1.1. Sociological approaches to innovation diffusion
1.1.1 Review of sociological models
The study carried out by Ryan and Gross (Ryan and Gross 1943) about the diffusion of hybrid corn in two Iowa communities is often considered as the starting point of the research on innovation diffusion. Their main problem was to understand why some farmers adopted earlier than others, among all who had an economic interest to adopt the innovation. They tried to capture these differences with variables such as social participation, education, cosmopoliteness, and media consumption which were more highly correlated with the time to adopt than the size of the farm and the length of farming experience. Moreover, the data of cumulative adoption showed the typical S-shaped curve for the diffusion of innovation. This type of curve is explained by the risk of innovation adoption, and the uncertainty about the outcome of adoption. The observation of earlier adopters is a means for reducing this uncertainty. Therefore risk tends to force individuals to turn to their peers to gain more information and/or reassurance about the outcome of the adoption. (Rogers 1983) refers to contagion as the diffusion effect. When the number of adopters increases in the network of peers, the pressure for adoption increases. However, the capacity to resist to this pressure vary among individuals. This led to the definition of the threshold models (Granovetter 1978) . An individual’s threshold is the proportion of adopters in the considered group necessary to convince him to adopt. Originally this model was applied to the case where the group was considered to be seen globally by all individuals (the example of riots). Innovation diffusion theory has been applied to very different subjects : farming innovations, family planning practices, medical technology, policy innovation, language (see (Rogers 1983) for a comprehensive review). A connected concept is the idea of critical mass, which is the necessary number of adopters to propagate the innovation to the rest of the population. The introduction of the network analysis techniques (see (Burt and Minor 1983) for a review) in the innovation diffusion opened new directions of research ((Coleman, Katz et al. 1966) , (Rogers and Beal 1958) ). The network analysis techniques are very important for innovation diffusion because they help to determine who influences whom. The network approach led to refine the models and to define the personal network threshold models, corresponding to the proportion of adopters in the personal network (also called personal exposure) leading to adoption. The exposure can be calculated for direct links, or for indirect links divided by their length using the flow matrix (Freeman, Borgatti et al. 1991) . The exposure at the first level is used to define a local threshold for adoption, corresponding to the proportion of adopters in this one step network which led to adoption. The threshold lag corresponds to the time necessary for adoption when the threshold is reached (in general the adoption does not occur right when the threshold is reached). The more formal definition of the threshold is the point at which the perceived benefits exceed the perceived costs ((Granovetter 1978) , p. 1422). The network approach also led to reconsider the critical mass concept, and connect it to the different versions of the personal network threshold. This research shows that the critical mass varies with the threshold distribution of the population and with the social network structure. Finally some empirical models mix the personal network approach and the system approach, postulating that both influences play a role (Valente 1995) . 1.1.2 Some results or stylised facts found by the sociological approaches
The relational and structural approaches are considered by (Valente 1995) as an important advance in the research, compared to the previous sociological researches. However, some qualitative results of previous studies (classic model), can be of importance to interpret the dynamic models involving more detailed representation of the decision making process. Classic model
The classic model (Rogers 1983) distinguishes five key stages in the innovation diffusion : · Awareness : the agent becomes aware of the existence of the innovation and gets a general · Persuasion : the agent becomes progressively interested by the innovation · Trial : the agent tests the innovation on a small scale · Adoption : actual decision of adoption · Consolidation : The agent seeks reinforcement on his decision or rejects it. In general since the AEMs are adopted through a five year contract, the categories of trial and consolidation seem less relevant in this context. Network approach
Two types of approaches are distinguished in the network approach : relational network diffusion, and structural network diffusion (Valente 1995) . Relational network diffusion focuses on direct ties among individuals. Structural network diffusion focuses on individual’s position in the social structure. In both cases the results are obtained by different statistical calculations applied to data sets of social networks and adoptions. The main results of relational network approaches can be summarised as follows : · Opinion leadership. The opinion leadership is considered in the two-step flow hypothesis model ((Katz 1957) , (Robinson 1976) , (Weimann 1982) ). In the first step of the model, opinion leaders are convinced by the media, and in the second the diffuse the information to their network. The idea is that when opinion leaders adopt, the uncertainty decreases for the others and they tend to adopt more easily. In practice, the leadership is evaluated by the number of nominations sent or received by the individual, indicating the size of his network. The correlation studies between connectedness and innovativeness (meaning early adoption of the innovation), realised on different sets of data (medical techniques, family planning, farming practices) show the following result (Valente 1995) : opinion leaders, measured in their number of connections, adopt innovations relatively early, and then based on the communications with these leaders, other imitate their behaviour and adopt later. However, (Becker 1970) showed that the role of leaders depends on the type of innovation : high potential innovationse adopted earlier by leaders who tend to increase the diffusion, whereas low potential innovation are more likely to be adopted earlier by marginal individuals, and the presence of leaders tends to decrease the level of adoption. · Personal network radiality. The personal network radiality of agent A increases when the people connected to A are not connected together. The empirical results on different data sets show that the radiality is positively correlated with early adoption. 1[1] The potential of the innovation is defined by the proportion of the population which finally adopts it (when the adoption process is totally stabilised). This is not a judgement about its intrinsic interest or value. The structural models focus on the structure of the network and the diffusion, using different network structure indices. In the data sets examined in (Valente 1995) , the correlation is calculated on the different communities involved in each data set. Typical example of structural indices are : · Centrality : Different measures of centrality can be defined. They indicate roughly the ability of an individual to reach the other agents of the network. From the individual measure of centrality, a global measure of centralisation of the network can be evaluated. The empirical results show that centralisation favour the diffusion of innovations which are initially perceived as non risky, but it slows the diffusion of innovations which are initially perceived as risky or not relevant. · Structural equivalence : It measures the degree two individuals have the same relations with the same others. (Burt 1987) used structural equivalence for a reanalysis of the (Coleman, Katz et al. 1966) study on medical innovation. The results show that people with high structural equivalence tend to adopt with the same network exposure. 1.1.3 Limits of the network approach results
The work of (Van den Bulte and Lilien 1999) tends to question some results of the network approach of adoption diffusion. They consider a probabilistic variant of the network threshold model, in which the probability of adoption is a parametric logit function. Classically variables of the parametric threshold are sociologic characteristics of the agent (age, status…).They also included parameters corresponding to the exposure to marketing effort from the companies selling the product. The parameters of this function are empirically evaluated on from the adoption data of medical innovation (Coleman, Katz et al. 1966) about tetracycline diffusion. The results show that in this case, the marketing effort was the most important factor explaining the adoption. The effect of contagion seems negligible, which contradicts the initial studies on this data set. This result invites to be careful about the empirical results found on various studies in which the exposure to marketing effort was not taken into account. On a more general stand point, one limit of these models is that the actors rationality is not explicitly expressed in the model, it is still a kind of statistical black box. The agent based models constitute a very appealing solution to go in this direction. The first step is the simulation approach of threshold networks.

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