The author has declared that no competing interests exist.

Complex innovations– ideas, practices, and technologies that hold uncertain benefits for potential adopters—often vary in their ability to diffuse in different communities over time. To explain why, I develop a model of innovation adoption in which agents engage in naïve (DeGroot) learning about the value of an innovation within their social networks. Using simulations on Bernoulli random graphs, I examine how adoption varies with network properties and with the distribution of initial opinions and adoption thresholds. The results show that: (i) low-density and high-asymmetry networks produce polarization in influence to adopt an innovation over time, (ii) increasing network density and asymmetry promote adoption under a variety of opinion and threshold distributions, and (iii) the optimal levels of density and asymmetry in networks depend on the distribution of thresholds: networks with high density (>0.25) and high asymmetry (>0.50) are optimal for maximizing diffusion when adoption thresholds are right-skewed (i.e., barriers to adoption are low), but networks with low density (<0.01) and low asymmetry (<0.25) are optimal when thresholds are left-skewed. I draw on data from a diffusion field experiment to predict adoption over time and compare the results to observed outcomes.

Central to the study of how innovations diffuse in society is an understanding of the processes by which individuals learn from and influence each other over time [

In this article, I examine how the formation of collective opinions within a community connected by a social network shapes the diffusion of complex innovations. My approach is to adapt the DeGroot model to study the diffusion of innovations for several reasons. First, recent behavioral experiments have shown that real opinion formation processes in social networks resemble the naïve learning assumed in the DeGroot model [

The existing literature that has applied the DeGroot model to opinion dynamics has focused on two primary areas of investigation: (i) how closely the model replicates the process of learning among people in real networks (i.e. sharing opinions and revising one’s personal opinion in light of the opinions of others), and (ii) under what parameters the model converges to an “equilibrium” consensus within social networks. Prior research, for example, has established that the DeGroot model comes closer to replicating opinion formation processes in behavioral experiments than Bayesian models of learning [

My aim in this article is twofold. First, I apply and extend the classic DeGroot model of consensus formation [

The proposed model and approach differ in the central question that they address as compared to existing research. While the majority of existing work examining the diffusion of innovations has focused on why innovations diffuse at different rates

I first describe the proposed model and explain how opinion formation about the value of a complex innovation shapes individual decisions to adopt the innovation within networks over time. I then simulate results from this model on Bernoulli random graphs with varying model parameters to explain how network structure, in particular the extent to which people are connected to everyone else (network density) and the extent to which the direction of influence is asymmetric (network asymmetry) affect the speed and breadth of diffusion. I then provide an application of the model to simulate the diffusion of microfinance, a complex innovation, as a process of opinion formation through the social networks of 43 village communities in India. I conclude by discussing the implications of the results for the study of socially influenced behavior and for individual and collective decision-making in groups, organizations, and markets.

Let us assume that complex innovations have some “true” but unknown value,

Let us now suppose that people differ not only in their subjective opinions, but also in their individual thresholds for adopting, _{i}. Thresholds denote a minimum value for an individual that is required for her to adopt an innovation. Differences in thresholds reflect differences in the perceived value that individuals need to have to be willing to adopt a complex innovation. A person adopts only if and when her perceived value of the innovation exceeds her threshold value,

Over time, people communicate and revise their subjective opinions about the value of the innovation. In the process of revising their opinions, people are socially influenced: they incorporate the opinions of their contacts in a network. For simplicity, let us assume that _{ij} = 1 denotes that actors _{ij} = 0 otherwise. Further, let _{ij}, denotes the weight that person _{ij} ≥ 0 and

Given this initial structure of social influence, let us now define the initial distribution of individual opinions and threshold values at the start of the process. I define _{1}, …, _{n})′ as a column vector of threshold values for adoption for ^{(t)} = (_{1t}, …, _{nt})′ as a vector of adopters at time _{it}(_{i}), then _{it} = 1 (“adopt”), else _{it} = 0 (“do not adopt”).

Let us introduce at this point dynamics into the model. Suppose that people communicate with each other over time and share their opinions about the value of the innovation. As people learn the opinions of others, they revise their own opinions (“social influence”). At time

Collective opinions about the value of an innovation at time

At time

The evolution of collective opinions at each time interval ^{(t)}. Given initial (t = 0) parameters for individual opinions and thresholds and a stochastic influence matrix

Set parameter values for ^{(0)} and

Compute the t-th powers of the influence matrix for each period t = 1, 2, …T.

Compute the adoption rate at time t, _{t}.

An analyst can use these steps to model the rate of diffusion in empirical networks or random graphs. To illustrate how this model can be applied in both cases, I examine diffusion dynamics on Bernoulli random graphs with varying network density and network asymmetry, and varying distributional assumptions for the vector of initial opinions

I begin with simulating influence within Bernoulli random graphs with

Patterns of influence among individuals (nodes) over time represented as networks in which a tie denotes the presence and direction of social influence between two individuals at a particular time.

Patterns of influence among individuals (nodes) over time represented as networks in which a tie denotes the presence and direction of social influence between two individuals at a particular time.

Patterns of influence among individuals (nodes) over time represented as networks in which a tie denotes the presence and direction of social influence between two individuals at a particular time.

To understand how variation in network structure affects diffusion, I conducted a set of simulations varying the structural parameters of the influence matrix. In these simulations, I drew random networks from a dyad census-conditioned uniform random graph distribution using different parameters for network density and network asymmetry. To explore how network density affected diffusion, I varied the tie probability of the network between 0 and 1. When the the tie probability is set to 1, for example, an agent in the network is connected to every other agent in the network (i.e., the network is maximally dense, in other words, it is a complete graph). When the tie probability is set to zero, all agents in the network are independent (unconnected) to other agents, and the density of the network is zero. To vary the asymmetry of the network (i.e., whether there are more relations from

Panel (B) in

To understand the role of collective opinions in the diffusion of innovations, I conducted a second set of simulations, this time varying the distribution of initial opinions. Note that the model allows distributional assumptions about this parameter. Individuals may have initial opinions that follow a normal distribution, in which some opinions occur with higher frequency than others. The distribution can be also be uniform, with different opinions occurring at about the same frequency. Alternatively, the distribution can be skewed, as in a beta distribution. A right-skewed beta distribution (

These distributional assumptions produce different rates of innovation uptake on a Bernoulli random graph, as shown in

Y-axis shows the fraction of nodes (individuals) adopting the innovation.

Panel (C) shows the rate of innovation uptake on a Bernoulli random graph with

Seminal work on the diffusion of innovations has shown that the adoption curve often mimics the lagged rate of information acquisition about a given innovation [

The proposed model differs from the Bass model of population-level new product adoption [

Second, the proposed model also differs in several ways from event-history models of individual adoption in the presence of temporal and spatial heterogeneity [

In relation to prior studies of opinion formation in networks, the results suggest that two parameters regulate the benefits of network structure for diffusion: network asymmetry and network density. Prior research has focused on the role of agents’ eigenvector centrality [

In this section, I provide an application of the proposed model to predict the diffusion rates of microfinance (MF), a complex innovation, among people connected by social networks in 43 village communities in India [

There are several features of this setting that make it particularly interesting for testing the proposed model. First, the villages in this study were similar in many dimensions: in the fraction of “at-risk” adopters, in their location in the same province in India, in their level of economic development, and in their financial need for MF. Variation in MF adoption therefore likely depended on differences in the social processes by which people formed opinions about the value of MF and were influenced to adopt it. Importantly, the intervention involved a 6-month period of information dissemination and opinion formation before MF was introduced. Following this period, MF was launched in each of the villages and MF participation data were collected over 30 months (in 10 quarterly intervals) between 2007 and 2010.

To simulate the diffusion of MF in the villages, I use the social matrices of money borrowing and lending in each village prior to the intervention in 2006. These social matrices provide the structure of social influence by which people could have communicated about money matters and formed opinions about the value of MF. I convert each of the binary adjacency matrices into stochastic influence matrices, where the “weight” of influence for each person is proportional to the number of contacts in her village. For the vector of initial opinions, I ran a logistic regression of the effects of individual demographic characteristics—gender, caste, religion, language, and economic need—on the probability of adopting MF. I then estimated the probability of adopting MF for each individual based solely on individual demographic characteristics. For the individual thresholds, I assumed a vector of zeros for all villagers. This assumption is based on the notion that if people perceived any value from MF at all, they would adopt.

Using these assumptions and procedures, I simulated the evolution of influence and MF diffusion in each of the 43 villages. I then plotted and compared the results from this simulation to the observed diffusion rates in the villages over time. To assess the statistical similarity in the distributions of the simulated and observed diffusion over time, I conducted Wilcoxon and Kruskal-Wallis rank sum tests.

Panel (D) presents the simulated diffusion curve from the proposed model, while Panel (E) presents the actual diffusion curve for this village. Comparing panels (D) and (E), we can see that dynamics of social influence in this village produced a diffusion curve that peaked around time t = 2.3 and tapered at around t = 2.7. Most of the convergence in opinion formation happened between t = 2 and t = 3, as influence became more concentrated. Further, the simulated patterns closely resemble the observed patters of MF adoption. The actual uptake is similar to the simulated uptake, both averaging around 15 percent and peaking at around t = 2.3. Thus, the model appears to closely reproduce both the speed of diffusion and the breadth of diffusion on this network.

Average percent of individuals adopting MF across 43 villages over 10 quarters (30 months) between 2007 and 2010. Visually weighted plots show the density-scaled values of the simulated adoption (

The diffusion of complex innovations—technologies, practices, and ideas that hold uncertain value for potential adopters—depends on the collective opinions that people form about the potential value of these innovations. Networks shape how people communicate, and how they form and change their opinions over time. In this article, I have proposed a model of innovation diffusion as a process of opinion formation through social networks. This model incorporates two stylized facts about innovation diffusion: adoption depends on collective opinions about the value of the innovation, and these opinions emerge from social interaction and social influence. The proposed model takes as its premise that innovations have unknown value when they are first introduced, and that individuals learn from others about the perceived value of an innovation by sharing their opinions and influencing others’ opinions. This process of social learning within networks affects how fast and how broadly a complex innovation diffuses over time.

Simulations of the proposed model on random graphs showed that both individual opinions and the structure of social influence affect diffusion patterns in networks. Individual differences in opinions and thresholds for adoption ensure that some fraction of individuals always adopts an innovation even before it receives any social validation, regardless of whether the innovation is objectively valuable or widely adopted in society. Heterogeneity in opinions means that some individuals will never embrace complex innovations even when prevailing social consensus suggests that the innovation is valuable. Asymmetry in the structure of social influence increases a network’s ability to aggregate atypical opinions and thereby increase learning in the process of reaching a consensus over time. Conversely, density in the structure of social influence appears to produce polarization in opinions, thus preventing learning. In the many cases when individuals do not reach a consensus about the value of an innovation, the opinions of a select few individuals can carry disproportionate weight and limit social learning.

These results have important theoretical and practical implications for the study of innovation adoption over time. The proposed model demonstrates how social influence affects opinion formation about the value of innovations in networks with different topologies. A key insight from the model is that the Bass curve is a special case of possible diffusion curves that can arise in networks. Results from the model reveal that the rates of adoption—and the timing of the peaks and troughs in the diffusion curve—depend critically on the distribution of initial opinions and on individual thresholds for adoption. These parameters thus present important initial conditions that enable faster and broader diffusion.

Practically, the findings show that the process of opinion formation in networks produces diffusion curves that closely resemble empirical patterns seen in real networks. Small changes in the influence structure can give rise to cycles of rapid adoption and abandonment that resemble innovation“fads” and fashions: new technologies that are popular one day and gone the next. Targeting populations of adopters with low thresholds for adoption or high perceived value is thus an important way to promote new technologies, ideas, and social norms. Yet, if these populations are not well integrated in broader, sparser, and more asymmetric networks of contacts, the influence of early adopters will likely fail to produce majority adoption. Future work can extend the proposed innovation diffusion model to cases where adoption thresholds are not externally set but rather change endogenously within a network, along with changes in opinions about the value of an innovation. Further work can also use the model to predict adoption through opinion formation in different types of networks, for example advice networks versus friendship networks, and in networks of different sizes. These extensions can help us to understand whether the observed tipping dynamics in majority adoption within a network depend also on the type of network and the size of the network under analysis.