Reinforced structural holes

Holes in social structure are variably reinforced by the social organization around the hole. The more reinforced the hole, the greater the difficulty in bridging it, but the more likely a successful bridge will carry information novel, and so potentially valuable, to people on the other side. To study how reinforcement varies with access to structural holes, and the achievement associated with access, I propose a measure of access to reinforced structural holes (RSH), and present results predicting achievement in an integrated banker organization and a balkanized supply-chain organization. In both study populations, the people who have access to structural holes also have access to reinforced structural holes, and all measures of access have a statistically significant association with achievement. However, there is no consistent prediction advantage from incorporating reinforcement in measures of access to structural holes. The reinforced-holes measure predicts compensation better or as well as network constraint and betweenness, but is weaker or no better than a count of nonredundant contacts. I do not infer from the results a rank-order of alternative measures so much as substitutability. I expect achievement to be associated with access to structural holes, but I expect the association to vary across alternative measures depending on how achievement is achieved in a specific population. © 2015 Elsevier B.V. All rights reserved. You are at a cocktail party. The hostess smiles, grabs you by the arm, and introduces you to someone, highlighting an interest she believes you two have in common. The hostess veers off to link up other people, leaving you and your new acquaintance to delight in hostess-highlighted mutual interest. You just experienced network brokerage. You have your social circle. Your new acquaintance has his. The hostess has facilitated connection across the structural hole between yours and his. This paper is about situations one step more complicated. Suppose the other person is engaged in animated conversation with two colleagues. The hostess interrupts their conversation to introduce you. The polite thing to do—in deference to the hostess—would be for your new acquaintance to disengage from his colleagues to strike up conversation with you. But suppose the pull of their conversation is such that he does not break away. You are now the odd man out; their conversation continues with you the peripheral observer. This second situation is an example of what can be termed brokerage across a “reinforced” structural hole; the disconnection between you and your new acquaintance is reinforced by connections among he and his colleagues, and their mutual disconnect from you. The hole would be more reinforced if you had your own colleagues with you, to whom you returned after being slighted ∗ Tel.: +1 312 953 4089. E-mail address: [email protected] by the new acquaintance, and still more reinforced if there was a history of such slights between your and his colleagues. The structural hole between groups is reinforced by coordination within each group to the exclusion of the other. At some point, each group becomes a reference point for the other, with stereotypes about the other group made concrete in stories about those people. Network models of bridging structural holes typically ignore reinforcement, despite the fact that the social dynamic of the odd man out is familiar in everyday life, and in academic discussion such as Durkheim (1933 [1893], p. 102) on group solidarity enhanced through shared distain of an outsider, or Caplow (1968) on “two against one.” Popular network predictors measure access to structural holes without regard to reinforcement. For example, given the network around a person, ego, Freeman’s (1977) betweenness measure is a count of the structural holes to which ego has exclusive access. Burt’s (1980, 1992) network constraint and effective size variables measure the concentration of connections in redundant contacts, thus measuring ego’s lack of access to structural holes. Reinforcement around the structural holes to which ego has access is defined by the network around each of ego’s contacts (e.g., those colleagues of the new acquaintance to whom the hostess introduced you), but those networks around contacts are not essential to Freeman’s betweenness measure, and typically ignored in Burt’s constraint and effective size measures. There are at least three reasons to continue ignoring reinforcement. First, current measures of access to structural holes ignore http://dx.doi.org/10.1016/j.socnet.2015.04.008 0378-8733/© 2015 Elsevier B.V. All rights reserved. 150 R.S. Burt / Social Networks 43 (2015) 149–161 reinforcement yet do well in predicting achievement. The gist of the network story is that information becomes homogeneous, tacit, and therefore sticky within clusters of densely connected people such that clusters disconnect, buffered from one another by structural holes between them, which gives information breadth, timing, and arbitrage advantages to people whose networks span the structural holes. Two people who have no connection with one another are more likely than connected people to work with different ideas and practices. The more disconnected the contacts in a network, the more likely the network spans structural holes. People who connect across the holes (call those people network brokers, connectors, hubs, or entrepreneurs) are exposed to the diversity of surrounding opinion and behavior so they are more likely to detect productive new combinations of previously segregated information, and more likely to see alternative sets of people whose interests would be served if the new combination were brought to fruition. Thus, a structural hole is a potentially valuable context for action, brokerage is the action of coordinating across the hole with bridge connections between people on opposite sides of the hole, and network entrepreneurs, or more simply, brokers, are the people who build the bridges. Network brokers are rewarded socially and materially for their work decoding and encoding information. Numerous research projects show that people with access to structural holes are paid more than peers, receive more positive evaluations and recognition, and get promoted more quickly to senior positions (see Burt, 2005; Burt et al., 2013, for review and contingencies; Aral and Van Alstyne, 2011, for an analysis of network structure as a proxy for information in predicting achievement; Aral and David, 2012, for replication; Rodan and Galunic, 2004, for a similar hypothesis tested with survey data; and Vilhena et al., 2014, for an innovative approach to measuring Pachucki and Breiger’s, 2010, image of “cultural holes” as information boundaries coincident with structural holes). There are second and third reasons to continue ignoring reinforcement. A second reason is that structural holes are reinforced in large part by coordination in the networks around each of ego’s contacts, and research in diverse organizations shows no effect from those neighbor networks on the achievement associated with direct access to structural holes (Burt, 2010). Third, the competitive advantage of brokerage does not depend on collaboration between people on opposite sides of a structural hole. Advantage can involve collaboration, but in general—and I suspect usually—need not depend on collaboration. The broker learns something here, and sells it to his advantage over there. Here and there need never connect directly. Indeed, there are situations in which brokerage is valuable precisely because here and there do not connect directly (Kellogg, 2014). On the other hand, argument can be made for bringing reinforcement into the analysis. As a concept grounded in the advantage-implications of cohesion around ego’s contacts, reinforcement is related to the concept of secondary structural holes (Burt, 1992, pp. 38–42, 56). Primary structural holes are between ego’s contacts. Secondary structural holes are between each contact and the people to whom ego could turn to replace the contact. Secondary structural holes have seen little application in empirical research on individual managers because analysis requires knowledge of the categories that define substitutes for ego’s current contacts (e.g., I go to a doctor, for whom there are three alternative doctors to whom I could go). The concept of secondary structural holes has been applied for many years in organization research, where Department of Commerce industry categories define substitutable organizations. Evidence accumulated since 1975 shows that secondary structural holes in organization networks have their hypothesized effect of weakening network constraint such that performance increases (Burt, 1992, Chap. 3, 2010, Chap. 5). However, reinforcement is not about weakening constraint so much as hardening it. It is not about the ease with which difficult contacts in a cohesive group can be replaced by substitutes. It is about the difficulty, the improbability, of brokerage across cohesive groups. Similarly, Krackhardt’s (1992) concept of Simmelian ties is related to, but distinct from, reinforced structural holes. Simmelian ties are relations reinforced by mutual contacts. Simmelian ties become noteworthy for brokerage when they are bridges in an adjacent structure. For example, Tortoriello and Krackhardt (2010) study Simmelian ties between organization units to draw inferences about information flow between units. Managers A and B, respectively in units A and B, are more strongly connected when they have mutual contacts (versus managers in separate units who have no mutual contacts). Tortoriello and Krackhardt show that the innovation associated with bridging structural holes is more likely for managers whose bridging ties are reinforced by mutual contacts (for similar results, see Hansen, 1999 and Reagans and McEvily, 2003, on information transfer; Cross and Cummings, 2004, on brokerage and performance; Centola et al., 2005, on innovation diffusion more generally). Tortoriello and Krackhardt’s analysis is in two ways distinct from the analysis here: (1) Their primary point is that reinforced relationships can facilitate information flow across structural holes, while the analysis here is about reinforced relations on either side of the hole inhibiting flow. (2) Tortoriello and Krackhardt rely on formal structure to define the structural holes, the boundries, between organization units (as in the early studies of boundary-spanning ties, Tushman, 1977). Here, to avoid the problem of defining which boundaries between organization units are structural holes and which are not, both bridges and holes are defined by the structure within one network (which could be defined by informal relations, or jointly defined by formal and informal). The deepest structural holes between organization units will be the ones most reinforced by strong internal cohesion within the respective units. In contrast, Vedres and Stark’s (2010) concept of a structural fold is very relevant to reinforcement. A structural fold exists where membership overlaps between two largely separate, cohesive groups. A person, ego, located in the fold between two groups bridges numerous structural holes between the groups. The concept of structural fold is closely related to the concept of structural hole, as illustrated by the network metrics to be presented—but there is also something new. The structural holes around the structural fold are reinforced by cohesion within each group and there is management evidence that peer pressure created in closed networks spills into adjacent networks (Burt, 2010, Chap. 6). If brought into a network analysis, it is not obvious whether reinforcement would increase or decrease the achievement association with bridging structural holes. Reinforcement could be expected to increase the association with achievement. Cohesion within groups, and separation between groups, increases the probability that the groups operate with different points of view, which decreases the probability of people in either group seeing brokerage opportunities across the groups, while increasing the probability that productive knowledge in either group will likely be novel in the other group. More, ego’s affiliation with both groups is an incentive for her to find synthetic understanding compatible across the groups. This positive effect of reinforcement is related to what has been discussed as the “depth” of a structural hole (Burt, 1992, pp. 42–44). Cohesion on either side of a structural hole increases its depth, making it easier for ego to play either side against the other, increasing ego’s control over the situation. Ego is better positioned to synthesize understandings across the groups than are individuals within either group. Although consistent with the positive implication of reinforcement, there is no empirical evidence on the achievement implications of more or less deep structural holes. On the other hand, reinforcement could weaken the brokerage association with achievement: Cohesive groups are more likely to R.S. Burt / Social Networks 43 (2015) 149–161 151 insist on the priority of their point of view, which increases the pressure on ego to conform to each, and increases the difficulty of coordinating across the groups. Simultaneously affiliated with both groups, ego can expect to be rip-sawed by conflicting pressures, in response to which ego can keep a low profile in either group, or try to segregate in time or space his affiliation with the groups (Merton, 1957, on role strain; Podolny and Baron, 1997, on difficulty bridging structural holes in formal organization; Burt, 2005, pp. 235–240, on active versus passive structural holes; Reagans and Zuckerman, 2008, on brokerage difficulties created by dense networks around a broker’s contacts). We do not know how or whether reinforcement matters, but there is reason to suspect it could matter. Vedres and Stark (2010) present evidence of achievement associated with structural folds. However, the concept of structural folds is confounded with the concept of structural holes, so it is impossible to determine, without measuring both, how much of achievement is due to the reinforcement provided by structural folds versus the familiar achievement association with access to structural holes. The purpose of this paper is twofold: (1) to propose a measure of the extent to which the structural holes in a network are reinforced to show how reinforcement duplicates, and differs from, currently popular measures of access to structural holes, and (2) to estimate the extent to which reinforcement affects the achievement association with access to structural holes. The next section introduces the proposed network measure as an extension to familiar measures of access to structural holes. Data are then introduced on two populations of senior business leaders, followed by results. Access to structural holes Ego’s access to structural holes is typically measured in terms of three characteristics: network size (many contacts increase the likelihood of brokerage opportunities), network density (strong connections between contacts lower the likelihood of brokerage opportunities), and network centralization or hierarchy (one or a few contacts connected to the other contacts mean that brokerage opportunities might not be available or have to be shared). Oftenused summary measures are illustrated in Fig. 1 for a selection of small networks. Ego’s contacts are indicated in Fig. 1 by gray circles. Lines indicate connections between contacts. Ego is of course connected with each contact, but to keep the sociograms simple, ego’s relations are not presented. Network density and hierarchy are low around brokers A network contains few structural holes to the extent it is small and the contacts in it are interconnected. Size increases down the networks in Fig. 1, from networks of three contacts at the top, to networks of five, to networks of ten at the bottom. Connectivity increases from left to right, from networks at the left in which none of ego’s contacts are connected (labeled “broker networks”), to the networks on the right in which all of ego’s contacts are connected (labeled “clique networks”). Network density is the average strength of connection between ego’s contacts, which in Fig. 1 is the number of connections divided by the number possible (multiplied by 100 to be a percentage). Density is zero for networks in the left column; no contact is connected with others. Density is one hundred percent for networks in the far-right column; every contact is connected with every other. A second way contacts can be connected, closing the network around ego, is by mutual connection with a central person other than ego. This is illustrated by the “partner networks” in the middle column of Fig. 1. Partners provide a substantively significant kind of network closure useful in detecting diversity and coordination problems in a population (Burt, 1998, 2010, Chap 7). The middle-column networks in Fig. 1 are characterized by no connections between contacts except for all being connected with contact A. The networks are centralized around A, making A ego’s “partner” in the network. This kind of network is detected with an inequality measure, such as the Coleman-Theil disorder measure in the third row of each panel in Fig. 1 (explained below). Hierarchy varies with the extent to which connections among ego’s contacts are all with one contact. There is zero hierarchy when contacts are all Fig. 1. Measuring access to structural holes. 152 R.S. Burt / Social Networks 43 (2015) 149–161 disconnected from one another (first column in Fig. 1) or all connected with each other (third column). Hierarchy scores are only non-zero in the middle column. As ego’s network gets larger, the partner’s central role in the network becomes more obvious and hierarchy scores increase (from 7 for the three-person network, to 25 for the five-person network, and 50 for the ten-person network). The graph in Fig. 1 provides a sense of the population distributions from which manager networks are sampled. The graph plots hierarchy scores by density scores for two thousand manager networks in six management populations. The populations, analyzed in detail elsewhere (Burt, 2010), include stock analysts, investment bankers, and managers across functions in Asia, Europe, and North America. The large, open networks of brokers are in the lower left of the graph, low in density and low in hierarchy. Closure can involve simultaneous hierarchy and density, but the extremes of each exclude the other. To the lower right are clique networks, in which there is no hierarchy because all contacts are strongly connected with each other. To the upper left are partner networks, in which density is low because there are no connections between contacts other than their mutual strong connections with ego’s partner. Network constraint: brokers have large, sparse, flat networks The three characteristics—size, density, and hierarchy—are brought together in summary measures such as network constraint, which measures ego’s lack of access to structural holes (Burt, 1980, 1992, pp. 54–56). Constraint decreases with the extent to which ego has many contacts (size), increases with the extent to which ego’s network is closed by strong connections among ego’s contacts (density), and increases with the extent to which ego’s network is closed by a partner strongly connected with all of ego’s contacts (hierarchy). A constraint score of 100 indicates no access to structural holes (ego had no friends, or all of ego’s friends were friends with one another). Across the networks in Fig. 1, network constraint increases from left to right with closure by hierarchy or density (e.g., 20 points for the five-person disconnected network versus 65 points for the five-person clique network), and decreases from top to bottom with increasing network size (e.g., 93 points for the three-person clique network versus 10 points for the ten-person clique network). Constraint begins with each of ego’s relations, measuring the extent to which ego, e, would have a difficult time avoiding contact k, either because ego’s relation with k is large or because everyone ego knows is connected to k: cek = (pek + ̇j pejpjk), where summation is across ego contacts j other than k (j / = k), pek is the proportion of ego’s network spent directly with contact k, pek = [zek + zke]/(!j [zej + zje]), where summation is across ego’s contacts j, and variable zej measures the strength of connection from ego to contact j. The contact-specific constraint term cek varies from zero to one with the extent to which ego cannot avoid contact k, either directly (pek) or indirectly (!j pejpjk). The term is squared to capture concentration in a single contact. Network constraint on ego is the sum of the squared terms across ego’s contacts: Network Constraint (C) = !k cek. Network constraint varies from zero to one—for all but very small networks—with the extent to which ego’s network time and energy is concentrated in a single source, indicating that ego has no access to structural holes.1 Scores are multiplied by 100 in Fig. 1 to indicate points of constraint. 1 The index is ill-behaved for social isolates and dense networks of less than four contacts. The index can exceed one in such small networks. Since such networks provide no access to structural holes, I round their constraint scores to one. Constraint is undefined for social isolates because proportional ties have no meaning (zero divided by zero). Some software outputs constraint scores of zero for isolates. That would mean that isolates have unlimited access to structural holes when in fact Contact-specific constraint scores, cek, are listed in Fig. 1 for the networks composed of three and five contacts. Note the equal levels of constraint posed by each contact for ego in the broker networks to the left and the clique networks to the right. Unequal levels appear when one contact is better connected than the others, illustrated by the partner networks in the middle column. The more unequal the contact-specific constraints on ego, the more ego’s network is co-owned with a partner. The network hierarchy scores in Fig. 1 are Coleman-Theil index scores measuring the extent to which contacts pose very different levels of constraint on ego (Burt, 1992, pp. 70–71): Network Hierarchy = [!k rek (ln rek)]/[Ne (ln Ne)], where summation is across ego contacts k, rek is the constraint posed by contact k relative to the average constraint posed by ego’s contacts (rek = cek/(C/Ne), and Ne is the number of ego’s contacts. Again, scores in Fig. 1 are multiplied by 100. Effective size: brokers have many nonredundant contacts Two other summary measures are given in Fig. 1. Both are attractively intuitive metrics proven in empirical research. Both avoid the small-network issues of the constraint index (footnote 2). Effective size is a count of ego’s contacts discounted for clustering—in essence, it is a count of the clusters to which ego is connected, or the number of nonredundant contacts in ego’s network (Burt, 1992, pp. 51–54). Begin with a measure of contact k’s nonredundancy with ego’s other contacts: 1 − !j pejmkj, where summation is across ego contacts j other than k, pej is the proportional strength of ego’s connection with j (defined above), and mkj is k’s marginal strength of connection with j (connection between k and j divided by k’s maximum connection in ego’s network). Sum the nonredundancy scores for ego’s contacts to define ego’s number of nonredundant contacts (!k nonredundance k). For networks of disconnected contacts, the first column in Fig. 1, network size equals effective size. Every contact is disconnected from the others, so each is nonredundant with the others. For the clique networks in the third column of Fig. 1, ego has only one nonredundant contact regardless of increasing network size, because each contact is redundant with the others. Betweenness: brokers have exclusive access to many structural