important role in the dynamics of opinion consensus, by setting thetime scale associated to opinion updates, and can be interpreted as an effective probabilitythat a node adopts the opinion of randomly chosen opposite-opinion neighbor. That is,35q. (A12) for the gain of a + node simply describes the process of selecting a − node i anda + neighbor j , which happens with probability ρ + − /
2, and then switching i ’s opinion witha probability ω that depends on the connection type and disease state of both i and j . This“effective copying probability” ω turns out to be an average copying probability over theentire social network, as shown in section IV.In order to find the equation for the average loss of a + node in a time step, correspondingto the sum of the third and forth terms of Eq. (A1), we can exploit the symmetry between+ and − opinion states and simply interchange signs + and − in Eq. (A12) dρ + dt (cid:12)(cid:12)(cid:12) + →− = 11 /N h ∆ ρ + (cid:12)(cid:12) +0 → − + ∆ ρ + (cid:12)(cid:12) +1 → − i ≃ − ωρ + − , (A14)where we used ρ − + = ρ + − . Finally, adding Eqs. (A12) and (A14) we obtain dρ + dt = 0 , (A15)quoted in Eq. (3a) of the main text. Therefore, the fractions of + and − nodes are conservedat all times: ρ + ( t ) = ρ + (0) and ρ − ( t ) = ρ − (0) = 1 − ρ + (0). Even though the abovecalculation leads to a very simple result, it serves as an introduction to the methodologyused for deriving rate equations for the other fractions ρ + − , ρ and ρ , as we show next. Appendix B: Derivation of the rate equation for ρ + − In analogy to the calculation for ρ + in the previous section, the average change of thefaction of + − social links ρ + − in a time step is given by the rate equation dρ + − dt = dρ + − dt (cid:12)(cid:12)(cid:12) −→ + + dρ + − dt (cid:12)(cid:12)(cid:12) + →− , (B1)with dρ + − dt (cid:12)(cid:12)(cid:12) −→ + = 11 /N h ∆ ρ + − (cid:12)(cid:12) − → +0 + ∆ ρ + − (cid:12)(cid:12) − → +1 i (B2) dρ + − dt (cid:12)(cid:12)(cid:12) + →− = 11 /N h ∆ ρ + − (cid:12)(cid:12) +0 → − + ∆ ρ + − (cid:12)(cid:12) +1 → − i = (cid:26) dρ + − dt (cid:12)(cid:12)(cid:12) −→ + (cid:27) −⇐⇒ + , (B3)where the symbol − ⇐⇒ + indicates the interchange of signs + and − in the expressionbetween braces. Equation (B3) means that the symmetry between + and − opinions allowsto find the second term in Eq. (B1) by interchanging signs in the first term. To calculate36he first term in Eq. (B2) we sum over all four types of interactions of a (cid:2) − (cid:3) node i with a (cid:2) + D (cid:3) neighbor j that lead to the (cid:2) − (cid:3) → (cid:2) +0 (cid:3) transition∆ ρ + − (cid:12)(cid:12) − → +0 = ∆ ρ + − (cid:12)(cid:12)(cid:12) − +0 0 → ++0 0 + ∆ ρ + − (cid:12)(cid:12)(cid:12) − +0 0 → ++0 0 + ∆ ρ + − (cid:12)(cid:12)(cid:12) − +0 1 → ++0 1 + ∆ ρ + − (cid:12)(cid:12)(cid:12) − +0 1 → ++0 1 . (B4)As explained in the previous section, the probabilities of interactions [ − +0 D ] and [ − +0 D ] are givenby the respective fractions N [ − +0 D ] /µ and N [ − +0 D ] /µ of each type of neighbor. The changein the number of + − social links after node i switches opinion is given by the expression µ − (cid:0) N [ − +0 0 ] + N [ − +0 0 ] + N [ − +0 1 ] + N [ − +0 1 ] (cid:1) , which takes into account the specific configurationof links and neighbors connected to i , as depicted in Fig. 10. We obtain∆ ρ + − (cid:12)(cid:12) − → +0 = P (cid:0) − (cid:1) µ X {N − } M (cid:0)(cid:8) N − (cid:9) , µ (cid:1) µ (cid:0) N [ − +0 0 ] + N [ − +0 0 ] + N [ − +0 1 ] + p o N [ − +0 1 ] (cid:1) × (cid:2) µ − (cid:0) N [ − +0 0 ] + N [ − +0 0 ] + N [ − +0 1 ] + N [ − +0 1 ] (cid:1)(cid:3) µN/ P (cid:0) − (cid:1) µ N (cid:26) µ h hN [ − +0 0 ] i + hN [ − +0 0 ] i + hN [ − +0 1 ] i + p o hN [ − +0 1 ] i i − h(cid:10) N [ − +0 0 ] (cid:11) + (cid:10) N [ − +0 0 ] (cid:11) + (cid:10) N [ − +0 1 ] i + p o hN [ − +0 1 ] (cid:11) + 2 (cid:0) hN [ − +0 0 ] N [ − +0 0 ] i + hN [ − +0 0 ] N [ − +0 1 ] i + hN [ − +0 0 ] N [ − +0 1 ] i (cid:1) + (1 + p o ) (cid:0) hN [ − +0 0 ] N [ − +0 1 ] i + hN [ − +0 0 ] N [ − +0 1 ] i + hN [ − +0 1 ] N [ − +0 1 ] i (cid:1)i(cid:27) , (B6)where the first and second moments of M (cid:0)(cid:8) N − (cid:9) , µ (cid:1) are hN [ − + D i D j ] i = µP [ − + D i D j ] , hN [ − + D i D j ] i = µP [ − + D i D j ] , hN [ − + D i D j ] i = µP [ − + D i D j ] + µ ( µ − P [ − + D i D j ] , hN [ − + D i D j ] i = µP [ − + D i D j ] + µ ( µ − P [ − + D i D j ] , hN [ − + D i D j ] N [ − + D i D′ j ] i = µ ( µ − P [ − + D i D j ] P [ − + D i D′ j ] , hN [ − + D i D j ] N [ − + D i D′ j ] i = µ ( µ − P [ − + D i D j ] P [ − + D i D′ j ] , hN [ − + D i D j ] N [ − + D i D′ j ] i = µ ( µ − P [ − + D i D j ] P [ − + D i D′ j ] . (B7)37ere D i = 1 , D j = 1 , i and j , respectively. Replacingthe expressions for the moments from Eqs. (B7) in Eq. (B6) and regrouping terms we obtain∆ ρ + − (cid:12)(cid:12) − → +0 = = 2 P (cid:0) − (cid:1) µN (cid:26) ( µ − h P [ − +0 0 ] + P [ − +0 0 ] + P [ − +0 1 ] + p o P [ − +0 1 ] i − µ − h(cid:0) P [ − +0 0 ] + P [ − +0 0 ] + P [ − +0 1 ] (cid:1) + p o P [ − +0 1 ] + (1 + p o ) P [ − +0 1 ] (cid:0) P [ − +0 0 ] + P [ − +0 0 ] + P [ − +0 1 ] (cid:1)i(cid:27) . (B8)Plugging the expressions for the probabilities P [ − +0 D ] and P [ − +0 D ] from Eq. (A8) into Eq. (B8),and after doing some algebra we finally obtain∆ ρ + − (cid:12)(cid:12) − → +0 = ρ + − µN (cid:26) ( µ − h (1 − q ) ρ + q (cid:16) ρ + p o ρ (cid:17)i − ( µ − ρ + − ρ − h ρ − (1 − p o ) q ρ i (cid:27) . (B9)We now follow an approach similar to the one above for ∆ ρ + − (cid:12)(cid:12) − → +0 and calculate the secondterm of Eq. (B2) as∆ ρ + − (cid:12)(cid:12) − → +1 = P (cid:0) − (cid:1) µ X {N − } M (cid:0)(cid:8) N − (cid:9) , µ (cid:1) µ (cid:2) N [ − +1 0 ] + N [ − +1 1 ] + p o (cid:0) N [ − +1 0 ] + N [ − +1 1 ] (cid:1)(cid:3) × (cid:2) µ − (cid:0) N [ − +1 0 ] + N [ − +1 0 ] + N [ − +1 1 ] + N [ − +1 1 ] (cid:1)(cid:3) µN/
2= 2 P (cid:0) − (cid:1) µ N (cid:26) µ h hN [ − +1 0 ] i + hN [ − +1 1 ] i + p o (cid:0) hN [ − +1 0 ] i + hN [ − +1 1 ] i (cid:1)i − h(cid:10) N [ − +1 0 ] (cid:11) + (cid:10) N [ − +1 1 ] i + p o (cid:0)(cid:10) N [ − +1 0 ] (cid:11) + hN [ − +1 1 ] (cid:11)(cid:1) + 2 hN [ − +1 0 ] N [ − +1 1 ] i + (1 + p o ) (cid:0) hN [ − +1 0 ] N [ − +1 0 ] i + hN [ − +1 0 ] N [ − +1 1 ] i + hN [ − +1 0 ] N [ − +1 1 ] i + hN [ − +1 1 ] N [ − +1 1 ] i (cid:1) + 2 p o hN [ − +1 0 ] N [ − +1 1 ] i (cid:1)i(cid:27) = 2 P (cid:0) − (cid:1) µN (cid:26) ( µ − h P [ − +1 0 ] + P [ − +1 1 ] + p o (cid:0) P [ − +1 0 ] + P [ − +1 1 ] (cid:1)i − µ − h(cid:0) P [ − +1 0 ] + P [ − +1 1 ] (cid:1) + p o (cid:0) P [ − +1 0 ] + P [ − +1 1 ] (cid:1) ++ (1 + p o ) (cid:0) P [ − +1 0 ] + P [ − +1 1 ] (cid:1)(cid:0) P [ − +1 0 ] + P [ − +1 1 ] (cid:1)i(cid:27) , (B10)where we have used the moments from Eqs. (B7). After substituting expressions (A10) forthe probabilities P [ − +1 D ] and P [ − +1 D ] we arrive to∆ ρ + − (cid:12)(cid:12) − → +1 = ρ + − µN (cid:26) ( µ − h (1 − q ) ρ + q p o (cid:16) ρ + ρ (cid:17)i − ( µ − ρ + − ρ ρ − [1 − (1 − p o ) q ] (cid:27) . (B11)38y adding Eqs. (B9) and (B11) we obtain the following expression for the change in ρ + − due to − → + transitions dρ + − dt (cid:12)(cid:12)(cid:12) −→ + = ωρ + − µ (cid:20) µ − − ( µ − ρ + − ρ − (cid:21) . (B12)Then, by interchanging sings + and − in Eq. (B12) we obtain the change in ρ + due to+ → − transitions dρ + − dt (cid:12)(cid:12)(cid:12) + →− = ωρ + − µ (cid:20) µ − − ( µ − ρ + − ρ + (cid:21) . (B13)Finally, adding Eqs. (B12) and (B13) we arrive to the following rate equation for ρ + − quotedin Eq. (3b) of the main text dρ + − dt = 2 ωρ + − µ (cid:20) ( µ − (cid:18) − ρ + − ρ + ρ − (cid:19) − (cid:21) . Appendix C: Derivation of the rate equation for ρ The average change of the fraction of infected nodes ρ in a single time step can be writtenas dρ dt = 11 /N h ∆ ρ (cid:12)(cid:12) +1 → +0 + ∆ ρ (cid:12)(cid:12) +0 → +1 + ∆ ρ (cid:12)(cid:12) − → − + ∆ ρ (cid:12)(cid:12) − → − i , (C1)where each term represents a different transition corresponding to a disease update on thecontact layer. The first term of Eq. (C1) corresponds to the recovery of a (cid:2) +1 (cid:3) node, andcan be estimated as ∆ ρ (cid:12)(cid:12) +1 → +0 = − P (cid:0) +1 (cid:1) (1 − β ) 1 N ≃ − (1 − β ) N ρ + ρ . (C2)That is, with probability P (cid:0) +1 (cid:1) ≃ ρ + ρ a (cid:2) +1 (cid:3) node is picked at random, and then recoverswith probability 1 − β , decreasing ρ in 1 /N . The second term corresponds to the infectionof a (cid:2) +0 (cid:3) node, while the last two terms are equivalent to the first two, but where a node withopinion − is recovered and infected, respectively. By the symmetry of + and − opinions,the last two terms are obtained by interchanging signs + and − in the first two.We now find an approximate expression for the second term of Eq. (C1). A susceptiblenode j in state (cid:2) +0 (cid:3) can be infected by a sick neighbor i with + or − opinion and connected39o j by a contact link or by both a social and a contact link. Thus, four possible contactinteractions lead to the (cid:2) +0 (cid:3) → (cid:2) +1 (cid:3) transition:∆ ρ (cid:12)(cid:12) +0 → +1 = ∆ ρ (cid:12)(cid:12)(cid:12) ++1 0 → ++1 1 + ∆ ρ (cid:12)(cid:12)(cid:12) ++1 0 → ++1 1 + ∆ ρ (cid:12)(cid:12)(cid:12) − +1 0 → − +1 1 + ∆ ρ (cid:12)(cid:12)(cid:12) − +1 0 → − +1 1 . (C3)The symbol (cid:2) O +1 0 (cid:3) represents a contact interaction between node i in state (cid:2) O (cid:3) ( O = + , − )and node j in state (cid:2) +0 (cid:3) . The state that changes in the interaction is now displayed on theright-hand side of the symbol, instead on the left-hand side as for the case of the socialinteractions described in the previous sections. This is because the chosen neighbor j of i changes state in the CP, while in the VM is node i who changes state. Taking into account theevents and their associated probabilities that lead to each of the four interactions describedabove, we can write Eq. (C3) as∆ ρ (cid:12)(cid:12) +0 → +1 = P (cid:0) +1 (cid:1) µ X {N +1 } M (cid:0)(cid:8) N +1 (cid:9) , µ (cid:1) βµ (cid:0) N [ ++1 0 ] + N [ ++1 0 ] (cid:1) N + P (cid:0) − (cid:1) µ X {N − } M (cid:0)(cid:8) N − (cid:9) , µ (cid:1) βµ (cid:0) N [ − +1 0 ] + p d N [ − +1 0 ] (cid:1) N . (C4)The first and third terms of Eq. (C4) correspond to selecting an (cid:2) O (cid:3) node i and a contactneighbor j with state (cid:2) +0 (cid:3) at random, which happens with probability P (cid:0) O (cid:1) N [ O +1 0 ] /µ , andthen i infecting j with probability β , given that they are not connected by a social link. Thesecond and fourth terms are similar to the first and second terms, respectively, but selectinga social/contact neighbor j . As both types of links are present in this case, i infects j with probability β p d when both nodes have different opinions (fourth term). In all cases ρ changes by 1 /N . Performing the sums of Eq. (C4) we obtain∆ ρ (cid:12)(cid:12) +0 → +1 = βµN h P (cid:0) +1 (cid:1) (cid:0) hN [ ++1 0 ] i + hN [ ++1 0 ] i (cid:1) + P (cid:0) − (cid:1) (cid:0) hN [ − +1 0 ] i + p d hN [ − +1 0 ] i (cid:1)i . (C5)Replacing the expressions for the first moments hN [ O +1 0 ] i = µP [ O +1 0 ] and hN [ O +1 0 ] i = µP [ O +1 0 ]in Eq. (C5), and using the following expressions for the conditional probabilities P [ ++1 0 ] ≃ (1 − q ) ρ + ρ ρ , P [ ++1 0 ] ≃ q ρ ++ ρ ρ + ρ , (C6) P [ − +1 0 ] ≃ (1 − q ) ρ + ρ ρ , P [ − +1 0 ] ≃ q ρ + − ρ ρ − ρ , (C7)we finally arrive to ∆ ρ (cid:12)(cid:12) +0 → +1 ≃ β ρ N h ρ + − q − p d ) ρ + − i , (C8)40here we have used the conservation relations from Eqs. (1a) and (1c).Now that we estimated the first two terms of Eq. (C1), the last two terms are obtainedby interchanging signs + and − in Eqs. (C2) and (C8):∆ ρ (cid:12)(cid:12) − → − ≃ − (1 − β ) N ρ − ρ , (C9)∆ ρ (cid:12)(cid:12) − → − ≃ β ρ N h ρ − − q − p d ) ρ + − i . (C10)Adding Eqs. (C2), (C8), (C9) and (C10), the rate equation (C1) for ρ becomes dρ dt ≃ γβ ρ − (1 − β ) ρ , (C11)with γ ≡ − q (1 − p d ) ρ + − , (C12)as quoted in Eqs. (5a) and (6) of the main text. Appendix D: Derivation of the rate equation for ρ The average change of the fraction of infected-susceptible pairs of nodes ρ in a singletime step can be written as dρ dt = 11 /N h ∆ ρ (cid:12)(cid:12) +1 → +0 + ∆ ρ (cid:12)(cid:12) +0 → +1 i + 11 /N h ∆ ρ (cid:12)(cid:12) +1 → +0 + ∆ ρ (cid:12)(cid:12) +0 → +1 i + ⇔− , (D1)where the first and second terms correspond to the change in ρ due to the recovery of a (cid:2) +1 (cid:3) node and the infection of a (cid:2) +0 (cid:3) node, respectively, while the last two terms are thecorresponding recovery and infections events of nodes with − opinion, and are obtainedby interchanging the symbols + and − in the first two terms. The recovery term can becalculated as∆ ρ (cid:12)(cid:12) +1 → +0 = P (cid:0) +1 (cid:1) (1 − β ) µ X {N +1 } M (cid:0)(cid:8) N +1 (cid:9) , µ (cid:1) (cid:2) µ − (cid:0) N [ ++1 0 ] + N [ ++1 0 ] + N [ + − ] + N [ + − ] (cid:1)(cid:3) µN/ , (D2)where the expression in square brackets is the change in the number of 10 links connected to anode i in state (cid:2) +1 (cid:3) when i recovers, given a specific configuration of node types connected to i [see Fig. 11(a)]. The summation in Eq. (D2) leads to the first moments of the multinomial41 − social link0 0 contact link1 0 contact link1 1 contact link+ + social link ∆ρ = 2(µ−2Ν )/µΝ + − N[ ]+N = + +
N[ ]+ + +
N[ ]+ + −
N[ ] + + + − − −+ + − − − + + prob = + − + + − i i µ N 0 0 contact linksN 1 0 contact links µ −N 1 1 contact links −N 1 0 contact links a) Recovery + + +− + − + + + + +− + − + + β prob = µ −1−N 0 0 contact links jj ∆ρ = 2[µ−2(1+Ν )]/µΝ j1 0 + + N[ ]+ + +
N[ ]+ + − + −
N[ ]N =
N[ ]+ − + i − + ij j + + µ −1−N 1 0 contact links j10j b) Infection FIG. 11: (Color online) Schematic illustration of two disease updates. (a) Recovery: a node i instate (cid:2) +1 (cid:3) recovers with probability 1 − β . (b) Infection: a node i in state (cid:2) +1 (cid:3) infects a contactneighbor j in state (cid:2) +0 (cid:3) with probability β . The change in the density of 10 contact links is denotedby ∆ ρ . probability M (cid:0)(cid:8) N +1 (cid:9) , µ (cid:1) , with single event probabilities P [ ++1 0 ] and P [ ++1 0 ] given by Eqs. (C6),and P [ + − ] ≃ (1 − q ) ρ − ρ ρ , P [ + − ] ≃ q ρ + − ρ ρ + ρ . (D3)Replacing these expressions for the probabilities and using the conservation relations fromEqs. (1a) and (1c) we obtain, after doing some algebra,∆ ρ (cid:12)(cid:12) +1 → +0 ≃ − β ) ρ + N ( ρ − ρ ) . (D4)We now calculate the second term of Eq. (D1) corresponding to the change in ρ afterthe infection of a node with + opinion:∆ ρ (cid:12)(cid:12) +0 → +1 = h P (cid:0) +1 (cid:1) µ X {N + i, } M (cid:0)(cid:8) N + i, (cid:9) , µ (cid:1) βµ (cid:0) N i [ ++1 0 ] + N i [ ++1 0 ] (cid:1) + P (cid:0) − (cid:1) µ X {N − i, } M (cid:0)(cid:8) N − i, (cid:9) , µ (cid:1) βµ (cid:0) N i [ − +1 0 ] + p d N i [ − +1 0 ] (cid:1)i × µ − X {N + j, } M (cid:0)(cid:8) N + j, (cid:9) , µ − (cid:1) (cid:2) µ − (cid:0) N j [ ++0 1 ] + N j [ ++0 1 ] + N j [ + − ] + N j [ + − ] (cid:1)(cid:3) µN/ P × C . (D5)42he term called P in Eq. (D5) –the two summations inside the square brackets– is theprobability that an (cid:2) O (cid:3) node i infects a (cid:2) +0 (cid:3) neighbor j , and is the same as the one calculatedin Eq. (C4) for ∆ ρ (cid:12)(cid:12) +0 → +1 , which is estimated in Eq. (C8) as P ≃ βρ h ρ + − q − p d ) ρ + − i . (D6)We notice that the extra 1 /N prefactor in Eq. (C8) comes from the change in ρ , which for ρ depends on the neighborhood of node j . The subindex i in the term P indicates thatthe infection probability term depends only on node i and its neighborhood [see Fig. 11(b)].The term called C corresponding to the summation outside the square brackets expresses thechange in ρ when node j gets infected [see Fig. 11(b)]. Here the subindex j refers to node j and its neighborhood. This term carries the information that the infection on j comesfrom one of its infected neighbors i , and thus it is known already that at least one of j ’sneighbors has disease state D i = 1. This is taken into account by running the summation onthe other µ − j is at least one, which is added to the total number of 10 links inside the parentheses.Using the conditional probabilities P [ ++0 1 ] ≃ (1 − q ) ρ + ρ ρ , P [ ++0 1 ] ≃ q ρ ++ ρ ρ + ρ , (D7) P [ + − ] ≃ (1 − q ) ρ − ρ ρ , P [ + − ] ≃ q ρ + − ρ ρ + ρ , (D8)and the conservation relations Eqs. 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