A New Bass Model Utilizing Social Network Data

A New Bass Model Utilizing Social Network Data

Tae-Hyung Pyo School of Business State University of New York at New Paltz Email: [email protected]

Thomas S. Gruca, Gary J. Russell Tippie College of Business University of Iowa Emails: [[email protected], [email protected]]

May 2017

A New Bass Model Utilizing Social Network Data

ABSTRACT Most models for the diffusion of innovation are deeply rooted in the work of Bass (1969). The Bass model, however, assumes all potential customers are linked to all other customers in a social network. In this research, we develop the Network-Based Bass model (NBB), a generalization of the Bass model that incorporates observed individual-level social network information. Based upon simulation evidence, we show that calibration of the classical Bass model on sparsely-connected social networks results in biased estimates of diffusion parameters: social influence (q) in the Bass model is biased downward, while external influence (p) is biased upward. In contrast, we show that the diffusion parameters of the NBB model are accurate regardless of the network structure. Moreover, the NBB provides superior fit compared to the traditional Bass model. We apply the NBB model to song adoption data from an online social network. Substantively, we provide strong evidence that the traditional Bass model underestimates the effects of social influence, particularly in commonly-observed sparse social networks.

Keywords: Bass model, Social Network, Diffusion of Innovation, Social Influence, Network Sampling

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INTRODUCTION The Bass (1969) model has been used extensively and globally to forecast first purchases of new products. It has been named by INFORMS as one of the Top 10 Most Influential Papers published in the 50-year history of Management Science. Most models of diffusion in marketing science are based upon the Bass model. Bass (1969) considered word-of-mouth (i.e., social interaction) to be a key driver of the diffusion process, a view that has been confirmed in myriad empirical studies (Goldenberg et al., 2001, Mahajan et al., 1990). The tremendous growth of online social networks has presented new opportunities for consumers to influence each other via word-of-mouth. Companies often attempt to motivate word-of-mouth promotion (often called “viral” or “buzz” marketing) using social media as the communications platform. These campaigns can be unusually effective due to the high level of trust that consumers place in recommendations from people they know (Nielsen, 2012). In this study, we attempt to blend these two streams of research: aggregate models of diffusion and social network structure. Network Structure and the Diffusion Process The network effect of word-of-mouth promotion can be especially important to the success of new products (Gupta and Mela, 2008). Moreover, the structure of the social network can have a large effect on the adoption pattern of a new product (Dover et al., 2012, Goldenberg et al., 2009, Hill et al., 2006, Katona and Sarvary, 2008, Katona et al., 2011, Muller and Peres, 2017, Newman et al., 2006, Shaikh et al., 2010). Two recent studies (Delre et al., 2006, Shaikh, Rangaswamy and Balakrishnan, 2010) model diffusion in scale-free networks, a network topology commonly associated with online social networks. It is now clear that different types

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of network structures yield predictable differences in aggregate diffusion curves (Dover, Goldenberg and Shapira, 2012, Trusov, Rand and Joshi, 2013). The availability of individual level data has spurred new interest into the previously unobserved micro-level process of adoption. The pioneering work of Chatterjee and Eliasberg (C-E) (1990) modeled the effect of individual characteristics on the micro-level diffusion process, reflecting the heterogeneity among individuals. The micro-process is then aggregated to predict the diffusion curve. Social network data is not necessary in the C-E model due to strong assumptions about the nature of social interactions among individuals. More recently, agentbased (AB) models have been used to study individual-level adoption processes. This has led to an examination of the impact of various network topologies on aggregated dynamics of diffusion (e.g., Rahmandad and Sterman 2008). Using the AB approach, researchers posit a model structure and then use simulation to examine the properties of the network. Network-Based Bass Model Because social network structure impacts the diffusion pattern, there have been recurring calls for the development of micro models of social influence that lead to Bass model generalizations when aggregated to the market level (Mahajan et al. 1990; Peres, Muller and Mahajan 2010). Marketing science research on social networks and the diffusion of the innovation literature have not yet provided the necessary tools to achieve this goal. Typically, information on individuals’ networks and behavioral data are aggregated to summary measures, such as the average or variance of the degree distribution (Delre, Jager and Janssen, 2006, Dover, Goldenberg and Shapira, 2012). These studies ignore the individual network and adoption data that may be available to the analyst. At best, they provide limited insights into the role that social network structure plays in the diffusion process. 4

In this research, we develop a generalized Network-Based Bass (NBB) model that allows the analyst to fully utilize all of the social network linkage data available directly in the model estimation. The NBB model incorporates individual-level network information into the original Bass formulation, while preserving its parsimonious form and ready interpretability of parameters. The basis of the aggregate NBB model is an individual-level adoption process. The individual level model leads to the classical Bass model when all consumers are linked to all other consumers (completely connected network). However, the NBB leads to a wide variety of generalized Bass models for other types of social network structures. Most notably, the NBB model yields accurate measures of the strengths of innovation and imitation processes even when social networks have a low density of connections. Such sparse networks are typically observed in empirical research. Overview of Research The goal of this research is to develop a generalized Bass model that takes into account the individual-level social network data. We begin by proposing a micro-model of the adoption process that models individual’s heterogeneity of adoption probability. Heterogeneous social influence is determined by each person’s different connection with other network members. By aggregating across customers, we obtain the Network-Based Bass (NBB) model, a generalized Bass model that adopts the diffusion curve to the network structure. We then compare the NBB and Bass models using simulated data. This analysis supports the notion that Bass model estimates of social influence can be severely biased in commonly observed sparse networks. An application of the NBB model to song adoption behavior in an online panel confirms the simulation findings and provides clear evidence for the superior measurement properties of the

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NBB model, relative to the traditional Bass model. The last section of the paper summarizes our key findings and offers suggestions for future research. NETWORK-BASED BASS MODEL Our model is based upon a micro-level adoption process which implicitly defines the concepts of innovation and social influence. By aggregating the process over consumers, we obtain a new diffusion model, which we call the Network-Based Bass (NBB) model. In this section, we derive the NBB model and compare it to the traditional Bass model. This analysis reveals that the traditional Bass model is a special case of the NBB under certain assumptions regarding the micro-level process and the social network. However, the NBB is a considerably more flexible model that can accurately reflect the imitation and innovation parameters for a wide range of social network topologies. Individual Adoption Process and Heterogeneous Social Influence The NBB model rests on a number of key assumptions about the consumer’s adoption decision. We assume that adoption is a stochastic process, conditionally independent over time, in which both innovation and imitation (social influence) processes operate simultaneously. Specifically, we assume that probability of consumer i adopting at time t is given by

sit  Bernoulli  p  q  rit  where i  a non-adopter sit  1 when a non-adopter i becomes an adopter at t, otherwise = 0 p = coefficient of innovation

(1)

q = coefficient of imitation rit 

# of i's friends who adopted as of t # of all of i's friends in the set of ultimate adopters

The binary random variable sit follows a (non-stationary) Bernoulli distribution dependent on the 6

number of the consumer’s friends who have previously adopted the product. Two consumers are considered friends if they have a direct (first-degree) connection within the social network. The model captures innovation and imitation effects through the constant (across consumer and time) coefficients p and q . Similar to the Bass model, the p coefficient captures the tendency of the consumer to adopt the product independently of others. The imitation coefficient parameter, q , measures the impact of social influence (also referred to as the wordof-mouth effect), which is proportional to the cumulative number of adopting friends. In other words, the value of q indicates how much of an increase in the adoption probability is attributed to social influence from friends who have already adopted the product. The increasing probability due to social influence or pressure q  rit varies across time and across individuals. Therefore, the social pressure on a non-adopter is heterogeneous. This individual-level model varies from the Bass model in several respects. In the Bass model, it is implicitly assumed that social pressure or social influence rit is constant across all non-adopters at given time period (because all consumers are friends of all others in the network). Stated differently, the Bass model assumes a homogeneous influence of adopters on nonadopters. In contrast, our model isolates social influence within the personal network only, not the entire network. The non-adopter adopts faster as she is exposed to a higher proportion of adopter friends among all of her friends. That is, rit measures the proportion of adopter friends. This assumption -- that the influence of one adopter friend will be different for an individual with a few friends relative to one with many friends -- is consistent with the literature on influences on individual level adoption behavior (Granovetter, 1973, Katona, Zubcsek and Sarvary, 2011). Aggregation of the Micro Process

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To understand the relationship between the NBB model and the Bass model, we next aggregate the micro adoption process. We assume that the events of being an adopter at the current time are independent among non-adopters. We also assume that each consumer obeys

the non-stationary Bernoulli process discussed earlier. Accordingly, the aggregate number of adoptions S t may be viewed as a draw from a probability distribution of the number of successes in n binary outcome trials with different success probabilities (  1 , 2 ,..., n ). In this notation, n refers to the size of the non-adopter population at particular time point. This is known as a Poisson-binomial process (Wang, 1993). The expected number of successes from the Poisson-binomial distribution is the sum over n

all different probabilities (     i ). Given our stochastic assumptions, it is clear that i 1

 i  E ( sit  1)  p  q  rit . Therefore, the expected number of new adopters over all non-adopters at t becomes:

E  St    E  sit  1 iA

where A= i | sit 1  1

   p  q  rit    p   q  rit  p 1  q  rit iA

iA

iA

iA

i A

(2)

 p  m  Yt   q  rit iA

where m = total number of adopters during time t St  total number of new adopters at t Yt  cumulative adopters at t The summation of rit over the whole market can be expressed in a simpler form using matrix notation. We represent the social network of consumers using an adjacency matrix, a 8

concept developed in the social network and spatial literatures (Bradlow et al., 2005, Jackson, 2010, Leenders, 2002). An adjacency matrix, C , is defined as a symmetric square matrix in which each element represents connections between all possible paired individuals in the network. More formally, each element of C is either 1 or 0 where a connection between consumer i and j is denoted by setting cij = 1. For technical reasons, the diagonal of C is set to zero. Using this notation, we show in the Appendix that

r i A

it

 a tC* 1m - at 

T

1 if i has adopted as of time t where at is 1 m row vector and ait  0 otherwise 1m is 1 m row vector with all entries equal to 1, and C* is the column standardized version of the C matrix1. This immediately leads to a macro representation of the micro diffusion process outlined above: T E  St   p  m  Yt   q at C* 1m - at    

where E  St  denotes expected sales. We call this formulation the Network-Based Bass (NBB) model. To calibrate the model on particular dataset, we modify the Network Based Bass (NBB) model to incorporate a random error process as T St  p  m  Yt   q at C* 1m - at     t  

where p and q are unknown parameters. We assume that the error terms  t are distributed

1

C* =Column-standardized matrix of C, where cij* 

cij

c

ij

j

9

(3)

independently over time as N  0,  2  . Accordingly, given the full network information of all ultimate adopters, we can use Ordinary Least Squares (OLS) methodology to estimate innovation and imitation parameters (p and q). It is possible to adopt more sophisticated calibration methodologies for the NBB model by making alternative assumptions about the error process (see, e.g., Srinivasan and Mason 1986). However, as we show in our subsequent empirical work, we obtained very good results using the OLS approach. Relationship Between the NBB and Bass Models

From a conceptual point of view, it is important to understand the relationship between the NBB model and the classic Bass model. There is no difference between the models in terms of model parameter interpretation. The Bass model specifies two key processes that drive new

Y  adoptions St  p  m  Yt   q  t  m  Yt   , namely innovation (p) and imitation (q). Assuming m  the same number of individuals in the market (m) as the Bass model, the NBB model uses the same concepts and the same parametric structure (p and q). However, as noted earlier, the key difference between the NBB and Bass models is the assumed structure of the social network. Clearly, the simplest network possible is a completely connected network. More formally, suppose all ultimate adopters in the market size of m are connected. Then, for any Yt the social interaction term in the Bass model, Yt  m  Yt  , can be expressed as Yt  m  Yt   a t CF 1m - a t  where CF is the m  m adjacency matrix for a fully T

connected network. That is, CF has zeroes on the diagonal and ones elsewhere. Next, we column standardize the adjacency matrix CF and denote it as C*F . Then

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C*F 

1 CF , since the sum of the column is the same across all columns of CF , m  1 2. Using m 1

this notation, we show in the Appendix that for large m T Y  E ( St )  p  m  Yt   q a t C*F 1m - a t    p  m  Yt   q  t  m  Yt     m 

(4)

In other words, the Bass model implicitly assumes a fully connected network among adopters and non-adopters. Clearly, real world social networks contain people who are not connected to all other consumers. For this reason, we may safely assume that a t C*F 1m - a t  is generally not equal to T

a t C* 1m - a t  , where C is the observed adjacency matrix in a social network. From the T

expression above, we can infer that estimates of the imitation coefficient (q) from the Bass model is likely to be biased downward in empirical applications. That is, it is very likely that the classical Bass model underestimates the amount of social influence in the diffusion process. In this next section, we use simulated data to test this conjecture. Network Density and the Bass Model

The density of a network is defined conceptually as the extent to which consumers are well-connected to others (Trusov et al. 2013). To analyze the potential biases found in Bass model parameter estimates, we simulate networks of various densities and then calibrate both the Bass and the NBB models on the simulated data. Results are consistent with the notion that the Bass model generally understates the amount of social influence in a real world social network. Based on our assumptions on the adoption process, we generate micro-level adoption

1 CF since ait 1  ait   0 for all i by its definition (i.e., same m individual cannot be an adopter and non-adopter at the same time).

2

If we redefine Cii  1, then C*F 

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data by the following steps. First, we set the true parameters for average density3, m , p , and q . Then, we generate a random network graph for m individuals with the given density. For individual random adoption, two adoption processes are applied to each non-adopter simultaneously for each time period t. For the given p and q  rit 1 4 , the probability of the nonadopter adopting innovation at the current time t is p  q  rit 1 . After both processes are applied simultaneously to all non-adopters at each time periods, we obtain at and compute St . In all our simulation experiments, the parameters of the Bass model are estimated using a non-linear least squares methodology detailed in Srinivasan and Mason (1986). For the NBB model, we use OLS methodology corresponding to the model presented earlier, namely T St  p  m  Yt 1   q at-1C* 1m - at-1     t  

(5)

In the first set of results, we vary the average density of the networks. For the high density networks, the average number of connections (avg.conn) was 8. For the low density networks, the average number of connections was 1.1. The diffusion parameters (p and q) were fixed. The value of p was fixed at 0.048 which is within the typical range of values reported in past research (Lilien, Rangaswamy and De Bruyn, 2007). The value of q was fixed at 0.6 which is also within the typical range of value reported in past research (Lilien et al., 2007). The network size (m) was set at 10,000. We report the results in Figure 1. [Figure 1 about here] In the high density network, the diffusion pattern follows the typical shape, gradually

3

4

For an undirected network, density  Note that rit 1 corresponds to

1T C 1 m( m  1)

where cij  (i, j )th element of matrix C .

Yt 1 in the discrete analog of the Bass model. m 12

increasing to the peak and decreasing adoptions after the peak. The diffusion curve is symmetric, indicating that our individual Bernoulli trial assumption replicates the macro-level diffusion pattern. For high density networks (left panel), the two models (Bass and NBB) fit the data equally well. The average estimate for q from the NBB model is almost equal to the true value (0.599). There is no meaningful difference for the estimates of from the Bass model (0.595). The estimate of p for the NBB is very close to the actual value (0.0488). The estimates obtained from the Bass model are slightly lower at 0.03. The effects of low network density are shown on right side of Figure 1. The fit of the Bass model is very poor early in the adoption cycle. In contrast, the fit of the NBB model closely follows the actual adoption curve. More important, the fitted Bass model suggests that there is no appreciable effect of the social influence on adoption. This is illustrated by a monotonic decrease in adoptions after a peak first period (Bass 1969). However, the actual peak in these simulated data is in the sixth period. The estimate of q for the Bass model is 0.01 which is substantially smaller than the underlying parameter (0.6). It is the underestimation of this parameter which leads to an incorrect shape of the diffusion curve and an incorrect prediction of the peak period for adoptions. In contrast, the NBB model recovers the true parameters well and correctly identifies the peak period of adoptions. Its estimates of q (0.587) and p (0.048) are almost identical to those driving the micro-level diffusion process. In the second analysis, we varied the density of the networks and the innovation parameter q. The density of the network (avg.conn) varied from an average of 0.1 to 30 connections. Each network (varying by density) was generated five times. Parameter estimates were obtained from each iteration. The value of q was varied from 0.01 to 0.6 which is the range 13

of the typical estimates reported in the literature. The size of the network m was fixed at 10,000. The imitation parameter p was also fixed at 0.021. As before, each network was simulated five times and the mean results are reported in Figures 2 and 3 with dotted lines representing true parameters of p or q. [Figure 2 about here] [Figure 3 about here] In Figure 2, the NBB model recovers the true p values for all density levels. For the Bass model, the estimates of p depend on the network density and values of q. For lower density networks (averaging fewer than 5 connections), the estimated p is higher than the true value and the deviation is worse as the innovation parameter q rises. For higher density networks, the estimate of p is best when the influence of the social network is low (q is small). Otherwise, the estimate of p is consistently too low. The estimates of q are generally unaffected by network density for the NBB model except at the very lowest levels. Depending on the true value of q, the estimate from the Bass model may be too low or high depending on the value of q and the density of the network. For lower levels of q, the estimates are consistent and reach an asymptote as the network density increases. However, for high levels of q and high density networks, the estimated values of q from the Bass model are biased upwards. Our results show that the estimates provided by the Bass model are highly dependent on the density of the network and the true innovation parameter (q). Furthermore, the estimated levels of p and q are not consistently higher or lower than the true values. Therefore, the analyst cannot simply deflate or inflate the estimated parameters to account for variations in network density or the underlying innovation parameter. In contrast, the results in Figure 2 and Figure 3 14

show that estimates provided by the NBB model are not affected by the level of the innovation parameter (q) or network density. The NBB model accurately correctly recovers the true parameters regardless of network density or the nature of the micro-level diffusion process (p and q). Calibrating the NBB Model: The Role of Social Network Sampling

In the simulation above and the Bass model (1969), the model is calibrated on a complete set of information from the network of customers. However, in practice, the required information from the entire network of consumers might not be available. For this reason, considerable research has focused on the role of network sampling methods on the accuracy of measurements of the amount of social interaction (Ahn et al., 2007, Chen et al., 2013, Ebbes et al., 2008, Gjoka et al., 2009, Lee et al., 2006, Leskovec and Faloutsos, 2006). In this section, we evaluate two alternative sampling methods: random node sampling (RNS) and snowball sampling (SBS) to ascertain which method better recovers the micro-level diffusion parameters (p and q) in Bass model applications. To increase the generality of our results, we evaluate the efficacy of these sampling approaches for two different network topologies: random networks and scale-free networks5. In a random network, there is no immediately identifiable hub or locally heavy cluster of highly interconnected individuals. This is due to there being little variation in the average number of linkages across network members. In a scale-free network, the number of connections of a given consumer follows a power law relationship. This yields a network in which a few

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In a scale-free network, the number of connections follows a power law distribution. That is,

P  # of connections of person i  k   k   where  is a parameter for power-law distribution

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consumers have many connections, while most consumers have few connections. That is, the network has a few clusters of high-density connections. Scale-free networks are of increasing interest (Barabási, 2009) since many social networks exhibit power law connection patterns (Barabási and Albert, 1999). This is especially true of online social networks (Mislove, Marcon, Gummadi, Druschel and Bhattacharjee, 2007). In RNS sampling, a sample is randomly drawn from the entire network. In SBS sampling, we first select one seed sample and recruit this seed node’s acquaintances (first layer of sampling), then further recruit seed node’s acquaintances’ acquaintances (2nd layer of sampling) and so on until we reach the predetermined sample size. For the simulation study, we fixed the population size at 20,000 and used fixed diffusion parameters (p and q) set at 0.025 and 0.40 respectively. We varied the sample size from 1% to 31% of the population. In the random networks, we vary the average number of connections in the network. In the scale-free networks, we vary the degree of the inequality in the distribution of connections. This level of inequality is determined by a power law parameter γ (see footnote 5) which we varied from 0.1 to 3. Scale-free networks with a low value of γ have a distribution of connections similar to a random network. In contrast, for high levels of γ, the connections in the network are highly concentrated in a small number of individual members. Each sampling with different sample size and method has been done 10 times. Here, we set the marketing size (m) as the observed number of the adopters in the samples by the end of time period (t=40) for the Bass model and the sample size for the NBB model. The diffusion parameters were estimated using the Bass and NBB models calibration methodologies discussed earlier.

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For the random networks, we first examine the effects of sample size, sampling method and model formulation (Bass vs. NBB) on the recovery of the true parameters. The results for p are presented in Figure 4 and the results for q are presented in Figure 5. [Figure 4 about here] [Figure 5 about here] For recovery of the innovation parameter p, the Bass model results are biased regardless of the sample size, network density or sampling method. For the NBB, random node sampling yields poor results for smaller samples. However, the results for snowball are consistently good no matter the sample size or network density. For the imitation parameter q, the estimates from the Bass model are good if the network is sufficiently dense, regardless of the sample size. For the NBB, random sampling yields poor results except for the most dense networks and highest sample sizes. On the other hand, the estimates for snowball sampling are very close to the underlying parameters except for the least dense networks. For the scale-free networks, we fixed the size and density of the network but varied the concentration of linkages. The resulting estimates organized by sampling method and model formulation are presented in Figure 6 and Figure 7 for the p and q parameters respectively. [Figure 6 about here] [Figure 7 about here] For the Bass model, the estimates for p are biased downwards as the network becomes less like a random network and there is greater inequality in the distribution of connections among network member (as γ rises). This occurs regardless of the sampling method. For the NBB model, the random node sampling approach yields poor results regardless of sample size or 17

network concentration (γ). However, for snowball sampling, the estimate is very close to the true value regardless of the sample size or network concentration. The results for the estimates of the innovation parameter q from the Bass model show almost no pattern after the network concentration rises above 1.0. For more highly concentrated networks, the estimate from the Bass model has no discernable relationship to the underlying parameter regardless of sample size or sampling method. The same is true for the NBB model when estimated using a random sample. In contrast, the parameters estimates for q from the NBB model are very close to the actual value when using snowball sampling. The fit is better for larger samples and for more highly concentrated networks (higher γ). Summary

Taken together, our results show that the NBB model is a robust generalization of the classical Bass model that takes into account different types of network structures. When the entire social network is available for analysis, the NBB model provides accurate measures of social influence regardless of network density. When only a sample of network connections is known, the NBB retains its ability to correctly recover micro-process parameters if snowball sampling (SBS) is used. Moreover, the NBB model is essentially identical to the Bass model when the social network resembles a completely connected network (high network density). Thus, the NBB provides the researcher with a flexible method of developing aggregate diffusion models that extends the conceptual framework of the Bass model in a natural way. APPLICATION To assess the robustness of the NBB model, we compare the NBB and Bass models in a real world setting. We compare the fit and parameter estimates of the Bass and NBB models for a dataset of adoptions for new songs released in 2011-2012. Our results provide strong support 18

for the NBB model. Data Description

Utilizing a web-crawling agent, we collected data for individuals’ music listening behaviors and friendships from the online music social network, Last.fm. The site publicizes friendship information on each user’s personal webpage. Most user track histories are also publicly accessible. The software developed by Last.fm, “Scrobbler,” extracts users’ music library histories from their electronic devices, such as cellular phones, iPods, and tablet PCs. The site also tracks web-streaming records. The entire library histories from these difference sources are available on user websites. The track history for each song includes title, performer, and a time stamp. Thus, we were able to determine the date when a user listened to a new song for the first time. The performer name and title identify a specific song within a user’s library. If a user’s track history does not include a given song, that user is excluded from the network of ultimate adopters of the song. Due to the large size of the Last.fm social network, we used snowball sampling to direct web-crawling algorithm. For a sample of 10,000 users, we collected 4 years of their music library history data from Jan. 2011 to Dec. 2014. New singles that were on Billboard’s Top 100 in 2011 and 2012 and released after Jan. 2011 were selected to this analysis. Songs with fewer than 500 ultimate adopters were eliminated from our sample as were those whose estimated Bass model coefficients were infeasible (e.g., negative innovation coefficients). A total of 25 newly released songs were analyzed in this study. Empirical Results

In sampling the social network, we treated each subsequent 4-week interval as one 19

observation period. The average number of connections in the networks of ultimate adopters (across all songs) is 2.3. This implies that the social networks of the Last.fm users are quite sparse. As noted earlier, low density networks are typical in online social network applications. Given our simulation results, we expect the NBB model to perform much better than the Bass model for these data. Estimates from the NBB and Bass models are summarized in Table 1. Table 1 about here To assess relative model fit, we first computed the mean squared error (MSE) for the Bass and NBB models. We then computed a relative fit measure6 which is 0 if the two models fit equally well. It is positive (negative) if the NBB (Bass) fits better. For all but two songs, the fit of the NBB model is superior. A sign test based upon these results favors the NBB model over the Bass in terms of model fit (p < .001). Since the social networks of ultimate adopters are sparse, we expect that the Bass model will have much lower estimates of the imitation parameter (q). This is borne out in our results. We report the ratio of the two estimates with the q from the NBB model in the numerator. The average ratio is 3.9 and only one ratio is less than one. Combined with the poorer overall fit results, the lower estimates of q from the Bass model suggests that using the traditional formulation will greatly underestimate the actual magnitude of social influence (i.e. the social networks) on new product adoptions. To illustrate the importance of these findings, we plot the actual and fitted results from two songs in Figure 8. The NBB model correctly captures the existence of social influence

6

Bass MSENBB 

MSEBass  MSENBB 100 where MSEBass = MSE from the Bass model MSEBass 20

(when it is not strong) whereas the Bass models do not. The fitted lines from the NBB model indicate that there are peaks at the early stage of these two songs. The Bass model, in contrast, indicates that sales never or only minimally increase after product launch suggesting that there is no evidence of social influence. We particularly interested in this finding since the word-ofmouth effects is an important determinant of new product success, especially in early stages of a new product’s lifespan (Gupta and Mela, 2008). This is a key issue since measuring the effect of social interaction is one of the major reasons Bass developed his model (Bass 1969). [Figure 8 about here] Sales Peak Forecasts

Obtaining an accurate estimate of when the initial sales of a new product will peak is a critical input to new product planning (Bass, 2004). The estimates of sales peak, t * 

ln q  ln p pq

(Bass, 1969), depends on accurate estimation of p and q. The Bass model provides a forecast estimate of peak sales when the estimated q is greater than p. If the estimate of q is biased downwards, as is the case with the Bass model applied in sparse networks, the peak period will be forecasted to be too early. In some cases, there will be no peak beyond the first period. For the 25 songs in our empirical analysis, the NBB model predicted 24 times that sales peak occurred after the first time period. For the Bass model, the forecast for the peak time for 6 songs was less than 1. A key question is whether these empirical findings based on song adoption data can be considered to be typical. Using simulated data on random networks, we calculated how often the Bass and NBB models yielded a negative or no sales peak ( tˆ*  1 or no estimates for q or p). There were 9,315 out of 10,000 sales simulated diffusion curves wherein the sales peak occurred after first time 21

period. The Bass model predicted a first period peak 1,139 times, while the NBB model predicted a first period peak 507 times. For the Bass model, the likelihood of such errors was significantly related to the density of the social network. There was no such pattern for the NBB model. For scale-free networks, we simulated adoption over 40 time periods with γ ranged from 0 to 3, average connections ranged from 1 to 31, and true q values ranges from 0.05 to 0.6. Out of all 1,792 simulated adoptions, peak sales occurred after first time period 1,786 times. Some 66 of peak estimates ( tˆ* ) from the Bass model are smaller than 1 (including negative or estimation is impossible since qˆ  0 ), while only one peak estimate from the NBB model is smaller than 1. Again, the NBB model performed better than the Bass model in predicting the diffusion curve peak. Obtaining accurate estimates of the innovation (p) and imitation (q) parameters has implications beyond generating a forecast the time period when new product adoptions will peak. One of enduring advantages of the Bass model is its simplicity and the large number of parameter estimates (p and q) available in the published literature and in the archives of consultants. Using estimates from “similar” products, a new product planner can generate periodby-period estimates of initial sales using only a few inputs (m, p and q) (Mahajan, Muller and Bass, 1990, Thomas, 1985). At the same time, an analyst can fit the Bass model to data from a number of new products to measure the relative effectiveness of the firm’s efforts to stimulate trial through external communications (affecting p) and its efforts to generate positive word-of-mouth (affecting q). If the analyst uses the Bass model in situations where the level connectivity is sparse, the estimates of q will be biased downwards. This may lead to a misallocation of future 22

resources between traditional “push” type promotions and “buzz marketing” efforts that try to enlist members of one’s social network. Our results (both with simulated data and real world data) provide strong evidence that researchers need to incorporate social network structures into aggregate diffusion models. The NBB model, which nests the Bass model as a special case, provides a simple but robust framework for avoiding biases in measures of social influence. CONCLUSION The Network-Based Bass (NBB) model developed in this paper is the first model framework to directly incorporate individual-level social network data into an accurate and interpretable aggregate model of new product adoption. It utilizes the data on connections among consumers facilitated by social networks, a source of data unavailable when Bass (1969) published his original results. Using the NBB model, analysts can leverage sources of “Big Data” while not abandoning the utility and parsimony of a proven modeling approach. Methodological Contributions

In developing the aggregate NBB formulation from a micro-level diffusion model, we propose a micro-model that underpins the aggregate Bass model. More significantly, we show that within the traditional Bass model is a previously unexamined assumption that the network of adopters and non-adopters is fully connected (i.e, all network members are directly connected to all other members). This assumption results in estimates of the imitation parameter (q) that are biased downwards. Therefore, for sparsely connected social networks, the impact of social interaction on adoption will be significantly underestimated by the classical Bass model. The NBB does not have this limitation and recovers the true micro-level estimates accurately regardless of network density. 23

In terms of model calibration, we showed that estimating the NBB parameters using snowball sampling (SBS) works well. SBS is consistent with web-crawling approaches used to capture data from online sources. In contrast, random sampling of nodes yielded poor results. Estimates from the NBB model are also robust to the type of social network. We reported on simulations using random and scale-free networks. Results from analyses of small-world networks (Watts and Strogatz, 1998) are comparable and available from the authors. Diffusion Process Forecasting

One limitation of the Bass model is that it requires two points (take-off and slowdown) on penetration to calibrate good parameter estimates (Heeler and Hustad, 1980, Srinivasan and Mason, 1986). Such requirement raised questions on effectiveness of diffusion models as forecasting tool (Kohli et al., 1999, Mahajan, Muller and Bass, 1990). By the time we observe these two points a firm already made most of risky investment decisions. In such circumstances, the model’s value as normative tool is significantly diminished (Hauser et al., 2006). It is critical for the diffusion model to provide good estimates before major decisions have been made. Since the NBB model shows better fits versus the Bass model at the early stage of diffusion, it should have better forecasting power. The Bass model also has been used to forecasting sales for the new products using historical sales data from similar products (Thomas, 1985, Trusov, Rand and Joshi, 2013). Empirical study on forecasting performance of the NBB model using only early stage of penetration is a promising direction for new research. Although the sales peak estimates from the NBB model show better performance than the Bass model for the scale-free network, a sales peak of scale-free network with high γ largely depends on the timing of adoption of hubs, i.e., sudden sale peak cannot be solely estimated by correct values of p and q. Future research might look at how we can develop the sale peak 24

forecasting tool for networks with highly skewed distributions of linkages among members based on the NBB model. This model should seek to incorporate the adoption timing forecast for the hubs as well. Limitations and Extensions

The new NBB model should open numerous research opportunities. We developed our NBB model from the original Bass model in 1969. The NBB model could be a baseline model to follow the numerous advances in the original Bass model over the year, e.g. including price (Dolan and Jeuland, 1981, Kalish, 1983, Robinson and Lakhani, 1975) or advertising (Dodson Jr and Muller, 1978, Horsky and Simon, 1983). Such an augmented NBB model may shed light on how the marketing mix affects new product growth while accounting for micro-level social influence. The NBB model is built on the assumption that a firm can identify all connections in the network (or at least can perform network sampling). However, this assumption might not be hold for many firms (Godes, 2011, Watts and Dodds, 2007). Better understanding of the conditions (e.g., network characteristics) under which the Bass and NBB model can recover relatively accurate parameter estimates without complete individual network information will be of great interest to the new product managers. Currently, the linkages between social network members are estimated as binary (linked or not). However, people have varying levels of interaction with members of their social network. They may be very similar to some members of their network on some dimensions and very different on others. Incorporating information about tie-strength into the model may yield some interesting insights into how the adoption of a new product by a friend may be more or less influential depending on which friend is doing the adopting. For example, when a friend with 25

similar tastes in music recommends (directly or indirectly through her play list) a new song, this information might be highly influential in spurring one’s own trial of that same new song. Since the model formulation does not depend on a binary (0/1) adjacency matrix, the NBB can be used to see which types of ties actually drive social influence on adoption. Expanding the NBB model to accommodate individual level data on exposure to external information (non-social messages) is another possible way to leverage the data available to today’s analyst seeking to understand innovation diffusion.

26

APPENDIX RELATIONSHIP BETWEEN NBB AND BASS MODELS Recall from equation (1) that rit is defined as the proportion of the friends of consumer i who have adopted by time t. Then,

m

r iA



it

iA

a

c

jt ij

j

m

c

ij

j

   m m m m cij     a jt cij*   a jt cij*    a jt cij*   a jt m  iA j i j iA j  iA j   cij  j   m

m

m

m

m

m

m

m

m

i

j

i

j

i

j

i

j

  a jt cij*   a jt cij* ait   a jt  cij*  cij* ait    a jt cij* 1  ait   a t C* 1m - a t 

T

For a fully connected network, we have m

m

i

j

 a c 1  a  * jt ij

it

m

m

m

m

m

i

j i

i

i

j i

  a jt cij* 1  ait    ait cii* 1  ait    a jt

m 1 1  ait    ait ci*i 1  ait  m 1 i



1 m m  a jt 1  ait   0 since ait 1  ait   0 for all i by defintion m  1 i j i



m m 1 m m 1 m 1   a 1  a  a 1  a  Y m  ait         jt it jt  it t  m 1 i j m 1 j m 1  i i 



1 Yt  m  Yt  m 1

Given these results, we now derive the relationship between the NBB and the classical Bass models. As we show below, the NBB model is equivalent to the Bass model when the social network is fully connected. We begin by stating the following theorem:

27

Theorem : For a fully connected adjacency matrix CF and a large m, T Y  St  p  m  Yt   q at C*F 1m - a t    p  m  Yt   q  t  m  Yt     m  q Proof: St  mp   q  p  Yt  Yt 2 (Frank M. Bass, 1969) m Y    p  m  Yt   q  t  m  Yt   m   1  if i is an adopter at time t Let a t be a 1 m row vector and ait  and 0 otherwise 1m be a 1 m row vector with all the entries equal to 1. Then: m

m

i 1

i 1

Yt =  ait and m  Yt   1  ait  1 if i and j are connected and i  j Let C denotes a m  m adjacency matrix , i.e. cij  0 otherwise Suppose all i and j are connected to each other, i.e. cij  1 for all i  j i.e. CF is the adjacency matrix for the fully connected network then: a t CF 1m - a t 

T

m m m m m m m m m    ait cij 1  ait    ait  cij 1  a jt    ait   cij  cij a jt    ait   cij   cij a jt  i j i j i j i j  j  m

m

i

j

m

m

m

m

m

i

i

j

  ait  cij   ait  cij a jt   ait  m  1   ait  cij a jt i

j

   Yt (m  1)    ait  a jt  ait2  since cij  1 for all i  j and cij  0 for all i  j j i  i  m m m   Yt (m  1)    ait  a jt  ait  since ait2  ait for all i by definition j i  i  m m   =Yt (m  1)   ait   a jt  1 =Yt ( m  1)  Yt Yt  1  Yt  m  Yt  i  j  c 1 Let the column-standardized matrix of C be C* , where cij*  ij then: C*F  CF for large m m 1  cij m

m

m

j

28

Table 1 -- Summary of Empirical Results Song Title  Somebody That I Used to Know  Someone Like You  We Are Young  Lights  Some Nights  Party Rock Anthem  Set Fire to the Rain  Call Me Maybe  Super Bass  Feel So Close  Take Care  The One That Got Away  Headlines  Give Me Everything  Stronger  Sexy and I Know It  Glad You Came  Good Feeling  What Makes You Beautiful  Wild Ones  Where Them Girls At  Stereo Hearts  Domino  Wide Awake  Everybody Talks 

Avg.  p.NBB  Conn  1.6  3.0  2.1  2.2  2.0  2.0  2.5  2.6  3.2  1.8  1.6  3.7  1.0  2.5  4.5  1.3  2.8  1.4  3.4  2.0  1.9  1.4  3.2  3.6  1.1 

0.03 0.06 0.03 0.02 0.08 0.02 0.03 0.01 0.01 0.03 0.04 0.06 0.10 0.06 0.05 0.04 0.01 0.05 0.02 0.07 0.07 0.06 0.05 0.10 0.02 

p.Bass 

q.NBB 

q.Bass 

m 

Bass MSE NBB

0.01 0.06 0.03 0.02 0.06 0.03 0.02 0.01 0.03 0.03 0.02 0.04 0.09 0.07 0.04 0.01 0.01 0.05 0.02 0.08 0.07 0.06 0.05 0.09 0.03 

0.80 0.21 0.58 0.27 0.17 0.68 0.47 0.92 0.60 0.28 0.47 0.05 0.21 0.25 0.29 0.70 0.61 0.45 0.52 0.50 0.16 0.40 0.24 0.21 0.63 

0.33 0.08 0.15 0.10 0.10 0.15 0.18 0.27 0.11 0.09 0.20 0.06 0.04 0.02 0.13 0.26 0.22 0.11 0.21 0.07 0.04 0.07 0.09 0.04 0.10 

1742  1659  1380  1290  1109  1084  1079  1047  918  903  793  749  662  658  658  642  585  581  564  560  553  546  529  519  517 

2.2  26.0  22.2  2.8  13.6  51.4  7.4  44.3  45.6  6.4  ‐3.5  3.0  9.9  16.1  3.4  ‐2.5  19.4  19.6  13.8  31.9  9.8  14.9  9.6  17.0  33.0 

t 

 

46 2.42 52 2.62 43 3.87 52 2.70 39 1.70 53 4.53 45 2.61 43 3.41 52 5.45 46 3.11 47 2.35 53 0.83 44 5.25 50 12.50 45 2.23 46 2.69 47 2.77 45 4.09 44 2.48 40 7.14 49 4.00 47 5.71 44 2.67 37 5.25 41  6.30 

Note: MSE BA  0 indicates model B fits better; q.Ratio= q.NBB/q.Bass; t = number of time points

         

 

29

q.  Ratio 

Figure 1. Fits from the High vs. Low Density Network (True p=0.048; True q=0.6)

30

Figure 2. p Estimates from the Models

Figure 3. q Estimates from the Models

31

Figure 4. p Estimates from RNS Vs. SBS on Random Network

Figure 5. q Estimates from RNS Vs. SBS on Random Network

32

Figure 6. p Estimates from RNS Vs. SBS on Scale-free Network

Figure 7. q Estimates from RNS Vs. SBS on Scale-free Network

33

Figure 8 Model Fits for Empirical Data

34

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