This is the accepted version of the article: Konc, Théo; Savin, Ivan. «Social reinforcement with weighted interactions». Physical Review E, Vol. 100, Issue 2 (August 2019), art. 022305. DOI 10.1103/PhysRevE.100.022305 This version is avaible at https://ddd.uab.cat/record/213304 under the terms of the license Social reinforcement with weighted interactions ∗ Théo Konc Institute of Environmental Science and Technology, Universitat Autónoma de Barcelona, Spain Ivan Savin Institute of Environmental Science and Technology, Universitat Autónoma de Barcelona, Spain; Graduate School of Economics and Management, Ural Federal University, Yekaterinburg, Russian Federation The speed and extent of diusion of behaviors in social networks depends on network structure and individual preferences. The contribution of the present study is twofold. First, we introduce weighted interactions between potential adopters that depend on the similarity in their preferences and moderate the strength of social reinforcement. The reason for the extension is the existence of a conrmation bias in the way agents treat information by prioritizing evidence conforming to their opinion. As a result, individuals become less likely to be inuenced by peers with relatively dierent preferences, reducing the overall diusion rate under clustered networks. Second, we en- rich our analysis by also considering a scale free network topology with a high degree asymmetry, motivated by its pervasiveness in online social networks. This network performs consistently well in terms of diusion for dierent parameter combinations and clearly outperforms clustered networks under weighted interactions. Our results show that more realistic assumptions regarding agents' interactions shift the focus from clustering to degree distribution in the study of network structures allowing for fast and widespread behavior adoption. I. INTRODUCTION found that participants were more likely to join as the number of invitations they received grew, demonstrating Individual decision-making is sensitive to conspicuous the role of social reinforcement. Furthermore, exploring peer behavior. This has been conrmed, among others, dierent network topologies Centola concluded that be- for energy consumption choices [1, 2], adoption of solar haviors subject to social reinforcement diuse at a higher panels [3] and greenhouse gas mitigation practices (in- rate in clustered networks. Several modeling studies re- cluding usage of public transport and recycling, see [4]). produced this nding. [9] build a simple model of infor- Social inuence is often the result of descriptive norms, mation diusion where the likelihood of accepting and i.e. a regularity in the behavioral pattern of peers that transmitting the information increases with the number signals a socially 'correct' behavior [5]. The adoption of of signals received. [10] use a threshold model, in which such behaviors is triggered by the percolation process  agents adopt a behavior depending on peer pressure and only those agents can adopt that are exposed to peers individual preferences dened as intrinsic cost to adopt that already adopted the behavior  and by the social a new behavior. reinforcement process  every further peer increases the We introduce two novelties to the model by [10]. First, likelihood of adoption. we consider that the peer pressure between two agents de- Many studies have modeled the diusion of a behav- pends on the similarity in their preferences. Second, we ior at the individual level in an explicit social network include the scale free network topology in our analysis. searching for a topology ensuring fast and widespread Both novelties are motivated by extensive empirical ev- adoption [6, 7]. The main topological features investi- idence documenting relevance of these assumptions. In gated are the average path length (average number of particular, as demonstrated by [11], agents put a higher edges along the shortest path between any two nodes) weight on opinions that conform to their existing beliefs, and the clustering (the extent to which peers of any node pointing out an important conrmation bias in the way tend to be also peers with each other) of the network [8]. agents treat information. Later the conrmation bias Shorter average path length allows for a faster spreading was demonstrated in dierent contexts [1214], with ex- of information. High clustering enhances a local diu- periments showing that agents are more likely to comply sion via the social reinforcement process. The role of with a norm communicated by a peer with similar prefer- clustering has been demonstrated in an experiment by ences [15]. To model the conrmation bias, we introduce Centola [7] where participants faced the decision to join weighted edges based on the distance in agents' prefer- a health forum. Every time a participant registered to ences. In other words, the social inuence one agent the forum, a message was sent to her social neighbor- exerts on another is not binary (signal the behavior or hood inviting the recipients to join as well. This study not), but mediated by their similarity in preference to- wards adopting that behavior. Our conjecture is that this bias can aect the performance of the studied networks in terms of diusion rates. Including the scale free net- ∗ theo.konc@uab.cat work is motivated by the increasing empirical evidence This is the peer-reviewed version of the article: Konc, T., & Savin, I. (2019). Social reinforcement with weighted interactions. Physical Review E, 100(2), 022305, which has been published in final format at https:// doi.org/10.1103/PhysRevE.100.022305 2 that online networks exhibit the power law degree distri- where Q ∈ [0, 1] is the quality of the behavior and mi,t bution [16, 17], while interactions occur with ever greater is the minimum quality requirement (or MQR) of agent extent online [18]. This particular topology is the result i.2 A zero quality means a completely unattractive be- of the preferential attachment process where few nodes havior that no one is willing to adopt, while a quality have a very large number of connections, and majority of one means a behavior that everybody wants to adopt of nodes have very few peers only [19]. While simple from the moment they are informed about it. The mini- diusion in the scale free network have been studied by mum quality requirement without conrmation bias is a [20, 21], this paper is the rst addressing the scale free decreasing function of the number of adopting peers: topology for complex diusion with social reinforcement −γ and weighted interactions. mi,t = πi × (gi,t ) (2) We demonstrate that, rst, the addition of insights from behavioral science on how agents asymmetrically with γ ∈ [0, 1] capturing the intensity of social reinforce- treat information undermines the performance of clus- ment in the decision-making, Gi,t ⊆ Ni being the subset tered networks. This is because the long average paths of adopters among agent i's peers at time t, gi,t ≤ ni between nodes in the network allow individuals to re- being the number of adopters among agent i's peers at sist social pressure from peers with very dierent prefer- time t, and πi ∈ [0, 1] being the intrinsic switching cost of ences precluding further spreading of the behavior. Sec- agent i towards a given behavior. The switching costs are ond, we show that the scale free network is robust to inversely related to intrinsic preferences in adopting the the weighting of interactions. It best ensures fast and new behavior. An agent with high switching cost may widespread diusion under social reinforcement and con- adopt only if several peers inuence her to do so. An rmation bias when the majority of the population has agent with low switching costs will adopt as soon as one a high resistance against adopting the behavior. These peer signals her the behavior. In the following, we will two results are novel and important for the literature interchangeably refer to preferences and switching costs in understanding the complex processes of social diu- that both capture the individual resistance in adopting a sion shifting the focus from network clustering to net- certain behavior (MQR). Heterogeneous switching costs work degree distribution as the critical factor for behav- between agents can be explained by dierences in income, ior spreading. They also suggest that the diusion of a education, environmental concern, available infrastruc- behavior is more eective on digital social platforms than ture and even amount of free time to adopt the new behavior. 3 In line with [10], we assume that switching on spatial oine networks where high degree asymmetry is less likely. costs between neighbors are independent. This means The remainder of this paper is organized as follows. that the generating process of the network is unrelated Section 2 describes our model of diusion with social re- to the particular behavior in question. The preferences inforcement under conrmation bias. There we introduce follow a Beta distribution: πi ∼ B(β1 , β2 ). This dis- weighted interactions and the scale free network topology tribution family is exible, allowing us to study several not considered in this context earlier. Section 3 presents settings (Figure 1): simulation results. Section 4 concludes. (i) a uniform distribution with β1 = 1, β2 = 1, (ii) a pseudo-normal distribution with β1 = 4, β2 = 4, II. THE MODEL (iii) a right skewed distribution (the majority of agents Percolation with social reinforcement have strong preference for the behavior) with β1 = 1, β2 = 4, We describe a simple model of behavior adoption with (iv) a left skewed distribution (the majority of agents social reinforcement and conrmation bias, in which the have strong preference against the behavior) with decision to adopt is based on personal preferences, aware- β1 = 4, β2 = 1. ness about certain behavior (percolation model) and peer pressure. Following [10], we consider agents interacting in a xed undirected social network N , where the set of agent i's neighbors is denoted by Ni and its degree by 2 Quality is an abstract aggregate term. Referring to the examples ni .1 The agent decides to adopt the behavior if and only from the introduction, quality captures the eectiveness of mea- if: sures mitigating greenhouse gas emissions or technical quality of solar panels. Instead of quality, one can also operate with prices: Q ≥ mi,t (1) Q as a market price and M QR as a reservation price of the agent [22]. Considering the more expensive electricity from the renew- able energy sources as a new behavior, social reinforcement raises the agent's willingness to pay and adopt the new behavior. 3 Consider, for example, the choice to adopt waste sorting be- 1 Upper case letters denote sets, and the corresponding lower case havior where the above-mentioned characteristics can serve as letters the cardinality of the respective sets. arguments in favor of or against adopting the new norm. 3 4 we extend the social interaction process underlying our Uniform model. While in [10] it was only about sharing infor- Pseudo-Normal mation about adoption decision (showing), here we as- 3 Right-skewed sume agents know and dierentiate between preferences Probability density Left-skewed of their peers (i.e. telling information about the reasons behind their adoption) [28]. 2 We proceed by extending MQR with the conrmation bias as follows: 1 !−γ e−ρ|πi −πj | P j∈Gi,t mi,t = πi × × ni (3) 0 e−ρ|πi −πk | P k∈Ni 0.0 0.2 0.4 0.6 0.8 1.0 πi with ρ ∈ [0, +∞) being the strength of conrmation bias, Ni being the set of peers of agent i, Gi,t ⊆ Ni being its FIG. 1. Dierent distributions of switching costs (πi ). subset that adopted the behavior at time t and ni the de- gree of agent i. The strength of the signal owing from −ρ|πi −πj | agent j to agent i is e . The ratio in Eq.(3) repre- In the rest of the paper, we will focus on the results for sents the relative weight of the signal from the adopting the latter specication of the Beta distribution (πi drawn peers. We multiply this ratio by the agent's degree ni so for the B(4, 1) with the majority of people having high that Eq. (2) without conrmation bias proposed by [10] switching costs. This setting corresponds best to the case can be seen as a special case of our Eq.(3) with ρ = 0. where social reinforcement makes a dierence with many Furthermore, MQR of agent i would be the same in both agents adopting because of social pressure [10]. 4 models if agent i is surrounded by agents with the same While network position (and the resulting number of preference (Eq. 4): adopters in the neighborhood) and switching costs are  −γ heterogeneous across agents, other parameters like the gi,t −γ intensity of social reinforcement and behavior quality, act mi,t = πi × × ni = πi × (gi,t ) if ∀πk = πi ni uniformly for all agents. (4) or if all peers of agent i adopted the behavior (Eq. 5): Weighted interactions and conrmation bias mi,t = πi × (1 × gi,t ) −γ if gi,t = ni . (5) Thus, introducing the conrmation bias we eectively The rst novelty of our approach is to dierentiate the transform our network from an unweighted to a weighted inuence of peers on the agent's decision to adopt de- one. pending on the distance in their preferences. In doing so, Note that MQR with conrmation bias from Eq.(3) we follow the literature on the conrmation bias show- being a function f (πi , gi,t , γ) can exceed the original ing that opinion dynamics between two agents depends switching cost πi if the distance in switching costs to on their opinion proximity [2325]. The bias arises due the adopters in agent i's neighborhood is larger than the to often unconscious selectivity in searching, interpreting average distance to all his neighbors: and remembering evidence supporting one's own opinion. In this framework, agents assign more weight to opinions e−ρ|πi −πk | P X k∈Ni −ρ|πi −πj | that already conform to their beliefs, while neglecting to e ≤ . (6) ni gather, or discounting, evidence that would tell against j∈Gi,t it [13, p.175]. While [26] captures this bias by allow- ing agents to regard only those peers who are suciently That means f (πi , 1, γ) = πi does not always hold any- close in their opinion, we follow [27] by allowing agents more (unlike in [10]), but f can be larger or smaller than being both close and far in their intrinsic preference to πi depending on the distance in preference to the rst adopting peer. 5 One should, however, not mix it with a aect their peers' decision but with dierent weights. For instance, observing someone having the same preference possible deterring eect in our model since πi is only an ingredient of MQR. In line with the fundamental prin- adopting certain behavior is more persuasive than ob- serving someone with a very distant preference. By doing ciple of a percolation model, f (πi , 0, γ) = ∞ (an agent cannot adopt a behavior if she is not aware of it), and this, we introduce heterogeneity in terms of peer pressure the agent can experience from dierent neighbors. Also, f (πi , gi,t , γ) is strictly decreasing in gi,t for γ > 0. 5 The inequality in Eq. (6) can hold even for two or more adopting 4 The results for other distributions can be found in Appendix C. peers. 4 For very large ρ, the agents only consider the decision evidence, small world and scale free topologies represent of their most similar neighbors. Let us consider the case structural properties of networks resulting from physical when agent l is the most similar (in terms of preference) and digital interactions in the real world. In particular, peer of agent i, i.e. |πi − πl | < |πi − πk | ∀k 6= l ∈ Ni . small world property is typical for professional networks Then: (movie actors, scientic collaborations [30] and frequently used to describe interactions leading to the adoption of lim ρ→∞ e−ρ|πi −πj | = 0 ∀j ∈ Ni (7) home-specic goods  such as solar PV in [32], while scale free networks are more suited to study information dif- and fusion in online social networks, such as in [17, 33]. We   lim −ρ|πi −πl | −ρ|πi −πk | −ρ|πi −πl | also deliberately set the density of the synthetic networks ρ→∞ e +e =e . (8) very low (approx. 0.0004), which is in line with empirical Therefore, for the strength of the conrmation bias ap- estimates by [34]. proaching innity, we have agents simplifying the behav- ior of their whole ego network to the behavior of their TABLE I. Network characteristics for 10 000 nodes and 20 closest peer: 000 undirected links −γ Average Average Degree  πi × (ni ) if l ∈ Gi,t mi,t = (9) clustering path length asymmetry 0 else. Regular lattice 50.00 % 1250.00 0.00 Small world 35.62 % 12.50 0.12 Network topology Random 0.04 % 6.76 0.50 Scale free 0.15 % 4.27 36.30 Note: Average clustering is measured as an average probability for Our second novelty is that we extend the set of net- every node that any two of her neighbors are connected. Average path works considered with the scale free network. The reason length measures the average number of links required to connect two is that many empirical networks present a very high de- nodes. Degree asymmetry of a network is measured by the skewness of its degree distribution. gree asymmetry. This holds particularly true for digital social networks due to the lower time constraint asso- Figure 2 is meant to illustrate our model of diu- ciated with maintaining online communication allowing sion with social reinforcement with and without the con- agents to have a very large number of connections [17]. rmation bias. The top two panels in Figure 2 ex- As a result, few star agents can have a high number of emplify the diusion processes with and without con- peers, whereas the majority of agents have a few con- rmation bias in a regular lattice with Q = 0.6 and nections only. We generate the network using the pref- γ = 1. The early adopter E activates her four neighbors erential attachment algorithm [19]. The probability that NE = {C, D, F, G}. Without conrmation bias (Eq.2) an agent has k connections in the network decays as a agent G has minimal quality requirement of: mG,1 = power law: P (k) ∼ k −α . The algorithm generates degree 1 distribution with α = 3 following empirical estimations 0.60 × 11 ≤ Q and will therefore adopt. Once two con- nected agents have adopted in a regular lattice, it is an [16, 30, 31]. Comparing our results with Tur et al. [10], we use the Watts-Strogatz algorithm [29] that starts with immediate property of Eq.(2) that for γ = 1 the diusion generating a network in which agents are connected to a will reach 100% for all Q greater than 0.5. few nearest neighbors (regular lattice), and rewires ev- With conrmation bias and ρ = 2 (Eq.3) the same agent has, in contrast, a mini- ery link with a probability µ. As the probability goes to 1, the topology resembles the random network. The so- mal quality requirement of: mG,1 = 0.60 ×  −1 called small world network topology with still high clus- e−2×0.23 e−2×0.23 +e−2×0.34 +e−2×0.14 +e−2×0.21 ×4 ' 0.609 ≥ Q tering (similar to regular lattice) but already low enough and therefore will not adopt, which prevents the further average path length (similar to random network) is ob- diusion process. 7 This illustrates how under conrma- served forµ ∈ [0.001, 0.1] (see Table I). In the following tion bias agents become less likely to be inuenced by 6 we adopt µ = 0.05. The networks created with the small peers with very dierent preferences. world algorithm all have a relatively small degree asym- The lower panel in Figure 2, in contrast, demonstrates metry and have been studied without conrmation bias that in a network with high degree asymmetry the behav- [10]. ior can spread equally well with and without conrmation While the regular lattice and random network are bias. This is because applying Eqs.(2) and (3) leads in mathematically convenient graphs with little empirical this example to the same diusion path with two, three and four more adopters in subsequent periods. 6 [10] tested alternative specication of the parameter but did not nd any qualitative dierence. We also checked dierent µ for 7 For ρ = 1 further diusion would also be prevented as mG,1 = the small world network in our model with conrmation bias and 0.602, while for ρ = 3 its value would have been even larger and came to the same conclusion. equal to 0.618. 5 FIG. 2. Example of a diusion process for Q = 0.6, γ = 1, B(4, 1) in (A) a regular lattice without conrmation bias (B) a regular lattice with conrmation bias, ρ=1 , (C) a network with high degree asymmetry with and without conrmation bias. Note: Scalars in parenthesis are the switching costs of each agent. Black nodes represent agents having adopted the behavior while gray nodes are agents that are subject to adopt; red links represent the signal owing from the adopting agent to its neighbors. H (d=0.14) strength of conrmation bias increases, the most similar ρ=0 (in terms of preferences) neighbor, agent H in this exam- ρ=1 ple, has an overarching inuence on the decision-making ρ=2 of agent G. ρ=3 ρ = 10 To have a more systematic comparison of the four net- works for a larger number of adopters, dierent scenarios of early adopters as well as alternative distributions of preferences, we proceed with the numerical experiment G in the next section. I (d=0.21) F (d=0.34) III. RESULTS Consistent with [10], we simulate the diusion process E (d=0.23) in undirected networks with 10 000 agents with mean degree of 4. The process starts with 10 random early adopters and runs until a steady state is reached, i.e. no FIG. 3. Relative inuence of neighbors on agent G, with new adopter is generated in any subsequent period. As dierent strength of conrmation bias given the distance in the model can generate dierent outcomes depending on preferences (d) the initial conditions, 8 we report average results over 50 Figure 3 shows how the relative inuence of the neigh- 8 Three features subject to random number initialization inuence bors of agent G changes when a conrmation bias is in- the outcome: the draw of individual switching costs, the choice troduced in the model. As noted in Eqs. (7-9), when the of early adopters and the topology of the network. 6 1.0 1.0 1.0 regular 0.8 small world 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate random 0.6 scale free 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 4. Adoption rate without conrmation bias, ρ = 0, and for π ∼ B(4, 1) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 1.0 1.0 1.0 regular 0.8 small world 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate random 0.6 scale free 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 5. Adoption rate with conrmation bias, ρ = 2, and for π ∼ B(4, 1) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 7 restarts for each combination of parameters: quality Q, (see Figures 8 and 9 in Appendix A). Combined with the intensity of social reinforcement γ , distribution of prefer- fact that a few resisting agents are sucient to isolate ences B(β1 , β2 ) and strength of conrmation bias ρ. In a large share of the network from early adopters (Figure Appendix B we also report the period in which the dif- 2), we observe the very high fragility of the performance fusion stops to make the potential trade-o between the of the highly clustered network towards presence of the rate of adoption (i.e. the share of adopters in the popu- conrmation bias. It is worth mentioning that already lation) and the duration of the diusion process explicit. few short path links present in the small world network We compare the four networks with the two specications solve the problem and make the network robust to the of MQR presented in Section II: without conrmation bias. Thanks to those links, agents bypass the resisting bias described in Eq.(2) and with conrmation bias from nodes and eectively prevent them from stopping further Eq.(3) with ρ = 2. This value of ρ corresponds to a mod- diusion. Increasing further the number of adopters later erate conrmation bias (see Figure 3). In this setting, can convert the resisting agent into an adopter anyway. agents with relatively dissimilar preferences have a lower but still substantial inuence on each other. Simulations To check the robustness of this result, we explore al- results for other values of ρ can be found in Appendix A. ternative values of the conrmation bias strength ρ (see Appendix A). We nd that the performance of the reg- In line with the description of our model, the higher ular lattice deteriorates even further when the similarity quality Q and the intensity of social reinforcement γ con- in preferences plays a stronger role in the diusion pro- sistently contribute to the rising rate of diusion of be- havior all else being equal. While higher Q -capturing cess. For higher value of ρ, we nd that the adoption rate in the regular lattice is close to 0 for virtually all combi- higher attractiveness of behavior- simply increases the nations of parameters. This conrms that the diusion chance to adopt under the threshold rule in Eq. (1), the process in clustered networks is highly contingent on the higher γ does the same by increasing the role of social weight agents assign to similar peers. reinforcement on potential adopters. From Figure 4 we see that without any conrmation Another important consequence of the introduction of bias when the majority of the population has strong pref- the conrmation bias is the fact that the scale free net- erences against adopting the behavior, the scale free net- work becomes the best performing graph in terms of work performs well, reaching a diusion rate higher or diusion for virtually all combinations of quality and equal to other networks for the majority of combinations strength of social interaction (Figure 5). Thus, unlike of quality and strength of social reinforcement without previous studies, we nd that high clustering is not nec- conrmation bias. The only exception is the interme- essary to reach high diusion rates when the majority diate level of quality (0.6-0.8) combined with the high of agents originally resists the adoption. The most con- intensity of social interaction (γ ≥ 0.6). Note that in line nected agents in the scale free network act like inu- with our illustration in Figure 2, 100% diusion rate in ence hubs: they are more likely to have early adopters the regular lattice is assured for certain combinations of among neighbors (hence, adopt the behavior themselves), Q and γ. This result is trivial considering the perfectly and subsequently distribute the signal to their numerous structured topology of the regular lattice where any two peers quickly increasing the strength of social pressure on peers share two more peers in common. According to other agents. Unlike clustering, this structural property 1 Eqs. (1-2), if 2γ ≤ Q then the presence of two connected is not sensitive to the introduction of weighted interac- adopters is a sucient condition to reach 100% diusion. tions. Although describing the important role of inu- One can compare this with the domino chain reaction, ence hubs is not novel in the case of simple diusion, the where agents cannot resist social pressure from a cer- presence of high degree asymmetry allows for a broad and tain point. In particular, having two connected adopters fast diusion under social reinforcement even with a low guarantees a full diusion for γ =1 under Q ≥ 0.5; for clustering and conrmation bias. γ =0.8 under Q ≥ 0.575; γ =0.6 under Q ≥ 0.66; γ =0.4 under Q ≥ 0.758; etc. Comparing diusion across networks in terms of speed Once we introduce the conrmation bias in Figure 5, (number of periods until diusion stops) supports the ad- we observe a deterioration of the rate of diusion in the vantage of the scale free network. Thus, it does not only regular lattice while performance of all other networks, reach highest diusion rates, but does so in the shortest particularly the scale free one, remains remarkably ro- number of periods (see Appendix B). The latter is impor- bust. This is made clear by Figure 6, which shows the tant if one is concerned not just with the nal outcome change in adoption rate when conrmation bias is in- but also the speed of its realization. A good example troduced. This deterioration happens because the bias can again be environmental behavior: a quick diusion raises the propensity of agents to resist social pressure would help to reduce CO2 emissions early enough to pre- from peers with relatively dierent preferences. Hence, vent global warming of 1.5 °C above pre-industrial levels the domino eect described in the regular lattice under [35]. Other network topologies, whenever they reach rel- no conrmation bias arises only for much higher values of atively high diusion rates, take much more time. This Q. For larger ρ, there is no threshold quality such that is particular true for the regular lattice. Taking its very 100% diusion rate in the regular lattice is guaranteed long average path length (Table I), however, this is not 8 0.8 0.8 1 1 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.4 0.4 0.0 0.0 γ γ −0.4 −0.4 −0.8 −0.8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Quality Quality (A) (B) 0.8 0.8 1 1 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.4 0.4 0.0 0.0 γ γ −0.4 −0.4 −0.8 −0.8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Quality Quality (C) (D) FIG. 6. Dierence in percentage points between the adoption rate with conrmation bias, ρ = 2, and without conrmation bias, ρ = 0, for π ∼ B(4, 1) (A) regular, (B) small world, (C) random, and (D) scale free. Note: Negative values mean that the adoption rate is lower under conrmation bias. The results are generated for dierent combinations of Quality and intensity of social reinforcement (γ ). surprising. 9 are open to adopting new behavior, networks with short- We also test the model under alternative distributions est paths are good enough to reach high diusion rates of preferences like the uniform distribution or majority of within a short amount of time. agents having low switching costs (see Appendix C). In The present paper makes the setting studied by [10] those cases the scale free network does not always rank more realistic by dierentiating the inuence of peers on rst, but is only slightly inferior to the best performing the agent's decision to adopt depending on the distance in network. their preferences. In doing so, we follow the experimental evidence that people put a higher weight on opinions that conform to their beliefs  the so-called conrmation bias. IV. CONCLUSION Furthermore, accepting the growing role of online social networks and their highly asymmetric degree distribution The mounting empirical evidence on the role of social exhibiting power law, we add the scale free topology in interaction in aecting individual decision-making and our network comparison. behavior spreading has drawn attention to the search of We demonstrate that the introduction of the bias network topology that best promotes the diusion pro- changes the ranking of networks in terms of the diu- cess, both in terms of adoption rate and the amount of sion rates they achieve. In particular, while the regular time needed. Earlier, the literature demonstrated that lattice was performing best when the majority of agents clustered networks best foster diusion if a majority of have a strong preference against adopting the behavior, it agents is resistant to adopt. This is because the seem- becomes last once agents start dierentiating their neigh- ingly redundant ties reinforce the probability of adop- bors based on the distance in their preferences and, con- tion of the most resisting agents. If, in contrast, agents sequently, resist the adoption. Such a fragility is due to the absence of short paths between agents allowing to bypass the resisting nodes. In contrast, other studied networks, and particularly the scale free one, are robust 9 Note that the curves of diusion times for dierent Q resemble an to the introduction of this assumption to the model. The inverted U shape. This is because for greater Q agents are more scale free network becomes the best performing one under likely to adopt without any social reinforcement. In fact, for Q → 1 any agent adopts once he has one peer among adopters. Hence, such a setting in terms of diusion for dierent intensities the speed of diusion for Q → 1 is increasing. Low diusion of social reinforcement. Unlike the existing literature, we time for Q → 0, in turn, is due to the low diusion rates where nd that high clustering is not necessary to explain high behavior spreading stops early in time. diusion rates when the majority of agents is reluctant 9 to adopt. This is because high degree nodes serve as in- innovation program (grant agreement Nr. 741087). IS uence hubs collecting and redistributing the signal of acknowledges nancial support from the Russian Science a new behavior raising social pressure in the network. Foundation [RSF grant number 19-18-00262]. We have Thus, our results shift the focus from network cluster- greatly beneted from discussions with Jeroen van den ing to network degree distribution as the key factor for Bergh, Elena Tur, Paolo Zeppini and two anonymous diusion. As highly asymmetric networks are pervasive referees. All remaining shortcomings are solely our re- in digital social platforms, our results suggest that a dif- sponsibility. fusion will be more eective in those platforms than in spatial oine networks characterized by higher clustering and lower degree asymmetry. ACKNOWLEDGMENTS This work was supported by an ERC Advanced Grant under the European Union's Horizon 2020 research and [1] H. Allcott, Journal of Public Economics 95, 1082 (2011). [18] E. Bakshy, I. Rosenn, C. Marlow, and L. Adamic, [2] W. Abrahamse and L. Steg, Global Environmental arXiv:1201.4145 [physics] (2012), arXiv: 1201.4145. Change 23, 1773 (2013). [19] A.-L. Barabási and R. Albert, Science 286, 509 (1999). [3] B. Bollinger and K. Gillingham, Marketing Science 31, [20] R. Pastor-Satorras and A. Vespignani, Physical Review 900 (2012). Letters 86, 3200 (2001). [4] Z. Babutsidze and A. Chai, Ecological Economics 146, [21] P. Zeppini and K. 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Valente, Industrial and Corporate [35] IPCC,  SR1.5: Global Warming of 1.5 °C - Chapter Change forthcoming (2019), 10.1093/icc/dty021. 4: Strengthening and implementing the global response, http://www.ipcc.ch/report/sr15/, (2018). 10 Appendix A RESULTS WITH DIFFERENT CONFIRMATION BIAS PARAMETERS 1.0 1.0 1.0 regular 0.8 small world 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate random 0.6 scale free 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 7. Adoption rate with conrmation bias, ρ = 1, and for π ∼ B(4, 1) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 1.0 1.0 1.0 regular 0.8 small world 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate random 0.6 scale free 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 8. Adoption rate with conrmation bias, ρ = 3, and for π ∼ B(4, 1) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 11 1.0 1.0 1.0 regular 0.8 small world 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate random 0.6 scale free 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 9. Adoption rate with conrmation bias, ρ = 10, and for π ∼ B(4, 1) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 0.8 0.8 1 1 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.4 0.4 0.0 0.0 γ γ −0.4 −0.4 −0.8 −0.8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Quality Quality (A) (B) 0.8 0.8 1 1 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.4 0.4 0.0 0.0 γ γ −0.4 −0.4 −0.8 −0.8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Quality Quality (C) (D) FIG. 10. Dierence in percentage points between the adoption rate with conrmation bias, ρ = 10, and without conrmation bias, ρ = 0, for π ∼ B(4, 1) (A) regular, (B) small world, (C) random, and (D) scale free. Note: Negative values mean that the adoption rate is lower under conrmation bias. The results are generated for dierent combinations of Quality and intensity of social reinforcement (γ ). 12 Appendix B TIME OF DIFFUSION regular 6 6 small world 6 Final period (in ln) Final period (in ln) Final period (in ln) random 4 scale free 4 4 2 2 2 0 0 0 0.5 1.0 0.5 1.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 8 8 10 6 8 6 Final period (in ln) Final period (in ln) Final period (in ln) 6 4 4 4 2 2 2 0 0 0 0.5 1.0 0.5 1.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 11. Diusion time without conrmation bias, ρ = 0, and for π ∼ B(4, 1) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. regular 6 6 6 small world Final period (in ln) Final period (in ln) Final period (in ln) random 4 scale free 4 4 2 2 2 0 0 0 0.5 1.0 0.5 1.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 6 6 6 Final period (in ln) Final period (in ln) Final period (in ln) 4 4 4 2 2 2 0 0 0 0.5 1.0 0.5 1.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 12. Diusion time with conrmation bias, ρ = 2, and for π ∼ B(4, 1) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 13 Appendix C DIFFERENT SWITCHING COSTS DISTRIBUTIONS 1.0 1.0 1.0 regular 0.8 small world 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate random 0.6 scale free 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 13. Adoption rate with conrmation bias, ρ = 0, and for π ∼ B(1, 1) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 1.0 1.0 1.0 regular 0.8 small world 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate random 0.6 scale free 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 14. Adoption rate with conrmation bias, ρ = 2, and for π ∼ B(1, 1) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 14 1.0 1.0 1.0 regular 0.8 small world 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate random 0.6 scale free 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 15. Adoption rate without conrmation bias, ρ = 0, and for π ∼ B(4, 4) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 1.0 1.0 1.0 regular 0.8 small world 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate random 0.6 scale free 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 16. Adoption rate with conrmation bias, ρ = 2, and for π ∼ B(4, 4) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 15 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate regular 0.6 small world 0.6 0.6 random 0.4 0.4 0.4 scale free 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 17. Adoption rate with conrmation bias, ρ = 0, and for π ∼ B(1, 4) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average. 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate regular 0.6 small world 0.6 0.6 random 0.4 0.4 0.4 scale free 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (A) (B) (C) 1.0 1.0 1.0 0.8 0.8 0.8 Average adoption rate Average adoption rate Average adoption rate 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 Quality Quality Quality (D) (E) (F) FIG. 18. Adoption rate with conrmation bias, ρ = 2, and for π ∼ B(1, 4) with social reinforcement intensity (A) γ = 0, (B) γ = 0.2, (C) γ = 0.4, (D) γ = 0.6, (E) γ = 0.8, and (F) γ = 1. Note: The dashed lines represent two standard deviations around the average.