A Weighted Statistical Network Modeling Approach to Product Competition Analysis

,


Introduction
Network modeling has emerged as a key method for statistical analysis of complex systems in a wide variety of social and engineering domains [1][2][3][4][5]. For example, networkbased models have been applied in design collaborations [6][7][8], design crowdsourcing [9], new technology adoption [10,11], and design and manufacturing systems [12]. As a complex socio-technical system, customer-product relations can be modeled with network analysis based on social network theory and techniques [13][14][15], where nodes represent individual entities and links represent their relationships. Among the existing network-based modeling techniques, the exponential random graph model (ERGM) is a unique method with the ability to model the influence of both exogenous effects (e.g., nodal attributes) and endogenous effects (network configuration/nodal relations) on the network formation. In recent studies, this model has been adopted in studying customers' consideration behaviors [13], forecasting the impact of technological changes on market competitions [16], modeling customers' consideration-then-choice behaviors [17], and predicting products' co-consideration relations [13,18]. Particularly, the studies on how products are co-considered relate to the focus of this paper: analysis and prediction of products' competition relations.
Depending on the level of complexity, different network structures have been explored in customer-product relations, including "unidimensional" network, "bipartite" network, and "multidimensional" network, as shown in Figure 1. Among these structures, both bipartite [17,19] and multidimensional [15] networks model customers and products as separate nodes and the relation between customer and product (customers' considerations and choices) as links. Unlike bipartite and multidimensional networks, unidimensional networks focus on product competition based on aggregated customer preference. In a unidimensional network, nodes represent products in the market, and links among them are formed based on whether customers have co-considered the products together. Prior work has shown the importance of modeling unidimensional networks for customer preference modeling. For example, Sha et al. [13] studied a binary unidimensional network to understand the influence of endogenous effects, such as the existing competition relations between car models, on the formation of new competitions in the market. Ahmed et al. [20] proposed a graph neural network approach to predict the binary unidimensional relationships between products. In this study, we adopt unidimensional network analysis to investigate product competition for two reasons. First, a unidimensional network represents competition as aggregated customer preferences and demand at the market level, from which the insights obtained would provide better decision support for enterprises than studying individual choice behaviors on competing products. Second, in a unidimensional network, customers' considerations and choices can be modeled jointly at the market level by the introduction of directed links. erefore, it enables the prediction of market shares of different products beyond merely studying product competitions, thereby serving for the design for market systems.
Despite earlier attempts at using network models and theories in understanding the driving factors in customers' consideration and choice behaviors, existing studies have several limitations. First, the networks are simplified as binary networks, meaning that the weights or the strength of links are neglected. However, the link strength is an important aspect of understanding product competition as well as customer preferences. is is because to probe into the question of how much a competition relation between two products could be changed because of the change of designs or customer preferences, the link strength must be explicitly modeled. Second, most, if not all, past research on network models on car competition analysis does not use directed networks for modeling the final choice decision of a product but instead focus on the first stage of choice-making, that is, customers' consideration decisions. Aiming to address these limitations in the past work (as illustrated in Figure 1) [14,[16][17][18][19]21], this study is the first, in our knowledge, to use weighted networks as well as both choice (directed networks) and consideration (undirected networks) to study product competition and customer preferences. Figure 1 summarizes the existing studies on using unidimensional, bipartite, and multidimensional network analysis in customer preference modeling and how this work differs from them.
e new approach proposed in this study is based on the valued ERGM models that allow a link between nodes to carry weights, and such a link can be either directed or undirected. Despite the applications of network modeling techniques in different research areas, the valued ERGM technique [22] has received little attention in engineering research. Our research aims at acclimating and transferring this statistical modeling knowledge into the engineering design field for further understanding product competition relations. In a unidimensional car competition network, we study both customers' consideration and choice behaviors by establishing two types of networks as illustrated in Figure 2, an undirected network, in which links represent the co-consideration relationship and a directed network, in which a directed link between the two products co-considered indicates the customers' aggregated preferences towards the final choice decisions.
As a summary, the objectives of this research are: (a) to develop an approach based on valued ERGM to model product competition, as exemplified by the study on both weighted undirected co-consideration network and weighted directed choice network and (b) to evaluate the performance of valued ERGM in link prediction (i.e., the competition strength prediction) when nodal attributes change in different years, for example, the change of product design features when a car model upgrades from one year to another. e primary contributions of this paper are: first, a new network-based approach using valued ERGM to explore product competition is proposed for the first time. Second, we demonstrate that valued ERGM models predict customer consideration behavior substantially better than binary ERGM models. ird, we show that valued ERGM effectively models both directed and undirected networks in analyzing aggregated customer considerations and purchasing behaviors.

Technical Background
2.1. Exponential Random Graph Models. Exponential random graph model (ERGM), a statistical analysis technique that serves as a formal representation of the network formation process [23], has been a popular choice in social network research. ERGM outputs a probability for every possible network that can be formed from a fixed number of nodes. is leads to a probability distribution on the set of all possible networks with the same number of nodes [23]. Mathematically, ERGMs can be expressed as a function of a set of input parameters (which can be node properties, link properties, network configuration attributes, etc.) [24], as shown in the following: 2 Complexity where network structure Y is treated as a random variable and an observed network Y is the network data the researcher has collected and regarded as one realization from a set of possible networks. e probability of the observed network structure is determined by network statistics g(y), which can include attributes of nodes, attributes of links, and network structural attributes, along with the corresponding model parameters θ. As θ and g(y) are vectors, T, a transpose operator, is needed to ensure a proper dot product operation. κ(θ) � y ′ ∈Y exp(θ T · g(y ′ )) is a normalizing constant, which is a summation of the numerator over all possible networks Y to make sure the function yields a realistic probability value. Equation (1) suggests that the probability of observing a specific network structure is proportional to the exponent of a weighted combination of network statistics [25]. To estimate the parameters (or learn the model from existing data), a Markov chain Monte Carlo (MCMC) procedure using maximum likelihood estimation is typically employed, and the details of the algorithm are documented in [26,27]. e estimated parameters indicate the importance of different statistics (such as node attributes and network structural effects) in the formation of links in a network. By analyzing the magnitude and statistical relevance of those parameters, one can find and interpret the factors that are important to the formation of the observed network. ERGM has several advantages over traditional statistical models. For instance, unlike traditional logit models [28], they allow the interdependence among network links, which is more realistic in many network formation processes. For example, in a friendship network, when two nodes have common partners, there could be a higher possibility to have a link connecting them. ERGM also provides a flexible statistical inference framework that can model the influence of both exogenous effects (e.g., nodal attributes) and endogenous effects (e.g., the triangular network configurations that represent the three-way product competition) on the probability of forming a connection between nodes.

Valued ERGM Model.
A limitation of traditional binary ERGM is that it cannot model networks with weighted links (e.g., the demand between two airports in an air transportation network). If one wishes to model a weighted network with the traditional ERGM, they have to first binarize the network with a link weight threshold. is dichotomization step may lead to biases and information  [13,16,18,20] This Paper [17,19] [15] Figure 1: Unidimensional, bipartite, and multidimensional networks in customer-product relation modeling. We show that past work in literature did not address weighted models for directed or undirected networks. is paper fills this gap for unidimensional networks.
Car attributes: price, power, fuel consumption, brand origin … (a) (b) Figure 2: We use valued ERGM network models to study product competition in both the consideration stage (the network in (a)) and the choice stage (the network in (b)). e nodes represent cars as an example in these network illustrations and links represent competition strength.
Complexity 3 loss, which can eventually affect network prediction. Valued ERGM [22], a technique recently developed by statisticians, addresses this limitation by modeling the strength of links rather than merely their presence or absence. For a given set of discrete variables, a valued ERGM is expressed as follows: where most of the parameters are the same as those in equation (1), and κ(θ, y) � y ′ ∈Y h(y ′ )exp(θ T · g(y ′ )) also works as a normalizing constant, to make the function output a feasible probability value. Two major distinctions between the valued ERGM and the regular ERGM are the support Y term and the reference distribution h(y) term. Different from binary ERGMs, the support of a valued ERGM is over a set of weighted networks, which is often infinite or uncountable [29]. One cannot enumerate all possible weighted networks with real-valued link strengths.
us, in a weighted network case, we need to consider what the strengths of connections are and how they are distributed.
is brings in the need of specifying a reference distribution, which determines the sample space and baseline distribution of link values. e sample space is a set of possible networks given the size and density of the observed network. A reference distribution simply answers the question of what the link distribution might look like in the absence of any ERGM terms. e ability to model valued links has greatly advanced network research as it enables researchers to conduct more nuanced examinations of network structures. Moreover, similar to traditional ERGMs, valued ERGMs are capable of modeling networks with both undirected links and directed links. Despite these benefits, valued ERGMs are still very much an exploratory area within statistical network analysis [30] due to computational difficulties.
Valued ERGMs have been employed in various applications ranging from policy studies [30] and organizational communication [31] to disease transmissions [32] and global migration [33]. An important step of using valued ERGM is to first define meaningful links and a way to measure the link strength. e definition of link strength often depends on the domain, and in the past, researchers have determined it based on factors ranging from the level of interaction between two nodes [30], the strength of friendship [31], or the total duration of human contact [32]. ese links, although valued, are typically discrete in a small range such as 0, 1, 2, 3 { }. Existing methods in the social science area cannot be directly used in our study to model the valued product competition networks because: (a) the link strength in a product competition network could have a substantially large range. is infinite sample space increases the complexity of the task of prediction and (b) existing studies mainly concentrate on interpreting the models, whereas we focus on both interpretation and prediction. e prediction of the network involves network simulation based on the estimated parameters, and it can also serve as a validation of the fitted model. Despite their complexity, there are two motivations behind using valued ERGM models in this work: (1) they can model the magnitude of competition strength between products, thereby supporting car manufacturers' strategic decisions on product positioning. As the valued ERGM will establish the functional relations between the car design features and the competition strength, the resulting model will be able to predict future market competition based on the change of certain car features, such as a design upgrade or design modification.
(2) With more information captured, the valued ERGM model should demonstrate a better link prediction accuracy compared to traditional binary ERGMs.

Methodology
In a product market, the number of customers considering a pair of products (u and v) or choosing one product over the other reflects the in-between competitive strength. To capture the product competition strength based on customers' considerations and choices, we build weighted product competition networks and model them with valued ERGMs. In this section, we outline the three main steps required for the statistical modeling of a weighted competition network: (1) Construct the weighted network (2) Train a valued ERGM model and interpret the estimated model parameters (3) Predict the future competition among products in the market under investigation Our contribution is to extend the valued ERGM to modeling product competition networks. We describe the step-by-step process of building a weighted network and how to analyze it in this section. We use car design as an example in this study, but our approach can be generalized to many other product design contexts.

Weighted Product Competition Network Construction.
To capture the multistage nature of a customer's decisionmaking process, we build two different unidimensional networks, "co-consideration network" and "choice network." e first is an undirected network that represents customers' choice set in the consideration stage, and the second is a directed network, which represents the customers' aggregated choice preferences.
In both networks, a product (in this case, a car) corresponds to a node. Each node is associated with a set of attributes such as price, fuel consumption, and engine power. We denote both networks as G � (V, ε, W), where V, ε, and W represent nodes, links, and weights, respectively. Figure 2 provides a simplified illustration for both the unidimensional consideration and the choice networks that we investigate. e thickness of the link between two nodes is proportional to its strength (i.e., the number of customers who co-consider the two products or choose one product over the other), and the size of the node is proportional to the popularity of the product (i.e., the number of customers who consider or purchase the product).

Defining Link Strengths in the Co-Consideration
Network. In the co-consideration network, we define an undirected link between node u and node v, if there exists at least one customer who considers both cars u and v together. e number of customers who consider the two cars together is denoted as the link weight (w u,v ) between nodes u and v.

Defining Link Strengths in the Choice Network.
In the choice network, a directed link from node u to node v is established if there exist customers who considered car u and v together but finally bought v instead of u. e total number of customers who bought car v despite considering car u denotes the link strength from u to v and vice versa.

Descriptive Network Analysis.
Descriptive network analysis helps researchers quickly explore some major characteristics of a network, such as which products are popular and how dense the network is, without going into the sophisticated statistical modeling process. It requires the computation of topological measures to assess the network structural characteristics and the implication of structural advantages [15]. e descriptive metrics adopted for analyzing a unidimensional weighted car competition network are "network weight distribution," "centrality," and "clustering coefficient." e values of weights w u,v , which measures the competition (while the methods we present in this paper generalize to many definitions of competition between two items, we have primarily used the term "competition" between two cars as a measure of the number of occurrences that two cars being co-considered.) strength between pairs of cars (u and v), can be considered as a fundamental element in the weighted network analysis. e "probability distribution of weights" P(w) indicates the overall competition strength, that is, the frequency of a pair of cars being coconsidered in an undirected network. We can also calculate the "centrality" of a node. In an undirected consideration network, the centrality is measured by the strength of a node, which is defined as s(i) � j∈V(i) w ij (V(i) is the node-set of node i's neighbourhood). It is a measure of how popular the car is. Note that as this is a measure of popularity in the consideration network, it is possible that a model is popular (i.e., considered by many people) in the consideration stage but still has a low market share. In the directed choice network, the in-strength of a node s in (i) equals the sum of weights of all directed inward links, which is a measure of the popularity in the final purchase decisions. Furthermore, the fraction of a node's in-strength to the total in-strength of all the nodes (s in (i)/ n j�1 s in (j)) for directed networks represents the market share of that car model. Finally, we are interested in observing if there exist cliques of cars with intense competition. To quantify this, we use the "weighted global clustering coefficient," which measures the overall network interconnected triplets [34]. A cluster in a weighted network is defined as a group of nodes with high-weight links between each other and with low-weight links to other nodes in the network. erefore, a high clustering coefficient indicates interconnected communities (car competitions within market segments) are more common in the network. While descriptive analysis provides broad insights about the network structure, it does not throw light on how different attributes quantitatively affect link formation. In what follows, we discuss the valued ERGM technique that complements the descriptive network analysis.

Network
Modeling and Interpretation. As described in equation (2), the inputs of valued ERGMs are a reference distribution h(y) and a vector of selected input terms g(y), such as car price and fuel efficiency, and the weighted network configurations, such as the network density.

Defining a Reference Distribution.
e reference distribution h(y) acts as our prior belief about the network based on the known distribution of link weights. erefore, the choice of reference distribution should reflect the prior knowledge about the link strength distribution. While binomial distribution is typically used for binary networks, other choices such as the Poisson, Geometric, Bernoulli, Uniform, and Standard Normal distribution are possible for a weighted network. e exact choice of prior belief depends on the application domain and the actual data.

Defining Input Variables.
Many of the variables used as input in a valued ERGM model are similar to those used in binary ERGM models, and they can be classified into three categories: network configurations, main effects, and homophily effects. Network configurations are metrics that can be used to measure network structures, such as the number of edges, triangles, stars, and degrees. e main effects correspond to nodal effects of product attributes (such as cars' price and fuel consumption), and the homophily effects are the similarity or difference between the attributes of two nodes (such as the difference of two cars' price and fuel consumption). e complete list of input variables is introduced in the case study section. Unlike any dyad-independent binary ERGM statistics expressed as g k � (i,j)∈Y x k,i,j y i,j , where x k,i,j denotes to (nodal/edge) covariate and y ij is allowed to have values either 0 or 1, in the valued ERGM, y ij has a larger range of choices (0,1,2,3, . . . in our case). As for the network configuration terms, valued ERGM can handle network sparsity, mutuality, individual heterogeneity, and triadic closure via various network structural terms [22].

Interpreting Valued ERGM Parameters.
e result of the valued ERGM is a set of estimated coefficients and associated P-values for all variables. Network configuration effects indicate link independence, that is, the formation of links due to the presence of other links [35]. e estimation of those effects can be seen as evidence of the prevalence or absence of certain structures (such as density, transitivity, and star effects) in a network. For example, a negative estimate of "edges" indicates that the competition network has Complexity 5 a low density. e impact of main attributes refers to how an attribute might influence a product's propensity to form a link in a co-consideration network or a choice network. For the car example, we examine the selected car attributes, and the result will help designers understand whether cars with certain attributes, for example, a higher price and lower fuel consumption, are more likely to be considered by customers and win a competition. e homophily effects test the hypothesis that cars with more similarities in different attributes are likely to be co-considered, which is a common explanation established in social relations.

Market Competition Prediction.
While statistical network models are typically used to interpret what factors lead to link formation or dissolution, predicting what a network will look like in the future is useful for manufacturers to make strategic decisions. In practice, if manufacturers can predict how the competition between car models would change when certain product design attributes are changed, they can use this knowledge to position their products in the market strategically against competitors. Using the estimated parameters of input variables in valued ERGM, we can predict competition networks in the future, with new car attributes as input.
Based on the valued ERGM equation (equation (2)), the distribution of network models is determined by a base network structure, estimated parameters, input variables, and a reference distribution. erefore, when predicting a future competition network, we substitute the old car attributes with new ones and derive the distribution of the predicted network structures based on the valued ERGM formula. en, we draw many samples from the network distribution (simulated networks) and take the averaged network structure as the aggregated network, which represents the central tendency (highest probable network) of all simulated networks. We use this aggregated network as our prediction and compare it with the known network in the future to show our model's accuracy.
Future predictions using aggregated simulations can be made for either the co-consideration network or the choice network. In the predicted co-consideration networks, the number of competitors and their strengths are predicted. In the predicted choice networks, the manufacturers will get an understanding of which car models are their main competitors. In the next section, we show how the methods and the process discussed above are applied to two real-world vehicle data sets.

Case Studies
In this section, we demonstrate the use of the valued ERGM approach to study the Chinese car market. We use data from a new car buyer survey as a test example. Weighted network modeling can be applied to different stages of decisionmaking of a customer, which corresponds to different types of network models (undirected and directed). We show two case studies covering different aspects of network structures and the decision-making process of customers. e first case study focuses on the initial stage of customer decisionmaking and uses an undirected co-consideration network model. e second case study focuses on the final stage of choice-making using a directed choice network model. In this case, one of the cars that are co-considered by the customers "wins" the competition, thus is finally purchased.

Data Description.
Our data set contains survey data from 2013 to 2014 in the China market. In the survey, there were around 53,000 and 60,000 respondents, respectively, in 2013 and 2014, who specified which cars they purchased and which cars they considered, before making their final choice. Each customer indicated at least one and up to three cars which they considered. e data set also contains many car attributes (e.g., price, power, brand origin, and fuel consumption) and customer-specific attributes (e.g., gender, age, etc.).

Case Study 1: Car Co-Consideration Network.
In this case study, we use valued ERGM models to study the competition between any pair of car models reflected by the number of co-considerations received between them.

Step 1: Network Construction and Characterization.
To study car co-consideration, we start by creating a car coconsideration network based on customers' survey responses in the 2013 survey data. For purpose of validation, we control the studied market size and a random sampling of 50,000 customers was made. It is noteworthy that customers who have only considered one car in the survey are removed because they do not provide valuable information about product competition, and our network currently has taken roughly 38,000 customers. e network consists of 296 unique car models as network nodes. e link between a pair of nodes carries the weight equal to the number of customers who considered both the car models together in their consideration set. e overview of the 2013 co-consideration network is shown in Figure 3. As the node size is proportional to the weighted degree of a car model, a larger node size depicts a more popular car model because it is considered by more customers. Similarly, a thicker link width displays a stronger co-consideration relationship (competition) between a pair of cars. Figure 3 also shows a glimpse of a three-way competition. In this example, cars "Great Wall Hover" and "Honda Dongfeng CRV" appear together in the consideration set of 18 customers in 2013 and 30 customers in 2014, showing that their competition has potentially increased in one year (note the sampled market size for 2013 and 2014 are the same). In contrast, cars "VW SVW Tiguan" and "Honda Dongfeng CRV" appear together in the consideration set of 201 customers in 2013 and 192 customers in 2014. is shows that their competition has decreased in one year, although both car models are still more popular than the "Great Wall Hover," as indicated by the sum of all link strengths connected to them. To build the valued ERGM network model, we select the set of most representative car attributes based on the selection criterion used in a previous study [13], including price, engine power, fuel consumption, market segment, import status, and car make origin. is selection allows us to use our prior work as a baseline for comparison purposes. We apply log transformation to price (in Chinese Yuan RMB) and engine power (in brake horsepower BHP) to normalize the range of attribute values and reduce the large outlier effects. Fuel consumption is calculated by the ratio of consumed gasoline (in liters) to driving distance (in 100 km), and a smaller fuel consumption value speaks for higher fuel efficiency. e market segment is a categorical variable that contains 17 car segment codes provided by Ford. e variables of import and make origin are related to the car's brand information, and 35.1% of cars are imported from Europe, the United States, Japan, and South Korea, and 64.9% of cars are domestically produced in China.

4.2.2.
Step 2: Network Modeling and Interpretation. In the implementation of the valued ERGM model, we assign the selected car attributes to network nodes and the occurrence of co-considerations to the link strengths. Based on the sample space of link strength (nonnegative, integer, and not bounded), the available reference distributions are Poisson distribution and geometric distribution. In an empirical setting, Poisson distribution provides a converged and legitimate result; therefore, we have chosen Poisson distribution as the reference distribution. e input variables can be divided into three categories: the network configuration effects, the main effects [13], and the homophily effects. e whole set of input variables can be found in Table 2. We use the statistical network analysis package "Statnet" in R programming, in which the valued  Complexity ERGM is integrated [36]. e second column of Table 2 (i.e., "Weighted") shows the estimated coefficients from fitting the valued ERGM models. e sum/intercept variable serves as a constant term in valued ERGM, and it estimates the likelihood of two cars' co-consideration strength without any knowledge about the cars' attributes. All the input variables, except the main effect of power and the homophily effect of the power difference, are statistically significant at the level of significance of 0.05. As all variables are normalized to a similar order of magnitude, the differences in the coefficients denote their relative importance in the model fit. Among the main effects, the coefficient of import effect is negative, but the coefficients of brand origin from different countries are positive. is implies that customers tend to consider domestically made cars with foreign brands, such as Ford Changan Focus, Honda Dongfeng Civic, and so on. Variables such as price, power, and fuel consumption are not as important as the other main effects. We observe that the coefficients corresponding to the homophily effects are mostly positive and significant. is indicates that the homophily effects may play an important role in forming the competitive relations between two car models, which verifies our common beliefs. Among the homophily effects, market segment matching and brand origin matching are significant. is may reveal that car models within the same market segment and the same brand origin tend to be co-considered by customers. Furthermore, a statistically significant large negative coefficient of price difference shows customers prefer to consider cars in a similar price range. is observation aligns with our intuition, as a customer may consider cars within his/ her budget range.
To validate the findings of our valued ERGM model, we compare it against binary ERGM models (where the network only considers the existence of the competition instead of the competition strength). We set the model terms, such as the attributes considered and MCMC termination criteria (i.e., the P-value of Hotelling's T 2 test for equality of MCMCsimulated network statistics exceeds 0.5 [37]), to be the same between all models. However, as the performance (as measured by network prediction) of the binary ERGM model is sensitive to the method of binarization, we compare our method with multiple binary networks. Specifically, we add results for three cutoff link strength values, that is, 1.0 (the first quantile), 2.0 (the median), and 4.0 (the third quantile), and we denote the link strength larger than the cutoff value as a link in the binary network.
e model results are shown in Table 2.
e estimations of different binary network models show that the effects of fuel consumption and power differences become nonsignificant when the cutoff value changes, which indicates that the different choices of the cutoff value for the binarized network can result in different model results and different interpretation of the model. When using valued ERGM, this inconsistency is not observed due to the lack of need for binarizing the network with an artificial cutoff value.
However, while binary networks with different cutoff values have some inconsistency among themselves, it is important to understand that overall the ERGM modeling package for binary and valued networks seems reliable for our problem. is is evident due to two reasons. First, the relative value of attribute coefficients in different binary models does not drastically change. Second, we see that most coefficients have a similar sign and relative value between valued and binary networks too, which shows the ERGM models are reliable in estimating the attribute effects and converge to similar values.
e valued ERGM provides a few new insights. For instance, the coefficient of market segment matching is larger than brand origin matching for valued ERGM in contrast to binary ERGM, which implies brand origin matching plays an important role during the link formation but market segment matching is more dominant in creating higher competition strength. We trust these interpretations To test the consistency of the model with different customer samples, multiple random samples have been run to sample 50,000 customers. e estimated parameters are stable and consistent across varied sampling results in 50 random samples. e mean and standard deviation of the estimated parameters in different samples are shown in Table 3.

Step 3: Validation and Prediction.
We perform three different types of validation to examine both the model fit and the predictive power, as elaborated in the following.
(1) Trained Model Prediction Matches the Link Strengths in the Training Data Pretty Well. We start the model validation by performing simulations with the current network configurations and the estimated coefficients of the selected model terms. More concretely, we create 100 simulated networks with the 2013 car co-consideration network configurations and the estimated parameters in Table 2 and then take the average of the link strength values from 100 simulations and denote it as the aggregated simulated car coconsideration strength. e comparison of the link strength between the simulated network and the original network reveals the goodness of the model fit. Figure 4(a) plots the link strengths of the true network compared to the aggregated simulated network along the diagonal. We observe that two sets of link strengths are positively correlated, where a perfect y � x line indicates a perfect fit. is is manifested by the Pearson coefficient of 0.988 and the coefficient of determination (R 2 ) of 0.976.

(2) Trained Model Predicts Link Strengths of Future Unseen
Data Reasonably Well. In practice, the benefit of training a statistical model is to predict the future state and behavior of networks that are unseen. While the market competition between different car models varies yearly, we test whether our fitted co-consideration model can be utilized to predict the co-consideration relationship in the future market. Figure 3 illustrates an example of the real market evolution. It can be observed that in 2014, Great Wall Hover gains more customers' consideration, and the strong co-consideration relationship between VW Tiguan and Honda CRV decreases slightly. Our examination of the model's predictive power uses a similar method of network aggregation as used in the above validation study but with the input of 2014 car attributes as the updated node attributes. With a similar simulation process, we derive the aggregated predicted coconsideration network for the 2014 market data and compare it with the actual co-consideration network. e scatter plot of the actual link strength and the predicted link strength is reported in Figure 4(b), with a R 2 of 0.794 and a Pearson coefficient of 0.893. More importantly, we observe that although there exist some deviations between the prediction and the true link strength in the lower range of the link strength values, the prediction is better when the link strength is larger. In practice, the ability to correctly predict large link strength values is more important because they indicate more intense competition where major players in the market are always involved. Moreover, the model's performance is robust and insensitive to the changes in the estimated coefficients. For example, on changing the coefficients by ±10%, the change in R 2 is always less than 2%.

(3) Valued ERGM Has Higher Precision and Recall Compared to the Baseline Binary Models.
We want to further compare the prediction results with the previous binary nonweighted network baseline. However, for comparison, we have to convert a simulated weighted network to a binary counterpart using a cutoff value of the link strength.
We choose three different cutoff values, 1.0, 2.0, and 4.0, for creating the binary network. ese cutoffs are determined based on the first, second, and third quantiles from the actual network link strength distribution. After that, we compare the predicted co-consideration network with the actual binary network. is comparison allows us to measure the false and true positive rates as metrics to evaluate the model performance. More specifically, we draw the receiver operating characteristics (ROC) curve for each cutoff value. ROC curve [38] is a performance measurement for classification problems at various threshold settings of the predicted probability, and the larger the area under the curve (AUC) is, the better is the model's predictability.
As shown in Figure 5, for all the ROC curves, AUC for the weighted network is larger, which indicates a better predictive performance of valued ERGM compared to binary ERGM. As the cutoff value increases, the performance of binary ERGM keeps, while the performance of valued ERGM becomes better and better.
is is because as the binary network becomes sparser, only links with higher strength are preserved and valued ERGM has better performance in predicting those links.

Case Study 2: Crossover SUV Choice
Network. In this case study, we use valued ERGM models to study the competition between crossover SUV cars in the final choice stage of a customer. 4.3.1.
Step 1: Network Construction. In the second case study, we focus on the market competition among crossover SUVs, such as Ford Edge and Mazda CX-7, which are designed with the body and space of an SUV but the platform of a sedan. is type of car models has gained increasing attention in recent years and has witnessed considerable growth in many countries, owing to the low cost, compact size, stylistic design, and better maneuverability.
ere are 14 crossover SUV models in the 2013 survey data, and we have collected all survey data of which customers have either considered or chosen a crossover SUV model in that year. is gives a total of 1,975 customer observations (different from case study 1 when 50,000 customers are selected from the entire market, we focus on the crossover SUV car segment and only select the customers who have considered or purchased a crossover SUV model.). e directed choice network is established based on the customers' purchase behavior as described in the previous section, and all competitors in the network are divided into four segmentation groups: Sedan, SUV, luxury or sport, and crossover SUV. e visualization of the choice network is plotted in Figure 6, where the node size of a crossover SUV reflects the number of customers who have purchased it.
Overall, there are 217 car models in the crossover SUV choice network. All the links are directed and point to the "winner" in a competition. e average link strength is 2.431 corresponding to the average number of customers' purchases among all co-considered cars. A unique feature of the choice network is that the in-strength of a node is correlated with its market share.

Step 2: Network Modeling and Interpretation.
e procedure of network modeling of a choice network shares many similarities with that of a co-consideration network using the valued ERGM approach. However, as the choice behavior is not symmetric between pairs of nodes, the model terms are further specified for inward nodes or outward nodes. Specifically, the main effects in Table 4 refer to the nodal attributes of the inward nodes; hence, we can learn the important attributes of the "winners" and find possible reasons behind the popularity of a car model. Besides, we have added two network structural effects, "cyclical weights" and "transitive weights," which measure the triadic closure and refer to the links from i ⟶ j that have two paths (two paths refer to a network structure that there are two edges connected from i to j: i ⟶ h ⟶ j) from j ⟶ i and from   10 Complexity i ⟶ j, respectively (Figure 7). More precisely, in the product competition market, it accounts for a hierarchical three-way competition. e cyclical weights refer to the case when customers prefer car k to car j and prefer car i to car k while preferring car j to car i. e transitive weights refer to the case when customers prefer car k to car j and prefer car i to car k, while preferring car i to car j. Table 4 shows the estimated coefficients from fitting three directed valued ERGM models with different model terms. e first model is a baseline model with main effects and homophily effects, and the second and the third models include network structural effects to further investigate the endogenous network effect influence. Among all three networks, the estimated coefficients are consistent with small variations. In the choice network, the car models with lower prices, higher power, and higher fuel consumption are more likely to be bought by customers. is result is consistent with our common sense. Please note that for the group of customers who have a preference for crossover SUVs, they possibly prefer a model with higher fuel consumption, which is usually in a company with a higher power. Meanwhile, imported cars are not always preferred by this survey population, but a car with foreign brands still shows a positive effect on customers' final choice. Furthermore, the homophily effects have significant positive effects on the choice decisions, and the underlying reason is similar to the first case study. Also, in models 2 and 3, the cyclical weights have a negative effect, while the transitive weights have a positive effect. is implies that in a threeway competition, the competition relations tend to be  transitive, meaning that if car A "wins" a competition over car B and car B "wins" a competition over car C, then car A is likely to "win" car C. erefore, it can be inferred that the directed network market is hierarchical. We have also reported Akaike information criterion (AIC) and Bayesian information criterion (BIC) values for three models, a lower AIC and BIC value indicates a better model fit [39], and the models with network configuration statistics fit slightly better than the baseline model, which indicates that those network configurations could play an important role in the competition network formation.

Step 3: Validation with Pairwise Competition
Comparison. We validate our model using two methods: (1) predicting pairwise competition and (2) estimating the market share of each car model. We first evaluate the model fit at the level of pairwise competition. Given the original network structure, one can identify the "winner" in each pairwise competition by counting the customers' choice prevalence. For example, among 25 customers who have considered both cars A and B, 15 customers bought car A, and 10 customers bought car B, and then car A is denoted as the "winner" in this "A-B" competition. After generating the simulated choice networks based on the fitted model, the aggregated (i.e., averaged) link strength is used to quantify the pairwise competition. e results show that the simulated choice networks obtained from three different models can correctly predict over 60% of the pairwise competitions (i.e., 62.10%, 64.08%, and 61.19%, respectively).
(1) Validation with Market Share Comparison. In a directed choice network, the in-strength of node s in (i) is related to its market share. Hence, we can further validate the choice network by comparing the simulated market share for each crossover SUV with its true market share. Specifically, the instrength fraction s in (i)/ n j�1 s in (j) is calculated based on an observed choice network for the actual market share of the crossover SUVs. en, the simulated market share is derived by averaging the in-strength of the nodes from 100 simulations. e comparison of actual market share, simulated market shares of three different models, and the uniform market share (which assumes all crossover SUVs have the same market share and serves as a baseline) is plotted in Figure 8. Even though there exists discrepancy for some car models (e.g., Mazda CX-9 and GM USA Buick Enclave), most of the predictions of car models show a consistent trend with the actual market share. Compared to the baseline of uniform market share, all simulated market shares have a R 2 value above 0.7, which indicates that more than 70% of the observed variation can be explained by the fitted choice network model. Among them, model 1 has a R 2 value equal to 0.77; model 2 has a R 2 value equal to 0.70; and model 3 has a R 2 value equal to 0.74. As a side note, the models adding more network attributes do not provide a better-simulated market share than the baseline model (model 1), which could be raised by the sparsity and less influence of the network structure.
While valued ERGM shows a reasonably good fit for the relative pairwise competition and the market share, it does not predict well the absolute value of weights in the choice network. is is true in predicting both the current market and the future market. We suspect that this is due to the sparsity and directionality of the network. e network constructed in this case study only contains crossover SUVs, thus leading to a very low network density of 0.02.

Discussion
While the valued ERGM model provides many advantages over existing statistical models, it is a relatively new model with a few theoretical and practical challenges that require attention and more research. In this section, we summarize the benefits and limitations of the valued ERGM models and discuss how they pave the path to future research directions.

Supporting Engineering Design Decisions Using Valued ERGM.
One of the goals in using the valued ERGM model is to demonstrate how the approach helps identify the important factors that influence product competition. ese factors can support stakeholders in making strategic decisions. However, it is important to note that while the theoretical model allows one to estimate the importance of any attribute, the analysis in specific case studies may also depend on what product data is available and whether there indeed exists any relationship between product attributes and customers' choice decisions. To understand this, let us consider three hypothetical situations. In the first situation, a customer decides to buy a car merely based on the size of the car engine. Using a valued ERGM model, the analysis results show that the size of the engine (or power that is correlated with it) has a significant positive coefficient. In such a case, the network models inform that increasing the engine size can help gain a larger market share. However, increasing the engine size will inevitably increase the manufacturing cost, thus leading to a higher price. is, on contrary, may negatively influence the market share. ere is obviously a trade-off decision the car manufacturer has to make; then the network model should help car manufacturers make decisions of choosing the right combination of design features.
In the second situation, we assume that a customer decides to buy a car merely based on the quality of its airconditioning (AC) system. If the data we analyze does not include the AC design attribute, the results will not be able to provide specific insights into the impact of AC design on customers' choice behaviors. e only remedy for this is to collect data that captures the relevant attributes for the choice analysis. In the third situation, we assume that customers' choice behaviors are only influenced by social and/or cultural factors but not car design features. In such cases, the coefficients of all design attributes may not have statistical significance. is indicates that improved design features may not help automakers gain more market share. Hence, the guidance provided to the manufacturer is to not waste resources on improving factors that do not have an impact.
In this paper, the customers' choice behaviors described in the two case studies are a mixture of the three situations. For example, we find that some design attributes have a statistically significant influence, but we also discover that this data set lacks information about certain car design attributes. Finally, many design attributes studied are not statistically significant, indicating that those attributes may not play a role in customer decisions.
From our current results for both case studies, we successfully identify a few factors that impact engineering design decisions for product consideration. Specifically, in the co-consideration network of case study 1 (Table 2), we observe that a car designer may want to reduce fuel consumption (which relates to engine efficiency) to increase the competitiveness of their car models. Although factors such as price, power, and fuel consumption are statistically significant, they do not directly provide actionable design guidance for a car manufacturer. In the choice network of case study 2 (Table 4), the model results help decisionmakers with strategic planning. For example, in the crossover SUV market, the improvement of fuel consumption may not increase the likelihood of a car being purchased. Instead, reducing the price and increasing the power could be helpful improve the market share. We notice that our data set lacks certain car design attributes that may be influential to customers' choices. In future work, we aim to address this issue using crowdsourced data inputs. Moreover, we have discovered the important effect of particular network configuration statistics, such as "cyclical weights" and "transitive weights" in Table 4 in the choice network. It manifests the advantages of (valued) ERGMs in utilizing network configurations to capture endogenous effects in a market. e insights into these endogenous effects help car manufacturers gain an in-depth understanding of the market and their competing opponents.

Trade-Off between Feature Engineering and Model
Interpretability. In valued ERGM models, we start with a large collection of features. ese features can be nodespecific (e.g., car fuel efficiency and price), link-specific (e.g., the price difference between two car models), or networkspecific (e.g., popularity and density). e choice of what features to use has a large impact on the goodness of fit of the model, the estimated coefficients as well as their statistical significance. While we use automated methods for feature selection (which largely select features that are uncorrelated), the process is often manual. In contrast, one can use modern deep learning models to learn hierarchical feature representations. Yet the deep learning models are largely black box and are hard to interpret, which is one of the key reasons for us to adopt the interpretable and theorygrounded statistical network models in this study. In the future, we will attempt to find the middle ground of reducing dependence on feature selection, while still retaining model interpretability by combining the two methods.

Numerical Issues with Valued ERGMs.
Existing literature reports two numerical issues of (valued) ERGMs: the reliability of model interpretation and computation issues for large networks.

Reliability.
In recent years, there have been critiques of using (valued) ERGM packages related to the accuracy of inference methods reported by the statistical software for ERGM. While some experiments suggested that the variants of ERGM models can work well even with a relatively small sample taken from the network [40], Shalizi and Rinaldo [41] have argued that ERGMs are designed for modeling the entire network. In many applications, the data used consists of a sampled sub-network, which could lead to inconsistency of interpretation due to the MCMC sampling process. However, our first case study is unlikely to suffer from the reported issues due to two reasons: (1) the subset of customers in our network for the first case study only changes the link strength magnitude and we still use all nodes and (2) we also test with different subsets samples of customers and find that the results are similar, which indicates the reliability of our network models. For the second case study, we use a particular market segment of cars to create the network, which may suffer from reported limitations. Hence, we are cautious in generalizing our findings from the study on crossover SUVs to other car segments.

Computation Issue for Large/Complex
Network. It is reported in the literature [42] that for large and complex network structures, the MCMC approach to estimate ERGM parameters may not converge. In our work, this limitation can be a problem for some stakeholders.
ere is some recent work on developing scalable binary ERGMs [43,44], and the extension of such methods to valued ERGMs can help alleviate the scalability problem for large data sets. Another approach that can improve the scalability of valued ERGMs is to use kernelized approximate Bayesian computation. It can improve computational efficiency and is being adopted by popular packages [45] as an alternative to MCMC.

Conclusion
In this paper, we enhance the network modeling approach for analyzing customer preferences and product competition by viewing customer-product relations in the context of a complex socio-technical system. With a focus on the unidimensional network as the aggregated result of customer preferences and the social and market environment, we exhibit how valued ERGM models can be used to model directed and undirected product competition networks with nonbinary link strengths. e method enables designers to estimate the major factors that affect customers' consideration and choice behavior and that can help predict the strength of future market competition when a manufacturer changes some product attributes.
is work has three main contributions. First, we extend the newly developed valued ERGM, which has traditionally been confined to social network modeling, to study competition between products. is network modeling approach enriches the knowledge base of product design modeling techniques. Second, by developing a procedure of weighted network construction, interpretation, and validation, we demonstrate that valued ERGM models provide a better model than binary ERGM, as measured by model fit and prediction accuracy for car competition. ird, this paper is the first work to study aggregated purchase preferences using a "directed" unidimensional network. e directed network we create is unique, as it encodes information from two stages of decision-making, both the final purchase decision as well as the items considered by the customers. e case studies in this paper show how network models are used to systematically analyze large real-world networks. For the first case study, which analyzes the co-consideration competition between 296 cars, we show that homophily effects, affecting the differences between two cars, are more important than the main effects in predicting link strength. Cars are generally found to compete more with other cars from the same market segment, same brand origin, and similar price range. In the second case study, which focuses on the crossover SUV market, we analyze a network of 217 cars and find that cars that are considered by more people are also purchased more often. In future work, we aim to analyze how valued ERGM can help study new domains and further investigate ways to integrate feature learning methods such as deep learning with valued ERGM models while retaining their interpretability. Improving the scalability of these models to larger data sets and using them for dynamically evolving car competition is another interesting avenue of research.

Data Availability
Anonymous data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.