Abstract
In the following, roots, levels of analysis and relevant measurement issues of SNA are discussed. The main purpose of these sections is to show (1) which are the main streams of development towards contemporary SNA, (2) which are possible levels of analysis and (3) which are relevant analytical instruments to capture social network structures as well as social capital issues.
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Notes
- 1.
According to Schenk (1984, pp. 2–3) or Freeman (2004, pp. 100–103), Alfred Reginald Radcliffe-Brown was the first introducing the term “social structure” as a metaphor for networks of actually existing relations.
- 2.
Following Moreno (1953, pp. 15–16), “[s]ociometry is the mathematical study of psychological properties of populations, the experimental technique of and the results obtained by application of quantitative methods.” Further, to him “[s]ociometry aspires to be a science within his own right” (Moreno (1953, p. 48)). The term itself can be regarded as a combination of “socius” (Lat.), meaning social or companion and “metrum” (Gr.) or “metron” (Lat.) standing for measurement.
- 3.
Main reason to this may be, that “field theory is in place indeterminate and resists formalization” (Martin (2003, pp. 42–43)) and therefore no progress into this direction occurred. Martin (2003) itself provides a good review of further development and discussion of contemporary field theory.
- 4.
See Rogers and Agarwala-Rogers (1976, pp. 36–48) or Freeman (2004, pp. 43–64) for a historical overview.
- 5.
Notably, Max Gluckman was the only resident in Manchester with the other three being regular visitors working at London School of Economics, see Freeman (2004, pp. 103–105). Moreover, anthropologists from other countries did also field studies of this style, see Mitchell (1974, 1979) or Wolfe (1978) for a more broad review.
- 6.
Cf. Hummon and Carley (1993). Notably, only a few contributions about methodological issues in social network analysis are being published in German, e.g. Kähler (1975), Schenk (1984), Pappi (1987) and Jansen (2003).
- 7.
See Jansen (2003, pp. 53–58) with a similar definition.
- 8.
Of course, there are more than the following measurement concepts existing as tools for social network analysis like triad analysis, structural equivalence, blockmodelling but these measurement concepts are (1) not suitable in the research context and (2) therefore beyond the scope of this piece of work. For further reference of these measurement concepts see e.g. Burt (1980a, pp. 96–130), Burt (1982, pp. 37–87), Knoke and Kuklinski (1982, pp. 56–86), Schenk (1984, pp. 83–108), Scott (1991, pp. 103–148), Wasserman and Faust (1994, pp. 227–721), Jansen (2003, pp. 163–220) or Knoke and Yang (2008, pp. 72–117).
- 9.
Cf. Burt (1983, p. 177), Scott (1991, pp. 71–72) or Jansen (2003, pp. 103–104).
- 10.
Cf. Jansen (2003, p. 104).
- 11.
This relationship was first formulated by Kephart (1950). The same formulae can also be used for undirected graphs if all mutual relations are counted twice.
- 12.
Cf. Barnes (1979, pp. 411–412), Burt (1980a, p. 90), Burt (1982, p. 32) or Diaz-Bone (1997, p. 59).
- 13.
Cf. Wasserman and Faust (1994, pp. 145–117), Diaz-Bone (1997, p. 46).
- 14.
Some authors called this measure homogeneityor individual diversity, see e.g. Weimann (1989, p. 190).
- 15.
Cf. Knoke and Kuklinski (1982, p. 41), or Jansen (2003, pp. 97–98).
- 16.
Cf. Lincoln (1982).
- 17.
Cf. Lincoln (1982, pp. 4–6), Schenk (1984, pp. 65–74), Weimann (1989, pp. 190–192).
- 18.
See Mitchell (1969, pp. 20–27), Kähler (1975), Marsden and Campbell (1984, pp. 483–484) or Weimann (1989, p. 191) with further reference.
- 19.
Cf. Shulman (1976).
- 20.
Cf. Lincoln (1982, p. 5).
- 21.
Cf. Knoke and Kuklinski (1982, p. 41) or Jansen (2003, pp. 97–98).
- 22.
Cf. Marsden and Campbell (1984, pp. 485–488).
- 23.
Cf. Niemeijer (1973), Mitchell (1974, p. 288), Shulman (1976), Barnes (1979, pp. 406-408), Scott (1991, pp. 72–84), Wasserman and Faust (1994, p. 181) or Jansen (2003, p. 108).
- 24.
See Kephart (1950) for a formal derivation of these figures.
- 25.
NDis, virtually by definition, a relative measure of network density. An absolute measure of network density, of course, was developed by Scott (1991, p. 98), but discussing this measure in depth is beyond the scope of this work. In case of an undirected graph, the same formulae can be employed with mutual relations counted twice.
- 26.
Cf. Burt (1980a), Scott (1991, p. 75), Jansen (2003, p. 108) or McCarty and Wutich (2005).
- 27.
See e.g. Jansen (2003, p. 109). Again, this formulae can be applied to an undirected graph with mutual relations counted twice.
- 28.
This is a well known attribute common to all network structures, usually called the “network effect.”
- 29.
Cf. Scott (1991, pp. 77-78).
- 30.
Cf. Knoke and Kuklinski (1982, p. 50), Witt (1996, p. 35) or Jansen (2003, p. 111).
- 31.
Cf. Knoke and Kuklinski (1982, p. 50) or Witt (1996, p. 35).
- 32.
Cf. Burt (1980a).
- 33.
Cf. Knoke and Kuklinski (1982, p. 51) or Jansen (2003, p. 111).
- 34.
Cf. Knoke and Kuklinski (1982, p. 41) or Jansen (2003, pp. 97–98).
- 35.
According to Barnes (1969, p. 239), these definitions are originally made by Harary et al. (1965), but several other classifications of connectedness on the network level are existing, too.
- 36.
See Knoke and Burt (1983, pp. 198–200), Wasserman and Faust (1994, pp. 169–178) or Koschützki et al. (2005) for a more thoroughly discussion of these two concepts.
- 37.
Scott (1991, pp. 95–96) questions this implicit assumption that a network structure has always an easily identifyable focal point being quite in the centre, and refers to Christofides (1975) concept of eccentricity, which builds on a distance matrix containing shortest path distances (or geodesics) between each pair of point. Then the eccentricity of a point is defined as the length of the longest geodesic in the distance matrix and the point with the lowest eccentricity is the absolute centre of a given graph. In some graphs, two or more points have equally low eccentricity. Then an imaginary point lying on a path between them is considered to be the absolute centre.
- 38.
Cf. Freeman (1979), Schenk (1984, pp. 51–56), Witt (1992) or Wasserman and Faust (1994, pp. 178–192).
- 39.
See Freeman (1979), Wasserman and Faust (1994, p. 178) or Witt (1996, pp. 19–28) for a more rigorous discussion.
- 40.
It is worth mentioning, that the magnitude of C D (p k ) is virtually identical to the actors degree d(p i ), because both are defined as a sum of adjancies, see Burt (1983, p. 190) for further discussion. In the case of digraphs, Wasserman and Faust (1994, p. 199) proposed to count the outdegrees of an actor.
- 41.
See Freeman (1979), Burt (1980a, pp. 91-93), Burt (1982, p. 33–35), Wasserman and Faust (1994, pp. 188–190) or Koschützki et al. (2005, pp. 28–32) with a similar discussion.
- 42.
See Anthonisse (1971) or Jansen (2003, pp. 148–149) for the digraph case.
- 43.
Following Wasserman and Faust (1994, p. 200), C B ′(p k ) must be multiplied by 2 for digraphs because \(\max {C}_{B}^{{\prime}}({p}_{k}) = (N - 1)(N - 2)\).
- 44.
Following Koschützki et al. (2005) this resembles to much earlier formulations by Wiener (1947) or Shimbel (1953). Further, Wasserman and Faust (1994, pp. 200–201) for the digraph case.
- 45.
Cf. Wasserman and Faust (1994, p. 185).
- 46.
Of course, there are a huge amount of other methods to solve this problem of obtaining eigenvalues and eigenvectors in a eigensystem in the field of network analysis proposed by other authors like Katz (1953), Hubbel (1965), Taylor (1969), Bonacich (1972b) and briefly discussed in Wasserman and Faust (1994, pp. 206–209) or more recent contributions by Richards and Seary (2000), Poulin et al. (2000), Bonacich and Lloyd (2001) or Koschützki et al. (2005, pp. 46–51).
- 47.
Cf. Wasserman and Faust (1994, p. 177).
- 48.
Cf. Freeman (1979) or Wasserman and Faust (1994, p. 186).
- 49.
See Snijders (1981) or Wasserman and Faust (1994, p. 181) for this and the following discussion.
- 50.
Cf. Wasserman and Faust (1994, p. 187). Further, Wasserman and Faust (1994, pp. 200–201) for the digraph case.
- 51.
Cf. Knoke and Burt (1983, pp. 200–202), Wasserman and Faust (1994, p. 202) or Jansen (2003, pp. 142–145). According to Knoke and Burt (1983, p. 201), Moreno (1934, pp. 98–103) was the first author who introduced such a degree-based measure of prestige.
- 52.
This sometimes called a relative indegree, according to Wasserman and Faust (1994, p. 203).
- 53.
Cf. Burt (1980a, pp. 93–95), Burt (1982, pp. 33–35), Knoke and Burt (1983, pp. 203–204), Wasserman and Faust (1994, pp. 203–204) or Jansen (2003, pp. 145-147). According to Wasserman and Faust (1994, p. 203), Lin (1976, pp. 340–349) was the first author who defined such an influence domain in order to construct a measure for proximity prestige.
- 54.
Cf. Knoke and Burt (1983, pp. 206–207), Wasserman and Faust (1994, pp. 207–209) or Jansen (2003, pp. 149–153).
- 55.
See Wasserman and Faust (1994, pp. 207–210) or Koschützki et al. (2005, pp. 46–51) for a brief review of this literature.
- 56.
See e.g. Burt (1980a, pp. 93–95), Burt (1982, pp. 33–35), Scott (1991, p. 55), Wasserman and Faust (1994, pp. 206-209), Richards and Seary (2000) and Poulin et al. (2000) for a detailed discussion of this and other more refined eigendecomposition methods.
- 57.
Cf. Knoke and Burt (1983, pp. 204–206) or Jansen (2003, pp. 149–149).
- 58.
Cf. Jansen (2003, p. 144).
- 59.
Cf. Wasserman and Faust (1994, p. 204) or Jansen (2003, pp. 142–143).
- 60.
See e.g. Bolland (1988).
- 61.
Cf. Bonacich and Lloyd (2004) or Bonacich (2007).
- 62.
These centrality measures are often called flow betweenness, see e.g. Newman (2005) or Borgatti and Everett (2006).
- 63.
Originally, Knoke and Burt (1983) named their measurement concepts VAR1 to VAR5, with VAR1 equivalent to degree-based measures, VAR3 to closeness- or proximity-based measures, VAR4 to betweenness-based measures and VAR5 to rank- or status-based measures. VAR2 are measures based on connectivity or reachability, a measurement concept quite similar to degree, which was excluded from investigation.
- 64.
The other two are the types of walks considered (such as only geodesics, only true paths, limited length walks, and so on) and the choice of summary statistic (e.g. sum or average).
- 65.
Or in the words of Borgatti and Everett (2006, p. 481): “Whereas radial measures assess group membership, medial measures assess bridging, reminiscent of the distinction in the social capital literature of bonding social capital and bridging social capital, or closed versus open ego networks.”
- 66.
See Burt (1992, pp. 54–55), Burt (1998, Appendix) or Burt (2000, Appendix) for further details.
- 67.
Cf. Burt (1998, Appendix, 2000, Appendix).
- 68.
Burt (1998, Appendix, 2000, Appendix) proposed two alternative measures of network hierarchy: (1) the Coleman-Theil disorder index applied to contact-specific constraint scores as developed first in Burt (1992, pp. 70–71) and (2) the betweeness-based measure of point centrality C b (p k ) by Freeman (1977) as already defined in the last section measuring the extent to which one contact stands in between all others.
- 69.
Strictly speaking, the terms sender and receiver imply to some extent directed relations from a sender to a receiver. So in the following, the terms first- and second-party instead of sender and receiver are used in order to indicate that these relations may be mutual.
- 70.
Cf. Gould (1989, p. 537) or Gould and Fernandez (1989, p. 98).
- 71.
Notably, this classification of brokerage roles draws heavily reference to individual communication roles in organizations as defined by Rogers and Agarwala-Rogers (1976, pp. 132–140) proposing a four-fold typology with (1) gatekeepers, (2) liaison, (3) opinion leaders and (4) cosmopolites or boundary spanners. Whereas the gatekeeper and liaison role is defined exactly in the same way as in Rogers and Agarwala-Rogers (1976) and the opinion leader corresponds to the local broker or coordinator, they defined the cosmopolite or boundary spanner as an individual who has a relatively high degree of communication with the outside environment of an organization so that it corresponds at best with the representative role as defined by Gould and Fernandez (1989). Further, as Rogers and Agarwala-Rogers (1976) set a focus on communication networks in organizations, they did not define a counterpart to Gould and Fernandez (1989) itinerant broker or cosmopolitan.
- 72.
This mathematical expression of the kth intermediary’s raw brokerage scores is adapted from Gould (1989), which is a refinement of the original formulation by Gould and Fernandez (1989). Further, it is worth to mention, that w 1k and w 0k are not expressed in the most parsimonious way, but this is done to show a consistent set of equations.
- 73.
The category of serial duplication refers to network flows of miotic reproduction, viral infection or gossip and is therefore not of interest in the context of this work.
- 74.
More definite, trails are paths, where a point may be reached more than once in a sequence of points but not adjacent ones, see e.g. Borgatti (1995).
- 75.
More precisely, they proposed measures for three different conceptions of social capital of which only one is of interest in this context.
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Schramm, HJ. (2012). A Closer Look on Contemporary Social Network Analysis. In: Freight Forwarder's Intermediary Role in Multimodal Transport Chains. Contributions to Management Science. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2775-0_5
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