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Mathematics LibreTexts

7.E: Connection (Exercises)

Review Questions

  1. Explain the differences among the "three levels of analysis" of graphs (individual, aggregate, whole).
  2. How is the size of a network measured? Why is population size so important in sociological analysis?
  3. You have a network of 5 actors. Assuming no self-ties, what is the potential number of directed ties? What is the potential number of undirected ties?
  4. How is density measured? Why is density important in sociological analysis?
  5. What is the "degree of a point"? Why might it be important, sociologically, if some actors have high degree and other actors have lower degree? What is the difference between "in-degree" and "out-degree"?
  6. If actor "A" is reachable from actor "B" does that necessarily mean that actor "B" is reachable from actor "A"? Why or why not?
  7. For pairs of actors with directed relations, there are four possible configurations of ties. Can you show these? Which configurations are "balanced"? For a triad with undirected relations, how many possible configurations of ties are there? Which ones are balanced or transitive?
  8. What are the differences among walks, trails, and paths? Why are "paths" the most commonly used approach to inter-actor distances in sociological analysis?
  9. What is the "geodesic"distance between two actors? Many social network measures assume that the geodesic path is the most important path between actors - why is this a plausible assumption?
  10. I have two populations of ten actors each, one has a network diameter of 3, the other has a network diameter of 6. Can you explain this statement to someone who doesn't know social network analysis? Can you explain why this difference in diameter might be important in understanding differences between the two populations?
  11. How do "weighted flow" approaches to social distance differ from "geodesic" approaches to social distance?
  12. Why might it matter if two actors have more than one geodesic or other path between them?

Application Questions

  1. Think of the readings from the first part of the course. Which studies used the ideas of connectedness and density? Which studies used the ideas of distance? What specific approaches did they use to measure these concepts?
  2. Draw the graphs of a "star", a "circle", a "line", and a "hierarchy". Describe the size, potential, and density of each graph. Examine the degrees of points in each graph - are there differences among actors? Do these differences tell us something about the "social roles" of the actors? Create a matrix for each graph that shows the geodesic distances between each pair of actors. Are there differences between the graphs in whether actors are connected by  multiple geodesic distances?
  3. Think about a small group of people that you know well (maybe your family, neighbors, a study group, etc.). Who helps whom in this group? What is the density of the ties? Are ties reciprocated? Are triads transitive?
  4. Chrysler Corporation has called on you to be a consultant. Their research division is taking too long to generate new models of cars, often the work of the "stylists" doesn't fit well with the work of the "manufacturing engineers" (the people who figure out how to actually build the car). Chrysler's research division is organized as a classical hierarchy bureaucracy with two branches (stylists, manufacturing) coordinated through group managers and a division manager. Analyze the reasons why performance is poor. Suggest some alternative ways of organizing that might improve performance, and explain why they will help.