Emergence of leaders and PP

To capture the influence of PP over the emergence of leaders in social groups, we consider that the pressure that an individual p receives from q deteriorates proportionally with the social distance between p and q. The social distance is captured by the number of links in the shortest path connecting p and q. Mathematically, we model the mobilizing power between two individuals at distance d as Δ d ~ f(d)−1, where f(d) represents a function of the social distance (see Methods equations (11) and (12)). The collective dynamics of the network under peers' mobilizing effects is described by the following generalization of the consensus model

where L d captures the interactions between individuals separated by d links in their social network, Δ d ~ 1/dα where the parameter α accounts for the strength of the PP pulling an individual into the consensus.

We now compare the hypotheses about the random emergence of good leaders—those who significantly reduce the time for reaching consensus in a network—to those in which leaders emerge from the most central individuals. Let us examine the emergence of leadership from five centrality criteria: degree, eigenvector, closeness, betweenness and subgraph (see Supplementary Information equations (S1)–(S5)). In general, we observe that the leaders emerging from the most central individuals are better in leading the consensus than those emerging randomly. However, when there is certain level of PP over the actors, the situation changes dramatically (Fig. 1a, b). First, the time to reach consensus significantly decreases to less than 20% of the time needed when no PP exists. Second, a leader emerging randomly in the network could be as good as one emerging from the most central actors when PP exists in the system. Due to the recent results about the role of low-degree nodes in controlling complex networks19 we have also tested the role of PP over these potential drivers. Our results show again that good leaders emerge regardless of their centrality in the network when PP exists in the system (Supplementary Information). In other words, under the appropriate PP any individual in a social group could emerge as a good leader independently of her position in the network. This result adds a new dimension to the problem of network controllability19,20,21,22 by demonstrating that PP is a major driving force in determining how potential controllers can emerge in the network independently of their centrality (Supplementary Fig. S1) and — in contrast with previous results19,23,24 — of the degree distribution of the network (Supplementary Fig. S2).

Figure 1 Random and centrality-based emergence of leaders. The emergence of leaders is analyzed according to randomness (Rnd), betweenness (BC), closeness (CC), degree (DC), eigenvector (EC) and subgraph (SC) centrality. The peer pressure is modeled by Δ d ~ dα, with α equal to −1.5 and −2.0. The third line corresponds to no peer pressure. (a) Communication network among workers in a sawmill. (b) Elite corporate directors. (c) Friendship network of injected drug users in Colorado Springs. (d) Random network having communities. Full size image

In roughly half of the 15 social networks studied (Supplementary Information) we observe the following anomalous pattern. Leaders randomly emerging in the network are better in leading the consensus than some emerging from the most central individuals (see Fig. 1c). This situation appears when the network has the leaders distributed through diverse communities in the network. A community is a group of individuals who are more tightly connected among themselves than with the other actors in the network25. Actors in one of these communities reach consensus among themselves easily, but it is difficult to reach consensus between different communities. Most central actors in such networks are frequently located in a single community. When they emerge as leaders, they drive consensus only in their community but not in the global network. In contrast, when leaders emerge randomly, they more likely emerge simultaneously in different communities, a situation that favors global agreement in the network. Constructing a random network with communities as illustrated in Fig. 1d corroborates this hypothesis (Supplementary Tables S6 and S7). These results suggest the necessity of considering community leaders in social networks as effective mobilizers of actors throughout the network. We have observed that the leaders emerging on the basis of their community positions exhibit greater success in reaching consensus than those randomly emerging in the network. However, when appropriate PP exists, leaders who effectively reach consensus emerge regardless of their position in their communities.

The leaders in a social group do not always exhibit a high level of cohesiveness. We posit that the leaders' capacity to lead the consensus in a network depends on their divergence of opinions. A cohesive group of leaders can more effectively lead the social group than leaders with larger divergences among their opinions. To model leader cohesiveness we introduce the divergence parameter ∇ L , which is the circumradius of the regular polygon comprising all the leaders. ∇ L = 0 indicates a very cohesive group of leaders. We now examine the influence of the leaders' cohesiveness on consensus. Figure 2 illustrates the results for the friendship network of workers in the sawmill with either no PP (left plots) or with PP modeled by Δ d ~ 1/d2 (right plots). The values of leader divergence range from 0.0 to 0.2. The lack of leader cohesiveness significantly increases the time to consensus when there is no PP. In fact, the time increases more than 33% when the divergence changes from 0.0 to 0.2 (it grows to 80.2% for ∇ L = 0.5, see Supplementary Figs. S3 and S4 and Supplementary Tables S1, S3–S5). In addition, the cohesiveness of the group—measured by the standard deviation at consensus ∇ G —is very poor for large values of ∇ L (∇ G = 154.6, 183.6 and 226.9 for ∇ L = 0.0, 0.1 and 0.2, respectively), which indicates highly heterogeneous group opinions. However, when PP exists, the situation dramatically changes. First, the time to consensus does not increase as drastically with the decrease of leader cohesiveness. Second, group cohesiveness at the consensus is very high even for the lowest leader cohesiveness (∇ G = 27.0, 35.4 and 33.0, for ∇ L = 0.0, 0.1 and 0.2, respectively). In short, when PP is absent, leader cohesiveness plays a fundamental role in the time needed to reach consensus and in group cohesiveness at the consensus. When PP is present, the time needed to reach consensus and group cohesiveness are largely independent of the degree of divergence in the leaders' opinions and the consensus is driven primarily by the influence of the nearest neighbors and PP.

Figure 2 Leaders' cohesiveness and consensus. Analysis of the influence of leaders cohesiveness on the time to reach consensus in the communication network among workers in the sawmill without (left plots) and with (right plots) PP. The leaders' divergences used are: 0.0 (top), 0.1 (middle) and 0.2 (bottom). The time to reach consensus (in blue) relative to a total time of 1,500 units (Insets). Full size image

Diffusion of innovations and PP

Another area that has received great research attention is the diffusion of innovations26,27,28,29. The diffusion of innovations refers to the process through which new ideas and practices spread within and between social groups. Here we consider the hypothesis that PP plays a fundamental role in innovation adoption or rejection. To test our hypothesis, we study two datasets in which diffusion of innovations was followed for different periods of time (Supplementary Information). The first study analyzed the diffusion of a modern mathematic method among the primary and secondary schools in Allegheny County (Pennsylvania, USA). Results revealed that innovation diffused through the friendship network of the superintendents of the schools involved. The study was followed for a period of six years, 1958–1963. The second dataset represents the second phase of a longitudinal study about how Brazilian farmers adopted the use of hybrid seed corns, examining personal factors influencing farmers' innovative behavior in agriculture. We consider here the social network of friendship ties and the cumulative number of adopters of the new technology in three different communities of the Brazilian farmers study (Supplementary Fig. S5). The study was conducted over the course of 20 years and we consider only the individuals in the largest connected components of the networks.

Figure 3 depicts the number of actors that adopted the respective innovations at different times. These values correspond to the number of adopters observed empirically in field studies. To simulate the process of innovation adoption, we study the consensus dynamics with equation (2), assuming Δ d ~ dα: no PP, moderate PP (−6.0 ≤ α ≤ −5.0), high PP (−4.0 ≤ α ≤ −3.0) (see Supplementary Information). The simulations follow perfect sigmoid curves, as Fig. 3 illustrates. Observe that when there is no PP effect, the diffusion curves predict slower rates of adoption than those empirically observed. For example, the empirical evidence demonstrates that 50% of schools adopted the new math method in roughly three years, whereas the simulation without PP predicts a period of four years of a total of six years. In the case of the Brazilian farmers, the empirical time for 50% of the farmers to adopt the innovation is roughly 12 years, whereas the simulation without PP predicts 16 years of a total of 20 years. When the model uses strong PP, the diffusion curves display very rapid adoption rates, which are far from the reality of the empirical evidence in both cases. However, using a moderate PP predicts very well the outputs of the empirical results in both studies. These PP values are found by a reverse engineering method, but the important message is that a certain PP level is necessary to describe the empirical evidence on the diffusion of innovations in social groups (see also Supplementary Information).

Figure 3 Diffusion of innovations under PP. (a) Adopters of a new mathematical method among US colleges in a period of 6 years. (b) Adopters of the use of hybrid seed corns among Brazilian farmers for a period of 20 years. Experimental values are given as stars and the simulation with no (broken red line), moderate (continuous blue line) and strong (dotted green line) PP are illustrated. Full size image

These results demonstrate that interpersonal communication alone cannot sufficiently explain the process of innovation adoption in a social group. The pressure exerted by the social group plays a fundamental role in shaping this important social phenomenon. Our model describes effectively PP's role in these and other important phenomena, consistent with our intuition and with the existing empirical evidence.