Opinion formation in multiplex networks with general initial distributions

We study opinion dynamics over multiplex networks where agents interact with bounded confidence. Namely, two neighbouring individuals exchange opinions and compromise if their opinions do not differ by more than a given threshold. In literature, agents are generally assumed to have a homogeneous confidence bound. Here, we study analytically and numerically opinion evolution over structured networks characterised by multiple layers with respective confidence thresholds and general initial opinion distributions. Through rigorous probability analysis, we show analytically the critical thresholds at which a phase transition takes place in the long-term consensus behaviour, over multiplex networks with some regularity conditions. Our results reveal the quantitative relation between the critical threshold and initial distribution. Further, our numerical simulations illustrate the consensus behaviour of the agents in network topologies including lattices and, small-world and scale-free networks, as well as for structure-dependent convergence parameters accommodating node heterogeneity. We find that the critical thresholds for consensus tend to agree with the predicted upper bounds in Theorems 4 and 5 in this paper. Finally, our results indicate that multiplexity hinders consensus formation when the initial opinion configuration is within a bounded range and, provide insight into information diffusion and social dynamics in multiplex systems modeled by networks.

networks. In recent years, agent-dependent multi-level confidence bounds have been incorporated into the model, which mirror the complicated physiological and psychological factors such as the disparity of people's knowledge, experience, and personality; see e.g. [12][13][14][15] . The persuasion capacity of the mass media has also been found to play a role in opinion formation 16 . It is worth noting that most of them are based on numerical simulations with only a few exceptions [17][18][19][20] due to the complicated nonlinear dynamics involved.
The opinion negotiation processes studied in the above works take place on networks containing edges of the same type and at the same temporal and topological scale. However, the real individuals in a society are usually simultaneously connected in multiple ways, which can make a non-additive effect on network dynamics 21,22 . People in a society, for example, interact through diverse relationships: friendship, partnership, kinship, vicinity, work-related acquaintanceship, to name just a few. Admittedly, a natural and more appropriate description of such systems can be given by using multiplex networks, where the networks are made up of different layers that contain the same nodes and a given type of edges in each layer. Some recent works have pointed out that multiplexity can result in intrinsically different dynamics from their equivalent single-layer counterparts. The irreducibility of the Ising and voter models on multiplex networks has been emphasized in 23,24 . Opinion competition dynamics on duplex networks has been studied in 25 , where coexistence of both opinions in the two layers has been found possible using mean-field approximation. In the context of culture dissemination, multiplexity is found to generate a qualitatively different dynamical behaviour for the Axelrod model, which produces a new stable regime of cultural diversity 26 . In addition to single information spreading process on multiplex networks, the coupling between different types of contact processes, such as opinion formation and disease spreading, has been investigated in the multiplex networks; see e.g. [27][28][29] . Synchronisation processes between different layers in multiplex networks featuring the interplay between distinctive topological structures and dynamics have been reported recently in 30,31 . An updated survey towards the spreading processes and opinion formation on multiplex networks can be found in 32 . To the best of our knowledge, little attention has been paid to the opinion evolution in the Deffuant model (featuring bounded confidence) in the context of multiplex networks. In 33 , the author first examined the Deffuant model in a multiplex network, which is modeled by an infinite line with multiple layers. The critical confidence threshold is analytically identified through probabilistic analysis and verified by numerical simulations.
In this paper, we aim to moving a step further in the direction of 33 by considering both general initial opinion distributions and general multiplex networks. In the standard Deffuant model, the initial opinions are assumed to be independently and uniformly distributed in the interval [0, 1]. General initial distributions have been independently introduced in 19, 34 . We first address the opinion formation with general initial distributions over 1-dimensional multiplex networks after introducing our model in the Model description section. We then generalise our results to higher-dimensional multiplex lattices and, to general multiplex networks satisfying some regularity conditions. We derive analytical expressions for the critical confidence bound, where both the structural multiplexity and the initial distribution play essential role. Interestingly, we show that multiplexity essentially impedes consensus formation in the situations when the initial opinion configuration is within a bounded range. On the other hand, if a substantial divergence exists in the initial opinions, whether it is bounded distributed or not, multiplexity is found to play no role in determining the critical confidence level. Extensive numerical simulations are provided with both constant and degree-dependent convergence parameters, and the paper is concluded with some open problems in the Discussion section.
where μ ∈ (0, 1/2) is the so-called convergence parameter. Therefore, if the two pre-meeting opinions lie at a distance less than a certain confidence bound from one another, the meeting agents will come closer to each other symmetrically, by a relative amount μ, where μ = 1/2 implies that the two agents meet halfway through. If not, then they will stay unchanged. It is worth noting that the model is well-defined since the bounded degree assumption ensures that almost surely (i.e., with probability 1) none of the Poisson events will be simultaneous for an infinite node set [35, p. 28].
The multiplexity in the above opinion model lies in two aspects. First, the interaction rates λp i in each layer can be different. Second, the confidence bounds α i d in each layer can be different too. We might as well consider distinct convergence parameters μ = μ i for the ith layer indicating different willingness to change one's mind. However, it has been confirmed analytically and numerically that μ plays no role in the qualitative behaviour of the opinion dynamics; it rather only affects the convergence time 6,8,18,19 .
Sharing a drink process. In the section, we briefly review the sharing a drink (SAD) process proposed in 18 , which is particularly useful in later analysis of the Deffuant model on ; see also 14,19,34 .
, is a deterministic process defined iteratively as follows: set For a given sequence of nodes … ∈ u u , , 1 2  and μ ∈ (0, 1/2], we obtain the configuration This procedure can be envisioned as a liquid-exchanging process on . A glass is put at each site  u ∈ . At k = 0 only the glass located at the origin is full (with value 1) while all others are empty (with value 0). At each subsequent time step k, one picks two neighbouring glasses at u k and u k + 1, and pouring liquids from the glass with higher level to that with lower level by a relative amount μ. This gives rise to the SAD process. The following lemma on unimodality can be easily proved.  (3) and (4).
. This lemma implies that the constructed SAD process resembles the dynamics of the corresponding Deffuant model backwards in time so that the state X t (0) in the model at any time t > 0 can be expressed as a weighted average of states at time 0 with weights given by an SAD configuration. See 18,33 for a proof.
Data availability. All data are generated by numerical simulations and they have all been reported in the paper.

Results
Opinion dynamics in 1-dimensional multiplex networks. In this section, we will consider the multiplex opinion model on the integers , focusing on the general initial opinion distributions. More specifically, we  33 .
To appreciate this, we first present the results for the case  = 2 (see Theorem 1) and then extend it to the general multiplex case (see Theorem 2), as illustrated in Fig. 1. To this end, we take =  2, = p p 1 , and 2 α α = . With these assumptions, the main result concerning the critical confidence threshold for the 1-dimensional duplex model reads as follows.

Theorem 1. (1-dimensional duplex networks) Consider the above continuous opinion model (
(a) Suppose that the initial opinion follows some bounded distribution X ( ) 0  with expected value E X ( ) 0 , whose support is contained in the smallest closed interval [a, b] ) 0 0 ∈ = . Then, is the critical confidence threshold in the following sense: , then with probability 1, there will be (infinitely many) finally blocked edges, namely, e u u { , . Then, for any d > 0, with probability 1, there will be (infinitely many) finally blocked edges, namely, e u u { , for all t large enough.
Before proceeding with the proof, we provide a couple of remarks. Firstly, when the initial distribution X ( ) t t for sufficiently large t and all ∈ u , and hence, the integers split into (infinitely many) finite clusters of neighbouring agents asymptotically agreeing with each other, with no global consensus achieved. Secondly, in the special case of X ( ) 0  being the standard uniform distribution in [0, 1], we readily reproduce Theorem 1 in 33 . A general  X ( ) 0 has been considered both theoretically and via simulations in 19,34 for simplex networks (i.e.,  = 1). Theorem 1 can be thought of as an extension to multiplex networks. Finally, the assumption α μ > is required here for technical reasons as in 33 , which does not have a counterpart in the case of simplex networks where μ only influences the convergence time of the negotiation process.
The crucial technique adopted here is the SAD process introduced in 18 . The SAD process and its basic properties are briefly reviewed in the Method section. Another key concept from that paper is the flat-points concept. To accommodate the general distributions considered in the present paper, a slight extension of the definitions therein can be provided as follows. Given ε > 0 and the initial opinion configuration Likewise,  ∈ u is said to be an ε-flat point to the left if for all ≥ n 0, We also define that u ∈  is an ε-flat point to the right at time t if for all n 0 ≥ , Step is the critical confidence threshold in the same sense as in Theorem 1(a) (with a 0 = and = b 1). To prove this claim, we need to show that the essential ingredients in the proof of Theorem 1 in 33 still hold true. We mention here an obvious correction that the critical threshold separating the subcritical and supercritical regimes therein should be d min { ,1} c instead of d c . For the subcritical regime, note that the fact that the mass is around the expected value, i.e., >0 for all ε > 0 and u  ∈ by similarly applying the coupling trick and the strong law of large numbers. At time t when a Poisson event occurs, define a Boolean random variable A t by A t = 1 with probability p and A t α = with probability 1 − p so that the opinion model constitutes a marked Poisson process with rate λ 33 . We can then mimic the proof for Propositions 1 and 2 in 33 verbatim, which employs the condition For the supercritical regime, we need to note that the property P(u is two-sided ε-flat) > 0 for any ε > 0 and ∈  u can now be established by keeping in mind that following the same reasoning as in 18 ; see also 34 . Now the proof for the supercritical regime in 33 can be used, which concludes the proof of Step 1.
Step 2. Suppose that the initial opinion distribution  X ( ) 0 is again confined in [0,1] with expected value = X E( ) 1/2 0 . For any ε > 0, as in Step 1 we assume that X P( ) 0 0 ∈ = . Then, we claim is the critical confidence threshold in the same sense of Theorem 1(a) (with a = 0 and b = 1).
, we have initial opinions both below and above 1/2 with probability 1. Therefore, any edges which are blocked due to initial incident opinions lying on different sides of the gap I will remain blocked for all t. By ergodicity, there will be infinitely many such blocked edges, and thus consensus can not be reached in this case.
When d h > , we need to show that u t u t P( is flat to the right at time ) P( is flat to the left at time ) 0 (10) and for some sufficiently large t, since an arbitrary flat point at time t = 0 no longer exists due to the gap. Following the reasoning of [34, Section 2], one can then establish Eq. (10). The only minor change that has to be made in order to accommodate the multiplexity is that the involved marked Poisson processes has rate p p (1 ) λ λ λ + − = instead of a unit rate, which does not affect the validity of the proof. Now, as in Step 1, we can mimic the proof of Propositions 1 and 2 in 33 verbatim to settle the subcritical case. Accordingly, Next, the two-sided ε-flatness at time t for any ε > 0 can be established similarly as in [34,Section 2]. Hence, the argument in the supercritical case in Step 1 leads to , completing the proof of Step 2.
Step 3. Now, everything is in place to prove Theorem 1(a) in its full generality.
to the dynamics of our multiplex Deffuant model. Using the result in Step 2 and the fact that the dynamics stays unchanged with respect to translations of the initial distribution and that parameter d can be re-scaled as per a scaling transformation of the initial distribution in order to recover the identical dynamics, we have  , where + X 0 and − X 0 are the positive and negative parts of X 0 , respectively. We may further assume that ≤ > X P( 0) 0 0 , otherwise a translation would transform the problem to this case (c.f. Step 3 above).
Fix d > 0. The same argument in [34,Section 2] can be used to show that the event ε = ∑ = + , for all } 0 for any  ∈ u happens with positive probability. Similarly, along the lines of [33, Proposition 1], we obtain that if ε happens and X u ( ) 0 0 ≤ (which happens with positive probability), then 0 for all t. Namely, there will never be an opportunity for node u + 1 to average with u. The same thing holds for u − 1 by symmetry. Since the initial opinions are i.i.d., with positive probability we have , leaving the edges − u u { 1, } and u u { , 1} + finally blocked. Since this happens at every u  ∈ with positive probability, by ergodicity, it happens with probability 1 at infinitely many nodes. □ For a multiplex network  with  layers, Theorem 2 is within easy reach by essentially using the same arguments as above.

Theorem 2. (1-dimensional multiplex networks) Consider the above continuous opinion model on
is the critical confidence threshold in the following sense: , where the equality holds if and only if  = 1 or . This indicates it is more difficult to reach agreement over multiplex networks than simplex networks in general. When there is a large h, the critical confidence threshold d c is dominated by h and is independent from the multiplexity; on the other hand, for relatively small h, the threshold is determined in turn by both the multiplexity and the initial distribution. When the initial distribution  X ( ) 0 is unbounded, consensus cannot be reached regardless of the multiplexity. Thirdly, if there exists some k satisfying p p in the case of bounded  X ( ) 0 . This suggests that the critical confidence is governed by a frequently interacted layer in the underlying network as one would expect.
Opinion dynamics in general multiplex networks. In this section, we deal with more general multiplex networks and adopt a similar strategy by first looking into a duplex model on higher-dimensional lattices, generalising it to multiplex models and discussig on further extensions.
Particularly, we take For  = 2, we denote = p p 1 and 2 α α = as in the above section. The main result in this duplex case reads as follows.

If the initial opinion is distributed on [a,b] with expected value E(X 0 ) and d X a b b a (E 2 )
Unlike the 1-dimensional case, here we are only able to establish an upper bound for the critical confidence level d c . In fact, as commented in Remark 3.5 in 34 , the case of D 2 ≥ is much more complicated then the 1-dimensional counterpart and it is even not clear if there exists a critical d c separating the supercritical and subcritical regimes since the ultimate consensus does not need be monotonic with respect to d. Furthermore, we note that the consensus result in Theorem 3 is weaker that in Theorems 1 and 2 (for the supercritical regime) in the sense that only the difference between the opinions of two neighbouring individuals is required to converge towards zero. It is to verify that this is equivalent to the convergence of each individual's opinion in a finite network. For infinite networks considered in this paper, however, the picture is quite different as one may imagine a situation where the opinion shows wave-like patterns on broader and broader spatial scales with non-vanishing amplitude as time increases.
To prove Theorem 3, we first define the energy of node u at time t as u f X u ∈ , let T be the sequence of arrival times of the Poisson events at e. The accumulated energy loss along e is defined as , where e u ∼ means u is an end-point of e. Following the same argument of [34,Lemma 3.2] and noting that the number of Poisson rings on a single edge in any time period of length ε is a Poisson random variable with parameter λε, we have the following lemma.

Lemma 3.
For any ∈  u D and time t 0 | − | by more than μd, which for sufficiently small δ, is always less than the span of the gap (δ, Ad) that needs to be crossed due to μ < α. Since there are only countably many edges, the proof of Lemma 4 is completed. Following a similar argument as in [34,Lemma 3.4], we obtain with probability 1 that  Recall that the condition of Theorem 3 implies that , which leads to a contradiction. Hence, the assumption (14) must not be true. The proof then follows from applying Lemma 4. □ Theorem 3 can be directly extended to the multiplex setting for a general  ≥ 2.

Theorem 4. (higher-dimensional multiplex networks) Consider the above continuous opinion model on
. Some remarks follow: firstly, it is easy to check that the lattice D  in Theorem 4 can be extended to any infinite, locally finite, transitive and amenable connected graph = G V E ( , ) ∂ is the edge boundary of S n . The following result can be established.

If the initial opinion is distributed on [a,b] with expected value
{ , }) 1 0 ∈ = and (ii) X 0 is not constant with probability 1. This indicates that the condition in Theorem 4 stand a good chance to be nontrivial even for multiplex networks in most meaningful situations. Thirdly, we have assumed throughout this paper that the initial opinions following  X ( ) 0 are i.i.d. However, Theorems 4 and 5 still hold if the initial opinions are stationary and ergodic with respect to the graph automorphisms because no other specific features of i.i.d. variables are used in the above proof. Finally, it seems that agents forming a multiplex network are more difficult to reach consensus for the same reason as remarked for 1-dimensional multiplex networks in the above section. Furthermore, as we have mentioned in the beginning of this section, it is generally even not clear if we can still speak of critical confidence level d c in D-dimensional (D > 1) multiplex networks and more general multiplex networks.
Numerical results. In this section, we conduct agent-based simulations on different finite multiplex networks, including regular ones such as D-dimensional lattices which can be viewed as a truncation of  D in Theorem 4, and irregular ones such as small-world and scale-free networks, which obviously violate the regularity conditions in Theorem 5 and are prominent examples of non power-law and power-law networks, respectively. Interestingly, we see that for all such networks, the critical thresholds of consensus tend to agree with the predicted upper bounds in Theorems 4 and 5 in the special cases of uniform X 0 and some choices of Poisson rates associated with the multiple layers.
Particularly, in Fig. 2, we plot the percentage of convergence of opinions for five network sizes N ranging from 8 to 256 with  4 = layers each, Poisson rate λ = . p 0 3 i and 0 5 μ = . to maximise the convergence rate. At t = 0, we initialise each agent u V ∈ by assigning an opinion value ∈  X u ( ) 0 from the uniform distribution in (0, 1). To check for convergence of opinions, we require that X u For each curve, we have run 100 simulations to compute the percentage, each time for a different set of α values in Eqs. (1) and (2). Panel (a) shows the results for regular lattices, whereas panel (b) for Watts-Strogatz (small-world) and (c) for Barabási-Albert (scale-free) networks. In all cases, we observe that the system reaches perfect consensus (i.e. 100% opinion convergence) or almost perfect consensus (i.e. >90% opinion convergence), independently of the network structure, and that this starts occurring for different d values. It is worth noting however that in all cases, convergence to consensus is reached for d > 0. 5  Note that the convergence parameter μ considered above is constant and independent of the node-degrees and thus, assumes that all agents have the same paces towards adjusting their opinions. As this is rather ideal and not in agreement with real-life situations, we consider in the following two archetypal variants given by featuring the degree-centrality dependent scenarios, where the max and min of u v deg( )deg( ) is computed among all edges u adjacent to v (i.e. ∼ u v). Namely, we replace the convergence parameter μ in Eqs (1) and (2) by μ + u v ( , ) indicating that higher-degree nodes are more willing to adjust their opinions (and u v ( , ) μ − indicating the opposite way). Clearly, μ + u v ( , ) and μ − u v ( , ) are within the interval (0, 0.5] for connected networks. μ + is close to 0.5 for a well-connected pair of nodes, whereas μ − is close to 0.5 for a poorly-connected pair of nodes. Our choice of degree-related convergence parameter here naturally reflects the idea of the number of neighbors/ contacts in social networks, and models the possible mechanisms of heterogeneous psychological, habitual and cultural backgrounds in opinion spreading. Although degree-centrality apparently depends on the structure of the layers of the multiplex network, our numerical results indicate that the convergence parameters do not affect the ultimate opinion configuration. It is worth mentioning that there are other measures of centrality as well, such as eigenvector-like centralities 37 and measures based on random walks 38 , that have been studied in the context of multiplex networks. However, as commented above, it is reasonable to expect similar results for other convergence parameters mediated by more complicated measures. The simulation results for the degree-centrality dependent parameters μ + and μ − are presented in Fig. 3, where we have used the same network sizes, number of layers, Poisson rate and number of simulations as in Fig. 2 to compute the percentage of convergence of the opinions. In particular, panels (a) and (b) are for Watts-Strogatz (small-world) networks with μ + (u, v) and μ − (u, v), respectively and panels (c) and (d) for Barabási-Albert (scale-free) networks with μ + (u, v) and μ − (u, v), respectively. It is found that these degree-dependent convergence parameters (i.e. μ + (u, v) and μ − (u, v)) do not alter the ultimate opinion configurations and that the confidence threshold still remains. This is in line with previous findings for Deffuant models in the case of single-layer networks 8,19 and our results extend this finding to multiplex networks.

Discussion
In this paper, we studied analytically and numerically opinion dynamics over multiplex networks with an arbitrary number of layers, where the agents interact with each other with bounded confidence. In the literature, agents are generally assumed to have a homogeneous confidence bound and here we sought to study analytically and numerically opinion evolution over multiplex networks with respective confidence thresholds and general initial opinion distributions. We explicitly identified the critical thresholds at which a phase transition in the long-term consensus behaviour occurs. We then discussed the interaction topology of the agents by using multiplex D-dimensional lattices and extended them to general multiplex networks under some regularity conditions. Our results reveal the quantitative relation between the critical threshold and initial distribution. We also performed numerical simulations and illustrated the consensus behaviour of the agents in regular lattices and, small-world and scale-free networks. We found that the numerical results agree with our theoretical ones and in particular, the critical thresholds of consensus tend to agree with the predicted upper bounds in Theorems 4 and 5 for all network topologies considered in the special cases of uniform X 0 and some choices of Poisson rates associated with the multiple layers.
Moreover, we used the Deffuant opinion model represented as a stochastic process for the evolution of opinions that includes heterogenous confidence bounds and features general initial distributions and, determined the critical threshold by employing probability methods. The main results of our work are Theorems 2 (for D = 1) and Theorem 4 (for D > 1) which extend previous results in 33,34 by considering both multiplex structures with  > 1 and general initial opinion distribution X ( ) 0  . We show that both the initial distribution and multiplex structure play an important role in the phase transition of opinion evolution in an infinite D-dimensional regular lattice in the sense that the critical confidence bound in the case of Theorem 2 (or an upper bound of it in the case of Theorem 4) is influenced by both factors. Our results indicate that multiplexity hinders consensus formation  (1) and (2). Panels (a) and (b) are for Watts-Strogatz (small-world) networks with u v ( , ) μ + and μ − u v ( , ), respectively and panels (c) and (d) for Barabási-Albert (scale-free) networks with μ + u v ( , ) and u v ( , ) μ − , respectively. The vertical dashed line at d = 0.5 corresponds to the point around which a sudden jump occurs for increasing network sizes N . Note that in these plots we have set the number of layers  to 4. when the initial opinion configuration is within a bounded range. This is numerically found to be true in more general networks including small-world and scale-free networks, which are ubiquitous in the real world. Our results provide insight into information diffusion and social dynamics in multiplex real-life systems modeled by networks. However, the theoretical proof of this is beyond the scope of this paper as it would require the development of substantially new techniques that we leave for a future publication.
It is worth mentioning that the networks considered here are static, and thus the connectivity remains fixed throughout opinion spreading. As a result, structural properties such as centrality, correlations, homophily, and assortativity, remain the same throughout opinion spreading. On the other hand, in networks of human social interactions, the interaction can be assorted according to, e.g., the channels used for communication such as face-to-face, mobile phone, and social network services 39 . Certain social mechanisms such as assortativity and homophily (namely, the tendency of individuals to align to behaviours of their friends) are popular in real social networks and may play a key role in opinion formation and its dynamics. For instance, it is shown in 40 that the higher the homophily between individuals in a multiplex network, the quicker is the convergence towards cooperation in the social dilemma. Multiplex social ecological network analysis unravels that node heterogeneity has a critical effect on community robustness 41 . However, as far as convergence of the opinion spreading is concerned, our numerical results, for three different characteristic types of multiplex networks (regular, small-world and scale-free), indicate the same qualitative and almost similar qualitative tendency to reach consensus as a function of d for different network architectures. This is in agreement with our theoretical results. In fact, assortativity and homophily are not included in our theoretical analysis because they usually subvert the transitivity and amenability conditions (see Theorem 5) that form the foundation of our mathematical technique. In a future work, we will focus on how to incorporate multiplex characterisations by means of structural measures, such as homophily and assortativity of the multiplex network, into analytically tractable opinion-formation models. Finally, temporal or co-evolving networks with random environments also seem appealing in this respect as they might lead to differences with respect to convergence to consensus.