Chapter 20 Exercises: Six Degrees — How Small-World Networks Open Big Doors

Level 1 — Comprehension and Recall

These exercises test your understanding of the chapter's core concepts.

1. Describe Milgram's original small-world experiment. What was the sample, the procedure, and the finding? What were at least two methodological limitations he or subsequent researchers identified?

2. Explain the Watts-Strogatz model. Start with a regular lattice and describe what happens when you introduce random "rewiring." What two properties define a small-world network?

3. What is a "hub" in network terms? What does it mean for a network to have a "power-law distribution" of connections? Why are hubs essential to small-world dynamics?

4. What did the 2008 Microsoft Messenger study find about average path length across 240 million users? How did its methodology differ from Milgram's, and what advantages did that provide?

5. The chapter says social media has acted as a "degrees-of-separation collapser." Identify three specific mechanisms by which social media platforms reduce effective network distance.

Level 2 — Application and Analysis

These exercises ask you to apply chapter concepts to new situations.

6. Priya traced a two-hop path to Daniel Osei through Marcus. Using the concepts of this chapter, explain: (a) why she hadn't found this path through her six weeks of cold-applying, (b) what specifically made the two-hop path traversable, and (c) what the difference is between topological path length (the path that exists) and navigational path length (the path you can find and use).

7. The chapter says hubs serve as "super-shortcuts" that dramatically reduce average path length. Using LinkedIn or another social network you're familiar with, identify one person in your extended network who seems to have hub-like properties — many connections across diverse industries. What would it take to build a genuine weak tie with that person?

8. The chapter discusses Gladwell's "Connectors" concept and then presents several research-based limitations. Evaluate: Which of Gladwell's claims about Connectors do you find supported by the chapter's evidence? Which seem overstated? What does the evidence suggest Gladwell got right and wrong?

9. Run the Kevin Bacon game manually for one actor of your choice. Using IMDb or your own knowledge, find the shortest path from your chosen actor to Kevin Bacon. How many hops? What does the existence and length of this chain tell you about the structure of the Hollywood collaboration network?

10. Design a network mapping exercise for your own professional network, using the five-step framework from the chapter. Complete at least steps 1 and 3 (cluster identification and chain tracing to a specific target). Write a 300-word reflection on what you discover.

Level 3 — Research and Evidence Evaluation

These exercises ask you to engage with evidence and research methods.

11. Milgram's study had completion rates as low as 20–30%. Explain why this is a serious methodological concern. How does it affect your confidence in the "six degrees" finding? What can you conclude from the completed chains despite this limitation?

12. The Watts-Strogatz model shows that a small number of random long-range edges dramatically reduces average path length. Why does this matter for the relationship between weak ties (Chapter 19) and small-world structure (Chapter 20)? What is the mathematical basis for Granovetter's "bridges" being essential to short path length?

13. Barabási and Albert showed that power-law distributions of connectivity arise from "preferential attachment" — new nodes preferentially connect to already well-connected nodes. What does this predict about inequality in network connectivity over time? And what does it predict about the persistence of hub advantage?

14. Kleinberg's navigability research showed that not all small-world networks are equally navigable — the distribution of long-range connections must follow a specific pattern for efficient routing to be possible. What real-world factors might determine whether human social networks follow this pattern? What would make them more or less navigable?

15. The Microsoft Messenger study analyzed 240 million users but found that approximately 48% of user pairs had no path between them (disconnected components). What does this tell us about the limits of "six degrees"? Who is most likely to be in disconnected components, and what does this imply about the applicability of small-world theory to structurally disadvantaged populations?

Level 4 — Synthesis and Critical Thinking

These exercises require you to integrate multiple ideas and form original arguments.

16. Chapters 18, 19, and 20 form a progression: structural luck (who you start connected to), weak ties (which connections generate new opportunity), and small-world structure (why everyone is reachable). Write a 500-word essay synthesizing all three chapters around the following question: "Is network-based luck more equally distributed than structural luck — or does small-world structure reproduce the same hierarchies that Chapter 18 documented?"

17. The chapter argues that social media has "collapsed" degrees of separation. But critics might argue that the appearance of closeness through social media connections masks continued social distance in terms of trust, relationship quality, and willingness to act on behalf of others. Write a 400-word analysis: When social media makes someone "reachable," what does that actually mean — and what does it not mean?

18. The hub strategy (deliberately cultivating relationships with highly connected nodes) is strategically rational but raises ethical questions. If everyone tries to access hubs, what happens to hubs' time and attention? Is the hub strategy sustainable as a collective strategy, or does it only work as an individual strategy when few people use it? What does this imply about the ethics of deliberate network cultivation?

19. The chapter includes a Python network visualization. Extend this exercise: build a more complete network graph of at least 15 nodes representing your actual professional and quasi-professional connections. Calculate the clustering coefficient and the average path length. Does your network look like a small-world network? What specific connections (edges) would you add to improve its small-world properties?

20. Priya was "two handshakes away" from her dream company's hiring manager but had been cold-applying for six weeks. Write a 600-word case analysis that: (a) diagnoses why she was operating with incorrect network assumptions, (b) explains the specific insight from small-world theory that changed her approach, (c) analyzes the risks and uncertainties in the chain-traversal strategy she adopted, and (d) proposes what she should do if the chain fails (Marcus's introduction doesn't lead to a response from Daniel).

Level 5 — Creative and Integrative Projects

These exercises ask for original creative or research work.

21. The Full Network Mapping Project. Using the five-step framework from the chapter, complete a full network mapping of your professional and quasi-professional connections. You may use LinkedIn's "export connections" feature (Settings > Data Privacy > Get a copy of your data) to download a list of your connections, which you can then import into a spreadsheet or Python script. Identify: your three primary clusters, your five most valuable bridges, your two or three most hub-like connections, and at least three targets you could reach in two or three hops. Write a 500-word reflection on what you discover and what strategic actions you will take.

22. The Chain-Tracing Exercise. Identify one person or organization you would like to reach professionally — a company, a potential mentor, a professional community. Trace the full chain of connections between you and that target, using LinkedIn, mutual contacts, or any other network information available to you. Document: the chain length, the identity of the key bridge node(s), and the specific ask you would make to traverse the chain. Do not actually send the messages unless you are genuinely comfortable doing so — the exercise is the tracing and planning, not the execution.

23. Run a replication of Milgram's small world experiment in a modern, digital version. Identify a target (someone you don't know personally) and challenge five friends to find a chain connecting them to the target through people they know personally on a first-name basis. Document the chains that are found, the chain lengths, and whether any chains converge on the same intermediate nodes. Write a 400-word reflection on what your informal experiment suggests about small-world dynamics in your social environment.

24. Research the "Oracle of Bacon" website (oracleofbacon.org), which calculates Bacon numbers for any actor in its database. Look up 10 actors of your choice. What is the distribution of Bacon numbers? What does this distribution tell you about the structure of the Hollywood collaboration network? Identify which actors in your sample have the highest Bacon numbers — what do they have in common? Write a 300-word analysis.

25. Using Python and the NetworkX library, build a small-world network simulation. Start with a regular lattice of 30 nodes (each connected to its 4 nearest neighbors). Then iteratively add random long-range edges and calculate the average path length after each addition. Plot average path length as a function of number of long-range edges added. At what point does the dramatic reduction in path length occur, and what does this reveal about the threshold for small-world emergence? Write a 300-word explanation of what your simulation demonstrates.