A two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems
AA two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems
A two-layer model for coevolving opinion dynamics and collectivedecision-making in complex social systems
Lorenzo Zino, a) Mengbin Ye,
2, 1, b) and Ming Cao c) Faculty of Science and Engineering, University of Groningen, 9747 AG Groningen, the Netherlands Optus–Curtin Centre of Excellence in Artificial Intelligence, Curtin University, Perth 6102, WA,Australia (Dated: 19 February 2021)
Motivated by the literature on opinion dynamics and evolutionary game theory, we propose a novel mathematicalframework to model the intertwined coevolution of opinions and decision-making in a complex social system. In theproposed framework, the members of a social community update their opinions and revise their actions as they learn ofothers’ opinions shared on a communication channel, and observe of others’ actions through an influence channel; theseinteractions determine a two-layer network structure. We offer an application of the proposed framework by tailoring itto study the adoption of a novel social norm, demonstrating that the model is able to capture the emergence of severalreal-world collective phenomena such as paradigm shifts and unpopular norms. Through the establishment of analyticalconditions and Monte Carlo numerical simulations, we shed light on the role of the coupling between opinion dynamicsand decision-making, and of the network structure, in shaping the emergence of complex collective behavior in socialsystems.
Mathematical models have emerged as powerful tools todescribe and study the behavior of complex social systems.Here, we focus on the emergent behavior of a social com-munity whose members dynamically revise their opinionand take collective decisions. Motivated by the empiricalevidence of an interdepencency between these two socialdynamical processes, and the lack of mathematical tools toeffectively describe it, we establish a modeling frameworkfor the interdependent coevolution of opinions and deci-sions, extending and unifying the separate literature bod-ies on dynamic opinion formation and collective decision-making. We specialize the model to offer a realistic ap-plication of the proposed framework in which we studythe introduction of an advantageous innovation in a socialcommunity, and focus on the factors of coupling strengthbetween the opinion dynamics and decision-making, andthe network structure. Depending on how such factorscombine, a range of different complex real-world phenom-ena can be captured in our framework, enabling us toelucidate whether the society will see a paradigm shift inwhich individuals overwhelmingly adopt the innovation,the emergence of an unpopular norm where individualsfail to adopt the innovation despite opinions being over-whelmingly in favor of it, or a community that persistentlysupports the status quo over the innovation.
I. INTRODUCTION
The use of mathematically- and physically-principled mod-els to represent and study social systems has become increas-ingly popular in the last decades . Researchers from a wide a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] range of communities, including physics, applied mathemat-ics, systems and control engineering, computational sociol-ogy, and computer science have devoted their efforts to cap-ture the complexity of collective behavior within mathemati-cal models that allows one to accurately predict the evolutionof a social system, shedding light on the role of the individual-level dynamics on the emergence of complex collective be-havior at the population level.Since the 1950s, mathematical models have been widelyadopted in social sciences to capture the complex phenom-ena that may emerge when members of a community inter-act and share their opinions. Among the literature, we men-tion the seminal works by French, DeGroot, Friedkin andJohnsen, which paved the way for the development of themathematical theory of opinion dynamics and social influence . Recently, these classical works have been extended to in-corporate features of complex networks, such as antagonisticinteractions and the emergence of disagreement , boundedconfidence , the external influence of media , the hetero-geneous and time-varying nature of human interaction pat-terns , and the coevolution of opinions and network struc-ture .Collective decision-making is another real-world phe-nomenon that has been extensively studied by means of math-ematical models. Typically such models describe how an in-dividual’s decision between a set of possible actions evolvesas he or she takes into account the decisions of other in-dividuals that he or she interacts with on a social network.Since its formalization in the 1970s, evolutionary game the-ory has emerged as a powerful paradigm and sound modelingframework for collective decision-making in social communi-ties .The social-psychological literature provides clear evidencethat the two processes of opinion dynamics and collectivedecision-making are deeply intertwined and readers may in-tuitively appreciate such a coupling. On the one hand, it is in-disputable that an individual’s opinion has a key role in his orher decision-making process. On the other hand, many social- a r X i v : . [ c s . S I] F e b two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 2psychological theories support that the converse is also oftenobserved; the actions that one individual observes from theothers can shape his or her opinion formation process. Ex-isting literature reporting this include the social intuitionistmodel , norm interiorization processes , as well as exper-imental evidence . Surprisingly, few efforts have been madetoward generating a rigorous modeling framework for com-plex social systems with such a coupled coevolution of opin-ion and decision-making dynamics. We mention some worksin which actions are modeled as quantized outputs of the in-dividuals’ own opinion, evolving independently of others’ ac-tions . Other efforts assume that each individual has a pri-vate opinion that is fixed and influences his or her decision-making process , that may vary according to external factorswith a decision-making process that coevolves with the net-work structure , or that coevolves along with an expressedopinion, but in the absence of a decision-making process .Motivated by these preliminary works, the first key contri-bution of this paper is the development of a general modelingframework for the coevolution of opinion dynamics and col-lective decision-making in complex social systems. In the pro-posed model, the opinion dynamics of an individual evolvesnot only as a consequence of opinion sharing with other indi-viduals, but also due to the influence from observing the ac-tions of other individuals. The individual’s decision-makingprocess is governed by a coordination game to select betweentwo alternative actions , which is a classical framework tomodel the social tendency to conform with the actions of oth-ers, but is now additionally shaped by the individual’s ownopinion. In general, the opinion sharing process and the socialinfluence from observed actions can occur between differentpairs of individuals, and follow diverse interaction patterns.For this reason, we define our coevolutionary model on a two-layer social network, where a communication layer is used torepresent how individuals share their opinion, and an influencelayer captures the social influence due to observing the actionsof others. Similar two-layer techniques have been used to rep-resent epidemic processes and the simultaneous diffusion ofawareness on the disease , or to model complex synchro-nization dynamics .In the last few years, several works have examined the keyrole played by the topology of a complex network in shap-ing the evolution of dynamical processes occurring on its fab-ric. Paradigmatic examples can be found in different fields,ranging from agreement dynamics and emergence of socialpower to epidemic outbreaks in human populations and synchronization of power grids . Besides deepening ourunderstanding of the mechanisms that governs complex phe-nomena on networks, these results been used to inform the de-velopment of methodologies to control their evolution, such asin optimal vaccine allocation problems , or the implementa-tion of pinning control to synchronize coupled oscillators .In our second key contribution, we use the proposed model-ing framework to study the effect of network topology on theformation of social norms, in particular focusing on the emer-gence of a paradigm shift (in which an innovation replacesthe status quo norm), and on the phenomenon of unpopularnorms , in which a social community exhibits a collec- tive behavior that is disapproved by most of the members ofthe community. A classical example is on alcohol abuse bycollege undergraduates in Princeton university campus at thebeginning of the 1990s; it was found that even though mostof the students were privately uncomfortable with the alco-hol practices on campus but publicly continued to partake inheavy drinking . We model the formation of social norms bystudying the introduction of a social innovation in a commu-nity, represented as one of the two actions, and supported bya stubborn innovator individual .A theoretical result is derived for a necessary condition toobserve the diffusion of the innovation when the decision-making process of all individuals is fully rational, dependenton the structure of the influence layer and on the role of anindividual’s opinion in his or her decision-making process.Then, we put forward an extensive simulation to study the caseof bounded rational individuals. We find that the diffusionof the innovation is strongly influenced by the network struc-ture and the coupling strength between the two coevolution-ary dynamics. Phase transitions are identified between threedifferent regions of the parameter space in which we observei) a paradigm shift, ii) the emergence of an unpopular norm,and iii) the persistence of a popular but disadvantageous statusquo, respectively. We demonstrate that the network structureplays a key role in determining in a nontrivial way the shapeof these three regions and the sharpness of the phase transitionbetween them. For instance, network topologies that seem tofavor the occurrence of a paradigm shift when an individual’sopinion is only slightly influenced by the actions of others, areinstead strongly resistant to the introduction of the innovationwhen this influence increases in strength.The rest of the paper is organized as follows. In Section II,we propose and discuss the coevolutionary modeling frame-work. In Section III, we introduce our model for the adoptionof innovation. Section IV is devoted to presenting our mainfindings. Section V presents discussion of our findings andoutlines avenues for future research. II. MODEL
In this section, we propose a novel modeling frameworkto capture the coevolution of the opinions and decisions ofindividuals interacting on a complex social network. Afterthe model is formally introduced, we explain the intuition andmotivation of the model by providing details on its compo-nents.We consider a population of n ≥ V = { , . . . , n } . Each individual i ∈ V is character-ized by a two dimensional state variable ( x i , y i ) . The firstcomponent of the state variable represents a binary actionx i ∈ {− , + } made by the individual, while the second com-ponent y i ∈ [ − , ] , models his or her continuously distributed opinion . The opinion measures the individual’s preference foran action, so that y i = − y i =
0, and y i = i has maximal preference for action −
1, is neutral,and has maximal preference for action +
1, respectively.The individuals update their actions and opinions after in- two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 3FIG. 1: The coevolutionary dynamics occurs over a two-layernetwork. In the communication layer (in green), with edgeset E W , individuals exchange opinions with each other. In theinfluence layer (in violet), with edge set E A , individuals areable to observe the actions of other individuals. The edge setsof the two layers are not necessarily the same. For example,an individual may choose to only share his or her opinionwith a few close friends and family, but is able to observe andbe influenced by the actions of many others in his or hercommunity. Similarly, he or she may not be able to observethe action of individuals with whom he or she shares his orher opinion (e.g., due to long-distance interactions).teracting with their peers on a two-layer network : the firstlayer models how individuals observe and are influenced byothers’ actions, while the second layer models how individu-als communicate and exchange opinions with one another. Weterm these as the influence layer and communication layer ,respectively. In general, the two layers are characterized bytwo different topologies, as illustrated in Fig. 1. The influ-ence layer is characterized by the undirected edge set E A , withan associated (unweighted) adjacency matrix A ∈ { , } n × n ,having entries a i j defined as: a i j = (cid:26) ( i , j ) ∈ E A , ( i , j ) / ∈ E A . (1)We assume that no self-loops are present, that is, all diagonalentries of A are equal to 0, and denote by d i = |{ ( i , j ) ∈ E A }| (2)the degree of individual i in the influence layer. The com-munication layer is characterized by the undirected edge set E W and a weighted adjacency matrix W ∈ R n × n , with entries w i j (cid:54) = ⇐⇒ ( i , j ) ∈ E W . Self-loops are allowed in E W , andoccurrence of a negative w i j would result in a signed net-work . Even though E W is undirected, W is not necessarilysymmetrical, since w i j and w ji may be different. Although thiswork assumes that both layers are undirected, the proposedmodel easily admits a generalization to directed topologies oneither layer, which may be investigated in future works. The states of the individuals (i.e., opinions and decisions)evolve over discrete time-steps t = , , . . . . At each time t ,a single individual i ∈ V , selected uniformly at random andindependently of the past history of the process , is activatedand updates his or her opinion and action simultaneously, ac-cording to the following mechanisms. Opinion dynamics: the opinion of individual i ∈ V evolvesas y i ( t + ) = ( − µ i ) n ∑ j = w i j y j ( t ) + µ i d i n ∑ k = a ik x k ( t ) , (3)where the parameter µ i ∈ [ , ] , called susceptibility ,measures the influence of his or her neighbors’ actions x k ( t ) of the individual’s opinion. Decision making: the action of individual i ∈ V evolves ac-cording to a stochastic process. Specifically, the prob-ability for individual i to take action x ∈ {− , + } attime t + P ( x i ( t + ) = x ) = e β i π i ( x ) e β i π i ( x ) + e β i π i ( − x ) , (4)where β i > π i ( x ) = π i ( x | y i , x − i ) is the payoff for individual i to take action x , given hisor her current opinion y i ( t ) and the actions of the oth-ers, x − i ( t ) : = [ x ( t ) , . . . , x i − ( t ) , x i + ( t ) , . . . , x n ( t )] (cid:62) ∈{− , + } n − . We define the following payoff function: π i ( x | y i , x − i ) = λ i xy i + − λ i d i n ∑ j = a i j (cid:20) + x − x (cid:21) (cid:62) (cid:20) + α
00 1 (cid:21) (cid:20) + x j − x j (cid:21) , (5)where α ≥ + − λ i ∈ [ , ] ,called commitment , measures the importance that indi-vidual i gives to his or her own opinion in the decision-making process.The remainder of this section is devoted to a detailed dis-cussion and motivation on the two intertwined componentsthat compose the novel coevolutionary dynamics of opinionsand decisions in our proposed model, which are illustrated inthe schematic in Fig. 2. A. Opinion Dynamics Component
According to (3), the opinion of individual i at time t + i in-teracts on the communication level; the second term capturesthe influence of the actions observed by individual i on theinfluence level. Such a convex combination is regulated bythe susceptibility µ i ∈ [ , ] , which measures the influence of two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 4FIG. 2: Schematic of the coevolutionary dynamics, and theinterdependence between the two mechanisms.the actions observed on the individual’s opinion so that µ i = ( − µ i ) w i j and µ i a i j are the weights that in-dividual i assigns to the opinion and action, respectively, ofindividual j . Since the two layers of the network may havedifferent edge sets, it is in general possible that one of the twoweights is nonzero and the other is zero. A standard assump-tion, which we shall adopt in the rest of this paper and oftenmade in opinion dynamics models , is that ∑ j = | w i j | = i ∈ V . This assumption guarantees that the opinionsin the coevolutionary model are always well defined, as ex-plicitly stated in the following result, whose proof is given inAppendix A. Proposition 1.
Let W be such that ∑ j = | w i j | = , for all i ∈ V , and let the initial opinions satisfy y i ( ) ∈ [ − , ] , for alli ∈ V . Then, y i ( t ) ∈ [ − , ] , for all i ∈ V and t ≥ . If the weights of the communication layer are nonnegative,that is, w i j > ( i , j ) ∈ E W , then the updated opinionis a weighted average of i) the actions x k ( t ) of his or herneighbors on the influence layer, and ii) the opinions y j ( t ) of his or her neighbors on the communication layer. Begin-ning with the classical French–DeGroot model , weightedaveraging is seen as a classical approach to modeling the wayan individual processes, and is influenced by, external opin-ions; the French–DeGroot model can be recovered by setting µ i = w i j < w i j are allowed, then setting µ i = . Hence, our model encompasses and generalizes stan-dard models used in opinion dynamics.A stubborn node s ∈ V can be introduced by setting µ s = w ss = w si =
0, for all i (cid:54) = s ). Then,opinion of individual s remains constant for all time, i.e., y s ( t + ) = y s ( ) for all t ≥ . A keyresult, which will be used in the sequel, is the following. Proposition 2 (Theorem 2 from Chen et al. ) . Let W be suchthat w i j ≥ for all ( i , j ) ∈ E W and ∑ j = w i j = for all i ∈ V .Suppose that there is a single stubborn node s that is reach-able from all other nodes on the communication layer , andlet µ i = , for all i ∈ V (cid:114) { s } . Then, under (3) , y i ( t ) → y s ( ) for all i ∈ V almost surely. That is, the opinion of every in-dividual converges to the opinion of the stubborn node withprobability . B. Decision-Making Component
The decision-making mechanism is developed within theframework of evolutionary game theory . Specifically, eachindividual’s action is updated according to a noisy best re-sponse which evolves according to the log-linear learningrule in (4), regulated by the level of rationality β i ≥
0. Inthe limit of no rationality, that is, β i =
0, actions are cho-sen uniformly at random and independent of the payoff, since P ( x i ( t + ) = + ) = P ( x i ( t + ) = − ) = /
2. The case β i = ∞ , instead, models the fully rational scenario, in whichindividuals always choose to maximize their payoff so that (4)reduces to a deterministic best response dynamics: P ( x i ( t + ) = + ) = π i (+ | y i , x − i ) > π i ( − | y i , x − i ) , if π i (+ | y i , x − i ) = π i ( − | y i , x − i ) , π i (+ | y i , x − i ) < π i ( − | y i , x − i ) . (6)For bounded levels of rationality, β ∈ ( , ∞ ) , individuals areallowed to choose both actions, but select the one that maxi-mizes their payoff with higher probability.We now elucidate how different factors impact the payoffof an individual, including an individual’s opinion, the actionsof his or her neighbors, the individual’s commitment, and theevolutionary advantage determine his or her payoff. Observethat the payoff for taking action + − π i (+ | y i , x − i ) = λ i y i + ( − λ i ) d i n ∑ j = a i j ( + α )( + x j ) , (7a)and π i ( − | y i , x − i ) = − λ i y i + ( − λ i ) d i n ∑ j = a i j ( − x j ) , (7b)respectively. The first term accounts for the opinion y i , so thatindividual i receives an increased payoff for taking the actionthat individual i prefers. For instance, an individual with anegative y i (that is, a preference for action −
1) will receivea component with a negative payoff λ i y i / − λ i y i / + −
1, respectively. The secondterm captures the social pressure to coordinate with neighbors.For each neighbor j , individual i receives a positive contribu-tion to the payoff if and only if he or she takes the same actionas individual j . The parameter α models the possible evo-lutionary advantage for taking one of the two actions with re-spect to the other. In a general formulation of the model, α can two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 5assume any real value. Without loss of generality, in this pa-per we assume that, if there exists an evolutionary advantage,then action + −
1, yielding α ≥
0. The commitment λ i measures how muchindividual i values and is committed to his or her own opinionduring the decision-making process relative to a desire to co-ordinate with the neighbors’ actions; setting λ i = . A stubborn node s can be modeled by setting λ s = β s = ∞ , so that he or she will always take the sameaction x s ( t ) = x s ( ) , for all t ≥ + − d i n ∑ j = a i j x j > − + α (cid:18) α + λ i − λ i y i (cid:19) , (8)as explicitly computed in Appendix B. In other words, a fullyrational individual i ’s best response is action + d i ∑ nj = a i j x j ∈ [ − , ] measures the (nor-malized) influence on individual i of the actions of his orher neighbors. Setting the commitment λ i =
0, individual i receives a higher payoff for taking action + + − α / ( + α ) ∈ ( − , ] , consistent with the results in the lit-erature on network coordination games . With λ i >
0, thisthreshold is shifted whenever individual i prefers one actionover the alternative. As y i increases or decreases, the frac-tion of neighbors taking action + i ’s best response to be action + i ’s threshold for selecting an action can beshaped by his or her preference for that action. Interestingly,if λ i > / ( + y i ) or λ i > ( α + ) / ( α + − y i ) , then action + −
1, respectively, always yields a better payoff thanthe opposite action, irrespective of his or her neighbors’ cur-rent actions. In other words, if individual i is simultaneouslystrongly committed to his or her opinion and has a strong pref-erence for one of the two actions, then he or she will alwaysfavor that action irrespective of the social pressure. III. ADOPTION OF ADVANTAGEOUS INNOVATION
For the rest of this paper, we specialize the proposed frame-work to model and predict whether or not a social networkwidely adopts an advantageous innovation, and if such anadoption occurs, whether or not the innovation is actually pop-ular among the individuals. In this section, we describe howour model is tailored to represent such a real-world processand we illustrate the different phenomena that can be typi-cally observed as an outcome of the proposed model. In thenext section, we investigate more closely the various factorsthat determine which phenomena is observed.We consider a population where all the individuals start bytaking the status quo (action − s ∈ V is introduced in the network. The innovator is modeled as astubborn node with fixed action and opinion equal to x i ( t ) = y i ( t ) = +
1, for all t ≥
0, where the innovative action + α > β i = β , λ i = λ , and µ i = µ , for all i ∈ V (cid:114) { s } . We further assume that the communication layeris connected and that W is a simple random walk on thecommunication layer, that is, the nonzero entries of any row of W are all positive and of equal value, and sum to 1. I.e., we areconsidering a specific implementation of a French–DeGrootmodel. This implies that in updating opinion y i , individual i gives the same weight to the opinion of each one of his or herneighbors on the communication layer.The goal of our study is to explore the role of the cou-pling between the opinion dynamics and the decision-makingmechanisms — determined by the commitment λ and the sus-ceptibility µ — and of the network structure on the emergingbehavior of the system. To help elucidate this goal, we definethe following two quantities: (cid:104) x (cid:105) : = n n ∑ i = x i , and (cid:104) y (cid:105) : = n n ∑ i = y i , (9)which are the average action and opinion in the population,respectively. In Fig. 3, we offer three paradigmatic samplepaths of the coevolutionary dynamics at the population level,exhibiting the different phenomena that can occur, which wenow describe in further detail. Unpopular norm (Fig. 3a): after a short transient, the aver-age of the individuals’ opinions shows a preference forthe innovation, that is, (cid:104) y (cid:105) >
0. However, an over-whelming majority of the individuals still takes the sta-tus quo action, that is, (cid:104) x (cid:105) ≈ −
1. While the ergodicnature of (4) ensures that the innovation will eventuallydiffuse across the entire network, we will see that theunpopular norm may be meta-stable for a long periodof time. In the real world, this would imply that thewidespread adoption of the innovation fails to occur.
Popular disadvantageous norm (Fig. 3b): the status quo(which is disadvantageous with respect to the innova-tion) remains the predominant action in the network( (cid:104) y (cid:105) ≈ −
1) and it is on average the preferred actionamong the individuals’ opinions ( (cid:104) y (cid:105) < Paradigm shift (Fig. 3c): after a short transient, a tipping-point is reached and the advantageous innovation isadopted and supported by almost the entire popula-tion ( (cid:104) y (cid:105) ≈ + (cid:104) y (cid:105) > (cid:104) x (cid:105) ≥ (cid:104) y (cid:105) . two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 6 (a) λ = . µ = .
001 (b) λ = . µ = .
01 (c) λ = . µ = . FIG. 3: Possible outcomes of the proposed coevolutionary dynamics when modeling the diffusion of an advantageous norm.The blue solid curve is the average action of the population, (cid:104) x (cid:105) , and the red dotted curve is the average opinion, (cid:104) y (cid:105) .Depending on the model parameters, (a) unpopular norms, (b) popular disadvantageous norms, or (c) a paradigm shift can beobserved. All sample paths are generated on networks with n =
200 individuals, evolutionary advantage α = . β =
20. Both layers are regular random graphs, with degree equal to 4 for the communication layer and 8 for the influencelayer. Parameters λ and µ differ from one simulation to the other and are reported in the corresponding captions.A fourth phenomenon should be in principle possible, beingthe establishment of a meta-stable state in which the advanta-geous innovation is widely adopted as the norm, but is un-popular. However, this was never observed in our numericalsimulations. An intuitive reason can be found in the followingconsideration. If a large majority of the individuals adopt theinnovation, then their opinion will drift toward + IV. EFFECT OF THE NETWORK STRUCTURE ON THEADOPTION OF INNOVATION
In this section, we aim to understand how the model param-eters and the network structure may determine the emergenceof one of the three different collective phenomena describedin Section III, during the adoption of an advantageous innova-tion.We begin our analysis by considering the limit case of fullyrational individuals, that is, β = ∞ . We again consider thecase in which a single stubborn node (termed the innovator) s ∈ V is introduced in the network, taking a fixed action andhaving opinion equal to x i ( t ) = y i ( t ) = +
1, for all t ≥
0. Wefurther assume that the network is connected. In this scenario,the following result can be established, with the proof foundin Appendix C.
Theorem 3.
Let us consider a coevolutionary dynamics ofopinions and decisions. Let us defined ∗ : = min { d i : i ∈ V , ( i , s ) ∈ E A } . (10) In the limit β = ∞ , if α < d ∗ − and λ < λ ∗ : = − + α d ∗ − − α , (11) then (cid:104) x ( t ) (cid:105) = − + / n, for all t ≥ . That is, a paradigm shiftcannot occur. Theorem 3 yields a necessary condition for a paradigmshift; if either the evolutionary advantage or the commit-ment in the decision-making process is sufficiently large ( α ≥ d ∗ − λ > λ ∗ ), then a paradigm shift is possible. Note thatboth conditions depend on the network structure through theminimum degree of the neighbors of the innovator d ∗ .Such conditions are not sufficient, however. In fact, onecan easily produce simple examples in which even though theconditions in Theorem 3 are satisfied, a paradigm shift doesnot occur since the diffusion of action + β < ∞ , which has been demonstrated to be moreconsistent with real-world decision-making processes . Inthis case, we will see that paradigm shifts may occur evenfor levels of commitment λ < λ ∗ . In order to focus on theeffect of the coupling between the two mechanisms and ofthe network on the system’s evolution, the following numeri-cal studies will fix a moderate level of evolutionary advantage α = . β = λ and µ , and for different network topologies. These pa-rameters aim to capture a realistic scenario in which the evo-lutionary advantage of the innovation action + − %). The quantitative results of the following numericalsimulations may depend on the precise choice of the param-eters α and β . However, we have observed that the salientfeatures of the observed phenomena of the system are robust two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 7 (a) RR, average decision (b) ER, average decision (c) WS, average decision (d) BA, average decision(e) RR, variance (f) ER, variance (g) WS, variance (h) BA, variance FIG. 4: Estimation of the threshold value for λ for transitioning from meta-stable unpopular norm to paradigm shift. Averageaction of the population (in (a)–(d)) and variance in the fraction of + at time T = n over 100 independent runs of the coevolutionary dynamics with α = . n = β =
20, and both layers with averagedegree 8 (influence layer) and 4 (communication layer), generated according to (a) a random regular graph, (b) an Erd˝os-Rényi random graph,(c) a Watts-Strogatz small-world network with rewiring probability p = .
2, and (d) a Barabási-Albert scale-free network. to different choices of the parameters α and β that representthe described scenario. A. Opinions not directly influenced by actions
In the first part of our analysis, we will assume that the evo-lution of the opinion is not influenced by the actions, i.e., withsusceptibility µ =
0. Since we assume that the communicationlayer is connected, Proposition 2 establishes that the opinionsof all individuals converge almost surely to +
1. Hence, onlytwo phenomena can occur: unpopular norm or paradigm shift.Before starting our analysis for bounded rational individuals,we briefly report a straightforward consequence of Theorem 3and Proposition 2 for the behavior of the coevolutionary dy-namics with fully rational individuals when µ = Corollary 4.
Let us consider a coevolutionary dynamics ofopinions and decisions with µ = . Let W be such that w i j ≥ for all ( i , j ) in E W , ∑ j = w i j = for all i ∈ V . In the limit β = ∞ , if d ∗ > + α and λ < λ ∗ , then (cid:104) x ( t ) (cid:105) = − + / n, for allt ≥ , and (cid:104) y ( t ) (cid:105) → . That is, a paradigm shift cannot occur,and rather, an unpopular norm is almost surely observed. One can intuitively conjecture that, if opinions play a suf-ficiently dominant role in the decision-making process (thatis, λ is sufficiently large), then the whole network will adoptthe innovation, while, in the opposite scenario, the social pres-sure outweighs the individual’s commitment to his or her ownopinion, thus ensuring the population continues to choose thestatus quo action, even though the opinion of the overwhelm- ing majority shows preference for the innovation. Indeed, ev-idence of a phase transition depending on the commitment λ can be observed in Fig. 4a.We investigate the presence of such a phase transition bymeans of Monte Carlo numerical simulations, following amethod similar to the ones proposed to numerically estimatethe epidemic threshold in epidemic models . Specifically,we run repeated independent simulations of the process fordifferent values of commitment λ , keeping track of the frac-tion of adopters of the innovation in each run at the end ofa fixed observation window of duration T , which is equal to ( (cid:104) x ( T ) (cid:105) + ) /
2. Then, the threshold ˆ λ is estimated as the valueof λ that maximizes the variance of such a quantity within theindependent runs. Sharp peaks of the variance are evidenceof an explosive phase transition between a regime where un-popular norms are meta-stable (if λ < ˆ λ ), to a regime whereparadigm shift is observed in almost all the simulations (if λ > ˆ λ ) . We fix a sufficiently long time-window T = n (eachindividual thus revises his or her action and opinion on aver-age 4 n times) to allow the innovator to steer the whole popula-tion to an opinion close to +
1. If an unpopular norm persistseven after T = n , then it is meta-stable, and implies that theinnovation will never realistically be adopted in the real world.To better elucidate the role of the network topology indetermining such a threshold, we apply the Monte Carlo-based technique on four classical network models with dif-ferent features . Specifically, we considered random regular(RR) graphs, Erd˝os-Rényi (ER) random graphs (which havea slight heterogeneous degree distribution), Watts-Strogatz(WS) small-world networks (which are characterized by a two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 8clustered structure), and Barabási-Albert (BA) scale-free net-works (which have a strongly heterogeneous degree distribu-tion with a few hubs with high degree). In all our simulations,both layers of the network are generated according to the samenetwork model, one independent of the other, and the innova-tor is placed in the first node, that is, s =
1. In the case ofBA networks, this would imply that the innovator is almostsurely placed in a hub. In order to avoid possible confoundingdue to the network density when comparing different networktopologies, we keep the average degree to be the same be-tween different network structures: in the influence layer allnetworks have average degree equal to 8 and 4 in the influencelayer and communication layer, respectively. More details onthe generation of the networks can be found in Appendix D.The results of our Monte Carlo simulations, presented inFig. 4, confirm our conjecture and suggest the presence of aphase transition, which is estimated to occur at the value ˆ λ indicated by the peak (vertical dashed line). When comparingthe numerical estimations in Fig. 4 with the necessary con-dition to achieve paradigm shift from Corollary 4, it appearsthat bounded rationality favors the emergence and establish-ment of the innovation, thereby leading to a paradigm shift forvalues of commitment λ that are smaller than the necessaryvalue λ ∗ for fully rational individuals. In fact, for RR (where d ∗ = λ ∗ = . λ = .
16. Similarly, for the BA (where,by construction, d ∗ ≥ λ ∗ ≥ . λ = .
1. For the the othertwo cases (ER and WS), λ ∗ depends on the specific realizationof the network, since the degrees are nonuniform and randomvariables. However, using their expected values, we obtain E [ λ ∗ ] = . E [ λ ∗ ] = . λ = .
18, while the onefor WS is estimated as ˆ λ = .
08 and seems to vanish (sinceeven for λ = λ , the network structure plays an importantrole in shaping the phase transition. Notice that in RR and ABnetworks (Figs. 4(a) and (d)), the phase transition seems to beextremely sharp: if λ is below the threshold, then an unpopu-lar norm is observed in almost all the simulations, while abovethe threshold, a paradigm shift is almost always observed. Onthe contrary, in ER and WS networks (Figs. 4(b) and (c)), thethreshold seems to be less sharp as λ increases, suggestingthe existence of a region for the commitment λ where bothunpopular norms and paradigm shifts are possible, depend-ing on the specific realization of the network. We believe thatsuch a phenomenon might be caused by the variability in thedegree of the innovator, which in ER and WS networks de-pends on the specific realization. In contrast, all the nodes inRR networks have the same degree, and the innovator is al- FIG. 5: Average action and opinion of the population in asample path of the coevolutionary dynamics onWatts-Strogatz small world networks with rewiringprobability p = .
2. Other parameters are α = . n = β = λ = .
1, average degree 8 in the influence layer and4 in the communication layer.most always placed in a hub in BA networks, by construction.For WS networks, we also observe that the region of the pa-rameter space in which paradigm shifts can never occur seemsto vanish. This may be due to the high levels of clustering insmall-world networks, which helps the spread of innovation .In fact, the sample path in Fig. 5 of a WS network showsan interesting transient phenomenon; an increasing nonzerofraction of the population adopts the innovation in steps, andpersists in the adoption even though remaining in the minor-ity. We conjecture that this occurs because the high clusteringstructure results in certain clusters where the individuals havemostly adopted the innovation and remain meta-stable, whilein other clusters, the status quo is still widely adopted.Our theoretical findings in Corollary 4 for fully rational in-dividuals suggest that the density of the influence layer hasa detrimental effect on the diffusion process, hindering theemergence of a paradigm shift. In fact, the threshold λ ∗ in-creases as d ∗ increases, and approaches 1 /
2. For boundedrational individuals, we investigate the effect of the density ofthe influence layer by repeating the numerical estimation ofthe threshold ˆ λ performed in the above, doubling the averagedegree of the influence layer to 16.Consistent with the intuition coming from our analytical re-sult in the limit of fully-rational individuals, the results of ournumerical study, reported in Fig. 6, indicate that denser net-works lead to an increased threshold ˆ λ . However, the mag-nitude of such an increase seems to be strongly dependent onthe network topology. For ER and RR networks, the increaseis quite moderate (10% and 18 . . λ is small, then unpopular norms are observed in almost all two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 9 (a) RR (b) ER (c) WS ( p = .
2) (d) BA
FIG. 6: Estimation of the threshold for different network topologies: (a) regular random, (b) Erd˝os-Rényi, (c) Watts-Strogatz,and (d) Barabási-Albert. Each data point is the variance of the fraction of + T = n over 100 independentruns of the coevolutionary dynamics with α = . n = β =
20, and both layers with average degree 16 (for the influencelayer) and 4 (for the communication layer), generated with the four network models denoted in the corresponding caption.the simulations.Depending on the communication layer topology, conver-gence of the individuals’ opinions to + . To sum up, the topology of bothlayers may be key in predicting whether the spread of an inno-vation will fail, even though it has an evolutionary advantagewith respect to the status quo, and further, even in scenarioswhere the majority of the population’s opinions favor it. B. Feedback between opinion and actions
In the previous section we have extensively analyzed thelimit case of µ =
0, in which the actions of an individual’sneighbors do not directly influence the opinion dynamics (3).However, this assumption may be overly simplistic in real-world scenarios, where evidence of such an influence has beentheorized in the social-psychological literature and ob-served in empirical studies .In this section, we will study the general case of suscepti-bility µ >
0. In this case, we immediately notice that the in-novator is not necessarily always able to steer the opinions ofall individuals to +
1. As a consequence, all three phenomenareported in Section III and illustrated in Fig. 3 can be observeddepending on the value of the commitment λ and susceptibil-ity µ . In particular, we will now explore in detail the interplaybetween susceptibility and commitment, and the role of thenetwork structure in determining the outcome of the coevolu-tionary dynamics. Specifically, we choose the two topologiesanalyzed in the previous subsection that produced the great-est differences in observed outcomes, i.e., the RR and WSnetworks. For each one of these topologies, we estimate theaverage opinion and action of the population at time T = n by means of 100 independent simulations, while varying thevalues of both parameters λ and µ .The results of our numerical simulations are reported inFig. 7. Comparing the average action in (a) and (c) withthe corresponding average opinion in (b) and (d), we identifythree regions corresponding to the three possible phenomenathat can occur, highlighted in (e) and (f), respectively. In the green region denoted by the roman number I, the commitment λ is sufficiently large and the susceptibility µ is small, and wethus observe paradigm shifts ( (cid:104) x ( T ) (cid:105) ≈ + (cid:104) y ( T ) (cid:105) > (cid:104) x ( T ) (cid:105) ≈ − (cid:104) y ( T ) (cid:105) > (cid:104) x ( T ) (cid:105) ≈ − (cid:104) y ( T ) (cid:105) <
0, signifyingthe presence of meta-stable disadvantageous popular norms.The shape of these regions and the sharpness of the phasetransition between each region is strongly influenced by thenetwork structure. In RR networks (Figs. 7 (a), (c), and(e)), we mostly observe sharp phase transitions between theregimes. In particular, a popular disadvantageous norm is al-most always observed if µ > .
007 (region III), regardless ofthe commitment λ . For intermediate values of susceptibil-ity, that is, 0 . < µ < . λ (cid:48) < λ (cid:48)(cid:48) . Specifically, if λ < λ (cid:48) , then we observe theemergence of a disadvantageous popular norm (region III);for λ (cid:48) < λ < λ (cid:48)(cid:48) , we have an unpopular norm (region II); if λ > λ (cid:48)(cid:48) , a paradigm shift is observed (region I). Finally, ifthe susceptibility is small, that is, µ < . λ , we observeeither an unpopular norm (region II) or a paradigm shift (re-gion I).In the case of WS networks (Figs. 7 (b), (d), and (f)), weimmediately observe that all the phase transitions appear tobe less sharp, similar to what was already reported in Sec-tion IV A on the case of µ =
0. The three regions describedabove also appear to have a different shape with respect tothose observed in RR networks. In fact, if the commitment λ > .
25, then the precise value of λ seems not to play anyrole in determining the outcome of the process, and rather,the outcome is instead uniquely determined by the suscepti-bility: if µ < . . < µ < . µ > . λ < .
25, the behavior issimilar to the one already described for RR networks.When comparing the two topologies, we can conclude thatthe introduction of a direct feedback of the observed actions two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 10 (a) RR network: average action (cid:104) x (cid:105) (b) WS network: average action (cid:104) x (cid:105) (c) RR network: average opinion (cid:104) y (cid:105) (d) WS network: average opinion (cid:104) y (cid:105) (e) RR network: regions (f) WS network: regions FIG. 7: Outcome of coevolutionary dynamics on (a,c,e) RR and (b,d,f) WS graphs for different values of the parameters λ and µ . The average action is plotted in (a–b), the average opinion in (c–d), and in (e–f) we highlight the three distinct regions of theparameter space associated with the emerging phenomenon observed in the simulations. Each data point is the average over100 independent runs with α = . n = β =
20, over a time-window of duration T = n . Both types of graphs haveaverage degree 8 (for the influence layer) and 4 (for the communication layer). The rewiring probability of the WS networks is p = .
2. The green region I, violet region II, and orange region III correspond to regions in which a paradigm shift, anunpopular norm, and a popular disadvantageous norm, is observed, respectively.on an individual’s opinion has a different effect, dependingon the network structure. For instance, topologies that seemsto favor the occurrence of paradigm shifts in the absence ofsuch a feedback (e.g., WS netwoks), are instead less proneto promote the diffusion of innovation when the feedback ispresent, thereby favoring the emergence of popular disadvan- tageous norms. We believe that the presence of clustering inWS networks may explain this phenomenon. In fact, as µ increases, the ability of the innovator to shift others’ opinionsmay remain restricted to individuals in his or her own immedi-ate cluster, while in the other clusters at a longer path distancefrom the innovator, the influence of observed actions on an two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 11individual’s opinion may ensure that the majority of the indi-viduals’ opinions remains firmly in support of the status quoaction. V. DISCUSSIONS AND CONCLUSIONS
In this paper, we have proposed a novel modeling frame-work for capturing the intertwined coevolution of individu-als’ opinions and their actions; individuals share their opin-ions and are influenced by the actions observed from the othermembers of their community on two distinct layers of a com-plex social network. The first key contribution of this work isthe formal definition of the coevolutionary model itself, whichis grounded in, and intertwines, the theories of opinion dy-namics and evolutionary games.Then, we have tailored the proposed framework to studya real-world application, concerning the introduction of aninnovation (such as a novel advantageous product or behav-ior) in a social community. In Section III, we have provideddetails of such a model, illustrating three very different real-world phenomena that can be observed within our single uni-fied modeling framework: the formation of i) an unpopularnorm, ii) a popular disadvantageous norm, or iii) a paradigmshift. The possible formation of either unpopular norms orpopular disadvantageous norms has been rarely considered inagent-based models, even though substantial empirical dataand studies from the social-psychology literature indicate nei-ther phenomenon is especially rare . If indeed aparadigm shift does occur, it interesting that the opinions arefirst to change, followed by the actions. In a real-world exam-ple of such a phenomenon, Iowa farmers in the 1930s beganwidespread adoption of a new hybrid corn; it was during theprior years that farmers gradually learned about the new hy-brid corn and slowly shifted their opinion toward supportingits adoption .A preliminary analysis established a necessary conditionon the model parameters and network structure to observea paradigm shift when individuals actions are fully ratio-nal. However, real-life human cognitive processes have beendemonstrated to be only partially rational , and the caseof bounded rationality was then studied by means of MonteCarlo numerical simulations. We started from a simplifiedscenario, in which individuals’ opinions are not susceptibleto the observed actions of the neighbors. Evidence of aphase transition between two regimes of a meta-stable un-popular norm and a paradigm shift was identified, based onthe strength of individuals’ commitment to their own opin-ion, and further shaped by the network structure. This accordswith intuition: if an individual’s decision-making is primarilygoverned by the desire to coordinate with his or her neigh-bors, then it becomes unlikely that the social system collec-tively breaks out of the meta-stable state in which individualslargely select the status quo action, even though the innova-tor may shift the opinions of the community to support theinnovation.Our analysis was then extended to consider the more real-istic scenario in which an individual’s opinion is susceptible to the influence of the observed actions of others in their so-cial community. The results illustrated that the three socialphenomena mentioned above could all be observed, depend-ing on the model parameters. The range of parameter valuesfor which each phenomena could occur as well as the sharp-ness of the phase transition between the different regimes wasfound to be strongly dependent on the topology of the socialnetwork. An important general conclusion was also drawn. Ifindividuals’ opinions are strongly susceptible to being influ-enced by the actions of others, then independent of the net-work topology and of the individuals’ commitment to theirown opinion, the status quo will persist as a popular disad-vantageous norm. The model can thus shed light on whysome norms persist even though they are clearly disadvanta-geous to both the individual and the wider population. For in-stance, footbinding was a disadvantageous norm among Chi-nese women for several centuries prior to a rapid disappear-ance in the 20th Century, persisting in part because individu-als’ opinions were heavily influenced by the observed actionsof others .In contrast, having a large commitment to one’s own opin-ion is a necessary, but not sufficient, condition to observe aparadigm shift (see Fig. 7). This illustrates the importanceof individuality, or the role of an individual’s evolving pref-erence for/opinion on an action, in promoting the spread ofan innovation. Such a role, despite being intuitive, has beenlargely overlooked in most diffusion literature. For all topolo-gies, the range of parameter values for which an unpopularnorm could occur was nontrivial (region II in Fig. 7). Thishelps support the observations from the literature that unpop-ular norms, while not extremely common, are also not rare.The high clustering nature of small-world networks has beenlinked to paradigm shifts occurring rapidly when consider-ing just a decision-making process . In the coevolutionarymodel, we found that if an individual’s susceptibility to hav-ing his or her opinion influenced by the observed actions ofothers is small, then small-world networks favor the adop-tion of novel advantageous norms, consistent with . How-ever, as the susceptibility to social influence increases, small-world networks become more resistant to the diffusion of in-novation than other network structures (e.g., random regulargraphs), thus leading to the emergence of popular disadvan-tageous norms. Thus, we confirm that the network structureitself plays a non-negligible role in shaping the collective dy-namics, but when decision-making and opinion dynamic co-evolve, the impact can be unexpected and counter-intuitive.We hope to have convinced the reader that the proposed co-evolutionary modeling framework is of interest to the variousscientific communities that study social systems using dynam-ical mathematical models. The general formulation of ourmodeling framework and the promising preliminary resultsobtained have paved the way for several avenues of future re-search. On the one hand, further efforts should be devotedtoward a rigorous theoretical analysis of the model, beginningwith a comprehensive convergence result for the fully rationalcase. Further analysis of the network topology may be consid-ered, including the impact of introducing directed interactionson either layer, or the effect of negative interaction weights on two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 12the emergence of polarization phenomena . The role of clus-tering in favoring or hindering the occurrence of a paradigmshift should also be investigated, especially in the presenceof strongly connected components in directed networks orcommunities with negative weights. Time-varying networksare recognized as being more realistic, with several possi-ble directions including activity-driven networks , adaptivetopologies , or state-dependent weights . Moreover, whilea coordination game was used for the decision-making pro-cess, the proposed framework can easily be adjusted to con-sider other network games, such as anti-coordination, Pris-oner’s Dilemma, etc. Among the several fields in which theproposed framework may find application, we want to men-tion marketing and financial markets. In marketing and prod-uct promotion, it has often been observed that the mere factthat a novel product is superior to the competitors may be notsufficient for it to succeed, even if the superiority is widelyacknowledged. The proposed framework can offer mathemat-ical tools to represent realistic diffusion of a new product andpredict its outcome.Existing literature has recognized that in financial mar-kets, there is a coevolution of a trader’s (individual) reputationand trading strategies; while the reputation can be generallymodeled through a continuous variable (similar to opinions),the trading strategies can either be represented as edge cre-ation/deletion operations , or as a complex decision-makingprocess . Ideas drawn from these existing works may en-able our proposed framework to better describe the phenom-ena studied in this work and suggest their possible extensionto the particular application of financial markets. DATA AVAILABILITY STATEMENT
The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.
ACKNOWLEDGMENTS
This work was partially supported by the European Re-search Council (ERC-CoG-771687) and the Netherlands Or-ganization for Scientific Research (NWO-vidi-14134). M. Yewas also partially supported by Optus Business.
Appendix A: Proof of Proposition 1
We prove that if y i ( ) ∈ [ − , ] for all i ∈ V , then y i ( t ) ∈ [ − , ] , for all i ∈ V and for all t ≥
0. That is, the opinionsare always well defined if the initial opinions are well defined.We proceed by induction. Let us assume that the opinions arewell posed at a generic time t , that is, y i ( t ) ∈ [ − , ] , for all i ∈ V . Let i ∈ V be the individual that is activated at time t . Clearly, all the opinions of the individuals j ∈ V (cid:114) { i } arewell defined, since they remain the same. For the opinion of individual i , we obtain the following bound: | y i ( t + ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( − µ i ) n ∑ j = w i j y j ( t ) + µ i d i n ∑ k = a ik x k ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( − µ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n ∑ j = w i j y j ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + µ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d i n ∑ k = a ik x k ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( − µ i ) n ∑ j = | w i j || y j ( t ) | + µ i d i n ∑ k = a ik | x k ( t ) |≤ ( − µ i ) n ∑ j = | w i j | + µ i ≤ . (A1)Since we have selected a generic time t and the bound holdsfor all i ∈ V , the opinions at the next time step are well de-fined: y i ( t + ) ∈ [ − , ] . This yields the proof. Appendix B: Analytical derivation of Eq. (8)
We compute the condition for which the payoff for choos-ing action + −
1. Using (7a) and (7b), we observe that the inequality π i (+ | y i , x − i ) ≥ π i ( − | y i , x − i ) holds if and only if12 λ i y i + − λ i d i ( + α ) n ∑ j = a i j ( + x j ) ≥ − λ i y i + − λ i d i n ∑ j = a i j ( − x j ) (B1)which, after rearranging and recalling that d i = ∑ j = a i j ,yields1 − λ i d i ( + α ) n ∑ j = a i j x j ≥ − (cid:18) ( − λ i ) α + λ i y i (cid:19) . (B2)The inequality in (8) can then be recovered from the above byfurther rearranging and simplifying. Appendix C: Proof of Theorem 3
Consider a generic node i . According to (6), a neces-sary condition for node i to change action to + π i (+ y i , x − i ) ≥ π i ( − y i , x − i ) . Using their explicit expres-sion in (7a) and (7b), we bound π i (+ y i , x − i ) ≤ λ + ( − λ )( + α ) d i n ∑ j = a i j ( + x j ) , (C1) π i ( − y i , x − i ) ≥ − λ + − λ d i n ∑ j = a i j ( − x j ) . (C2)In order to start the diffusion, one individual has to adopt + −
1. We study separately the case in which the first adopter ofaction + i : ( i , s ) / ∈ E A , two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 13a necessary (but not sufficient) condition for node i to be thefirst adopter is derived from the inequalities above as12 λ ≥ − + − λ = ⇒ λ ≥ . (C3)For a generic individual i : ( i , s ) ∈ E A , the bounds above yieldthe following necessary condition for i to be the first adopterof action + λ + ( − λ )( + α ) d i ≥ − λ + ( − λ ) d i − d i , (C4)yielding λ ( d i − − α ) ≥ d i − − α . (C5)If d i > + α , then λ ≥ − + α d i − − α . (C6)If d i ≤ + α , then the necessary condition above is alwaysverified. We observe that the necessary condition for a neigh-bor of the innovator is always less restrictive than the one forthe other individuals, independent of the evolutionary advan-tage α and of the degree d i . The necessary condition is ob-tained by minimizing over all the neighbors of the innovator. Appendix D: Network models and their implementation
In the numerical simulations of this paper, we use differentnetwork topologies generated according to four different al-gorithms to obtain a network with n nodes and average degree d . Details on the properties of the generated networks can befound in the book by Newman , while more details on thespecific implementation of these algorithms in this paper arereported in the following. Regular random (RR): the network is generated using aconfiguration model, that is, each node is given d half-links. A pair of half-links is selected uniformly at ran-dom and, if the pair consist of nodes that are not alreadyconnected through an edge, then the two half-links areremoved and an edge between the two nodes is added.The procedure is repeated until all the half-links are re-moved. Erd˝os-Rényi (ER): the network is selected uniformly at ran-dom from the ensemble of graphs with n nodes and dn / dn / Watts-Strogatz (WS): the network is generated as follows.First, a regular ring lattice where each node is con-nected to the d nearest neighbors is constructed. Then,each edge is randomly rewired with probability equal to p , independently of the other edges. Edge rewiringis performed by randomly chose one of the two nodesconnected by the edge and substituting it with anothernode, chosen uniformly at random among the other n − p = . Barabási-Albert (BA): the network is generated followingthe preferential attachment algorithm. First, a com-plete network with d + d / d / i is proportional to the degreeof node i . The procedure is repeated until the node setcontains all the n nodes. C. Castellano, S. Fortunato, and V. Loreto, “Statistical physics of socialdynamics,” Reviews of Modern Physics , 591–646 (2009). J. Shao, S. Havlin, and H. E. Stanley, “Dynamic opinion model and inva-sion percolation,” Physical Review Letters , 018701 (2009). J. Ratkiewicz, S. Fortunato, A. Flammini, F. Menczer, and A. Vespignani,“Characterizing and modeling the dynamics of online popularity,” PhysicalReview Letters , 158701 (2010). F. Baumann, P. Lorenz-Spreen, I. M. Sokolov, and M. Starnini, “Model-ing echo chambers and polarization dynamics in social networks,” PhysicalReview Letters , 048301 (2020). J. R. P. French Jr, “A Formal Theory of Social Power,” Psychological Re-view , 181–194 (1956). M. H. DeGroot, “Reaching a Consensus,” Journal of the American Statisti-cal Association , 118–121 (1974). N. E. Friedkin and E. C. Johnsen, “Social Influence and Opinions,” Journalof Mathematical Sociology , 193–206 (1990). C. Altafini, “Consensus Problems on Networks with Antagonistic Interac-tions,” IEEE Transactions on Automatic Control , 935–946 (2013). D. Acemo˘glu, G. Como, F. Fagnani, and A. Ozdaglar, “Opinion fluctu-ations and disagreement in social networks,” Mathematics of OperationsResearch , 1–27 (2013). R. Hegselmann and U. Krause, “Opinion dynamics and bounded confidencemodels, analysis, and simulation,” Journal of Artificial Societies and SocialSimulation (2002). J. Lorenz, “Continuous opinion dynamics under bounded confidence: Asurvey,” International Journal of Modern Physics C , 1819–1838 (2007). W. Quattrociocchi, G. Caldarelli, and A. Scala, “Opinion dynamics on in-teracting networks: media competition and social influence,” Scientific Re-ports , 4938 (2014). L. Zino, A. Rizzo, and M. Porfiri, “Consensus over activity-driven net-works,” IEEE Transactions on Control of Network Systems , 866–877(2020). J. Hasanyan, L. Zino, D. A. Burbano Lombana, A. Rizzo, and M. Por-firi, “Leader-follower consensus on activity-driven networks,” Proceedingsof the Royal Society A: Mathematical, Physical and Engineering Sciences , 20190485 (2020). P. Holme and M. E. J. Newman, “Nonequilibrium phase transition in the co-evolution of networks and opinions,” Physical Review E , 056108 (2006). J. Smith,
Evolution and the Theory of Games (Cambridge University Press,1982). H. Peyton Young, “The dynamics of social innovation,” Proceedings of theNational Academy of Sciences , 21285–21291 (2011). A. Montanari and A. Saberi, “The spread of innovations in social net-works,” Proceedings of the National Academy of Sciences , 20196–20201 (2010). P. Ramazi, J. Riehl, and M. Cao, “Networks of conforming or noncon-forming individuals tend to reach satisfactory decisions,” Proceedings ofthe National Academy of Sciences , 12985–12990 (2016). J. Haidt, “The emotional dog and its rational tail: a social intuitionist ap-proach to moral judgment,” Psychological Review , 814–834 (2001). two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems 14 S. Gavrilets and P. J. Richerson, “Collective action and the evolution ofsocial norm internalization,” Proceedings of the National Academy of Sci-ences , 6068–6073 (2017). B. Lindström, S. Jangard, I. Selbing, and A. Olsson, “The Role of a “Com-mon Is Moral” Heuristic in the Stability and Change of Moral Norms,”Journal of Experimental Psychology: General , 228–242 (2018). F. Gargiulo and J. J. Ramasco, “Influence of Opinion Dynamics on the Evo-lution of Games,” PloS One , e48916 (2012). A. C. R. Martins, “Continuous opinions and discrete actions in opinion dy-namics problems,” International Journal of Modern Physics C , 617–624(2008). D. Centola, R. Willer, and M. Macy, “The Emperor’s Dilemma: A Compu-tational Model of Self-Enforcing Norms,” American Journal of Sociology , 1009–1040 (2005). C.-F. Schleussner, J. F. Donges, D. A. Engemann, and A. Levermann,“Clustered marginalization of minorities during social transitions inducedby co-evolution of behaviour and network structure,” Scientific Reports ,30790 (2016). P. Duggins, “A Psychologically-Motivated Model of Opinion Change withApplications to American Politics,” Journal of Artificial Societies and So-cial Simulation , 1–13 (2017). M. Ye, Y. Qin, A. Govaert, B. D. O. Anderson, and M. Cao, “An InfluenceNetwork Model to Study Discrepancies in Expressed and Private Opinions,”Automatica , 371–381 (2019). C. Granell, S. Gómez, and A. Arenas, “Dynamical interplay betweenawareness and epidemic spreading in multiplex networks,” Physical ReviewLetters , 128701 (2013). W. Wang, Q.-H. Liu, J. Liang, Y. Hu, and T. Zhou, “Coevolution spreadingin complex networks,” Physics Reports , 1 – 51 (2019). L. V. Gambuzza, M. Frasca, and J. Gómez-Gardeñes, “Intra-layer synchro-nization in multiplex networks,” Europhysics Letters , 20010 (2015). V. Nicosia, P. S. Skardal, A. Arenas, and V. Latora, “Collective phenomenaemerging from the interactions between dynamical processes in multiplexnetworks,” Physical Review Letters , 138302 (2017). A. Barrat, A. Baronchelli, L. Dall’Asta, and V. Loreto, “Agreement dy-namics on interaction networks with diverse topologies,” Chaos: An Inter-disciplinary Journal of Nonlinear Science , 026111 (2007). M. Jalili, “Social power and opinion formation in complex networks,” Phys-ica A: Statistical Mechanics and its Applications , 959 – 966 (2013). Q.-H. Liu, F.-M. Lü, Q. Zhang, M. Tang, and T. Zhou, “Impacts of opin-ion leaders on social contagions,” Chaos: An Interdisciplinary Journal ofNonlinear Science , 053103 (2018). R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani,“Epidemic processes in complex networks,” Reviews of Moderns Physics , 925–979 (2015). M. Rohden, A. Sorge, D. Witthaut, and M. Timme, “Impact of networktopology on synchrony of oscillatory power grids,” Chaos: An Interdisci-plinary Journal of Nonlinear Science , 013123 (2014). C. Nowzari, V. M. Preciado, and G. J. Pappas, “Optimal resource allocationfor control of networked epidemic models,” IEEE Transactions on Controlof Network Systems , 159–169 (2017). X. F. Wang and G. Chen, “Pinning control of scale-free dynamical net-works,” Physica A: Statistical Mechanics and its Applications , 521 –531 (2002). P. DeLellis, M. di Bernardo, and M. Porfiri, “Pinning control of complexnetworks via edge snapping,” Chaos: An Interdisciplinary Journal of Non-linear Science , 033119 (2011). R. Willer, K. Kuwabara, and M. W. Macy, “The False Enforcement ofUnpopular Norms,” American Journal of Sociology , 451–490 (2009). D. Smerdon, T. Offerman, and U. Gneezy, “’everybody’s doing it’: on thepersistence of bad social norms,” Experimental Economics (2019). D. A. Prentice and D. T. Miller, “Pluralistic ignorance and alcohol use oncampus: Some consequences of misperceiving the social norm,” Journal ofPersonality and Social Psychology , 243–256 (1993). M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A.Porter, “Multilayer networks,” Journal of Complex Networks , 203–271(2014). We observe that more realistic activation rules which account for temporaland inter-individual heterogeneity may be simply implemented by associ-ating a (possibly inhomogeneous) Poisson clock to the activation of eachindividual. A. V. Proskurnikov and R. Tempo, “A tutorial on modeling and analysisof dynamic social networks. Part I,” Annual Reviews in Control , 65–79(2017). A. V. Proskurnikov and R. Tempo, “A tutorial on modeling and analysis ofdynamic social networks. Part II,” Annual Reviews in Control , 166–190(2018). Y. Chen, W. Xia, M. Cao, and J. Lü, “Random asynchronous iterations indistributed coordination algorithms,” Automatica , 108505 (2019). A node s ∈ V is reachable from r ∈ V if and only if there exists a sequenceof nodes ( i = r , i ,..., i (cid:96) = s ) such that ( i i , i i + ) is an edge, for any i ∈{ ,...,(cid:96) − } . L. Blume, “The statistical mechanics of best-response strategy revision,”Games and Economic Behavior , 111–145 (1995). S. Morris, “Contagion,” The Review of Economic Studies , 57–78(2000). A layer is connected if every node is reachable from all the others. H. Simon, “Bounded rationality in social science: Today and tomorrow,”Mind & Society: Cognitive Studies in Economics and Social Sciences ,25–39 (2000). A. Moinet, R. Pastor-Satorras, and A. Barrat, “Effect of risk perception onepidemic spreading in temporal networks,” Physical Review E , 012313(2018). L. Zino, A. Rizzo, and M. Porfiri, “Modeling memory effects in activity-driven networks,” SIAM Journal on Applied Dynamical Systems , 2830–2854 (2018). M. Newman,
Networks: An Introduction (Oxford University Press, 2010). K. Abbink, L. Gangadharan, T. Handfield, and J. Thrasher, “Peer punish-ment promotes enforcement of bad social norms,” Nature Communications (2017). D. A. Prentice and D. T. Miller, “Pluralistic Ignorance and Alcohol Use onCampus: Some Consequences of Misperceiving the Social Norm,” Journalof personality and social psychology , 243–256 (1993). G. Mackie, “Ending footbinding and Infibulation: A Convention Account,”American Sociological Review , 999–1017 (1996). B. Ryan and N. C. Gross, “The Diffusion of Hybrid Seed Corn In Two IowaCommunities,” Rural Sociology , 15 (1943). P. Curty and M. Marsili, “Phase coexistence in a forecasting game,” Journalof Statistical Mechanics: Theory and Experiment , P03013 (2006). J. da Gama Batista, J.-P. Bouchaud, and D. Challet, “Sudden trust collapsein networked societies,” The European Physical Journal B , 55 (2015). P. DeLellis, A. DiMeglio, F. Garofalo, and F. L. Iudice, “The evolving cob-web of relations among partially rational investors,” PloS One12