Complex Contagions and the Diffusion of Innovations: Evidence from a Small-N Study

Abstract

The recent literature on “complex contagions” challenges Granovetter’s classic hypothesis on the strength of weak ties and argues that, when the actors’ choice requires reinforcement from several sources, it is the structure of strong ties that really matters to sustain rapid and wide diffusion. The paper contributes to this debate by reporting on a small-N study that relies on a unique combination of ethnographic data, social network analysis, and computational models. In particular, we investigate two rural populations of Indian and Kenyan potters who have to decide whether to adopt new, objectively more efficient and economically more attractive, technical/stylistic options. Qualitative field data show that religious sub-communities within the Indian and Kenyan populations exhibit markedly different diffusion rates and speed over the last thirty years. To account for these differences, we first analyze empirically observed kinship networks and advice networks, and, then, we recreate the actual aggregate diffusion curves through a series of empirically calibrated agent-based simulations. Combining the two methods, we show that, while single exposure through heterophilious weak ties were sufficient to initiate the diffusion process, large bridges made of strong ties can in fact lead to faster or slower diffusion depending on the type of signals circulating in the network. We conclude that, even in presence of “complex contagions,” dense local ties cannot be regarded as a sufficient condition for faster diffusion.

Appendices

Appendix 1. Ethnographic appendix

Appendix 1.1

In the Jodhpur region, we visited 18 rural villages and 1 urban center (Jodhpur). Some of these villages are inhabited only by Hindu potters (Bisalpur, Jodhpur, Sathin, Pachpadra); some others only by Muslim potters (Sar, Salawas, Rudakali, Tanawra, Sangasni, Ramasni, Rohat, Dhava, Banjara, Boranada Basni); and still others are mixed (Siwana, Mokalsar, Banar, Palasni). The villages’ sizes range from those where only one potter household is present (as in Bipalsur) to larger villages with 40 active potters (such as Banjara, Ramasni, or Pachpadra)—on average, 14 potter households were active in the villages under scrutiny. Among mixed villages, the Muslim/Hindu ratio also differs: in Banar, Muslim potters are in the overwhelming majority (14 versus 1); in Mokalsar, the reverse holds (3 versus 10); Siwana and Palasni present a much more balanced ratio (2[1] Muslims versus 3[2] Hindus, respectively).

Appendix 1.2. Excerpts of interviews with Indian potter

(From interview with BIK in Sar) In 1987, BIK, Talu’s son, had a brother-in-law who used to work in Ahmedabad and who informed him about possible well-paid seasonal work. So, he went to Ahmedabad and worked for a few months in workshops run by Hindu potters. He discovered how to use the kiln there. When he came back, he decided to build one of them because it was less effort and more rapid than the open firings. (From interview with Fakhir in Salawas) FAK was the first one to adopt the kiln in 1987. He saw the kiln when he went to Sar to meet relatives. BIK had just built it. (From interview with BIE in Sangasni) The adoption of the kiln started in Sangasni in 1990 with two potters, BIE and HAN, who had relatives in Salawas. After discussing with CHA, brother-in-law of HAN and cousin of BIE, CHA came to Sangasni to help them to build the kiln. (From interview with RAM in Banar) In 1990, RAM, son of NUR, 20 years old, went to Ahmedabad to sell pots. Over there, he saw the kiln and the Hindu potters explained to him how it worked. However, he did not adopt it even though he could see advantages. In 1996, he went to Salawas for the wedding of the youngest sister of his father. During this social event, he met CHA who agreed to come to Banar to help him build the kiln. (From interview with GAN in Ramasani) The kiln was adopted in 1995 by four households at a time (SAD, MOH, UZI, and GAN). They went to Salawas for a family event. They saw the kiln and found that it was much faster than open firing. They asked IDU from Salawas to come to Ramasani to make the kilns. IDU came, made the kilns, and showed them how to make the firing. He made a demonstration attended by dozen potters.

Appendix 1.3. Marriage rules

Both the Muslim and the Hindu potters are endogamous communities, patrilineal and virilocal (see Saraswati 1979; Ganguli 1983). They practice village exogamy, with the brides usually leaving their natal communities to join their grooms’ households. As a general rule, these brides originate from families which do not have a patrilineal link with the groom’s family—the families the daughter can, and cannot, marry with are called genait and bhaipa, respectively. Despite these common rules, significant differences exist in the marriage rules between the two communities. As for the Muslim potters, they descend, in each village, from a common, well-identified ancestor, who was usually born in one of the neighboring villages—for instance, Boranada Basni is the village ancestor of Salawas (11 km away) which is the village ancestor of Sar (10 km away). Muslim potters preferentially marry maternal first cross-cousins, a general rule in the Indian-Muslim communities (Bittles and Hussain 2000). This preferential marriage rule goes hand in hand with the exchange of women between genait villages, which enables Muslim potters to maintain symmetrical alliances and avoid differences of status between wife-givers and wife-takers. Hindu potters are embedded in a different social structure and have different marriage rules. They belong to three main sub-castes, the Banda, the Purubiya, and the Maru (the latter being present nowadays only in Jodhpur). These three sub-castes are endogamous, implying that there cannot be kinship ties between Purubiya and Banda (and Maru). Each of these sub-castes is in turn subdivided into gotras, or clans, which are exogamous. Several sub-castes can inhabit the same place, but, as a general rule, marriage alliances are made in different villages and sometimes with remote villages where only one alliance will occur (for instance, the brothers of a same family will marry women from different places) (Kramer 1997). Among the surveyed villages, Pachpadra is a good illustration. Two sub-castes, the Purubiya and the Banda, each of them sub-divided into three gotras, co-exist, and a sample of 31 brides we could carefully investigate show that the brides come from 24 different villages, the majority located more than 60 km away. As a consequence, kinship connections within the Hindu community typically involve large marriage networks at a regional level but with no concentration of ties within a given village, and sparse and weak family relationships across villages (Rowe 1960). The absence of regular visits between families, which contrasts with the frequent encounters at family events within the Muslim community, testifies of distended relationships between wife-givers and wife-takers, with no reciprocal relationships and perhaps the desire in some cases for no interference from in-laws (Miller 1985).

Appendix 2. Methodological appendix

Appendix 2.1. Bridge width

To capture the width of the bridges, we implemented (in NetLogo) the following procedure: for each pair of potters i and j who are not directly related, (1) we select the direct (i.e., one-step) neighbors of i; (2) we select the direct (i.e., one-step) neighbors of j; (3) we select the neighbors of the neighbors of i (i.e., neighbors of i at distance 2); (4) we count how many neighbors of the neighbors of i are neighbors of j. This last operation indirectly makes it possible to retrieve the number of links between the neighborhoods of the two disconnected potters because, when two-step neighbors of i are also one-step neighbors of j, only one of the two cases is verified: either the one-step neighbor of i has a link to one of the one-step neighbor of j or the one-step neighbor of i is also a one-step neighbor of j (which is the case of the common neighbor appearing in Centola and Macy’s definition, 2007: 713).

Appendix 2.2. Data limitation—family networks

Complete family trees were directly reconstructed with all interviewees, and, when ties beyond the limits of the surveyed households were mentioned, they were systematically double-checked, sometimes through quick questions, with potters living in the village concerned by these declarations. Our ability to travel within the Jodhpur region over a period of 3 years thus considerably limited observation biases usually present in the mapping of kinship connections (for a deep analysis of these problems, see Hamberg et al. 2011; Roth et al. 2013; Hamberg and Gargiulo 2014). The only simplification we introduced in our analysis is that we did not distinguish between different types of family ties so that, for instance, a cousinship and a brother-in-law relationship are regarded as equally relevant for the circulation of information about the kiln. Our qualitative evidence was indeed insufficient to formulate more specific hypotheses in this respect.

Appendix 2.3. Bridge fraction

Unless we are mistaken, Centola (2015: 1318) did not make explicit how he computed the “fraction of neighborhoods in the network with at least one ‘wide bridge’ to another neighborhood.” More specifically, the problem is how to compute the denominator of this “fraction.” For us, the most coherent way to compute that “fraction” is by making the denominator the potential number of not directly connected pairs of nodes in the network, which corresponds to the total number of possible dyads (N * (N − 1)), minus the number of observed ties in the network. The computation of the total number of possible dyads does not include the usual division by 2 for symmetric ties for the following reason: although the kinship network is symmetric, the width of the bridge between a given pair of not directly connected nodes i and j is not necessarily the same for the two nodes. In particular, as long as the number of neighbors at distance 1 of i and j differs, the node with the smallest number of neighbors at distance 1 is structurally likely to have the narrowest bridge.

Appendix 2.4. Data limitation—advice networks

Compared to our family networks, it must be acknowledged that information about advice is more fragile. The advice networks we reconstructed are a rough approximation of the unobservable influence process to which potters were likely to be exposed. In this respect, let us first note that we did not have access to the complete ego-centered networks (of a randomly selected sample of potters) on several dimensions (for such an exceptional design, see Banerjee et al. 2013). Second, we did not have access to fine-grained time-stamped advice exchanges. For instance, when a potter is described as being invited to give a public demonstration attended by a certain number of potters, as in the excerpt above from Ramasani (see Appendix 1.1), we attribute to this potter the role of main “influencer,” though some of the potters attending the demonstration adopted several months (or years) later, thus suggesting that in fact interactions following the demonstration may (also) have played a role in the decision process. Because of these limitations—which indeed are common in network studies relying on historical or qualitative data (see, for instance, Böhm and Hillmann 2015: 168; Mitschele 2014; Hollstein 2011)—, we decided not to estimate complex multivariate statistical models for network configuration (Robins 2011) and/or evolution (Snijders 2011) or individual-level models with network covariates relating potters’ time of adoption to their positions in the advice network (van Duijn and Huisman 2011; see also Valente 1999: ch. 8).

Appendix 2.5. QAP correlations

Taking inspiration from analyses on multiplex networks (for two remarkable illustrations with respect to marriage and business ties, and advice and friendship connections, see, respectively, Padgett and Ansell 1993, and Lazega and Pattison 1999), we first quantify the proportion of pair of potters for which an advice and a family tie was observed at the same time. Among Indian-Muslim potters, 96% of advice ties correspond to a family tie; among Indian-Hindu potters, 60% of advice ties are also family ties; among Kenyan-Mukurino and other-religion potters (as before considered as a single network because mixed-religion diffusion ties are present), 55% of advice ties are also family ties. Then, quadratic assignment procedure (QAP) correlations were used to test for the robustness of these counts (for a description of this method, see Borgatti et al. 2013: 128–129; Robins 2015: 190–191). QAP correlations for Muslim, Hindu, and, on the other hand, Mukurino and other-religion potters are, respectively, 0.254 (p < 0.001), 0.247 (p < 0.001), and 0.08 (p = 0.07). If one considers the meaning of the p value within the quadratic assignment procedure—that is, the proportion of randomly permutated networks for which one finds the same correlation observed on the empirical networks—, then it seems reasonable to consider that, although the result is slightly less spectacular for Kenyan networks, it is highly unlike that the observed overlap in both social contexts between advice and kinship ties is due to chance or structural constraints only. (QAP correlations were computed through the function “qaptest” in R package “sna” and double-checked through a program we wrote in Java from scratch.)

Appendix 2.6. Simulated time and interaction rate (IR)

In our simulations, one iteration represents 1 day so that 6 months, the temporal unit over which our empirical diffusion rates were computed (see Figs. 1 and 2), amount to 180 iterations. The simulations last for 52 (Indian case) or 32 semesters, the temporal window for which we have empirical data, namely from 1987 (Indian case) or 1997 (Kenyan case) to the second half of 2013. To get the total number of interactions for a given model variant, it suffices to multiply the interaction rate by 180 (the number of days in one semester) by 52 (the total number of semesters over which the simulation unfolds). For simulations for the Indian case, when the interaction rate is equal to 1, 2, 3, or 4, this means that 1, 2, 3, or 4 adopter/potential adopter pairs interact at each iteration. When the interaction rate is equal to 0.25 or 0.5, this means that, at each iteration, there is only a probability of 0.25 or 0.5 that a pair of adopter/potential adopter gets in contact. For simulations for the Kenyan case, a lower number of adopter/potential adopter pairs to be sampled at each iteration is required because the Kenyan potter population is smaller, the diffusion process spans over a shorter while, and pottery only occupies a month a year. In particular, when the interaction rate is equal to 0.005, this means that, at each iteration, there is only a probability of 0.005 that a pair of adopter/potential adopters gets in contact, thus implying that, over a semester (180 iterations), only one interaction takes place. Following the same reasoning, readers can deduce that 0.02, 0.03, 0.05, 0.09, and 0.18 respectively correspond to 4, 6, 10, 16, and 32 interactions per semester.