Notes on Prestige

Prestige in directed networks (chapter 9)

Definitions, basics

• Prestige can be defined in two different ways:
  • Structural prestige. As a function of social network structure. Positive choices are generally a sign of prestige, and being chosen by a prestigious vertex is generally a sign of greater prestige. SNA uses this ideas as a means of defining prestige.
  • Social prestige. Other measures of prestige simply depend on combining what many different people think of a particular vertex (representing a person or association), without any explicit reference to social networks. This social prestige thus defines an attribute variable that could be represented in Pajek by a partition or vector.
• As usual, the task of SNA lies in determining the relation between network and attribute properties - here between structural and social prestige.
  • Is it possible that social prestige in fact derives from structural prestige? (choices result in prestige)
  • Or perhaps structural prestige derives from social prestige? (prestige results in choices)
  • Either is likely, but the relation may not be simple - it's a topic for exploratory SNA!
• Popularity of a vertex is its indegree (analogous to degree centrality).
  • Note that this procedure presumes no multiple arcs (simple network)
  • Multiple arcs and line values suggest a different operation
• Note that the choice relation must be "positive" (e.g. "likes")
  • if negative ("dislikes"), prestige may be interpreted as a kind of "infamy"
  • if reversable ("lends money to") use outdegree instead (or transpose the network matrix, reversing all arc directions)
• Pajek techniques:
  • Net>partitions>degree>input to generate input degree partition and normalized input degree vector
  • Net>transform>transpose to transpose network

Variable relations

• Relating two variables: two variables are related (or associated) if their values don't occur independently of one another: the value of one variable biases the possible values of the other (for instance: "height" and "weight" are certainly not independent, whereas "height" and "musical preference" probably are). Statistical tests are used to measure the significance of these relations or associations. The kind of test used depends on the kind of variable.
  • Variable types:
    • Nominal (categorical): variable values cannot be ordered; are fundamentally non-numeric (e.g. red, blue, green)
    • Ordinal or rank: variable values are ordered, without implication of distance (e.g. first, second, third)
    • Interval: variable values occur on a number line, but the zero point is arbitrary (e.g. temperature Celsius) so the ratio between two values is meaningless (20 degrees isn't twice as hot as 10 degrees)
    • Ratio: variable values on number line with a significant zero point, so that ratios are meaningful (e.g. weight: 20 kg is really twice as heavy as 10 kg)
  • Pajek and variable representations
    • Partitions: Nominal and ordinal
    • Vectors: Interval and ratio
  • Different statistical tests are required to detect relations between different kinds of variables
    • Comparing two nominal variables: Chi-squared and cross tables - use Partitions>info>Cramer's V, Rajski
    • Comparing two rank variables: Spearman rank correlation - use Partitions>info>Spearman's rank
    • Comparing two interval or ratio variables: Pearson correlation - use Vectors>info
    • Spearman's takes only rank into account; few cases should have equal rank. Pearson's measures the degree of linear association between two variables and is more precise for interval or ratio data, however may miss relationships when the linearity condition isn't fulfilled for rank data. Generally, if Pearson's suggests a relation, so will Spearman's - but the former is more precise. But the latter may suggest a relation when the former doesn't.

• San Juan Sur data
  • Network of visits, partitions identifying (a) social prestige rank and (b) prestige leaders (families 23, 39, 47, 61, 66).
  • Relation between input degree and prestige leadership: draw partition-vector ... no clear relation?
  • Relation between indegree and social status groupings
    • Run Spearman rank test on two partitions: social status and indegree
    • Run Pearson's test on two vectors: social status (converted to vector) and indegree