1. Introduction
We are living in an increasingly connected society (Travers and Milgram, 1967; Backstrom et al., 2012; Leng et al., 2018b, c). The connections among individuals foster information diffusion and enable the interdependencies in decisionmaking among peers. Therefore, understanding and modeling how hidden social influence changes individuals’ decisionmaking are essential and critical for many practical applications, such as viral marketing, political campaigns, and largescale health behavioral change (Fowler and Christakis, 2008; Pan et al., 2012; Leng et al., 2018a, c).
Homophily, the tendency of similar individuals to associate together, widely exhibits in various types of social networks, and governs the outcomes of many critical networkbased phenomena (McPherson et al., 2001; Kossinets and Watts, 2009; Currarini et al., 2010). Salient features for homophily come from a wide range of sources, including age, race, social class, occupational, and gender (McPherson et al., 2001). The complex nature of social relationships and highdimensional characteristics of individuals thus determine the multidimensionality of homophily (Block and Grund, 2014). Homophily results in locally clustered communities and may affect network dynamics, such as information diffusion and product adoption. The Block Model has been applied to lowdimensional, predefined homophilous features and provides a building block to uncover underlying community structures^{1}^{1}1In this paper, we use community and block interchangeably. with highdimensional homophily empirically (Abbe, 2017).
Social influence is widely studied in economics and computer science literature due to its importance in understanding human behavior. In economics, researchers focus on causally disentangling social influence from homophily with randomization strategies, such as propensity score matching (Aral et al., 2009), behavioral matching (Leng et al., 2018a) and regression adjustment (Angrist, 2014). In the computer science literature, researchers focus on maximizing the likelihood of the diffusion path of influence by proposing different generative processes (GomezRodriguez et al., 2012; Gomez Rodriguez et al., 2013; Myers et al., 2012; Yang and Leskovec, 2010). These works focus on the strength or the pathways of social influence, and they do not link social influence to the underlying homophilous communities and the network formation process.
There exist two theories explaining how local communities affect information diffusion (Weng et al., 2013) and contagion in decisionmaking (Leng et al., 2018a; Golub and Jackson, 2012). On the one hand, homophily and the requirement of social reinforcement for behavioral adoption in complex contagion theory indicate that influence tends to be localized in homophilous communities (McPherson et al., 2001; Centola and Macy, 2007). In other words, behavioral diffusion and network formation are endogenous, explaining the phenomenon of withincommunity spreading (Pin and Rogers, 2016; Weng et al., 2013). On the other hand, the weak ties theory (Granovetter, 1977) implies that bridging ties between communities facilitate the spreading of novel ideas. As empirical evidence, Ugander shows that reinforcement from the multiple communities, rather than from the same communities, predicts higher adoption rates (Ugander et al., 2012). With these two competing theories, we seek to understand whether social influence spreads locally within each homophilous community or globally to other communities taking advantage of the long ties.
Role theory posits that “the division of labor in society takes the form of interaction among heterogeneous specialized positions” (Biddle, 1986). That is to say, depending on the social roles and the behavior of interest, the underlying interactions and norms for decisionmaking are different. Motivated by this proposition, we aim to develop a method to associate social influence with the underlying communities, which are associated with the behavior of interest. To formalize this idea, we propose a generative model to understand how social influence impacts decisionmaking by inferring the spreading of influence across empiricallyidentified blocks. Our framework jointly uncovers the underlying blocks and infers two types of relationships across these blocks: social interaction and social influence. Different from the Stochastic Block Model, the observed individual decisions are used to inform the communities, as complementary to the observed network. Along with this, we infer an influence matrix as the social influence across different communities. This influence matrix reveals the hidden social influence at the community level, which would otherwise be impossible to observe and generalize.
As a case study, we experiment on the diffusion of microfinance in an Indian village and perform extensive analysis on the influence matrix estimated from the model. We find that even though social relationships are denser within communities, social influence mainly spreads across communities. This may be explained by the importance of crosscommunity weak ties
(Granovetter, 1977) and the strength of structural diversity (Ugander et al., 2012). Our generative framework and subsequent understanding of how social influence operates are informative for practical applications, such as viral marketing, political campaigns, and largescale healthrelated behavioral change (Fowler and Christakis, 2008; Pan et al., 2012; Leng et al., 2018a).Contributions
To summarize, the Stochastic Block Influence Model (SBIM) developed in our study makes the following contributions to the literature:

SBIM integrates networks, individual decisions, and characteristics into the generative process. It jointly infers two types of relationships among empiricallyidentified communities: social connection and social influence. Moreover, our model flexibly accommodates both positive and negative social influences.

Our model is motivated by role theory, which posits that individuals make decisions depending on the context of the decision type (Biddle, 1986), e.g., adopting microfinance as opposed to adopting healthy habits. To achieve this, we allow the underlying community to vary with the behavior of interest.

We perform a case study on the adoption of microfinance in an Indian village. Moreover, we demonstrate the interpretability of our model with a detailed analysis of the influence structure.

The analysis from our study can be used for designing network interventions and marketing strategies. For example, we show that communities with smaller overlaps in characteristics exert negative influences on one another. Therefore, marketing firms should encourage individuals to communicate with neighbors in the same community, such as inviting these individuals together to an informational event to promote the positive influence among them.
The remaining sections are organized as follows. We describe the literature in Section 2. In Section 3, we introduce the proposed Stochastic Block Influence Model. Then, we test the method in Section 4 and analyze the results on a realworld data set in Section Analysis and discussions. In Section 5, we summarize the paper with practical applications and future work.
2. Related literature
Contagion models
There are two prominent theories in the literature for explaining the propagation of social influence (Ugander et al., 2012; Bond et al., 2012; Aral et al., 2009; Leng et al., 2018a), i.e., simple contagion and complex contagion. Simple contagion theory assumes that individuals will adopt the behavior as long as they have been exposed to the information (Granovetter, 1977), which is a sensible model for epidemics and information spreading. Complex contagion theory, on the other hand, requires social reinforcement from neighbors to trigger the adoption (Centola and Macy, 2007). Many studies have shown that complex contagion explains behaviors such as registration for health forums (Centola, 2010).
These exposurebased models bear analytical simplicity, however, do not allow social influence to be negative, i.e., the adoption decision of one’s neighbors might decrease, rather than increase, the likelihood of one’s adoption decision. Moreover, they typically are not able to capture the heterogeneity of social influence (Leng et al., 2018c). In this paper, we propose a model to account for negative and heterogeneous influence.
Stochastic Block Model
The Stochastic Block Model is a statistical model for studying latent cluster structures in network data (Abbe, 2017)
. SBM generalizes the ErdosRenyi random graph model with higher intracluster and lower intercluster probability. The traditional SBM only infers the community structures from network connections. However, when contextual information on nodes is available, leveraging information from different sources facilitates the inference. In recent statistics literature, there has been some interesting work on utilizing covariates to infer the block structures. For example, Binkiewicz et al. present a covariateregularized community detection method to find highly connected communities with relatively homogeneous covariates
(Binkiewicz et al., 2017). They balance the two objectives (i.e., the node covariance matrix and the regularized graph laplacian) with tuned hyperparameters. Yan et al. propose a penalized optimization framework by adding a kmeans type regularization
(Yan and Sarkar, 2019). This framework enforces that the estimated communities are consistent with the latent membership in the covariate space.Though these variations to SBM utilize auxiliary information on individual nodes, they specify the importance of recovering the network and the smoothness of covariates on the network, on an adhoc basis. Different from these models, we take advantage of role theory (Biddle, 1986) and utilize the decisionmaking process on the network that could also inform community detection. For example, let us assume professional communities are more useful for the adoption of technologies at work, and social communities are more useful for the adoption of social apps. The underlying communities depend on the role and behavior of interest because social influence spreads through some specific network links in different applications.
3. Methodology
3.1. Stochastic Block Influence Model
Notations
Assume a random graph with individuals in node set and edge set . It is partitioned into disjoint blocks (), and the proportion of nodes in each block is , and . represents the adjacency matrix. if and are connected, and otherwise. Let matrix denote the interblock and intrablock connection probability matrix. Let be the block assignment of individual and summing over C blocks, we have
. Together, we combine the block vector of all individuals in the matrix
. Therefore, the probability of a link between and between two separate blocks and as . is a binary vector representing individuals’ adoption behaviors. Let represent demographic features, where is the number of covariates. We use to represent the blocktoblock influence matrix. Finally, is a binary vector, capturing whether or not each individual is aware of the product at the beginning of the observational period. For a new product, is sparse, while for a mature product, is dense.Model formulation
Extending SBM to utilize the network, adoption decisions, and sociodemographic features, we propose the Stochastic Block Influence Model, abbreviated as SBIM. Linking the latent communities to their sociodemographic composition, we reveal the underlying nature of highdimensional homophily in a datadriven fashion rather than using predefined communities using observed sociodemographics, e.g., race or occupation. Solely using predefined homophilous characteristics does not aptly capture the multiplex characteristics that define individuals and their social ties. In other words, individuals are associated with different communities, each of which is formed by various homophilous characteristics. Neighbors belonging to different communities may influence the focal individuals differently.
Let us illustrate this using the adoption of microfinance in an Indian village. It is reasonable to posit that several traits define the diverse nature of individuals  different professions, castes, education levels, and a variety of other demographic features. Let us take one particular individual, who is an educated worker of a lower caste, for example. This individual belongs with varying degrees of affiliation to different communities: perhaps most strongly affiliated to a group of a certain level of education and less strongly affiliated with another group of a majority of a lower caste. This mixed membership captures the realistic nature of our social relationships and characteristics. Within such a village with multidimensional homophily, how can we understand who influences this individual and what processes are involved in that individual’s decision making? Specifically, she could be influenced both by neighbors belonging to different communities characterized by specific educational backgrounds, professions, and castes. The datadriven multidimensional block aspect of the model allows us to capture these critical, hidden relationships.
Next, we formalize our model. To jointly infer how influence spreads within and across communities, we desire a model with the following properties:

The model leverages both the observed friendship network structure and the adoption behavior to infer the underlying communities.

The link formation and social influence between two individuals are jointly determined by their underlying communities.
For each individual pair , depending on their community assignment vectors, the predicted link is generated according to the connection probability matrix, . In particular, the probability of the existence of a link between and is,
(1) 
Next, we discuss how our model incorporates individual characteristics and adoption decisions. The adoption likelihood depends on individuals’ characteristics and on the influence of their neighbors who have already adopted (Katona et al., 2011). The generative model builds upon the communities a particular individual , and ’s neighbors belong to, as well as the communitytocommunity matrix . Each individual makes a decision on whether or not to adopt in order to maximize her utility. The utility of depends on her own preferences and the aggregated influence from neighbors. The pairwise influence depends on the communities and her neighbors belong to. We illustrate how influence and communities affect one’s decisionmaking in Figure 1. Let us consider individual A, who has three friends, B, C, and D, belonging to a lower socioeconomic status (SES) group (as colored in red), and one friend, E, belonging to a higher SES group (as colored in blue). The adoption likelihood of A is a function of her own preferences as well as the influence from her friends B, C, D, and E. The strength of the influence depends on the corresponding communities of A and her friends (B, C, D, and E).
More generally, the adoption likelihood of a user, , is defined as,
(2) 
where is the elementwise matrix multiplication. The first term, , measures the adoption decision conditioned on ’s sociodemographic features if there were no social influence, where and is the dimension of the covariates. The second term aggregates the influence of ’s neighbors. is the idiosyncratic error term. Without loss of generality, we assume .
For a mature product that everyone is aware of, we can simplify Equation (2) as,
(3) 
Equation (2) only accounts for the influence among direct neighbors. Note that in a smallscale network, it is reasonable to assume that there does not exist higherorder social influence. In a largescale network, Leng et al. show that social influence spreads beyond immediate neighbors (Leng et al., 2018a). For these applications, our model can be easily adapted to higherorder influence by summing up the powers of the adjacency matrix to account for multiple degrees of separation (Leng et al., 2018c).
3.2. Generative process
For the full network, the model assumes the following generative process, which defines a joint probability distribution over
individuals, based on nodewise membership matrix , blocktoblock interaction matrix , blocktoblock influence matrix , attributes’ coefficients , observed friendship network , observed attributes , observed adoption decision .
For each node , draw a dimensional mixed membership vector .

For the connection probability from community to in the blocktoblock connectivity matrix, draw .

For the influence from community to in the blocktoblock influence matrix, draw .

For each attribute in indexed by , draw the coefficient ,.

Draw the connection between each pair of nodes and , , according to Equation (1).

Draw the adoption decision , according to Equation (2).
For abbreviation, we denote as set of the hidden variables, and
as the set of hyperparameters, where
.The posterior distribution defined by the generative model is a conditional distribution of the hidden block structure and relationships given the observed friendship network and adoption behavior, which decomposes the agents into overlapping blocks. The posterior will place a higher probability on configurations of the community membership that describe densely connected communities as well as stronger (positive or negative) influences. We present a visualization in Figure 3, which illustrates that the posterior superimposes a block structure on the original network. The details of the data we use are described in Section 4.
Inference
The posterior of SBIM is intractable, similar to many hierarchical Bayesian models (Bayarri et al., 2003)
. Therefore, we use the Markov Chain Monte Carlo (MCMC) algorithm as an approximate statistical inference method to estimate the parameters. MCMC draws correlated samples that converge in distribution to the target distribution and are generally asymptotically unbiased.
There are different MCMC methods, including Gibbs sampling, MetropolisHastings, Hamiltonian Monte Carlo, and NoUTurn Sampler (NUTS). Gibbs sampling and MetropolisHastings methods converge slowly to the target distribution as they explore the parameter space by random walk (Hoffman and Gelman, 2011). HMC suppresses the random walk behaviors with an auxiliary variable that transforms the problem by sampling to a target distribution into simulating Hamiltonian dynamics. However, HMC requires the gradient of the logposterior, which has a complicated structure in our model. Moreover, it requires a reasonable specification of the step size and a number of steps, which would otherwise result in a substantial drop in efficiency (Hoffman and Gelman, 2014).
Therefore, we apply NUTS, a variant to the HMC method, to eliminate the need for choosing the number of steps by automatically adapting the step size. Specifically, NUTS builds a set of candidate points that spans the target distribution recursively and automatically stops when it starts to double back and retrace its steps (Hoffman and Gelman, 2014). We use the NUTS algorithm implemented in Python PyMC3 (Salvatier et al., 2016).
4. Experiments
Data description
We study the adoption of microfinance in an Indian village collected by the Abdul Latif Jameel Poverty Action Lab (JPAL) (Banerjee et al., 2013)^{2}^{2}2The village we study is indexed by 64.. In 2007, a microfinance institution introduced a microfinance program to some selected Indian villages. In early 2011, they collected information about whether or not the villagers had adopted microfinance. Because the village is fairly small (257 villagers) and microfinance had been on the market for four years when JPAL collected individuals’ adoption decisions, it is reasonable to assume that everyone in the village was aware of microfinance, which is hence a mature product. Therefore, we use Equation (3) as the decisionmaking function. The data contains information about selfreported relationships among households and other amenities, including village size, quality of access to electricity, quality of latrines, number of beds, number of rooms, the number of beds per capita, and the number of rooms per capita. These types of demographic features are used as the independent variables. The outcome variable is the adoption decision of = microfinance. The microfinance institution asked the villagers to selfreport other villagers they considered as friends.
Baseline
We use the Random Forest with sociodemographics and the hidden community learned by spectral clustering on the adjacency matrix as the independent variables. In this way, we use the same information in SBIM and the baseline. Spectral clustering uses the second smallest eigenvector of the graph laplacian as the semioptimal partition
(Ng et al., 2002).Model training
To train our model and evaluate the performance for a particular , the number of block, we crossvalidated by randomly splitting the data into 75% training samples and 25% test samples. We repeat this process ten times. With NUTS, we obtain the point estimates for all latent variables in ^{3}^{3}3Some critical hyperparameters for NUTS are the number of burnin samples, the number of samples after burnin, the target acceptance probability, and the number of chains. For all of our NUTS sampling runs, we burn 3,000 samples to ensure that MCMC mostly converges to the actual posterior distribution. The number of samples after burnin is 500; usually, only less than ten samples (among the 500) are diverging. Next, we select the target acceptance probability to be 0.8. At the end of each run, we average across the 500 samples to derive point estimates for all latent variables.. We then rerun our model (as previously described) with all latent variables fixed to the estimates on the test dataset. This step returns the predicted adoption probability for each villager in the test data.
To choose the optimal number of block, we first tune the model for and then calculate the average loss. We observe a negative parabolic trend with the loss peaking at its lowest at blocks, so we use this optimal number of block for further evaluation.
Model evaluation
Since the dependent variable in our data is imbalanced, we evaluate our method using the AUC, which is the area under the ReceiverOperatingCharacteristics curve plotted by the false positive rate and correct positive rate for different thresholds. We define a loss metric during the training period to select the best configurations. It is formulated by the negative of the standard improvement measure, which is the absolute improvement in performance normalized by the room for improvement. This measure captures the improvement of our method compared to the baseline. Since we have a small test set, a randomlydrawn test set may be harder to predict than others. Measuring the relative improvement ensures that the composition of the test set does not bias the performance due to sample variation. This metric is formulated by
(4) 
where the AUC of the baseline and SBIM on the test split in crossvalidation are represented as and , respectively.
Our model has seven hyperparameters, ^{4}^{4}4The ranges from which these hyperparameters were sampled are as follows: , , , , and . We let
for a reasonable and nonskewed prior.
. Since the parameter space is large, we adapt a banditbased approach to tune the parameters developed called Hyperband (Li et al., 2017). The Hyperband algorithm adaptively searches for configurations and speeds up the process by adaptive resource allocation and earlystopping. Our adaptation of this algorithm allows each configuration tested to run with full resources due to the sampling procedure used in our methodology, allowing NUTS to run consistently across all configurations.Performance
We compare the performance of our model with the baseline in Table 1. We observe that our method outperforms random forest in the test set by 13.8% by the improvement metric in Equation (4). Both models overfit the training set and the baseline overfit comparatively more.
Mean  Standard deviation  

Baseline train AUC  0.901  0.010 
SBIM train AUC  0.805  0.022 
Baseline test AUC  0.610  0.095 
SBIM test AUC  0.664  0.062 
Analysis and discussions
Size of communities and interaction matrix
We present the size of each social block in Figure 2. Social block two is larger than the other blocks, and the sizes of the rest are similar. This aligns with our intuition that many individuals belong to a majority group while several niches, minority communities also exist. We represent the adjacency matrix sorted by this inferred block index from smallest to largest block in Figure 3. We see that there are many links within all of the blocks along the diagonal, demonstrating that the block model is meaningful and captures more links within than across blocks. The largest block, furthest along the diagonal, is comparatively sparser.
Block type
We can associate individuals’ sociodemographic characteristics with the individuals who belong to each block to generalize block type as consisting of characteristics such as high or low SES, homogeneous or diverse, and skilled or less educated, as depicted in Table 2. In this example, each block is associated with a qualitative type, and the attributes within that block leading to such characterizations are described. Lower or higher SES blocks are designated by caste composition, education levels, and profession types. Homogeneous or diverse blocks are designated by some professional composition, caste types, mother tongue language composition, gender imbalance, and what fraction of village inhabitants are natives.
We also use diversity and gender ratio to evaluate block characteristics for a specific example in Table 2 and Figure 5, in addition to being used to evaluate the group attributes that are associated with different types of influence in Table 3. More analysis in Figure 5 is covered in the following section.
We use normalized entropy to measure the diversity of different attributes. Normalized entropy is a metric used to capture the number of types of characteristics within each category while accounting for the frequency of each entity type within a category. It can be formulated by, , where refers to the number of types within a category, refers to the probability of each type , refers to the number of occurrences of each type .
The gender ratio () is measured within a block and is formulated by , where and refer to the number of occurrences of males and females respectively. Thus, since is the ratio of males to females in a block, both a high or low gender ratio correspond to a high gender imbalance.
Influence matrix and attributes
The blocktoblock influence, sorted by increasing block size, is displayed in Figure (a)a, where the strength of social influence, allowed to be either positive or negative, is shown. We can see some blocks influence other blocks ranging from strong negative influence to no influence, and to strong positive influence.
The total influence into and out of each block is depicted in Figure (b)b, which allows us to evaluate the aggregated influence a block receives and spreads (net positive, negative, or neutral). For example, we can see diverse, lowSES block five and senior, lowclass block six with high output levels of positive influence, and diverse, middleSES block eight receives a net high level of negative influence. We observe that some blocks have a stronger outgoing influence than other blocks and can perceive these as positive and negative influence leaders. Similar reasoning applies to characterize blocks that receive a high level of influence as follower blocks, furthermore observing the difference in net incoming and outgoing influence within each block as relating to its role in the blocktoblock network. We refer to this to interpret different dynamics between social blocks, in addition to then pairing this information with demographic information to make further evaluations about block characteristics associated with different types of influence.
In Figure 5, a subset of the sociodemographic features are displayed for each block, where the network of blocks is connected with varying degrees of influence between them. For example, we can see that lower medianage block four negatively influences the older medianage block six. The equal gender ratio block ten positively influences the similarly equal gender ratio block nine. Block ten influences block nine, where both blocks have similarly high caste diversity. Highly language diverse block six positively influences low language diverse block one. Lower professionally diverse block one negatively influences higher professionally diverse block three.
Block  Block Type  Attributes 

1  Homogeneous, lowSES  only one disadvantaged caste and one language spoken 
low profession diversity and education levels  
2  Diverse, skilled, highlyeducated  several different castes from many levels 
diverse languages and diverse, highskilled professions  
3  Senior, lowSES  majority disadvantaged caste 
majority low skilllevel professions in agriculture  
4  Young, lowSES  younger average age, gender imbalanced block 
majority lowest caste members, mostly natives  
higher education  
5  Diverse, lowSES  diverse number of disadvantaged castes 
moderate language diversity, moderate education  
majority of jobs in agriculture  
6  Senior, lowSES  older average age, diverse in low castes 
two languages spoken, very low education  
lowerskilled professions  
7  Homogeneous, lowSES  gender imbalanced, mostly disadvantaged caste 
one language majority  
majority professions in agriculture and sericulture  
8  Diverse, middleSES  mostly one language 
caste diverse but mostly lower castes  
diverse professions  
9  Diverse, highlyeducated, lowSES  disadvantaged caste majority 
diverse jobs, higherSES professions (teacher, priest)  
high education level, diverse languages  
10  Homogeneous, lowSES  genderbalanced 
majority disadvantaged caste, only one language spoken  
majority of professions in agriculture and sericulture 
Attribute  Positive influence  Negative influence  Positive selfinfluence 

Gender  similar gender distribution  genderimbalanced block is more open to negative influence from genderbalanced block  large gender imbalance 
Caste  overlapping majority castes  lack of overlap in caste composition  majority village natives 
Profession  profession overlap, in specialty jobs specifically; large professions diversity  professionally diverse block receives negative influence from a less professionally diverse block; lack of professional overlap causes a negative influence  high job diversity and higherskilled jobs 
Education  large overlap in higher education level  higher educated block receives negative influence from less educated block  higher education level 
Language  overlapping language  lack of overlap in language  language diversity 
Age  none  olderage block can receive negative influence from youngerage block  younger age 
By analyzing several examples in this manner using block characteristic composition and observing the types and patterns of influence, several general trends arise, as depicted in Table 3. The block attributes most frequently associated with different types of influence are summarized into key trends. Positive influence occurs when two blocks overlap in the following characteristics: gender distribution, majority castes, professions, high profession diversity, highly educated, highlyskilled jobs, and mother tongue languages. Negative influence frequently occurs when two blocks have a lack of overlap in the following characteristics: gender distribution, caste composition, profession diversity level, education levels, and average age. Furthermore, the direction of negative influence is most frequently observed from a lowSES block to a highSES block. Additionally, we frequently observe positive selfinfluence, which is from a block to itself, and this occurs when a block is characterized by a younger average age, highlyeducated, high job diversity, higherskilled jobs, high language diversity, large gender imbalance, and having a large number of village natives.
These trends, when paired with block type characterizations, lead to interesting associations, such as blocktoblock perceptions of lower or higher SES groups with influence. Blocks of the higher SES group designation more frequently received negative influence from lowerSES blocks. Blocks of similar SES, especially higher SES, had a more frequent positive influence between them. HighSES blocks also had more frequent positive selfinfluence.
These findings suggest some marketing strategies that take into account the underlying communities. For example, the microfinance institution could organize separate information sessions for the highSES and lowSES groups to take advantage of the positive influence between groups that share similar characteristics, while avoiding the negative influence that occurs across the different communities. Moreover, if the microfinance institution is to introduce the product into other villages (as a new product), they should send the information to individuals with the following characteristics: (1) highSES with less lowSES neighbors, (2) individuals who speak a diverse set of languages, and (3) communities with similar gender ratios.
5. Applications and future works
Role theory postulates that the interactions of individuals depend on their roles and behaviors of interest. To conceptualize this idea, we use the underlying community structures to capture the “roles”, which affect the particular decisionmaking processes of individuals. Specifically, we develop the Stochastic Block Influence Model, which infers two types of hidden relationships: (1) blocktoblock interaction, and (2) blocktoblock influence on decisionmaking. Moreover, our model flexibly allows for both positive and negative social influence. The latter is more common in practice but has been ignored by the contagion models in the literature (Centola and Macy, 2007; Kempe et al., 2003; Banerjee et al., 2013). In the adoption of microfinance examples we present, the inferred blocktoblock influence offers insights into how different social blocks exert influence on individuals’ decisionmaking. The framework has farreaching practical impacts for understanding patterns of influence across communities and identifying the crucial characteristics of influential individuals for several applications. To name a few:

Practitioners and researchers can identify the most influential communities (e.g., leaders and followers) and understand the dynamics among different communities that are not available nor observable without our model.

Marketing campaigner can investigate in which sociodemographics predict positive or negative social influence, and utilize this information when introducing the product to a new market.

Marketing firms can use the influence of each individual to decide whom to target for campaigns (Leng et al., 2018c). For example, in marketing campaigns, we should advertise to individuals who spread positive aggregate influence.

For policymakers, the behavioral model in our paper can be used to perform counterfactual predictions for network interventions to predict responses to new policies.
Our method is not without limitations and hence opens up several directions for future studies. First, future research can easily adapt SBIM to accommodate a more complicated stochastic block model, such as a degreecorrected SBM or a powerlaw regularized SBM. Second, a scalable inference method as an alternative to NUTS sampling will help to improve the efficiency and scalability of SBIM. Third, future research can extend SBIM to a dynamic model, where the influence matrix varies with time and distances from the source of information. Lastly, for computer scientists and social scientists who have access to similar types of data, but in different settings (e.g., different behaviors and collected in different countries), it will be interesting to apply and compare the influence matrices to see if there exists any generalizable pattern to support existing contagion and decisionmaking theories.
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