Graphs and matrices

Hanneman and Riddle chapters 3 through 5.

Introduction: Representing Networks with Graphs

• Graphs are simplest way of seeing patterns for small data sets
• Graphs have powerful mathematical theory
• Graphs can be translated easily into numerical form

Graphs and Sociograms

• First network diagrams were sociograms of who chose whom in a group; circles and arrows
• But many kinds of relations can be represented with graphs
• Basic tools: nodes and edges or actors and relations

Kinds of Graphs:

• Levels of Measurement: Binary, Signed, and Valued Graphs
• Directed or "Bonded" Ties in the Graph
• Simplex or Multiplex Relations in the Graph

What is a Matrix?

• Simple mathematical representation of a graph as a 2 dimensional numerical array.
• Useful because we can define precise operations (algorithms) to describe features
• Can use computers to calculate measures

The "Adjacency" Matrix

• Rows = sources; Columns = receivers
• Binary, can be adapted to ordinal or interval

Doing Mathematical Operations on Matrices

• Matrix Permutation, Blocks, and Images
• Transposing a matrix
• Taking the inverse of a matrix
• Matrix addition and matrix subtraction
• Matrix correlation and regression
• Matrix multiplication and Boolean matrix multiplication

Review Questions

1. What are "nodes" and "edges"? In a sociogram, what is used for nodes? for edges?

2. How do valued, binary, and signed graphs correspond to the "nominal" "ordinal" and "interval" levels of measurement?

3. Distinguish between directed relations or ties and "bonded" relations or ties.

4. How does a reciprocated directed relation differ from a "bonded" relation?

5. Give and example of a multi-plex relation. How can multi-plex relations be represented in graphs?

6. A matrix is "3 by 2." How many columns does it have? How many rows?

7. Adjacency matrices are "square" matrices. Why?

8. There is a "1" in cell 3,2 of an adjacency matrix representing a sociogram. What does this tell us?

9. What does it mean to "permute" a matrix, and to "block" it?

Application Questions

1. Think of the readings from the first part of the course. Did any studies present graphs or matricies? If they did, what kinds of graphs and/or matrices were they (that is, what is the technical description of the kind of graph or matrix).

2. Suppose that I was interested in drawing a graph of which large corporations were networked with one another by having the same persons on their boards of directors. Would it make more sense to use "directed" ties, or "bonded" ties for my graph? Can you think of a kind of relation among large corporations that would be better represented with directed ties?

3. Think of some small group of which you are a member (maybe a club, or a set of friends, or people living in the same apartment complex, etc.). What kinds of relations among them might tell us something about the social structures in this population? Try drawing a graph to represent one of the kinds of relations you chose. Can you extend this graph to also describe a second kind of relation? (e.g. one might start with "who likes whom?" and add "who spends a lot of time with whom?").

4. Take one of the relations that you graphed from the previous question, and represent it as a matrix. Does it make sense to leave the diagonal "blank," or not, in your case? Try permuting your matrix, and blocking it.

5. Can you make an adjacency matrix to represent the "star" network? what about the "line" and "circle." Look at the ones and zeros in these matrices -- sometimes we can recognize the presence of certain kinds of social relations by these "digital" representations. What does a strict hierarchy look like? What does a population that is segregated into two groups look like?