Opinion competition dynamics on multiplex networks

Multilayer and multiplex networks represent a good proxy for the description of social phenomena where social structure is important and can have different origins. Here, we propose a model of opinion competition where individuals are organized according to two different structures in two layers. Agents exchange opinions according to the Abrams–Strogatz model in each layer separately and opinions can be copied across layers by the same individual. In each layer a different opinion is dominant, so each layer has a different absorbing state. Consensus in one opinion is not the only possible stable solution because of the interaction between the two layers. A new mean field solution has been found where both opinions coexist. In a finite system there is a long transient time for the dynamical coexistence of both opinions. However, the system ends in a consensus state due to finite size effects. We analyze sparse topologies in the two layers and the existence of positive correlations between them, which enables the coexistence of inter-layer groups of agents sharing the same opinion.


Introduction
Human interaction often gives rise to different phenomena of collective agreement ranging from common language or religion to opinion formation. Various models have been proposed [1][2][3][4] to describe the process of social consensus, with the so-called voter model being one of the simplest and most studied [5]. It is a model with two equivalent states, first introduced to describe competition in biological species [5], and later named as the voter model in [6]. Voter dynamics is based on the mechanism of imitation, in which individuals change their opinion by imitating their randomly selected neighbors [1,7,8]. The model has two absorbing states of consensus (collective agreement in one of the states) and a critical dimension. In dimension d = 1, 2 there is coarsening with the unbounded growth of domains of each of the absorbing states. For d 2 > , including typical complex networks, there is no coarsening and the system only reaches an absorbing state through finite size fluctuations. In the infinite system limit, the system remains in an active dynamical state with the two states coexisting [1,7]. A population model which aimed to describe the competition of two languages was proposed by Abrams and Strogatz (AS) [9]. It turns out that the AS model is a mean-field approximation of an individual based model, which is a modification of the voter model [10]. The two options here are two non-equivalent languages which compete in a bilingual society. The languages are not perceived by the individuals in the same way, they have complementary prestiges reflecting the difference in the social status of spoken languages. An additional parameter, the volatility, was introduced in the latter model to indicate the tendency to switch the use of a language. The AS model fits the real aggregated data of endangered languages such as Quechua (in competition with Spanish), and Scottish Gaelic and Welsh (both in competition with English) [11,12]. Moreover, as has been found by Vazquez et al [13], the AS model can support steady states where both languages coexist for small values of volatility. Several studies [1,2,[13][14][15] have been inspired by the AS model, which tests how different formalization of the interactions-e.g. different network topology-can determine dominance by one state or, in contrast, when configurations in which two states coexist is possible.
In recent years, similar studies have been performed on more complex structures: the so-called multiplex networks [16,17]. A multiplex network [18][19][20][21][22] consists of two or more interconnected networks lying in distinct layers. The layers have the same number of nodes, which are connected to their counterparts across the layers, and in general have a different connectivity structure within them [18,20]. This framework allows for a more realistic approach in the study of the interaction of individuals, which can communicate through different types of channel. Multiplex networks have been used to analyze public transportation systems [23,24], the spreading of awareness and infection [25,26], the dynamics of ecological populations [27,28], cultural dynamics [29] and the evolution of social networks [17,30].
More recently, Diakonova et al [16] demonstrated the irreducibility of a multiplex version of the voter model. In this approach, a multiplex network was considered, where a fraction of nodes and links can be present in both layers and any change in the state of those nodes in a layer is instantly replicated by the other. This mechanism affects voter model dynamics, and significant differences from the classical single layer case were found.
Here we propose a modification of the voter model organized in a multiplex network, where the dynamics occurring on the nodes is irreducible by nature. Specifically, each node can appear in a different state and may receive concurrent social pressures in distinct layers. Additionally, the state of a node in one layer influences its own state in the other layer. This system paves the way for new scenarios for the coexistence of opposing options, which are discussed in this article. Starting from microscopic dynamics, we develop a mean field theory which shows that in addition to consensus, there is a non-trivial steady state (the non-consensus states), where the two options coexist simultaneously, emerge and are linearly stable. Our theoretical findings were verified by numerical simulations, where, however, no-consensus states can lose their stability due to finite size effects and the system is eventually driven to consensus. Nonetheless, the time needed for the individuals to consent is longer than in single-layer networks. We analyze this behavior by developing a probabilistic macroscopic description. We show that the coupling of two layers, fully connected, or Erdös-Rényi networks, with two different preferred options can generate various kinds of coexistence options. In addition, we analyze how the topology of the two networks and the correlations between layers can influence the distribution of states among nodes. Specifically, we observe that links overlap the correlation, together with degree-degree correlation between the layers, promoting the formation of inter-layer groups of nodes in the same state.

Competition of options on multiplex networks
We propose here a model for describing the competition between two abstract options, A and B (they can be languages, opinions, voting intention, etc) in a multiplex network. The model is based on a modification of the AS model with volatility equal to one, and keeping the idea that the two options have different perceived status. In this model social interactions occur within distinct layers that may have originated from different contexts like family or business networks, Facebook or Twitter, etc. Nodes and their counterparts across layers correspond to the same individuals participating in different networks. Intra-layer links denote the individuals' connections within each network, while inter-layer links indicate the mutual influence of the individuals' state across layers (see figure 1).
We assign a state i s a to each node i (i N 1, 2, = ¼ ) in the layer α ( I, II a = ), such that 1 i s = a (or 0) if the node is option A (or B) in layer α. We also endow prestige S a to option A differently in each layer; the corresponding option B has a complementary prestige S 1a . We restrict the values of the prestiges to S 0.5, 1 in order to guarantee that the two layers do not have the same preferred option. The state of a node in a layer influences its own state in the other layer with strength γ, where 0 1   g . The limited values of γ correspond to opposing situations so, 0 g = denotes that the two layers are independent, while 1 g = represents a situation in which individuals are not influenced by their neighbors.
The dynamical evolution of this multiplex-organized system is described below. A randomly chosen node i in one layer can change its option according to the transition probabilities, where G ij a is the adjacency matrix of layer α, with elements G 1 ij = a , if nodes i and j are connected in layer α and G 0 ij = a otherwise, k i a is the degree of the node i in layer α, and with i s ā we denote the state of the node i in the other layer.
System equation (1) supports steady states of full consensus where all individuals appear to have a single option A or B. However, the multiplex structure of this model induces a new steady state, which has never been observed in the classical voter model, where the two options coexist. In the following we present the stability analysis of these steady states starting with a mean field (MF) description of the dynamics. Then, we build a theory for the ordering dynamics to analyze the system's evolution towards a steady state. We also perform a numerical simulation on complete (all-to-all) and Erdös-Rényi networks in order to verify our theoretical findings.

Mean field approach and master equation
One of the central problems in the analysis of opinion dynamics is understanding the conditions in which a collective agreement occurs. The dynamical evolution of system equation (1) is analyzed by means of an MF approach, where a previously used order parameter for the single-layer networks [13,31] is employed. The state of the system is characterized by the option's polarization (often called magnetization) and is defined as the difference between the fractions X a of nodes in state 1 (option A) and the fractions X 1a of nodes in state 0 (option B). Therefore, for layer α we obtain the polarization option The dynamics of the system is governed by the presence of active links, namely of links connecting nodes in different states, because the probability of a node switching to another state depends on the density of its active links. Two types of active link are associated with each node: active intra-layer links if the node in layer α does not consent with its neighbors in α, and active inter-layer links if it has a different state in the different layers. The density of active intra-layer links in layer α is given by the expression, , δ is Kronecker's delta and L a is the total number of links in layer α. The density of the active inter-layer links reads In the employed MF approximation, each node in a layer is connected with all the other nodes of that layer. Therefore, we can naturally express the densities of intra-layer and inter-layer active links as a function of m a as, In the appendix we derive the master equation for the probability Q m t , a ( )that the system has polarization option m a at time t We notice that the influence of each layer on the other appears not only in the potential but also as an additive term in the diffusion coefficient. We can say that term r^controls the diffusion of the two options in the two layers. The two potentials felt by the two layers have an opposite minimum because of the setting in the prestiges; for 0 g = each layer would reach the full consensus in opposite options.
In the thermodynamic limit the diffusive term is canceled and the potential V m m , I II ( )defined in equation (8) has three extrema: 1. Two corresponding to the states of full consensus: One that stands for the non-consensus steady state where options A and B coexist and is given by, ) . The stability of the fixed points is analyzed by imposing V 0 m ¶ = a and by studying the eigenvalues of the corresponding Jacobian matrix. We find that in the range of parameters defined by the relation: the steady state of coexistence, given by equation (9), is linearly stable while the state of full consensus is unstable. Out of this region, instead, the state of coexistence vanishes and the states of full consensus become stable; for b a a 1 > -+ the system consents to option A, while for b  (10), the two solutions of full consensus and coexistence coincide. The coexistence solution varies continuously from −1 to 1 generating a second order absorbing phase transition. The steady states of full consensus are absorbing, frozen states and the switch probabilities vanish. The steady state of coexistence, instead, is an active dynamical state (see [32]), where individuals continue switching and the system visits a set of configurations which are macroscopically equivalent in terms of ordering. Therefore, by varying parameters according to equation (10) we find an absorbing transition in which the system goes from an active state to a frozen configuration state. The complementary of the density of active links, 1 r a , gives what is called the coherent domain [33], namely the density of links between nodes in the same state. Figure 2 shows the landscape of steady states of system equation (9) in the parameter space S II gfor a fixed S I . For 0 g = the two layers do not communicate and the multiplex is reduced to two independent AS systems (see equation (1)). Each layer reaches a steady state of consensus; however, the states are complementary, i.e. one layer consents to option A while the other consents to option B. For 1 g = , both a and b are equal to zero and each solution of the form m m is a potential stable solution. The two layers communicate with the stronger possible coupling but the individuals are not affected by their neighbors (see equation (1)). A randomly chosen node replicates its state from the other layer, resulting in a frozen steady state where every node has the same state across the layers, while none of the layers reaches consensus. For other values of γ, the previously found condition equation (10) determines three different dynamical regimes displayed in figure 2 (for 0 1  g < ).  figure 2(c), while the density of coherent nodes between the layers, namely the nodes in the same state in both layers, is depicted in figure 2(d). We verify the results obtained from our MF approximation by constructing multiplex networks of different sizes composed of complete networks in their layers. In the coexistence regime, the options are distributed between the nodes in two different ways presented in figure 3. In figure 3(a), the parameter values allow the existence of both nodes in different states in the different layers and nodes in the same state across layers. The densities r a and r^of the active links are different from zero. In figure 3(b), γ is strong enough to drive the system in a steady state, where the nodes have the same state in both layers. However, small fluctuations from this steady state are

Finite size effects in multiplex networks with complete layers
As mentioned above, the coexistence state is an active one, namely the probabilities of switching the state do not vanish and the system keeps fluctuating around the fixed point. In finite size networks, however, these fluctuations can drive the system to an absorbing state of full consensus. This is a finite size effect and has been observed in our stochastic simulations. In our settings, the two layers have opposite prestiges; therefore for 0 g = the polarization option m I would go to 1 while m II would go to −1. = is a solution. In this particular setting we can study the effects purely induced by the multiplex structure, because we can avoid the effect generated by the competition between the two options in the two layers. From the related theory for single-layer networks [31] we know that in this case the system will reach an absorbing state with a characteristic time that scales with N. In the multiplex networks considered here we find by fitting (see figure 5) a factor arising from the inter-layer interaction, c , for which the characteristic time to reach consensus takes the form This relation is consistent with the studies presented in [31], where the scaling factor is the inverse of the prefactor of the active links in the diffusion coefficient. The factor N 2 results from the total number of nodes in the whole two-layer system. From the time evolution expressed in [31], we can approximate t e t r » t -( ) . Figure 5(a) shows the characteristic time equation (12) for a different size and different γ, using complete networks. Figure 5(b) shows the evolution of the density r^of the active inter-layer links. We can thus conclude that in the case of equal prestige, the multiplex effect translates into an extension c g ( ) in the lifetime of the coexistence option state.

Erdös-Rényi networks
Here we extend the ansatz of [31] by considering Erdös-Rényi networks. In [31], it was found for the density of active links that the relation m 1 . We test this assumption for multiplex networks consisting of different Erdös-Rényi layers of various size and mean degree, discovering that it is also valid in our case (see figure 6).
Then, the density of intra-layer active links reads The inter-layer density, instead, does not depend on the topology of the network. In order to extend the Fokker-Planck equation (7), for the case of two Erdös-Rényi networks we consider that if a node with k a changes its state at a time step, the polarization option changes by k, Since each node has an inter-layer degree equal to one, the inter-layer change remains Previous studies of the voter model (prestige equal to 1/2) in complex networks [31], have revealed the relation between the characteristic time it takes to reach the full consensus and the topology of the network. Previously in [31], for an Erdös-Rényi network with a mean degree k á ñ a scaling factor was found of the form   = we check the same relation for the scaling factor by considering that the prefactor of the active links term ends up being a one layer term multiplied by 1 g -, As in the case of equation (12), a factor two accounts for the total number of nodes N 2 . In figures 6(a) and (b), we observe the time evolution of the average density r^of inter-layer active links for a fixed mean degree k á ñ a and different values of γ, showing the validity of the assumption expressed by equation (16).

Impact of correlation
With a fully connected population we have shown how the coupling of two layers with two different preferred options can generate various kinds of coexistence options. Here we are interested in how the topology of two networks and the correlations between layers can influence the distribution of the states among the nodes. Previously, it was shown that the relation between the layers in a real multiplex can be characterized by geometric correlations in hidden metric spaces underlying each layer of the system [34][35][36]. There are two kinds of correlations: popularity correlations, which are correlations between the degrees the nodes have in the two layers, and similarity correlations, which control the probability of the links overlapping between layers. To understand the impact of correlations, we perform numerical simulation using the geometric multiplex model (GMM) developed in [34].
The GMM is based on the (single-layer) network construction procedure of the Newtonian 1  [37] and hyperbolic 2  [38] models. Here we present the treatment for the 2  version. To construct a network of N nodes, first it is requirement of the procedure for each node i = 1, K, N to be assigned its popularity r i and similarity i q coordinates, and accordingly, to connect each pair of nodes i j , with the probability p x , where x ij is the hyperbolic distance between the nodes and R N lñ . The connection probability p x ij ( )is the Fermi-Dirac distribution, where the temperature parameter T controls the level of clustering in the network [39]. The multiplex composed of these single-layer networks allows for radial and angular coordinate correlations across layers. The level of these correlations is regulated by the parameters 0, 1 d Î [ ]and 0, 1 n Î [ ], without affecting the single-layer topologies, where δ stands for the radial (also called popularity) correlation and ν denotes the angular (also called similarity) correlation. The popularity correlation relates to the probability of finding a node with the same degree in the different layers. The similarity correlation is related to the probability of the links overlapping between layers.
We compare the GMM with different correlation settings and with the ER networks. In all cases the multiplex networks are composed of layers of N=2000 nodes with a mean degree of k 6 á ñ = . The most significant effect is observed in the distribution of the states between the nodes. The similarity correlation, which increases the probability of links overlapping between layers, promotes inter-layer groups of nodes in the same state and connected in both layers, namely coherent islands. If the whole system has a favorite state For the ER and uncorrelated GMM, r^fluctuates around the MF solution. Instead, in the strongly correlated system, the size of the coherent islands grows, generating a slower decay of r^. The top row of figure 7 shows that none of the three systems has reached an absorbing state, for which the basic difference lies in the distribution of the states between nodes. We can conclude that the strongly correlated system is in a state of coexistence of different coherent islands. This feature becomes more evident for high values of the coupling, as shown in figure 8 where 0.8 g = . In this case, for some realizations, the system reaches full consensus in two different ways: in the uncorrelated case it is thanks to a single fluctuation from the finite size effects, while in the fully correlated case one of the coherent islands grows and incorporates the entire system. Another important measure reveals how the similarity correlation acts on the distribution of the states between nodes. In figure 9 we set S 0.5 = a , for which neither the layers nor any of the system has a favored state (the coupling is 0.3 g = ).
In figure 9(a), the system is uncorrelated (i.e.   density of links overlaps the nodes in state B in both layers. We notice that C B increases considerably when we set a similarity correlation ν equal to one. By comparing figure 9(a) with figures 9(b) and (c) we notice that the action of the popularity correlation alone does not produce significant effects, while the similarity correlation increases the coherent islands.

Conclusion
In this article we studied the consensus and coexistence of two opposing options in a discrete system organized in multiplex networks. For this purpose, we proposed a modification of the well-known AS model. Individuals correspond to the nodes of a multiplex and participate in different social networks in distinct layers. Social interaction within a given social context is denoted by intra-layer links, while inter-layer links represent the tendency to maintain the same option across different social networks. Although similar models have previously been studied in multiplex networks [4,16,17], the novelty of our study lies in the fact that individuals can have different options in different layers. This naturally reflects a common situation in which an individual can possess different opinions in different social contexts as a result of consensus with other individuals in one context but not in the other.
Our analysis shows that the latter property enriches the system dynamics and allows not only a consensus on a single state for both layers, but also for active dynamical states of coexistence for both options. This can be described by two layers having opposite preferred options, which generate potentials with the opposite minimum (there is no state that satisfies both layers). Each layer 'feels' the other to be an additive noise, so that even if individuals instantly consent in one layer, they preserve the chance of switching due to the influence of the other. In the MF approximation we found a coexistence phase: there is a wide range of parameters where the coexistence of the two options in each layer is in a stable steady state. An absorbing transition exists when going from this active phase of dynamical coexistence to the absorbing state of consensus. The transition lines are the ones indicated in figure 2, so that it can be induced by changing the coupling parameter between the layers. Beyond the mean field approximation we need to take into account finite size fluctuations. These fluctuations can drive the system from the active dynamical state of coexistence to an absorbing state of consensus. In particular, we considered the case of equally prestigious options, as in the voter model. For a single decoupled layer, the characteristic time it takes to reach an absorbing state is proportional to the size of the system [8]. In our case, indeed, because we have additive noise induced by the mutual influence between the layers, this characteristic time also depends on the coupling parameter γ. Therefore, in the presence of finite size fluctuations, the multiplex structure of our system can affect and lengthen the lifetime of the transient state of dynamical coexistence.
Mean field results are verified by numerical simulations in multiplex networks consisting of complete graphs, Erdös-Rényi networks and geometrical multiplex networks. For the Erdös-Rényi networks we find the same qualitative findings, but local effects modify the transition lines for the absorbing transition and lifetimes of active states depending on the distribution of degrees in the network. With geometrical multiplex networks we examined both the impact of network topology and the correlation between layers on the dynamics. We find that high correlations between layers promote the coexistence of different inter-layer islands of nodes in the same