Knowledge Sharing in Organizations

Organizations and Social Networks

With a limited number of recent exceptions (Rank, Robins, & Pattison, 2010), studies of social networks in organizations have implicitly treated participants as members of social groups—that is, “groups that are relatively small, informal and involve close personal ties” (Freeman, 1992: 152). This assumption typically leads to empirical studies of single interpersonal networks where organizational structure plays little or no role. Even the few studies examining multiple types of ties tend to focus almost exclusively on informal social relations (Lazega & Pattison, 1999—but see Dahlander & McFarland, 2009, for a partial exception).

The implicit tendency to consider organizations as informal social groups implies the empirically implausible assumption that informal relations in organizations are autonomous from the more formal elements that define organizations as structured social settings (Dokko et al., 2013). As McEvily et al. (2014: 4) recently put it, “the surge in scholarly attention to informal social structure (…) has created a sort of amnesia about the role of formal elements in explaining the functioning, performance, and nature of organizations.” One consequence of this “amnesia” is that we can now rely on a detailed understanding of how, when, and why social networks affect organizational outcomes such as innovation, knowledge transfer, product development, and innovative performance (Argote et al., 2003; Hansen, 1999; Tortoriello & Krackhardt, 2010; Tsai, 2001; Tsai & Ghoshal, 1998). However, we still know relatively little about how organizations affect social networks and hence about how network in organizations differ from other kinds of social networks that have been extensively studied (Newman & Park, 2003). More specifically, we still know little about how formal relations of hierarchical dependence interact with informal social networks to shape knowledge sharing and transfer activities within organizations. Understanding and predicting the outcomes of this interaction is particularly important when knowledge and information have to be shared and transferred between organizational members who are both separated by administrative boundaries, as well as connected by relations of interpersonal subordination defined by corporate hierarchies linking individuals across different units. This is a common problem for matrix and multi-unit organizations similar to the one examined in this article (Kleinbaum & Tushman, 2007; Nohria & Ghoshal, 1997).

Perhaps the most obvious implication of adopting organizations as settings for studying social relations is that formal hierarchical structures constrain interaction opportunities and frequency among organizational members (Marsden & Campbell, 1984). As we discuss in the next section, this is particularly the case for social relations—such as advice relations—that are systematically associated with core processes of knowledge transfer, sharing, and development within and across formal organizational boundaries.

Advice Relations in Organizations

Organizations routinely rely on their members’ willingness and ability to mobilize their social capital for transferring capabilities (Zander & Kogut, 1995), sharing knowledge (Argote, Beckman, & Epple, 1990; Reagans & McEvily, 2003), diffusing best practices (Brown & Duguid, 2001), generating new ideas (Burt, 2004), and developing new products (Hansen, 1999). Advice relations play a central role in these and related cases where organizational participants expect to “get action” from their network partners (White, 1992).

Networks of task advice relations are generally understood as informal social conduits through which resources, knowledge, and information flow within organizations (Lazega et al., 2012; Nebus, 2006; Podolny & Baron, 1997). In their phenomenology of advice, Cross, Borgatti, and Parker (2001) argue that intra-organizational networks of advice relations are important as much as they are unavoidable because they relate directly to five fundamental organizational knowledge transfer and sharing activities. First, advice relations provide essential information relating to problems requiring integration of different kinds of knowledge and expertise. The presence of informal advice relations is not unusual because they are routinely activated during the course of regular organizational problem-solving activities (Hansen, 2002). Second, advice relations provide meta-information about the location of relevant knowledge in organizations. Advice ties, therefore, produce richer and more complex information than the resolution of the problem at hand may require. Third, advice networks are not only the “pipes” through which material and symbolic resources flow but are also “prisms” that shape perception and meaning (Podolny, 2001). As a consequence, the network of advice relations in which organizational members are embedded affects their understanding of organizational activities and may help them to make sense of decision problems they face. Fourth, advice networks encourage exchange of opinions among individuals that may be working in different organizational units, divisions, or functions (Argote et al., 1990). Fifth, advice ties may help to garner legitimation and diffuse consensus for preferred solutions to contentious organizational issues (McDonald & Westphal, 2003).

The various organizational activities that advice relations support and facilitate are common but tend to become problematic when they involve knowledge transfer and sharing between individuals separated by boundaries defined around organizational subunits (Argote, 1999; Hargadon & Sutton, 1997; Tsai, 2001). Knowledge transfer relations across the boundaries of organizational subunits are as problematic as they are important. Reagans and McEvily (2003), for example, find that the presence of connections to diverse knowledge pools (network range) facilitates knowledge transfer within organizations. Kleinbaum and Tushman (2007) suggest that social networks play an essential role in initiating cross-line-of-business relations that support innovation in multidivisional firms. According to Cohen and Levinthal (1990: 133), “interactions across individuals who each possess diverse and different knowledge structures will augment the organization’s capacity of making novel linkages and associations—innovating—beyond what any individuals can achieve.”

At least four main families of factors contribute to the difficulty of establishing and sustaining crosscutting ties within organizations. The first is that shared membership in organizational subunits provides enhanced opportunities and stronger incentives to form within-unit ties. This is the case because organizational subunits provide common social foci or social settings (Feld, 1981; Pattison & Robins, 2002) that encourage the development of familiarity (Hinds, Carley, Krackhardt, & Wholey, 2000), supply a repertoire of shared experiences (Marsden, 1988), and facilitate the shared interpretation of past events (March & Olsen, 1975). The second family of factors is related to the explicit objective of organizational design to confine major interdependences within purpose-built subunits (Thompson, 1967). The fragmentation of knowledge inherent in the successful implementation of organizational design solutions is systematically reinforced by staffing practices and internal resource allocation mechanisms (Dokko et al., 2013; Reagans & Zuckerman, 2001) that crystallize the boundaries of subunits and decrease their permeability to processes of assimilation and integration of heterogeneous resources (Tortoriello & Krackhardt, 2010). Third—and directly connected to the second—information and ideas are harder to exchange and integrate when the parties involved do not share a common knowledge base or language (Reagans & McEvily, 2003; Tortoriello et al., 2012). Consequently, organizational units may experience considerable difficulty in understanding and absorbing knowledge and ideas generated elsewhere within the organization (Cohen & Levinthal, 1990). Finally, the fourth family of factors is inherent in the cost of supporting relations across boundaries. Hansen (1999) argues that such costs are induced by the time needed to cultivate relations across subunits and the attention necessary to process information generated in different and distant units. This subjective element of cost reduces further the permeability of subunit boundaries to knowledge and information that might be available in other subunits. Our first hypothesis summarizes this discussion and provides the baseline for the predictions we develop next:

The image of organizations portrayed by Hypothesis 1 is that of a “cavemen world” described by Watts (1999: 102) where members live in dense isolated clusters (or “caves”) of strong, frequent, and redundant relations. Despite the clear-cut predictions implied by Hypothesis 1, organizations may be rarely decomposed into isolated caves: The formal boundaries of organizational units might contain the majority, but typically not all of the observed ties among their members (Cummings & Cross, 2003). As Brass et al. (2004: 801) put it, “ties between people in different units are especially intriguing, because they create ties between organizational units. … When two individuals interact, they not only represent an interpersonal tie, but they also represent the groups of which they are members.” As we have discussed, ties between people in different organizational units are intriguing not only because they connect separate units but because they involve costs, risks, and uncertain benefits. What make these “distant” ties possible? Our attempt to address this question emphasizes reciprocity as a conflict resolution device (Powell, 1990) and as an uncertainty reduction strategy that may be adopted to alleviate the problems associated with the various forms of information asymmetries that we have identified (Uzzi, 1996). Like “trust,” reciprocity may be viewed as a “policy for handling the freedom of other human agents’’ (Dunn, 1988: 73).

In the specific case of the advice network, Cross et al. (2001) suggest that reciprocity in advice relations within organizations is consistent with the interpretation of advice as an expression of social solidarity, rather than status differences. But reciprocity in advice may be expected also on the basis of more “local” (i.e., dyadic) considerations. Paraphrasing Fehr and Gächter (2000), reciprocity in advice relations may be viewed as a form of “conditional kindness” whereby advice is given under the expectation that it will be received. In network terms, this kind of restricted exchange may be one mechanism behind the assortativity (or positive degree correlation) that—according to Newman and Park (2003)—is one of the distinctive characteristics of social networks.

By creating expectations of repeated interaction within the organization, reciprocity supports joint problem solving activities and arrangements, promotes trust (Uzzi, 1997), improves the understanding of complex problems (Tortoriello & Krackhardt, 2010), reduces the risk of opportunism (Coleman, 1988), and facilitates the transfer of private information and critical knowledge resources (Gulati, Dialdin, & Wang, 2002). For these varied reasons, we would expect ties between individuals situated in different organizational units (distant ties) to be more likely if supported by general norms of reciprocity.

Thus far, we have focused on reciprocity as a specific form of dyadic network dependence that may support knowledge sharing and transfer relations across organizational subunit boundaries. But reciprocity—a structural characteristic of social relations—is not the only factor that may affect cross-cutting ties in organizations. More than anything else, organizations are hierarchical social systems explicitly designed to focus the attention of their members and shape their social interaction (March & Simon, 1958; Simon, 1962). How does the presence of hierarchical dependence affect the propensity of informal advice ties to share and exchange knowledge across the boundaries of organizational units? The limited number of available empirical studies (systematically and comprehensively reviewed in McEvily et al., 2014) does not seem to offer a unique answer. Much seems to depend on how hierarchical dependence is actually represented (again see McEvily et al., 2014: 18-19, on the different ways in which formal organization structure may be conceptualized).

In a study predating much of current debate, Ibarra (1992) proposes a broad conceptual framework for articulating possible answers to our question. Her argument hinges on the well-established distinction between relations of formal hierarchical dependence (which she terms “prescribed”) and formal social relations (which se terms “emergent”). She argues that the latter kind of relations are produced, in part, by the former as a natural consequence of the activities that interdependent individuals must perform to manage task uncertainty, make sense of their work experiences, and get work done. If this view of “formal” organizational structure as a possible source of social organization were correct, then we would expect the presence of interpersonal relations of hierarchical subordination to be positively associated with the presence of informal advice relations. Rank et al. (2010) find evidence in support of this conjecture in a recent empirical study of social networks between managers in two German multinational companies. According to this study, the presence of a hierarchical relation between two managers creates the setting for the development of informal social relations (Robins, Pattison, & Wang, 2009). More specifically, Rank et al. (2010) find that the existence of a formally mandated cooperative tie between managers enhanced the probability that these managers actually exchanged information and advice. This result is generally supportive of Ibarra’s (1992) conjecture about the (positive) association between hierarchical and social ties in organizations. Such association may be expected also because relations of hierarchical dependence tend to induce the need for activities of information search, knowledge identification and integration, opinion formation and diffusion, sense making, and consensus building. As we have discussed, these are precisely the organizational activities that, according to Cross et al. (2001), are typically performed by social networks of advice relations. Extending this reasoning, we predict that advice relations will play a particularly important role in supporting interpersonal relations of hierarchical subordination across the boundaries of organizational units.

Clearly, it may well be the case that, in certain circumstances, hierarchical relations subsume the functions of advice relations, thus making them unnecessary. Or it could be the case that cultural norms make advice relations between “bosses” and “subordinates” inappropriate or impossible outside the context of hierarchical subordination. These and other possibilities ultimately lead to new empirical questions that can be entertained only if “mandated” and “emergent” relations in organizations are represented jointly, and if hypotheses can be tested about the effect of their interaction. This is precisely what we do in the empirical part of the article.

Empirical Setting

The empirical part of the study is based on original fieldwork and data that we have collected on relations among members of the top management team and corporate consultants in an international multi-unit industrial group (Lomi, Lusher, Pattison, & Robins, 2014). The effectiveness of multi-unit organizations hinges crucially on their ability to facilitate information sharing, knowledge transfer, relational coordination, and mobilization of dispersed human resources across units separated by formal organizational boundaries (Bartlett & Ghoshal, 1989; Hansen, 2002; Tsai, 2001). For this reason, understanding social networks is of considerable importance for understanding how multi-unit organizations can be managed effectively (Nohria & Ghoshal, 1997).

The corporate group selected for study includes five separate, quasi-independent companies involved in the design, manufacturing, and sale of high-quality products in the global market for leisure motor yachts. The central company in the group plays the double role of independent company and corporate headquarters for the whole group. In external corporate communications, membership of the companies in the corporate group is not hidden, but the companies are presented as independent, each with its distinct organizational and product identity, target market segment, customer base, dealer network, and management.

Sample and Data

We started our study by asking the President and the CEO of the group to examine the corporate organizational chart and identify the individuals they considered members of the “top management” team. We arrived at a list of 47 people distributed across the five different companies. Five individuals in the list were corporate consultants working in internationally prominent yacht design and architecture firms. Following the CEO’s suggestion, they are included in the sample because of their direct personal involvement in important product design and development decisions. We collected relational and demographic information by means of a questionnaire personally and individually administered to all the 47 top managers in the group (including the five consultants). Examples of demographic information collected include individual educational experience, age, professional experience, formal status, and membership in functional groups and in organizational functions. The research team visited each company. During the data collection, the research team spent a considerable amount of time in close contact with the management of each of the five companies in the group. The response rate was 100%.

We focus on task advice relations because extant research has demonstrated that advice ties provide a meaningful basis for understanding important aspects of knowledge sharing and knowledge transfer in organizations (Podolny & Baron, 1997). Although almost completely independent, the subsidiary companies in the group share frequently at least three kinds of information: technical, commercial, and accounting. Sharing technical information involves discussions among the engineers in different subsidiaries about the possibility of standardizing phases of the production process, or aspects of boat design. Sharing commercial information involves, for example, cross-line-of-business discussions within marketing and sales about the trustworthiness of specific dealers or about the value of potential customers. Sharing accounting information was obviously necessary because of the common requirements imposed by the need to establish and control transfer pricing policies and consolidate the organization-specific accounting systems at the corporate level. More generally, sharing information through advice relations between managers in different subsidiaries is facilitated by the small degree of overlap in the market segments covered by the individual subsidiaries.

As it is common in research on interpersonal relations within organizations, we collected information on social networks among managers using the so-called roster method (Kilduff & Krackhardt, 2008). Each respondent was presented with a list containing the names of the other 46 individuals in the sample arranged in alphabetical order and asked to indicate the existence of help and advice relations with each of them. The structured questionnaire was administered personally and individually to each manager. Interviewers, however, also prepared a list of concrete “questions and problems” that would help to root “advice relations” more firmly in the specific business and organizational context. Examples of issues included the evaluation of potential customers, pricing, and flexibility with terms of payment, production delays, and the relation between production costs of product customization. The advice network may be represented as a 47 × 47 binary adjacency matrix recording the presence or absence of advice relations for each possible pairs of individuals in the sample. The advice activity observed is relatively intense: On average, every manager entertains approximately 10 advice ties. Approximately 25% of all the advice ties observed are reciprocated.

To test our hypothesis that knowledge transfer across organizational units is facilitated by the presence of relation hierarchical subordination (Hypothesis 3), we also collected information on the formal organizational hierarchy in the form of formal interpersonal hierarchy of boss–subordinate relations at the dyadic level. For example, the managers responsible for Marketing and Sales working in individual subsidiary companies within the group all reported to the Corporate Vice-President for Marketing and Sales. Information on the network of official interpersonal hierarchical relations was provided directly by the CEO of the group during a series of interviews.

Model Specification

To link our arguments to appropriate statistical models, we consider each potential network tie between organizational members as a random variable. By considering each individual network tie as a random variable, we link our data structure directly to a class of ERGMs (Robins et al., 2009; Snijders et al., 2006; Wasserman & Pattison, 1996).

ERGMs assume that the topological structure of an observed network y can be explained in terms of the relative prevalence of a set of overlapping subgraph configurations s(y) called graph or network statistics (see Robins, Snijders, Wang, Handcock, & Pattison, 2007, for a recent review). Each configuration is assumed to have a particular probability of being observed in the given network: the higher the probability of being expressed in the graph is, the higher the chance of that statistic to occur is and vice versa. The probability of a configuration being present in the network is expressed in terms of parameters or “effects.” Configurations with a positive parameter value have a greater than chance probability of being observed in any graph represented by the model and vice versa. Configurations and parameters are at the core of ERGMs, and from a statistical point of view, the challenge is to estimate the parameters for each statistic such that the model is a good fit for the given data.

From a statistical point of view, networks are relational data that may be represented mathematically as graphs. A graph consists of a set of n nodes and a set of m ties that define some sort of relationships between pair of nodes called dyads. The connectivity pattern of a graph can be described by an n × n adjacency matrix y encoding the presence or absence of a tie between node i and j: Y_ij = 1 if the dyad (i, j) is connected, Y_ij = 0 otherwise. The likelihood of an ERGM represents the probability distribution of a random network graph and may be expressed as:

p ( y | θ ) = exp { θ t s ( y ) } z ( θ ) .

This equation states that the probability that an observed graph y given the set of parameters θ is equal to the exponent of θ multiplied by the observed graph statistics s(y) divided by a normalizing constant z(θ)—a term by which the exponential function in the numerator must be multiplied so that the distribution defined in equation 1 is a proper probability density function. In the ERGM context, the normalizing constant is calculated as the sum over allpossible network graph configurations on n nodes (which are 2 ( n 2 ) for undirected networks). This sum is therefore extremely difficult to evaluate in practice for networks with more than 20 nodes due to computational complexity. Standard inferential techniques cannot be used in this context.

The basic units for exponential random graph analysis are the subgraph configurations that are a subset of nodes and all the ties between them. Dyads and triads are important structures as they provide local components—or “motifs”—underlying the entire network graph (Milo et al., 2002). In a directed network, for example, an important dyadic network measure is the density of ties and mutual ties between the actors. Dyadic analysis provides important information regarding the study of fundamental effects such as connectivity, popularity, reciprocity, and transitivity that are known to represent general features of social networks (Newman & Park, 2003). ERGMs are unique in that they allow accounting for structural features of social networks, while at the same time providing an inferential framework to test hypotheses about how characteristics of the nodes affect network ties (Lusher, Koskinen, & Robins, 2013). This specific feature makes ERGMs particularly useful in research in interorganizational as well as interpersonal networks (Lomi & Pattison, 2006; Srivastava & Banaji, 2011).

Bayesian Models for Social Networks

The Bayesian approach to statistical inference—named after Reverend Thomas Bayes (1763)—is based on the estimation of posterior probability that is the conditional probability of unknown quantities given the observations. The posterior distribution extracts the information in the data and provides a complete summary of the uncertainty about the unknowns θ (Howson & Urbach, 1993). Prior knowledge about the parameters is summarized by the density p(θ), the likelihood is p(y|θ), and the updated knowledge is contained in the posterior density p(θ|y) as defined by the Bayes theorem:

p ( θ | y ) = p ( y | θ ) p ( θ ) p ( y ) .

The relative influence of the prior and data on updated beliefs a posteriori depends on how much weight is given to the prior (how “informative” the prior is) and the strength of the data. The marginal likelihood or model evidence p(y) is the probability of observing the data y given a specific model under study—that is, the probability that the stochastic structure of the observed network data is explained by the specified model. Unfortunately, marginal likelihoods are generally difficult to compute and exact solutions are only known for a small class of distributions. Bayesian analysis has many advantages: It provides rich diagnostic information about the unknown quantities, controlling for exploration of prior assumptions about model parameters. Bayesian inference uses probability intervals (quantile-based highest posterior density) called credible intervals to state the probability that the parameter is between two points. Classical or “frequentist” approaches are not able to predict that the true value of the parameter has a particular probability of being in the estimated confidence interval given the data actually obtained. In summary, the three steps of Bayesian inference may be summarized as follows: (i) specify a probability model for unknown parameter values that includes some prior knowledge about the parameters, (ii) update knowledge about the unknown parameters by conditioning this probability model on observed data, and (iii) evaluate the fit of the model to the data and the sensitivity of the conclusions to the assumptions. The purpose of Bayesian inference for ERGMs (Koskinen, Robins, & Pattison, 2010) is to learn about the posterior distribution (equation 2) of the ERGM parameters θ of an observed network graph y on n nodes:

p ( θ | y ) = exp { θ t s ( y ) } z ( θ ) p ( θ ) p ( y ) .

Equation 3 provides a probabilistic statement about how likely parameter values are after observing the data y. The likelihood p(y|θ) is translated into a proper probability distribution that can be summarized by computing expected values, standard deviations, and quantiles. In order to carry out Bayesian inference for ERGMs, we make use of the approximate exchange algorithm that circumvents the problem of computing the normalizing constants of the ERGM likelihoods (Caimo & Friel, 2011). This analysis has been carried out (and may be replicated) using the freely available Bergm package for R (Caimo & Friel, in press). In this article, we use vague prior distributions (i.e., densities with high spread) for all the ERGM parameters—namely, Gaussian distributions with mean 0 and large variance 100—in order to allow the network data to drive inference. The reason for this choice is to ensure that the results produced by our Bayesian approach will not be excessively influenced by the choice of the prior distribution so that they can be compared to classical (“frequentist”) results.

Variables and Measures

We specify and estimate models that express the probability of observing a network tie between two individuals as a function of (i) membership in various organizational units (“companies”), (ii) individual characteristics, and (iii) characteristics of the advice network constructed by the social interactions of the managers. Table 1 reports the description of each network’s statistics that we compute in order to model the observed network. The parameter of main theoretical interest is associated with joint membership in the various firms within the corporate group.

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Table 1

Description of the Network Statistics Used in the Models Proposed

Statistic (effect)	Description	Interpretation
s1 (Ties)	Number of ties in the network	Overall density of ties
s2 (Reciprocity)	Number of mutual ties yij, yji	Overall density of mutual ties
s3 (Reciprocity—different membership + hierarchy)	Number of mutual ties yij, yji, for which membership(i) ≠ membership(j) and i reports to j	Density of mutual ties between nodes with hierarchical relations and same membership
s4 (Sender—different membership + hierarchy)	Number of out-ties for which membership(i) ≠ membership(j) and i reports to j	Activity of the nodes with hierarchical relations and same membership
s5 (Receiver—different membership + hierarchy)	Number of in-ties for which membership(i) ≠ membership(j) and i reports to j	Popularity of the nodes with hierarchical relations and same membership
s6 (Reciprocity—subunit membership)	Number of mutual ties yij, yji, for which membership(i) = membership(j)	Density of mutual ties between nodes with same membership
s7 (Same gender)	Number of ties yij, for which gender(i) = gender(j)	Density of ties between nodes with same gender
s8 (Homophily—subunit membership)	Number of ties yij, for which membership(i) = membership(j)	Density of ties between nodes with same membership
s9 (Same nationality)	Number of mutual ties yij, yji, for which nationality(i) = nationality(j)	Density of ties between nodes with same nationality
s10 (Same location)	Number of ties yij, for which location(i) = location(j)	Density of ties between nodes with same location
s11 (Same prior experience)	Number of ties yij, for which experience(i) = experience(j)	Density of ties between nodes with same experience
s12 (Same function)	Number of ties yij, for which function(i) = function(j)	Density of ties between nodes with same function
s13 (Popularity)	Geometrically weighted in-degree distribution with weight parameter set to log(2)	Tendency toward centralization in in-degree distribution
s14 (Activity)	Geometrically weighted out-degree distribution with weight parameter set to log(2)	Tendency toward centralization in out-degree distribution
s15 (Multiple connectivity)	Geometrically weighted nonedgewise shared partner distribution with weight parameter set to log(2)	Tendency of nondirectly connected nodes to be connected through multiple others
s16 (Multiple closure—clustering)	Geometrically weighted edgewise shared partner distribution with weight parameter set to log(2)	Tendency to be connected through multiple triadic relations simultaneously

We want to arrive at a characterization of the effects that joint membership has on the propensity to constrain knowledge-sharing relations within the units. In our empirical model specifications, we also control for a number of known effects of individual differences on individual propensities to create network ties. More specifically we control for difference between the members of our organization observed in terms of (i) same gender (statistic s₇), (ii) subunit membership (statistic s₈), (iii) same nationality (statistic s₉), (iv) same location (statistic s₁₀), (v) some prior experience (statistic s₁₁), and (vi) same function (statistic s₁₂).

Participants are distributed across five organizational subunits. In the sample, 85.1% of the managers are males, 87.2% are Italian (approximately 66% of the overall productive capacity of the group is concentrated in Italy), and the average number of years of prior experience in the group was approximately 8 years. Approximately 38% of the participants worked in production and engineering, 15% in finance and accounting, and 10% in marketing. The remaining participants were members of smaller professional families represented within the group. Together, these potential sources of homophily control for the main individual factors that may affect the tendency of knowledge flows in organizations to be contained within intraorganizational boundaries (McPherson, Smith-Lovin, & Cook, 2001). Rivera, Soderstrom, and Uzzi (2010) provide a comprehensive review of the literature on homophily as a basis for the formation of dyadic social relations.

Following established empirical ERG modeling practice (Srivastava & Banaji, 2011), characteristics of the advice networks that we explicitly represent in our model include the baseline tendency of social networks toward reciprocity (statistics s₂, s₃, s₆), centralization (statistics s₁₃, s₁₄), multiple connectivity (statistic s₁₅), and clustering (statistic s₁₆). Centralization is often the result of preferential attachment processes and is associated with differential propensities to be the source, or target, of particular types of ties leading to star-like local configurations. Because the number of network “stars” is a function of the degrees, including parameters corresponding to star-like configurations is equivalent to modeling the degree distribution (Snjiders et al., 2006). We capture core features of the (in and out) degree distributions by controlling for heterogeneity in relational activities revealed by differences in the propensity of individuals to be selected as partner by many others (popularity effect described by s₁₃) and in the propensity of individuals to select multiple others as partners (activity effect described by s₁₄).

Clustering (multiple closure effect described by s₁₆) will occur in a social structure where the individuals tend to collaborate in small informal groups with team-like structures. Tendencies toward closure that may be present in the data are captured by (i) (generalized) cyclic closure where cycles of three arcs tend to be present in the network and (ii) a generalized effect for transitive closure where different types of transitive triads tend to be present in the network. The rationale behind various forms of triadic closure is discussed in Robins et al. (2009). As a counterpart to the various mechanisms of triadic closure, we also include a parameter for nonclosure (multiple connectivity effect described by s₁₅) where two actors are connected by multiple open two-paths, an effect we call multiple connectivity. Multiple connectivity effects may indicate the presence of many opportunities for brokerage in the network (Burt, 2005). As explained in Robins et al. (2009), inclusion of a multiple connectivity parameter sharpens inferences about the presence of closure in the network. Again, it may be worth noting that multiple connectivity implies the presence of extratriadic dependencies in the data. A complete list of mathematical definitions of the network statistics is reported in the Appendix.

Our empirical analysis is based on two models: Model 1 includes the first 12 statistics defined in Table 1. This is called a “dyadic independence” model, meaning that the statistics included in the model consider each dyad of the network independent of the others. Model 2 includes all the statistics of Model 1 plus four “endogenous” network statistics (s₁₃, s₁₄, s₁₅, s₁₆) that relax the dyadic independence assumption underlying Model 1 as they concern relations between three or more nodes of the network.

Model Estimation and Interpretation

The main objective of Bayesian inference is the estimation of the posterior parameter distribution. This probability distribution provides information about the parameters after observing the data. We can then calculate the credible intervals of the estimated posterior distribution for each parameter. For a given significance level (that is generally set to 0.95), credible intervals represent the range of values that are likely to be taken by the parameters of a model given the data we have observed. In other words, credible intervals are constructed so that the probability of a certain parameter to take value within the interval (marked by the 2.5% quantile and 97.5% quantile of the posterior distribution) is 0.95. Credible intervals and posterior means are quantities describing the impact of the effects included in the ERGM and can take into account the model-specific information provided by the prior distribution. The value of an ERGM parameter offers an indication on whether certain relational effects (expressed by the network statistics) are more or less frequent than expected by chance. This means that, for example, a credible interval for a certain parameter that includes only positive values indicates that the effect (or statistics) associated to that parameter is common in the observed network. Moreover, given the posterior distribution, it is possible to calculate the posterior correlation between the parameters in order to get additional insights about the statistical dependence between the estimated variables. The possibility of calculating correlations among the parameters is a unique feature of a Bayesian approach according to which “parameters” are “variables.”

Unfortunately, the posterior distribution (equation 3), as discussed above, is “doubly-intractable,” as both the likelihood normalizing constant z(θ) and model evidence p(y) cannot be evaluated analytically. This makes the use of standard inferential algorithms (such as naïve MCMC algorithms used in ERGMs) infeasible. In order to carry out Bayesian inference for ERGMs, we rely on the approximate exchange algorithm that circumvents the problem of computing the normalizing constant of the ERGM likelihood (Caimo & Friel, 2011). From the posterior density estimates, it is possible to calculate the posterior correlation matrix between each pair of parameters. This correlation helps to understand the statistical dependence between the parameters of the model. Because in Bayesian analysis parameters are themselves variables, the posterior correlation matrix provides further information concerning the uncertainty about the dependence between network effects included in the model.

Results

The figures reported in Table 2 and Table 3 show that both models provide evidence of a strongly negative density effect (parameter θ₁associated to number of ties). This result is consistent with the relatively low density of the advice network and with the fact that advice ties are costly and hence relatively infrequent. It is important to notice that given the definition of the model in equation 1, the value of an ERGM parameter represents the conditional log-odds of a tie. For example, the log-odds that a tie between two nodes will form in the network equal the parameter θ₁. This is equivalent to saying that the probability of a tie between any pair of nodes i and j (having fixed all the other parameters values) is p ( y i j = 1 | θ 1 ) = e x p ( θ 1 ) 1 + e x p ( θ 1 ) If θ₁ < 0, the probability of observing a tie is p ( y i j = 1 | θ 1 ) < 0 . 5 Vice versa, if θ₁ > 0, then p ( y i j = 1 | θ 1 ) > 0 . 5 In Model 1 (Table 2), we obtained a negative credible interval for Ties (θ₁: –3.44, –2.26) meaning that, all other conditions being equal, the probability of observing an advice tie varies between 0.03 and 0.09.

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Table 2

Summary of the Posterior Distribution of the Parameters for Model 1

Parameter (effect)	M	2.5% quantile	Mdn	97.5% quantile
θ1 (Ties)	−2.84	−3.44	−2.83	−2.26
θ2 (Reciprocity)	1.52	0.98	1.52	2.07
θ3 (Reciprocity—different membership + hierarchy)	2.32	0.48	2.24	4.60
θ4 (Sender—different membership + hierarchy)	1.27	−0.38	1.25	3.15
θ5 (Receiver—different membership + hierarchy)	1.45	−0.18	1.41	3.39
θ6 (Reciprocity—subunit membership)	−0.49	−1.45	−0.49	0.50
θ7 (Same gender)	0.33	−0.04	0.33	0.72
θ8 (Homophily—subunit membership)	1.30	0.78	1.30	1.87
θ9 (Same nationality)	0.25	−0.29	0.25	0.78
θ10 (Same location)	0.53	0.07	0.53	0.99
θ11 (Same prior experience)	−0.14	−0.48	−0.14	0.21
θ12 (Same function)	0.45	0.04	0.45	0.86

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Table 3

Summary of the Posterior Distribution of the Parameters for Model 2

Parameter (effect)	M	2.5% quantile	Mdn	97.5% quantile
θ1 (Ties)	−6.31	−7.81	−6.29	−4.93
θ2 (Reciprocity)	0.87	0.18	0.86	1.61
θ3 (Reciprocity—different membership + hierarchy)	2.77	0.33	2.71	5.50
θ4 (Sender—different membership + hierarchy)	1.18	−0.83	1.12	3.51
θ5 (Receiver—different membership + hierarchy)	1.67	−0.28	1.61	3.99
θ6 (Reciprocity—subunit membership)	−0.30	−1.44	−0.31	0.81
θ7 (Same gender)	0.22	−0.22	0.21	0.66
θ8 (Homophily—subunit membership)	0.83	0.21	0.82	1.48
θ9 (Same nationality)	0.18	−0.46	0.18	0.78
θ10 (Same location)	0.27	−0.27	0.27	0.82
θ11 (Same prior experience)	−0.03	−0.44	−0.03	0.40
θ12 (Same function)	0.51	0.04	0.51	1.02
θ13 (Popularity)	3.94	−0.29	3.76	9.16
θ14 (Activity)	1.70	−0.71	1.71	4.11
θ15 (Multiple connectivity)	−0.13	−0.19	−0.13	−0.06
θ16 (Multiple closure)	1.95	1.39	1.95	2.57

Positive credible intervals are obtained for reciprocity (θ₂: 0.98, 2.07), reciprocity – different membership + hierarchy (θ₃: 0.48, 4.60), subunit membership (θ₈: 0.78, 1.87), same location (θ₁₀: 0.07, 0.99), and same function (θ₁₂: 0.04, 0.86). The posterior distributions of the effects of same gender (θ₇), same nationality (θ₉), and same prior experience (θ₁₁) lace a significant amount of probability to 0 (as 0 falls inside their credible intervals), meaning that with significant probability the network effects associated with these parameters are not less or more frequent than what would be expected by chance. In a non-Bayesian statistical framework, these effects would be declared not statistically significant.

In terms of our hypotheses, estimates of subunit membership (θ₈) provide clear support for Hypothesis 1: Sharing knowledge by establishing advice relations is much more likely to be observed between managers within, rather than across, organizational units. More specifically, ties between members of the same subsidiary unit are 3.2 times more likely to be observed than ties between managers in different subsidiary units. The results obtained also suggest that reciprocity (expressed by θ₂) is an important general feature of the network but not between members of the same groups (expressed by θ₆). This result deserves attention because it reveals a tendency toward hierarchization of the advice network within the units. Considered together, the estimates of θ₂ and θ₆ provide clear support for Hypothesis 2: Reciprocity is an important feature of knowledge sharing relations between members of different organizational units—over and above a generic tendency toward reciprocity. Hypothesis 3 is also supported: Members of different organizational units are more likely to be connected by reciprocated advice relations if they are also connected by formal relations of hierarchical subordination. The corresponding effect of reciprocity – different membership + hierarchy is expressed by parameter θ₃. This parameter captures the joint effect on the presence of network ties between managers of having a reciprocated advice relations (reciprocity), being in different subsidiary units (different membership), and being connected by hierarchical dependence (hierarchy).

As a robustness check, we distinguish further between “sending” an advice tie, or asking advice across organizational boundaries (θ ₄ ), and “receiving” an advice tie, or being asked for advice (θ₅). The credible intervals for either parameter include zero, which means that the corresponding (directional) effects are indistinguishable from zero. In other words, managers connected by a relation of hierarchical subordination extending across organizational subunits are unlikely to establish directed advice relations. Managers do not ask for advice to colleagues in different units (and neither are they being asked) unless the relation is reciprocated. We take this as further evidence of the fundamental importance of reciprocity in sustaining boundary-crossing ties within organizations, particularly under conditions of hierarchical dependence.

In Model 2 (Table 3), the presence of the endogenous network effects has an important impact in explaining the overall network structure. In particular, parameters corresponding to multiple connectivity (θ₁₅) and multiple closure (θ₁₆) have a negative and positive credible interval, respectively. This means that a multiple closure effect is a common feature of the network structure: Knowledge transfer relations tend to form through closure of multiple triads simultaneously.

The multiple connectivity effect is less common than would be expected by chance: It is relatively unlikely to find triadic relations that are not closed at the base. More qualitatively, brokerage is a relatively uncommon feature of the knowledge sharing network that we have observed. The posterior density plots of the main effects of interests are displayed in Figure 1.

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Figure 1

Posterior Parameter Distributions for the Main Effects Under Study in Model 2

The inclusion of endogenous network statistics (Model 2) entails different estimates for the dyadic independent statistics. For instance, the credible interval for the parameter θ₁ in Model 2 is now (–7.81, –4.93). This estimate implies that the probability of observing a tie in Model 2 is 0.002. The decrease in the probability of observing a tie from Model 1 to Model 2 is determined by the fact that, in the latter model, the structure of the network is well explained by the endogenous network effects. In other words, the presence of random ties not embedded in network subgraph configurations defined by the endogenous network statistics is comparatively less likely in Model 2. We can also notice that the credible interval for parameter θ₁₀ (same location) in Model 2 is now (–0.27, 0.82) placing a significant probability on 0. This means that the covariate effect of same location can be expressed through network ties involved in endogenous configurations. This result provides an important insight because it suggests that knowledge sharing in organizations is affected by social rather than physical distance. Once social distance is properly accounted for, the effects of proximity on the presence of advice relations disappear. We note that out hypotheses continue to receive solid support after controlling for endogenous network structure (Model 2).

A unique aspect of our analysis that is specific to the Bayesian approach concerns the correlation between the estimated parameters. This information provides a better understanding of the structure of dependence existing between the various effects included in the model. The correlation matrix is computed on the estimated posterior distributions of the parameters. In Table 4, we can observe the correlations between the parameter estimates for Model 1. The strongest (negative) correlations are between θ₆ and θ₈ (–0.64) (representing, respectively, reciprocity and subunit membership) and between θ₉ and θ₁₀ (–0.60) (representing, respectively, same nationality and same location).

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Table 4

Parameters Correlation Matrix for Model 1

	θ1	θ2	θ3	θ4	θ5	θ6	θ7	θ8	θ9	θ10	θ11
θ2	−0.19
θ3	−0.03	−0.04
θ4	−0.01	0.03	0.49
θ5	−0.02	0.01	0.59	0.37
θ6	0.20	−0.53	−0.05	0.06	0.05
θ7	−0.56	−0.04	0.01	−0.01	0.00	−0.03
θ8	−0.19	0.23	−0.06	−0.05	−0.09	−0.64	0.03
θ9	−0.52	0.03	−0.02	−0.02	0.00	−0.09	−0.04	0.15
θ10	0.01	−0.05	0.04	0.03	0.01	0.05	0.10	−0.25	−0.6
θ11	−0.47	0.02	0.03	−0.02	0.00	−0.07	0.10	0.05	0.2	−0.23
θ12	−0.21	−0.08	−0.06	−0.03	−0.03	0.01	0.07	0.08	0.04	0.02	−0.02

In Table 5, we report the correlations between the parameter estimates for Model 2. In this case, we note the presence of an additional strong negative correlation is between parameters θ₁ and θ₁₆ (–0.87), meaning that the sign of the multiple closure effect depends negatively on the density of the network. In other words, there is a tendency of establishing a few advice ties across the whole network. In spite of this, there is a considerable tendency of these ties to self-organize into clusters. This result suggests that the observed structure of the knowledge sharing network within the organization is driven more strongly by closure, rather than preferential attachment mechanisms. The credible intervals for the effects associated to popularity (θ₁₃) and activity (θ₁₄) contain zero. A possible interpretation of these estimates is that the advice network shows no significant tendency toward centralization in the in- and out-degree distribution. Asking and being asked for advice do not seem to be activities that are disproportionately accounted for by a limited number of members within the top management team.

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Table 5

Parameters Correlation Matrix for Model 2

	θ1	θ2	θ3	θ4	θ5	θ6	θ7	θ8	θ9	θ10	θ11	θ12	θ13	θ14	θ15
θ2	0.13
θ3	−0.01	−0.04
θ4	−0.01	0.05	0.48
θ5	−0.05	0.03	0.60	0.35
θ6	0.04	−0.55	−0.06	0.00	0.06
θ7	−0.24	−0.01	−0.01	0.00	0.01	−0.01
θ8	0.02	0.20	−0.08	−0.05	−0.11	−0.59	0.00
θ9	−0.24	0.03	−0.02	0.01	−0.03	−0.05	−0.04	0.10
θ10	−0.01	−0.02	0.03	0.02	0.06	0.02	0.12	−0.20	−0.63
θ11	−0.27	0.00	0.03	0.01	0.00	−0.03	0.13	0.02	0.17	−0.13
θ12	−0.15	−0.16	−0.03	−0.05	−0.05	0.06	0.00	0.03	0.01	0.01	0.00
θ13	−0.28	0.02	0.04	0.01	0.04	−0.04	−0.01	−0.03	0.09	0.00	−0.03	0.07
θ14	−0.46	−0.07	−0.09	−0.04	0.01	0.01	0.05	−0.15	0.02	0.02	−0.03	0.03	0.23
θ15	0.12	−0.06	−0.09	0.01	−0.06	0.05	−0.02	0.15	0.00	0.02	0.10	−0.05	−0.36	−0.44
θ16	−0.87	−0.24	0.02	−0.01	0.04	0.04	−0.02	−0.10	0.02	−0.02	0.02	0.09	0.25	0.51	−0.25

Goodness-of-Fit Diagnostic Analysis

An important contribution of this article is to show how a Bayesian approach to the analysis of advice networks in organizations reveals salient (global) aspects of network structure that alternative approaches are unable to discover. In the Bayesian approach that we develop to the evaluation of the model goodness of fit, the observed network y is compared to a set of networks simulated from the estimated model (i.e., the estimated posterior distributions obtained from the empirical parameter estimates). This comparison is carried out in terms of general distributions of network statistics in order to check how well the estimated model is able to reproduce networks resembling the general structural features of the observed network. Network statistics commonly used in the statistical analysis of social network to describe network structure include (i) the in-degree distribution—the distribution of incoming ties for each node of a directed network; (ii) out-degree distribution—the distribution of outgoing ties for each node of a directed network; (iii) the minimum geodesic distance distribution—the distribution of the length of shortest path distance between two nodes; and (iv) the edgewise shared partner distribution—the distribution of the number of unordered pairs of connected nodes having exactly k common neighbors (Hunter, Goodreau, & Handcock, 2008). The diagnostic goodness-of-fit analysis consists of comparing the goodness-of-fit distributions of the observed network with the set of goodness-of-fit distributions calculated on networks simulated from the estimated model.

Bayesian goodness-of-fit diagnostics plots are displayed in Figure 2 and Figure 3. The red lines represent the goodness-of-fit distributions of the observed data. The boxplots represent the goodness-of-fit distributions calculated on 100 network graphs simulated from the estimated posterior distribution. In other words, a sample of 100 parameter values is drawn from the estimated posterior distribution, and then one network graph is simulated from each of the 100 parameters of the posterior sample drawn. The solid light grey lines displayed in Figure 2 and Figure 3 mark the 95% interval. An estimated model is fitting perfectly a certain observed network if the red line falls inside this interval—a result that is very difficult to obtain in practice.

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Figure 2

Bayesian Goodness-of-Fit Diagnostics for Model 1

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Figure 3

Bayesian Goodness-of-Fit Diagnostics for Model 2

Model 2 does a much better job than Model 1 at fitting the observed network. This is particularly clear when we compare the fit of the two models in terms of the minimum geodesic distance distribution and edgewise shared partners distribution: The red line in Figure 3 (Model 2) is almost always inside the 95% interval, whereas in Figure 2 (Model 1) the red line falls outside the 95% interval for many network statistics. As a numerical basis for the evaluation of the goodness of fit of the two models, we count the number of observed network statistics (red line) that are falling outside the 95% interval (grey lines). The lower the number of outliers, the better the model fit. Figure 2 (Model 1) illustrates that the number of statistics of the observed network falling outside the 95% interval is 7 for the in-degree distribution, 6 for the out-degree distribution, 2 for the minimum geodesic distance distribution, and 17 for the edgewise shared partners distribution.

Figure 3 (Model 2) shows that the number of statistics of the observed network falling outside the 95% interval is considerably lower: 3 for the in-degree distribution, 1 for the out-degree distribution, 0 for the minimum geodesic distance distribution, and 4 for the edgewise shared partners distribution. The better fit of Model 2 to high-level transitivity measures (described by the edgewise shared partner distribution) is due to the presence of the endogenous network statistics s₁₃, s₁₄, s₁₅, s₁₆ that are included in the model. In more general terms, the results of the goodness-of-fit diagnostic analysis that we have reported demonstrate that the estimates of Model 2 support our hypotheses and allow us to reconstruct with accuracy important structural features of the global network of knowledge sharing and transfer relations that we actually observed.