Threshold model of diffusion: An agent based simulation and a social network approach

Transcription

1 Threshold model of diffusion: An agent based simulation and a social network approach Suk-ho Kang, Wonchang Hur. Jeehong Kim, and Daeyoung Kim Abstract Innovation diffusion is a social process in which an innovation is adopted by the members of a social system. But, why some diffusion processes do precipitate bandwagon dynamics and others fail to do? In addressing this question, we considered a threshold model of diffusion. We particularly focus on the patterns of threshold arrangement: how individuals are arrayed over a given network topology. Employing a multi-agent simulation based on a threshold model of diffusion, we found that the model always results in the one of the two following states irrespective of the randomly varying diffusion networks; almost all the adopters will participate in adoption or only the small part of them will do so. In addition, we proposed 10 measures capturing the patterns in which individual threshold levels are associated with its topological features in the given network. From the regression analysis, it turns out that the nodes located closer to the innovators play a critical role in promoting diffusion. Keywords Agent based simulation, diffusion, social network, threshold model, centrality I I. INTRODUCTION NNOVATION diffusion is a social process in which an innovation is adopted by the members of a social system. Diffusion studies have provided many substantive examples showing that most successful innovations have an S-shaped rate of adoption [1]. The increasing rate of diffusion in the early period and the subsequent decreasing rate in the later period are characterized as snowball effect or chain reaction [2]. The chain reaction indicates the effect of interpersonal communication on the process of diffusion, the nature of which is a positive feedback loop: increases in the number of adopters create stronger pressures, and stronger pressures, in turn, cause increases in the number of adopters [3]. An interesting concept regarding this chain reaction of diffusion process is the critical mass. The notion of critical mass originated in physics, where it was defined as the amount Suk-ho Kang is with the Department of Industrial Engineering, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul , Republic of Korea (corresponding author. phone: ; fax: ; shkang@snu.ac.kr). Wonchang Hur is with College of Business Administration, Inha University 253 Yonghyun-dong, Nam-gu, Incheon , Republic of Korea ( wchur@inha.ac.kr). Jeehong Kim is with the Department of Industrial Engineering, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul , Republic of Korea ( jivong97@snu.ac.kr). Daeyoung Kim is with the Department of Industrial Engineering, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul , Republic of Korea ( kdy555@snu.ac.kr). of radioactive material necessary to produce a nuclear reaction [4], [5]. In the context of diffusion studies, the critical mass occurs at the point at which enough individuals in a system have adopted an innovation so that the innovation s further rate of adoption becomes self-sustaining [1]. Due to the chain reaction, there is, after the small number of system members adopts an innovation, relatively rapid adoption by the remaining members and then a period in which the holdouts finally adopt [2]. This paper raises the following questions on these features of diffusion dynamics; what are the key factors to instigate the positive feedback loop that makes diffusion self-sustaining? And how they are related to the other part of the system? Although these questions are fundamental to understanding why some diffusion processes do precipitate bandwagon dynamics and others fail to do, they are not largely explored in the diffusion literature. This paper employs a multi-agent simulation based on a threshold model of diffusion. Threshold models are important in diffusion studies because it can easily describe a complex interdependency of individual decisions over time. Based on the threshold hypothesis, we consider a large number of networks generated random rewiring of individuals of different characteristic. We particularly focus on the patterns of threshold arrangement: how individuals are arrayed over a given network topology. This is different from the topologies of network, which is only related to the shape of a network. Then we investigated under what conditions diffusion processes are able to reach the critical mass, that is, becomes self-sustaining. Our study reveals that the intricate combination of threshold heterogeneity and interdependency is a crucial factor affecting the bandwagon dynamics. The results carry interesting implications on the way different peoples are connected with each other and its effect on diffusion dynamics. II. THE RELATED LITERATURE An important diffusion model that can easily describe a diffusion mechanic like chain reaction is a threshold model. Threshold models assume that individuals take account of how many others are behaving in a particular way in making their own decisions about participating in the behavior. The crucial part of this assumption is that they have different thresholds that is, some individuals will adopt after only a small proportion of their alters has adopted, while others will not adopt until a large proportion of their alters has adopted [6]. As a result, the

2 distribution of thresholds in a population becomes a key factor closely linked to the mechanic of diffusion process and comes to affect significantly the final extent of the diffusion process. In fact, Reference [7] demonstrated mathematically that a slight perturbation of the threshold distribution sometimes generates entirely different level of participation. In this way, threshold models provide an important explanatory mechanism on why diffusion causes a chain reaction to start and various proportions of a collectivity s members to adopt. The threshold concept has been adapted, extended, and incorporated frequently into many later diffusion models as an important theoretical background for them. For example, [4] points out the problem of identifying individual thresholds which requires detailed knowledge of the individual expected rate of return on investments in public goods. In order to address this problem, the authors introduce a stochastic learning model in which individual thresholds are not deterministic. In the model, the structural configuration of thresholds is treated in two stylized cases, strong ties and weak ties, and their simulation results support the strength-of-weak-ties hypothesis. Reference [8] tries to relax the assumption of non-reciprocal communication of classic threshold models, and points out that a person learns about his/her neighbors preferences, their willingness to participate, and does not directly respond to their actions. The author argues that people with low thresholds, who are highly predisposed toward participation, are affected much more by social position than people with high thresholds. The author also shows that how strong links can be better for participation when thresholds are low and weak links can be better when thresholds are high. The model is well aligned with our intuition that whether a low-threshold person participates or not depends greatly on whether that person happens to have some sympathetic friends, while a high-threshold person participates only if a great mass of people participate. Reference [3], by employing a core-periphery network, highlights the role of boundary agents. The study introduces the two important concepts, boundary pressure point (agents with high threshold and high connectivity) and boundary weakness (agents with low threshold and low connectivity), and proposes that they have relatively greater effect on diffusion extent, particularly in lower-density networks. Most of these models relax the original model s assumption that an actor s decision to join a social movement has the same effect on all other actors in the social system. This relaxation of the restrictive assumption allows them to analyze the structural effect, the effect of preexisting social networks linking the actors in which the flow of influence is restricted and channeled. The difference between the models largely reflects differences in the theoretical understanding of how the structural arrangement of social ties into a network affects social influence. One recent work by [9] advances this issue further into the more sophisticated level. In contrast to previous work on the structural configuration of networks, the author focuses on how thresholds are arrayed across a network and shows that it can profoundly affect diffusion dynamics. The author, from abstract computational experiments, finds that a balance of similar and dissimilar thresholds is important in maximizing participation. That is, the optimal distribution of thresholds across networks would be a pattern where agents associate with a certain proportion of others with similar threshold values while keeping a certain level of friendship with others of discrepant thresholds. Our work starts from the same motivation of his work, but differs significantly from it in several points. First, we assume a normal distribution of threshold distributions rather than the quasi-uniform distribution adopted by many previous models. Although there has been little evidence on the exact distribution of threshold, it is generally accepted that thresholds are positively associated with the time-of-adoption which is found to be follow a bell-shaped distribution. Second, we consider scale-free network, which has more empirical supports as a stylized model for many real world social networks, but has not been treated much in the network threshold models. Most important, our work takes a more exhaustive approach than the previous works. We consider all the possible threshold arrangements over given network topologies rather than generate networks with intended arrangement patterns. We undertake statistical analysis for these exhaustive enumerations of arrangement patterns and seek to find any patterns that may be related with the diffusion dynamics. III. THE MODEL Consider a process in which a certain product diffuses across a social network of potential adopters. Each adopter in the network makes a decision of whether to adopt the product based on a simple condition; whether the proportion of adopters among his (or her) network peers at time t is above his or her threshold. Previous diffusion research has conceptualized the proportion of adopters among network peers as exposure, which represents a level of peer pressure imposed on a potential adopter. Thus the adoption condition implies that adoption occurs when peer pressure on an adopter exceeds threshold. The decision rule is described in Table 1. In the table, n i is the number of neighbors in the network, θ i the threshold of agent i, and n i_a_a (t) the number of adjacent adopters that have adopted the product A at time t. Unlike the original model, we assume that influence flow among agents is restricted by the structure of a preexisting social network. That is, each agent adopts the product when the proportion of agents who have already adopted among network peers is above his/her threshold. The agents threshold values are assumed to be determined from a normal distribution. We assume that the distribution parameters, μ and σ, are not independent. This is intended to have the fixed proportion of TABLE I DECISION RULE Adoption criteria Adoption result n i_a_a(t) n i_* θ i Adopt A n i_a_a(t) n i_* θ i Don t adopt

3 Fig. 1. Two same networks with different threshold arrangement 0-threhold agents in the population regardless of the threshold distribution. We consider 2.5% of the population as the initial adopters. This percentage is borrowed from the adopter categorization by [1], in which he reportedly says that the category that adopts at the earliest time makes up about 2.5% of the population. That is, diffusion starts from the 2.5% of the population (initiators, hereafter) and stops when there are no additional adopters. The model output, the diffusion extent, is thus the number of adopter when diffusion stops. The diffusion extent is determined from 3 variables; threshold distribution, network topology, and threshold arrangement. Threshold arrangement refers to a way in which agents are arrayed over a given network. It is important to understand that even when threshold distribution and network topology is fixed, the arrangement patterns can be varied substantially and the model produces a completely different outcome depending on the pattern. The following example demonstrates the effect of the arrangement pattern. In Fig. 1, there are two different networks of 4 agents; their thresholds are 0%, 30%, 30%, and 60%, respectively. Although the two networks are of the exactly same topological type, we can see that their diffusion outcomes are completely different diffusion extent of the left is 1, but the right is 4. Notice that although threshold re-arrangement (hereafter rewiring) does not change the network-level topological features, it does changes the micro-structure of how individuals interact with each other. That is, although rewiring does not change degree distribution, network centralization, clustering coefficients, or many other network-level features, it does change individual neighbors profile in terms of their threshold values. That is, the difference in diffusion extent will be caused by the difference in the relative position of adopters in the given network. This means that the factors affecting diffusion should be able to capture the characteristics in how adopters are distributed and positioned in a given network topology and the patterns that individual thresholds are correlated with the topological features. In order to examine the characteristics of the diffusion process, an agent-based model that simulates the specified adoption behavior has been built. For simulation, we first create a population of 1,000 potential adopters and their threshold values were assigned from a normal distribution. We vary threshold distribution by varying its μ from 0.25 to 0.4. For network topology, we considered 3 random networks and 3 scale-free networks with varying average degrees. For each network topology, simulation was conducted on 1,000 different threshold arrangements generated by rewiring the adopters randomly under a given network topological type. Note that only one outcome is obtained from one network setting since the adoption process is deterministic. Fig. 2 shows the effect of threshold distributions on the diffusion extent. The y axis denotes the average diffusion extents obtained from the different arrangement. Expectedly, as μ increases, the average of diffusion extents tends to decrease. When μ is large enough (e.g. μ > 0.4 in case of random networks), the diffusion extents become always 0%. Similarly, when μ is small enough, diffusion reaches 100% regardless of the thresholds arrangement. That is, there is a range of μ in which the diffusion extents can vary depending on how thresholds are arrayed over the networks. Then, we examine how the diffusion extent varies depending on the threshold arrangements when μ is given in the range. Fig. 3 shows the distribution of the diffusion extents resulted from simulation on 1,000 diffusion networks when μ is 0.35 and σ is From the previous graphs, we see that when population thresholds follow a normal distribution with μ=0.35 and σ=0.18, the average extents are about 20% ~ 40% depending on the network types. But Fig. 3 shows that its distribution is clearly bifurcated. That is, even though the population threshold has a fixed distribution, diffusion will result in one of the two extreme cases depending on how threshold are arranged over the given network; almost all the adopters (more than 95%) will participate in adoption or only the small part of them (less than 30%) will do so. It is surprising that there are no such cases that 50% ~ 60% of individuals adopt the behavior at the end of simulation although diffusion networks were rewired randomly. Fig. 2. Diffusion extent according to threshold distributions

4 Fig. 3. Distribution of diffusion extents (θ ~ N (0.35, 0.18)) The finding raises two significant questions. First, why does the diffusion extent always fall into one of the two extreme ranges, that is, above 95% or below 30%? It is certain from the result that once a small portion of adopters (about 20~30% in this setting) participates in the behavior, then all the remaining members will eventually follow them. This finding is consistent with the well-known theory of the critical mass from the literature on social dynamics. The theory postulates that collective actions characterized by population heterogeneity and interdependence, which are well captured in our behavior model, are often activated by the small portion of early movers. From the perspective of the critical mass theory, we can consider the 10~20% early adopters as the critical mass of participants that has to be crossed in order for perfect adoption to occur. Our simulation proves that the small part of the population does lead to the unanimous action, and shows how such behavioral dynamics can be derived from the simple behavior rules. The second question, which this paper lays more focus on, is what factors contribute to perfect diffusion? Regarding this question, it is important to understand that a network s initial topology remains unchanged during the rewiring process. That is, 1,000 diffusion networks, although their diffusion outcomes are completely bifurcated, are almost identical in terms of their topological features. For instance, when a network is given, we can create another network by simply relocating nodes in the given network. Notice that the new network generated this way is isomorphic to the original one, meaning that these two networks are exactly same in every topological feature. However diffusion outcomes on these two networks can be significantly different because the way nodes interact with each other is different in each network despite their topological isomorphism. IV. FACTORS CONTRIBUTING TO PERFECT DIFFUSION So far we have described the diffusion model and investigated its behavior. Although the proposed model is based on a quite simple behavior rule, it generates the results that cannot be easily explained. Why does the diffusion extents are bifurcated depending on the pattern of threshold arrangement? This result suggests that there are a certain proportion of agents that should be crossed over in order for diffusion to reach the whole population. In other words, once diffusion reaches the proportion, it then continues like a self-sustaining chain reaction process leading to the almost perfect adoption regardless of how the remaining agents are arranged. Therefore we need to explain when diffusion can establish the necessary initial contributors. The simulation results suggest that there are particular ways of arranging agents according to their thresholds, which are advantages for establishing the initial adopters sufficiently for precipitating the self-sustaining chain-reaction. There are some important facts that must be related with the role of threshold arrangement in diffusion. First, diffusion starts from the 2.5% initiators so the initial adopters should be located around them. Second, low-threshold nodes are likely to adopt earlier than others so it is advantageous for them to be placed near the initiators. Third, some nodes are topologically critical in promoting diffusion. From these facts, it is supposed that following factors are likely to be associated with promoting diffusion. A. Position of the starting nodes The simulation result shows how initial contributions can precipitate a chain reaction that may ultimately spread to every member of the group [4]. Since diffusion starts from the initiators, where and how they are located in a network should have a significant effect on the diffusion outcome. We assume that it would enhance diffusion if initiators are evenly dispersed over a network to cover a wider area in a cooperative way B. Threshold values of nodes located close to the starting nodes Since diffusion proceeds along the path from initiators, nodes closer to them are considered as candidates for prospective adopters earlier than others. Since exposure must be low in the early time, if they have high thresholds it is unlikely that they become adopters. This should blockade the path from initiators to other nodes in a network and then the overall diffusion process is likely to be deterred. In this respect, we can hypothesize that it would be advantageous that agents with low threshold are located near the initiators, and, similarly, agents with higher thresholds are located farther from them. If a diffusion network is structured in that way, then adopters thresholds becomes monotonically increasing from the nearest initiators to other nodes.

5 TABLE 2 MEASURES FOR A DIFFUSION NETWORK Factors Measures Meaning Thresholds according to a node s distance to the starting node Position of the initiators Thresholds according to a node s centrality M1. Average thresholds of nodes adjacent to the initiators M2. Average thresholds of nodes at 2-edges apart from the initiators M3. Average thresholds of nodes at 3-edges apart from the initiators M4. Average degrees of the initiators M5. Average closeness of the initiators M6. Average betweenness of the initiators M7. Average distances between the initiators M8. Correlation between degree and threshold M9. Correlation between closeness and threshold M10. Correlation between betweenness and threshold Whether early adopters are located close to the initiators? The coverage and distribution of the starting nodes Whether central positions play an expected role of promoting diffusion? C. Threshold values of nodes with high centrality In addition to nodes closer to initiators, central nodes must be also important for promoting diffusion. Central nodes are those that are extensively involved in relationship with others. These nodes have high level of access to others and play an important role of brokering information from one part of a network to the other. Hence, it is important that highly central nodes have low thresholds so that they adopt early and help adoption expand to a wider area. Based on the discussion so far, we consider the following measures for the abovementioned factors. Table 2 provides a brief description of those measures with the equations to calculate them. V. RESULTS Employing factors as independent variables, we performed a Probit regression to examine their effects on diffusion. As a dependent variable, we employ a binary variable P indicating whether the diffusion extent was over 90% or not. Table 3 shows the regression results, which clearly indicate that when the neighbors of initiators (M1) and these agents neighbors (M2) have low thresholds, the extent of diffusion can be significantly enhanced. The tables show that only M1 and M2 were consistently significant regardless of k and μ. This implies that the most significant factor affecting diffusion extents is the average threshold of the agents who are incident to or located closer to the initiators. Clustering of low-threshold agents around the initiators facilitates the formation of groups of early adopters, who spread their influence across the network. This implication pertains to both random networks and scale-free networks. Other variables, by contrast, turned out to be mostly insignificant. Two exceptions are M4 and M8 in some scale-free networks. It is known that in scale-free networks, the degree distribution follows a power law, which means that the vast majority of nodes are those with small degrees, with only a few having relatively high degrees. The highest-degree nodes are often called "hubs", and are thought to serve specific purposes in diffusion. Our results imply that whether hubs have low thresholds or not is a significant factor affecting diffusion extension. VI. CONCLUSION In this paper we considered a generic threshold model of diffusion and explored how it behaviors under the various diffusion networks by using agent-based simulation. Despite its simplicity, the model has some interesting characteristics worth exploring. First, adopters are heterogeneous in that they have their own varying threshold levels. As a result, each adopter behaves differently even under the same level of exposure. Second, more importantly, adopters behave based on local information that comes only from their network partners; the adoption behavior of their network partners in a population. This implies that diffusion will be largely affected by the network structure of who is connected to whom in the given population.

6 TABLE 3 PROBIT ANALYSIS OF DIFFUSION NETWORK Random Network Degree μ Variable Estimate Standard Error Estimate Standard Error Estimate Standard Error Intercept M *** *** *** 5.86 M *** *** *** 9.90 M *** *** M *** 0.07 M ** 0.64 M M *** M ** M M Avg. extent R square Scale-free network Degree μ Variable Estimate Standard Error Estimate Standard Error Estimate Standard Error Intercept M *** *** *** 6.50 M *** *** *** M *** *** 6.70 M M *** 0.19 M M M *** ** M M Avg. extent R square Notes: ***Significant at p<0.01; **Significant at p<0.05; *Significant at p<0.10 These features of individual behavior, heterogeneity and locality, are generally believed to contribute to the unexpectedness of collective behavior considerably. Consistent with this expectation, the simulation results showed that the model always results in the one of the two following states irrespective of the randomly varying diffusion networks; almost all the adopters (more than 95%) will participate in adoption or only the small part of them (less than 30%) will do so. This result suggests the model s behaviors are consistent with the well-known hypotheses from the theory of the critical mass. In addition, we proposed 10 measures capturing the patterns in which individual threshold levels are associated with its topological features in the given network. From the regression analysis, it turns out that the nodes located closer to the innovators play a critical role in promoting diffusion. ACKNOWLEDGMENT This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(mest) (No ) REFERENCES [1] E. M. Rogers, Diffusion of Innovations, 5th edition, New York: Free Press, [2] A Nypan, Diffusion of innovation and community leadership in east Africa, Acta Sociologica, Vol. 13, No. 4, pp , [3] E. Abrahamson and L. Rosenkopf, Social network effects on the extent of innovation diffusion: A computer simulation, Organization Science, Vol. 8, No. 3, pp , May [4] M. W. Macy, Chains of Cooperation: Threshold Effects in Collective Action, American Sociological Review, Vol. 56, No. 6, pp , Dec [5] P. Oliver, G. Marwell and R. Teixeira, A Theory of the Critical Mass. I. Interdependence, Group Heterogeneity, and the Production of Collective Action, American Journal of Sociology, Vol. 91, No. 3, pp , Nov [6] D. Krackhardt, Organizational viscosity and the diffusion of controversial innovations, Journal of Mathematical Sociology, Vol. 22, No. 2, pp , 1997.

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