A typology of social capital and associated network measures

For instance, see the discussion in Knack and Keefer (1997), Jackson (2019b).

One can find earlier mentions of the term—going back to the late 1800s—but often with different meanings from the more modern ones.

See, for instance, the many definitions in Portes (1998), Woolcock (1998), Dasgupta and Serageldin (2001), Sobel (2002), Glaeser et al. (2002), Dasgupta (2005). That important forms of capital are embodied in humans as well as their relationships is even mentioned in the writings of Alfred Marshall.

Marshall’s views were complex and changed over time, e.g., see the discussion of Blandy (1967).

The exclusion of land and labor as forms of capital is mainly made to stick with the historical distinction. That historical distinction was made mostly for convenience—it can be very useful to distinguish capital and labor in estimating production functions and in many modeling and policy applications. Given that land and labor play no role in the discussion here, I exclude them for ease of exposition and to stick with the classical roots of the definition, but not for some deeper logical reason.

Here I also separate distribution from production, just so that people understand that it is included, although one could fold it into the definition of production if that term is suitably understood.

This is also another source of confusion in the use of the term ‘social capital’, as it is often invoked in contexts relative to the improvement of an individual, which is certainly part of the story, but without reference to any production. Ultimately we care about how capital is instrumental in the “production” of something—a good, service, entertainment, or anything else that can be consumed or enjoyed by someone. Thus, something like favor capital can be possessed by an individual, but the value comes from the productive uses that it can eventually enable.

Note that some uses could also have negative value, for instance some forms of favor capital might include nepotism that has negative externalities. But many forms of production involve externalities, and that does not change the discussion of whether something is an essential and valued input to that production process.

Network measures of centrality, influence, and power are abundant (e.g., see Borgatti 2005; Jackson 2008; Bloch et al. 2016; Jackson 2019b). Although there is much discussion of centrality measures in the literature, the explicit discussion of how different ones may be used to identify different forms of social capital is new to this paper.

See also Lin (1999), Flap and Völker (2001), Kawachi et al. (2004), and Aldrich (2012).

Generally, I will consider a case in which \(g_{ii}=0\), so that there are no self-loops, but this is not consequential to the formal definitions.

For example, Arrow and Borzekowski (2004) discuss how having more access to job information can result in more productive matching between workers and jobs.

See Jackson et al. (2018). For another reason for such a decay, see Manea (2017) who examines a model of resale of information with bargaining.

See also the related notion of cascade centrality from Kempe et al. (2003, 2005).

Communication and diffusion centrality allow for heterogeneous weightings, while decay centrality does not. However, it similarly extends by tracking shortest weighted paths.

For instance, see Atkeson and Kehoe (1993) who consider the impact of the knowledge embodied in a group or organization.

See Katz (1953), Bonacich (1987), Jackson (2008), Laguna-Müggenburg (2017) for discussion. Laguna-Müggenburg (2017) shows that including values can make a big difference in who is most central, especially in networks with homophily.

This requires that the entries of \({\mathbf {p}}\) be substantially lower than one over the average degree, so that closer neighborhoods are accounting for relatively more of the calculations. More generally.

If one has an estimate for clustering, then one could further refine the estimate by adjusting this to account for the fact that some of those node’s friends may already be in i’s second neighborhood. So if average clustering is \({\overline{c}}\), then the estimation of nodes at distance 3 would be \( p^2 n_i^2 ({\overline{d}}-1)-2 {\overline{c}} \frac{n_i^2 (n_i^2-1)}{2}\), where the last expression accounts for the fact that if two nodes at distance two are connected to each other, then we should lower the count of further friends that we attribute to each of them by one (hence the factor 2). One could make further adjustments to account for the number of friends that nodes at distance 2 have that are at distance 1: that is if there are several paths of length 2 from i to some j, then several of j’s friends have already been counted.

An even better approximation is to adjust the \({\overline{d}}\) to be the average degree of neighbors in the network, which is generally higher than the average degree (e.g., see Jackson 2019a). An approximation is \(E[d^2]/{\overline{d}}\).

In that case, clustering and double counting is less of an issue, since all walks are counted, and so the approximations are even easier.

This is a very narrow use of the term leadership, as it stems entirely from a person’s positional ability to coordinate the activity of others, but does not involve other personal characteristics or other factors that determines whether people will actually pay attention to the individual.

See the discussion in Kent (1978), Padgett and Ansell (1993), Jackson (2008, 2019b).

For example, see the discussion in Brass and Krackhardt (1999) of the importance of network position and coordination in leadership. Being able to coordinate others has also been discussed in defining power (e.g., see Pfeffer 1981, 1992), as well as how unequal allocations can be (Kets et al. 2011).

For a discussion of truncated betweenness centrality measures to address this concern, see Ercsey-Ravasz et al. (2012).

One way to account for these issues is weight paths by their length. For instance, for some decreasing function \(f(\ell )\), \(\sum _{(j,k): j\ne i \ne k \ne j } f(\ell _{{\mathbf {g}}}(j,k) )\frac{\nu _{{\mathbf {g}}}(i:j,k)}{\nu _{{\mathbf {g}}}(j,k)}.\)

This differs from the sorts of measures that one finds in some network software, e.g. UCINET, which calculate measures based on group assignments and directed interactions. (For more discussion behind those measures see Gould and Fernandez 1989.) Those are actually closer in spirit, but still different from, what is proposed to measure bridging capital below.

This brings up a refinement of the measure of brokerage capital in cases where markets are bilateral. In the first example of countries, it could be that any two countries could need to enter negotiation and so the Godfather index properly captures the potential interactions. In an example of serving as an agent between buyers and sellers, then in counting the Godfather index, one would want to only count pairs of neighbors that involve one buyer and one seller, but discard pairs that involve two sellers or two buyers. That is, being a unique connector between two sellers might be useless since they do not need to transact.

Bridging capital is perhaps conceptually the most closely related to Burt’s (1992, 2000, 2005) notion of structural holes, although his discussions include aspects of all three of bridging, brokerage, and coordination capital.

See also Anderson (2017) for an account of how particular combinations of skills and knowledge can be more valuable than separate skills or superficial knowledge of many skills.

This is a much more direct measure than something like the effective size of Burt (1992), which does not capture exclusivity. Effective size computes a node’s degree and then subtracts the average degree of that node’s friends. Having many friends who are not well connected themselves may be correlated with bridging capital, but is still less directly tied to bridging capital than the measure above. Similarly, effective size is less directly related to brokerage or coordination capital than the Godfather Index.

For further discussion of how incentives for reciprocation and cooperation are dependent upon network structure, see Raub and Weesie (1990), Bloch et al. (2007, 2008), Karlan et al. (2009), Lippert and Spagnolo (2011), Ali and Miller (2016, 2013).

For a few starting references on measuring reputation see List (2006), Berens and van Riel (2004), Wartick (2002).

The fact that individuals in a community are more familiar with others in the community can also be productive, as it can help them predict how others will act and hence to coordinate their behaviors which can be productive (e.g., see Jackson and Xing 2018).